{"pdf": "arxiv_math/2503.04048_pg46.pdf", "url": "https://arxiv.org/pdf/2503.04048", "page": 1, "id": "2503.04048_pg46_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\mathcal{V}}(\\psi_m)\\rightarrow +\\infty"}
{"pdf": "arxiv_math/2503.07228_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07228", "page": 1, "id": "2503.07228_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\leq k \\leq 2^{N}-1"}
{"pdf": "arxiv_math/2503.04993_pg18.pdf", "url": "https://arxiv.org/pdf/2503.04993", "page": 1, "id": "2503.04993_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{d}{d\\epsilon} J_1(u_1 + \\epsilon v_1, u_2) \\Big|_{\\epsilon=0}"}
{"pdf": "arxiv_math/2503.04993_pg18.pdf", "url": "https://arxiv.org/pdf/2503.04993", "page": 1, "id": "2503.04993_pg18_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\frac{d}{d\\epsilon} J_1(u_1 + \\epsilon v_1, u_2) \\Big|_{\\epsilon=0}=I_1 + I_2"}
{"pdf": "arxiv_math/2503.06865_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06865", "page": 1, "id": "2503.06865_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "F(q,r)=\\gamma _{J\\nu (q)}(r)"}
{"pdf": "arxiv_math/2503.06865_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06865", "page": 1, "id": "2503.06865_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "F:N\\times \\mathbb{R}\\rightarrow \\mathbb{C}H^{2}"}
{"pdf": "arxiv_math/2503.06865_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06865", "page": 1, "id": "2503.06865_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{2}\\left( -4\\right)"}
{"pdf": "arxiv_math/2503.06865_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06865", "page": 1, "id": "2503.06865_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "B(\\gamma _{J\\nu (p)}(-t))=\\gamma _{J\\nu (p)}(t)% \\text{.}"}
{"pdf": "arxiv_math/2503.09550_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09550", "page": 1, "id": "2503.09550_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{f_j:X_n \\rightarrow \\mathbb{R}\\}"}
{"pdf": "arxiv_math/2503.09550_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09550", "page": 1, "id": "2503.09550_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "-1<\\beta_{\\vert X \\vert} \\leq \\ldots \\leq \\beta_2< \\beta_1=1"}
{"pdf": "arxiv_math/2503.08077_pg42.pdf", "url": "https://arxiv.org/pdf/2503.08077", "page": 1, "id": "2503.08077_pg42_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ \\tfrac{-1}{2} , 0 , \\tfrac{+1}{2} \\}"}
{"pdf": "arxiv_math/2503.04045_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04045", "page": 1, "id": "2503.04045_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "k\\geq k_{\\varepsilon}"}
{"pdf": "arxiv_math/2503.05614_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05614", "page": 1, "id": "2503.05614_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "R\\Gamma_{\\text{global}}(E, \\mathcal{D}) \\simeq R\\text{Hom}(R\\Gamma_{\\text{global}}(E, \\mathcal{D}), \\mathbb{Q}/\\mathbb{Z}(1))[1]"}
{"pdf": "arxiv_math/2503.05614_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05614", "page": 1, "id": "2503.05614_pg12_math_001", "type": "math", "max_diffs": 0, "checked": "verified", "math": "C^\\bullet(E) \\simeq R\\text{Hom}(C^\\bullet(E), \\mathbb{Q}/\\mathbb{Z}(1))[1]"}
{"pdf": "arxiv_math/2503.05614_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05614", "page": 1, "id": "2503.05614_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Lambda(E, s) = \\varepsilon_E \\cdot \\Lambda(E, 2 - s)"}
{"pdf": "arxiv_math/2503.05614_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05614", "page": 1, "id": "2503.05614_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "P, Q \\in E(\\mathbb{Q})"}
{"pdf": "arxiv_math/2503.05614_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05614", "page": 1, "id": "2503.05614_pg12_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E(\\mathbb{Q})/E(\\mathbb{Q})_{\\text{tors}}"}
{"pdf": "arxiv_math/2503.05614_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05614", "page": 1, "id": "2503.05614_pg12_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "R\\Gamma(\\mathbb{Q}_v, \\mathcal{D}_v) \\simeq R\\text{Hom}(R\\Gamma(\\mathbb{Q}_v, \\mathcal{D}_v), \\mathbb{Q}/\\mathbb{Z}(1))[1]"}
{"pdf": "arxiv_math/2503.05360_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05360", "page": 1, "id": "2503.05360_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\vdash \\phi \\quad \\text{iff} \\quad M \\vdash g \\tag{\\text{$\\dagger$}}"}
{"pdf": "arxiv_math/2503.05360_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05360", "page": 1, "id": "2503.05360_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi = \\chi_1 \\land \\chi_2"}
{"pdf": "arxiv_math/2503.08031_pg36.pdf", "url": "https://arxiv.org/pdf/2503.08031", "page": 1, "id": "2503.08031_pg36_math_001", "type": "math", "max_diffs": 0, "checked": "verified", "math": "h(x,x') = \\sum_{k=1}^{K} \\phi_k(x)\\phi_k(x')"}
{"pdf": "arxiv_math/2503.08031_pg36.pdf", "url": "https://arxiv.org/pdf/2503.08031", "page": 1, "id": "2503.08031_pg36_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma\\in \\mathbb{R}^{K\\times K}"}
{"pdf": "arxiv_math/2503.08031_pg36.pdf", "url": "https://arxiv.org/pdf/2503.08031", "page": 1, "id": "2503.08031_pg36_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "h: \\mathbb{R}^p \\times \\mathbb{R}^p \\to \\mathbb{R}"}
{"pdf": "arxiv_math/2503.08031_pg36.pdf", "url": "https://arxiv.org/pdf/2503.08031", "page": 1, "id": "2503.08031_pg36_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_1,\\dots,\\phi_K:\\mathbb{R}^p\\to\\mathbb{R}"}
{"pdf": "arxiv_math/2503.08031_pg36.pdf", "url": "https://arxiv.org/pdf/2503.08031", "page": 1, "id": "2503.08031_pg36_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi(X_1) = (\\phi_1(X_1),\\ldots,\\phi_{K}(X_1))"}
{"pdf": "arxiv_math/2503.05717_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05717", "page": 1, "id": "2503.05717_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta=0, \\,-10, \\,-20, \\,-30"}
{"pdf": "arxiv_math/2503.05717_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05717", "page": 1, "id": "2503.05717_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sqrt{2 \\pi r} T_{22}"}
{"pdf": "arxiv_math/2503.05717_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05717", "page": 1, "id": "2503.05717_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "10^4\\text{mm}^{1/2}\\text{Pa}"}
{"pdf": "arxiv_math/2503.04108_pg45.pdf", "url": "https://arxiv.org/pdf/2503.04108", "page": 1, "id": "2503.04108_pg45_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{g}\\supset \\mathfrak{g}^{\\prime}"}
{"pdf": "arxiv_math/2503.04108_pg45.pdf", "url": "https://arxiv.org/pdf/2503.04108", "page": 1, "id": "2503.04108_pg45_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{su}(4) \\supset \\mathfrak{su}(2) \\times \\mathfrak{su}(2)"}
{"pdf": "arxiv_math/2503.08522_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08522", "page": 1, "id": "2503.08522_pg23_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|v\\|_2 \\leq \\|v\\|_q"}
{"pdf": "arxiv_math/2503.08675_pg30.pdf", "url": "https://arxiv.org/pdf/2503.08675", "page": 1, "id": "2503.08675_pg30_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline r(t)-\\ell_t"}
{"pdf": "arxiv_math/2503.08675_pg30.pdf", "url": "https://arxiv.org/pdf/2503.08675", "page": 1, "id": "2503.08675_pg30_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\liminf_{i\\to\\infty}d(i)\\geq R"}
{"pdf": "arxiv_math/2503.08675_pg30.pdf", "url": "https://arxiv.org/pdf/2503.08675", "page": 1, "id": "2503.08675_pg30_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "K_0:=C(\\lambda^*+R)^{-1}"}
{"pdf": "arxiv_math/2503.05752_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05752", "page": 1, "id": "2503.05752_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon\\gtrsim0.5"}
{"pdf": "arxiv_math/2503.04620_pg35.pdf", "url": "https://arxiv.org/pdf/2503.04620", "page": 1, "id": "2503.04620_pg35_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\log(z) = \\log|z| + i(\\arg(z) + 2\\pi m), \\forall m \\in \\mathbb{Z}"}
{"pdf": "arxiv_math/2503.04620_pg35.pdf", "url": "https://arxiv.org/pdf/2503.04620", "page": 1, "id": "2503.04620_pg35_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "B_{kj} = \\begin{cases} \\delta_{k, j-1}, & \\text{for } k = 1, 2, \\ldots, 2r-1 ,\\\\ -\\dfrac{h_{j-1}}{\\tilde{a}_r - i \\tilde{b}_r}, & \\text{for } k = 2r, \\end{cases}"}
{"pdf": "arxiv_math/2503.04620_pg35.pdf", "url": "https://arxiv.org/pdf/2503.04620", "page": 1, "id": "2503.04620_pg35_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "x_t = (\\arg(z_t) + 2\\pi m) - i \\log|z_t|, \\quad t = 1, 2, \\ldots, 2r, \\quad \\forall m \\in \\mathbb{Z}"}
{"pdf": "arxiv_math/2503.04620_pg35.pdf", "url": "https://arxiv.org/pdf/2503.04620", "page": 1, "id": "2503.04620_pg35_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{B} = \\begin{bmatrix} 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ -\\frac{\\tilde{a}_2 + i \\tilde{b}_2}{\\tilde{a}_2 - i \\tilde{b}_2} & -\\frac{\\tilde{a}_1 + i \\tilde{b}_1}{\\tilde{a}_2 - i \\tilde{b}_2} & 0 & - \\frac{\\tilde{a}_1 - i \\tilde{b}_1}{\\tilde{a}_2 - i \\tilde{b}_2} \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.04620_pg35.pdf", "url": "https://arxiv.org/pdf/2503.04620", "page": 1, "id": "2503.04620_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "h_j = \\begin{cases} \\tilde{a}_{r-j} + i \\tilde{b}_{r-j}, & j = 0, 1, \\ldots, r-1 ,\\\\ 2 \\tilde{a}_0, & j = r, \\\\ \\tilde{a}_{j-r} - i \\tilde{b}_{j-r}, & j = r+1, r+2, \\ldots, 2r. \\end{cases}"}
{"pdf": "arxiv_math/2503.04620_pg35.pdf", "url": "https://arxiv.org/pdf/2503.04620", "page": 1, "id": "2503.04620_pg35_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{a}_k = \\begin{cases} 0, & \\text{for } k = 0\\\\ b_k k, & \\text{for } k = 1, 2, \\ldots, r \\end{cases} \\quad \\text{and} \\quad \\tilde{b}_k = -a_k k, \\text{for } k = 1, 2, \\ldots, r"}
{"pdf": "arxiv_math/2503.04620_pg35.pdf", "url": "https://arxiv.org/pdf/2503.04620", "page": 1, "id": "2503.04620_pg35_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "x_t = \\arg(z_t) + 2\\pi m, \\quad \\text{when} \\quad |z_t| = 1"}
{"pdf": "arxiv_math/2503.05276_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05276", "page": 1, "id": "2503.05276_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "V(s) \\approx \\hat{v}_w(s) \\coloneqq w^\\intercal \\psi(s)"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "3<p<\\infty, ~1<q<\\infty"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "S_F: U(b)\\subset L^q([0,b], H_p)\\multimap L^q([0,b],L^p(D)^3)"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "F(t,\\cdot): H_p\\multimap L^p(D)^3"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "F: [0,a]\\times H_p \\multimap L^p(D)^3"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta_F:[0,\\infty)\\to [0,\\infty)"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "S_F(u)=\\{f\\in L^q([0,b],L^p(D)^3): f(t)\\in F(t,u(t)),~\\text{a.a.}~t\\in [0,b]\\}"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "p\\in L^q([0,b], W^{1,p}(D)^3)"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(D,f_1,f_2)\\in \\Sigma\\times K\\times K"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "F: Z\\multimap L^1(I, X)"}
{"pdf": "arxiv_math/2503.08351_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08351", "page": 1, "id": "2503.08351_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\in L^q([0,b], L^p(D)^3)"}
{"pdf": "arxiv_math/2503.09178_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09178", "page": 1, "id": "2503.09178_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{ \\begin{aligned} \\varphi(0,\\mu) &= \\mu^{2} + const, \\quad \\text{if } \\mu > 0, \\\\ \\varphi(1,\\mu) &= \\mu^{2} + const, \\quad \\text{if } \\mu < 0. \\end{aligned} \\right."}
{"pdf": "arxiv_math/2503.09178_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09178", "page": 1, "id": "2503.09178_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi(x,\\mu) = \\mu^{2}\\cos^{4}\\pi x + const"}
{"pdf": "arxiv_math/2503.09178_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09178", "page": 1, "id": "2503.09178_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "s(x,\\mu) = -4 \\pi \\mu^{3} \\cos^{3}\\pi x \\sin\\pi x + \\Sigma_{t}(\\mu^{2}\\cos^{4}\\pi x + const) - \\Sigma_{s}(const + \\frac{\\cos^{4}\\pi x}{3})"}
{"pdf": "arxiv_math/2503.09178_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09178", "page": 1, "id": "2503.09178_pg16_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma_{t} = 22000, \\quad \\Sigma_{s} = 1"}
{"pdf": "arxiv_math/2503.09178_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09178", "page": 1, "id": "2503.09178_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Vert u-u_{N}^{M} \\Vert_{L^{2}}"}
{"pdf": "arxiv_math/2503.06379_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06379", "page": 1, "id": "2503.06379_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f:\\mathcal{P}\\rightarrow\\mathcal{Q}"}
{"pdf": "arxiv_math/2503.06379_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06379", "page": 1, "id": "2503.06379_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f : (\\mathcal{P},\\leq_\\mathcal{P})\\rightarrow(\\mathcal{Q},\\leq_\\mathcal{Q})"}
{"pdf": "arxiv_math/2503.06379_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06379", "page": 1, "id": "2503.06379_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|f|, |g|: |\\mathcal{P}| \\to |\\mathcal{Q}|"}
{"pdf": "arxiv_math/2503.06379_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06379", "page": 1, "id": "2503.06379_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "|f| : |\\mathcal{P}|\\rightarrow|\\mathcal{Q}|"}
{"pdf": "arxiv_math/2503.06379_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06379", "page": 1, "id": "2503.06379_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f \\circ g \\simeq \\text{Id}_Y"}
{"pdf": "arxiv_math/2503.06379_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06379", "page": 1, "id": "2503.06379_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "g \\circ f \\simeq \\text{Id}_X"}
{"pdf": "arxiv_math/2503.06379_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06379", "page": 1, "id": "2503.06379_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "f(x) \\leq_\\mathcal{Q} f(y)"}
{"pdf": "arxiv_math/2503.07251_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07251", "page": 1, "id": "2503.07251_pg28_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau_{1},\\ldots, \\tau_{L^{G}}"}
{"pdf": "arxiv_math/2503.07251_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07251", "page": 1, "id": "2503.07251_pg28_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta \\tau=\\tau_{i+1}-\\tau_{i}"}
{"pdf": "arxiv_math/2503.07251_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07251", "page": 1, "id": "2503.07251_pg28_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "G_l, l=1\\ldots,L^{G}=8"}
{"pdf": "arxiv_math/2503.03827_pg10.pdf", "url": "https://arxiv.org/pdf/2503.03827", "page": 1, "id": "2503.03827_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "I = \\langle f(x,y),~g(x,y) \\rangle"}
{"pdf": "arxiv_math/2503.03827_pg10.pdf", "url": "https://arxiv.org/pdf/2503.03827", "page": 1, "id": "2503.03827_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\vec{a}_1 = (0, \\frac{n}{2}), \\quad \\vec{a}_2 = (1, \\gamma), \\quad \\text{with } 0 \\leq \\gamma < \\frac{n}{2}"}
{"pdf": "arxiv_math/2503.03827_pg10.pdf", "url": "https://arxiv.org/pdf/2503.03827", "page": 1, "id": "2503.03827_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "[[254, 28, 14 \\leq d \\leq 20]]"}
{"pdf": "arxiv_math/2503.08770_pg37.pdf", "url": "https://arxiv.org/pdf/2503.08770", "page": 1, "id": "2503.08770_pg37_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "M_{z+w}\\otimes N_w\\otimes P_0"}
{"pdf": "arxiv_math/2503.08770_pg37.pdf", "url": "https://arxiv.org/pdf/2503.08770", "page": 1, "id": "2503.08770_pg37_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "M_{z+w}\\otimes (N_w\\otimes P_0)"}
{"pdf": "arxiv_math/2503.08770_pg37.pdf", "url": "https://arxiv.org/pdf/2503.08770", "page": 1, "id": "2503.08770_pg37_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(1\\otimes \\Delta_w) \\Delta_{z+w}=(\\Delta_z\\otimes 1)\\Delta_w"}
{"pdf": "arxiv_math/2503.08770_pg37.pdf", "url": "https://arxiv.org/pdf/2503.08770", "page": 1, "id": "2503.08770_pg37_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(M_z\\otimes N)_w\\otimes P_0"}
{"pdf": "arxiv_math/2503.08646_pg17.pdf", "url": "https://arxiv.org/pdf/2503.08646", "page": 1, "id": "2503.08646_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{u^\\perp}+\\mu_{v^\\perp}=0"}
{"pdf": "arxiv_math/2503.08646_pg17.pdf", "url": "https://arxiv.org/pdf/2503.08646", "page": 1, "id": "2503.08646_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "w\\in \\mathrm{T}_p\\mu^{-1}(0)/\\mathrm{T}_pL"}
{"pdf": "arxiv_math/2503.08646_pg17.pdf", "url": "https://arxiv.org/pdf/2503.08646", "page": 1, "id": "2503.08646_pg17_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{T}_p\\mu^{-1}(0)"}
{"pdf": "arxiv_math/2503.08646_pg17.pdf", "url": "https://arxiv.org/pdf/2503.08646", "page": 1, "id": "2503.08646_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\mathrm{Span}\\left(\\Bigl\\{ u_i \\Bigr\\}_{i=2}^n\\right)=\\mathrm{Span}\\left(\\Bigl\\{ v_i^\\ast \\Bigr\\}_{i=2}^n\\right)}"}
{"pdf": "arxiv_math/2503.08646_pg17.pdf", "url": "https://arxiv.org/pdf/2503.08646", "page": 1, "id": "2503.08646_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{T}_p\\mu^{-1}(0)\\simeq \\mathrm{T}_pL"}
{"pdf": "arxiv_math/2503.08646_pg17.pdf", "url": "https://arxiv.org/pdf/2503.08646", "page": 1, "id": "2503.08646_pg17_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "d\\mu[v]=\\omega(\\bullet, v)"}
{"pdf": "arxiv_math/2503.08646_pg17.pdf", "url": "https://arxiv.org/pdf/2503.08646", "page": 1, "id": "2503.08646_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_u^\\perp=\\sum_{i=2}^n\\,\\left(u_i\\otimes u_i^\\dagger-u_i^\\ast \\otimes u_i^t\\right)"}
{"pdf": "arxiv_math/2503.08646_pg17.pdf", "url": "https://arxiv.org/pdf/2503.08646", "page": 1, "id": "2503.08646_pg17_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(w, u^\\perp)\\mapsto \\Bigl(u, v, w\\Bigr)"}
{"pdf": "arxiv_math/2503.08646_pg17.pdf", "url": "https://arxiv.org/pdf/2503.08646", "page": 1, "id": "2503.08646_pg17_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Span}(\\mathrm{T}_pL, w)"}
{"pdf": "arxiv_math/2503.09588_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09588", "page": 1, "id": "2503.09588_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{\\alpha}\\colon S_0\\to S_1"}
{"pdf": "arxiv_math/2503.09588_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09588", "page": 1, "id": "2503.09588_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\iota_1\\circ c' = \\iota_0\\circ c\\circ \\alpha^{-1}"}
{"pdf": "arxiv_math/2503.09588_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09588", "page": 1, "id": "2503.09588_pg25_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{ac}\\colon X\\to S_{ac}"}
{"pdf": "arxiv_math/2503.09588_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09588", "page": 1, "id": "2503.09588_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{bd}\\colon X\\to S_{bd}"}
{"pdf": "arxiv_math/2503.09588_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09588", "page": 1, "id": "2503.09588_pg25_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{H_1,\\dots, H_{k-1}, K_k\\}"}
{"pdf": "arxiv_math/2503.09588_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09588", "page": 1, "id": "2503.09588_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{ab}\\colon X\\to S_{ab}"}
{"pdf": "arxiv_math/2503.09588_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09588", "page": 1, "id": "2503.09588_pg25_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "c'\\circ\\alpha = \\overline{\\alpha} \\circ c"}
{"pdf": "arxiv_math/2503.09588_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09588", "page": 1, "id": "2503.09588_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{cd}\\colon X\\to S_{cd}"}
{"pdf": "arxiv_math/2503.04448_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04448", "page": 1, "id": "2503.04448_pg10_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "d(X_1,X_2) + d(X_2,X_1) = 1"}
{"pdf": "arxiv_math/2503.04448_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04448", "page": 1, "id": "2503.04448_pg10_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_i/(\\sum_j \\lambda_j)"}
{"pdf": "arxiv_math/2503.05358_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05358", "page": 1, "id": "2503.05358_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v_{1,{\\max}} = \\sqrt{C3} + v^E = \\sqrt{C3} + \\sqrt {\\frac{\\mu^S}{r^E}}"}
{"pdf": "arxiv_math/2503.05358_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05358", "page": 1, "id": "2503.05358_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cos(\\theta_{12}) = \\frac{1}{e} \\left( \\frac{p}{r^F} - 1 \\right)"}
{"pdf": "arxiv_math/2503.05358_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05358", "page": 1, "id": "2503.05358_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "l^F_2 = \\Omega^F + \\omega^F + n^F (t_2 - t^F_p)"}
{"pdf": "arxiv_math/2503.05358_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05358", "page": 1, "id": "2503.05358_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\max} = \\frac{v^2_{1,{\\max}}}{2} - \\frac{\\mu^S}{r^E} \\to a_{\\max} = \\frac{-\\mu^S}{2E_{\\max}} \\to e_{max} = 1 - \\frac{r^E}{a_{\\max}}"}
{"pdf": "arxiv_math/2503.05358_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05358", "page": 1, "id": "2503.05358_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{\\min} = \\frac{r^E + r^F}{2} \\to e_{\\min} = 1 - \\frac{r^E}{a_{\\min}}"}
{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d(u_k,\\mathcal{M})\\to L\\in\\left[0,\\sqrt{\\mu_{s,t}}\\right]"}
{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "d(u_k,\\mathcal{M})<\\sqrt{\\mu_{s,t}}=\\|u_k\\|_{\\dot{H}^s}"}
{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "L=0<\\sqrt{\\mu_{s,t}}"}
{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{s,t}^{\\frac{1}{2_s^*(t)-2}}u"}
{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "d(u_k,\\mathcal{M})\\leq d(u_k,0)=\\|u_k\\|_{\\dot{H}^s}=\\sqrt{\\mu_{s,t}}<\\infty"}
{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{s,t}(u_n)\\to \\beta"}
{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u_k\\|^2_{\\dot{H}^s}=\\mu_{s,t}"}
{"pdf": "arxiv_math/2503.08597_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08597", "page": 1, "id": "2503.08597_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\Sigma} \\triangleq \\max_{(d_1,\\cdots, d_K)\\in \\mathcal{D}}(d_1+\\cdots +d_K)"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R_{\\bf T}=\\{e^{t T}, t \\in \\R \\}"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi \\in \\mathfrak{k}"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\R H \\oplus \\R L"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "p(\\R_{\\bf T}) = \\overline{p(\\Gamma)}"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Psi \\in \\mathfrak{k}"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lfloor \\frac{n}{2} \\rfloor"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bf T}=\\lambda+\\Phi+\\omega"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{p(\\Gamma)} = \\R_{\\bf H} \\times \\R_{\\bf L}"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{p(\\Gamma)} = \\R"}
{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R_{\\bf H} \\times \\R_{\\bf L} \\simeq \\R^2"}
{"pdf": "arxiv_math/2503.05558_pg18.pdf", "url": "https://arxiv.org/pdf/2503.05558", "page": 1, "id": "2503.05558_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "E = 41.9 log_2(B)^{-1}-1.92"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tilde \\Omega, \\tilde{\\mathscr{F}}, \\tilde{\\mathbb{P}})"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma: [0, T]\\times \\mathbb{R}^d \\times S \\mapsto \\mathbb{R}^{d \\times \\tilde{d}}"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "b:[0, T]\\times \\mathbb{R}^d \\times S \\mapsto \\mathbb{R}^d"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "r:=\\{r(t)\\}_{t\\geq 0}"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathbb{F}}^r := \\{\\tilde{\\mathscr F}_t^r\\}_{t\\geq 0}"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathbb{F}}^{B}:=\\{\\tilde{\\mathscr F}_t^{B}\\}_{t\\geq 0}"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "B:=\\{B(t)\\}_{t\\geq 0}"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{q}_{j_0 k_0}\\geq 0"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathscr F}_0^B"}
{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathscr F}_t:=\\tilde{\\mathscr F}_t^{B} \\vee \\tilde{\\mathscr F}_t^r"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f_\\pm(x_1,x_2)=\\pm \\Delta(x_1,x_2)(x_1,x_2,1)"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "m(u,v)=(u^2+v^2+1)^{-\\frac{1}{2}(n-1)}"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta(x_1,x_2)=(x_1^2+x_2^2+1)^{-\\frac{1}{2}}"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{Fix}(Q)\\subset\\mathbb{R}^m"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_i:U_i\\rightarrow\\mathbb{R}^2"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "Q(x_1,\\dots,x_m)=(x_1,\\dots,x_r,-x_{r+1},\\dots,-x_m)"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "H_+=\\{y\\in\\mathbb{S}^2:y_3>0\\}"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} &v^n\\;m(u,v)\\left(Q\\left(\\frac{1}{v},\\frac{u}{v}\\right)-uP\\left(\\frac{1}{v},\\frac{u}{v}\\right),-vP\\left(\\frac{1}{v},\\frac{u}{v}\\right)\\right) \\text{ in } U_1, \\\\ &v^n\\;m(u,v)\\left(P\\left(\\frac{u}{v},\\frac{1}{v}\\right)-uQ\\left(\\frac{u}{v},\\frac{1}{v}\\right),-vQ\\left(\\frac{u}{v},\\frac{1}{v}\\right)\\right) \\text{ in } U_2, \\\\ &m(u,v)(P(u,v),Q(u,v)) \\text{ in } U_3, \\end{aligned}"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{S}^2=\\{y=(y_1,y_2,y_3)\\in\\mathbb{R}^3:y_1^2+y_2^2+y_3^2=1\\}"}
{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "H_-=\\{y\\in\\mathbb{S}^2:y_3<0\\}"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\left (\\bigcup_{x\\in \\partial M} i |P_n\\left [ -\\sqrt{\\max(0,N^2)},\\sqrt{\\max(0,N^2)}\\right ]\\right )^c"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\cdot (\\rho_0 w_v) = 0"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\cdot (\\rho_0 z_v) + \\frac{g_0'}{c^2} \\cdot \\nabla \\times (\\rho_0 w_v) + \\frac{\\rho_0 g_0'}{c^2} \\cdot z_v = 0 , \\quad n \\cdot z_v|_{\\partial M} = 0"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\times (\\rho_0 z_v) = 0"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "w \\in \\operatorname{Ker}(T)"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\sigma_{ess}(L)^c"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "w = \\nabla \\times (\\rho_0 w_v) + \\nabla \\varphi_v"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_0 z_v = \\nabla \\varphi_v"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in \\operatorname{Ker}(T)"}
{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "u = \\nabla \\times (\\rho_0 w_u) + \\rho_0 z_u"}
{"pdf": "arxiv_math/2503.04438_pg16.pdf", "url": "https://arxiv.org/pdf/2503.04438", "page": 1, "id": "2503.04438_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "O^\\sigma = L(O_{h\\Sigma_n})\\times_{LB\\Sigma_n}\\{\\sigma\\}"}
{"pdf": "arxiv_math/2503.07105_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07105", "page": 1, "id": "2503.07105_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "i \\in \\set{1,\\ldots,n}"}
{"pdf": "arxiv_math/2503.07105_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07105", "page": 1, "id": "2503.07105_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "w_{k+1} := w_k + \\tau_k r(w_k)"}
{"pdf": "arxiv_math/2503.07105_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07105", "page": 1, "id": "2503.07105_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "2 (w_l - w_r) \\frac{q_{1,1}^e q_{2,2}^e - q_{2,1}^e q_{1,2}^e}{2} "}
{"pdf": "arxiv_math/2503.07105_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07105", "page": 1, "id": "2503.07105_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(w_{k+1} - w_k)/\\tau_k = r(w_k)"}
{"pdf": "arxiv_math/2503.03754_pg10.pdf", "url": "https://arxiv.org/pdf/2503.03754", "page": 1, "id": "2503.03754_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{XY}=\\begin{pmatrix} 1-s & 0 \\\\ sd & s(1-d) \\end{pmatrix}"}
{"pdf": "arxiv_math/2503.03754_pg10.pdf", "url": "https://arxiv.org/pdf/2503.03754", "page": 1, "id": "2503.03754_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "x_2=\\frac{m-svt}{1-st}"}
{"pdf": "arxiv_math/2503.03754_pg10.pdf", "url": "https://arxiv.org/pdf/2503.03754", "page": 1, "id": "2503.03754_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "s=\\frac{m-x_2}{t(x_1-x_2)}"}
{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{2}\\delta x_{s}^{1}(\\xi_i)}"}
{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{l}\\delta x_{s}^{l-1}(\\xi_i)}"}
{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{Z}^{W\\delta x}"}
{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{det}(C) \\neq 0"}
{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathring{\\chi}_{t, i}"}
{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{L\\top}dh_{s}^{L}(\\xi_i)}"}
{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{l\\top}dh_{s}^{l}(\\xi_i)}"}
{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{u}\\in V\\cap H^2(D)^2"}
{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "H_N = \\text{span\\;}\\{e_k : k = 1,\\dots, N \\}"}
{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} &|\\sigma(\\mathbf{u})|^2=\\sum_{k=1}^{m}|\\sigma_k(\\mathbf{u})|^2\\leq B_0,\\text{ for all }\\mathbf{u}\\in H;\\\\ &|\\sigma(\\mathbf{u})-\\sigma(\\mathbf{v})|^2=\\sum_{k=1}^{m}|\\sigma_k(\\mathbf{u})-\\sigma_k(\\mathbf{v})|^2\\leq L|\\mathbf{u}-\\mathbf{v}|^2,\\text{ for all }\\mathbf{u},\\mathbf{v}\\in H. \\end{aligned}"}
{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\sigma(\\mathbf{u})^{-1}(P_N\\mathbf{w})|\\leq C_0|P_N\\mathbf{w}|,\\text{ for all }\\mathbf{u},\\mathbf{w}\\in H"}
{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_005", "type": "math", "max_diffs": 0, "checked": "verified", "math": "V := \\{\\mathbf{u}\\in H^1(D)^2 :\\nabla\\cdot\\mathbf{u}=0,\\; \\mathbf{u}|_{\\partial D}=0\\}"}
{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "P_NH\\subset \\text{Range}\\;(\\sigma(\\mathbf{u})),\\text{ for all }\\mathbf{u}\\in H"}
{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_k^{-1}:P_NH\\rightarrow H"}
{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma=(\\sigma_1,\\dots,\\sigma_m)"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_t(\\cdot )=\\sigma(\\cdot ,t)"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x,T\\circ (\\sigma_t)^{-1}(x)"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\sigma(E,t)} \\rho_0(x)\\,dx=\\int_E \\rho_0(x)\\,dx"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{T^{-1}(E)}\\rho_0(x)\\,dx=\\int_E \\rho_1(x)\\,dx"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "T:(\\Omega_0,\\rho_0)\\to (\\Omega_1,\\rho_1)"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "T\\circ (\\sigma_t)^{-1}"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "T\\circ (\\sigma_t)^{-1}:(\\Omega_0,\\rho_0)\\to (\\Omega_1,\\rho_1)"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(y_1,t+0)=\\sigma(y_2,t+0)"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_0(\\sigma(z,t))\\,J_\\sigma(z)=\\rho_0(z)"}
{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "(t-\\delta,t+\\delta)\\subset J"}
{"pdf": "arxiv_math/2503.03861_pg30.pdf", "url": "https://arxiv.org/pdf/2503.03861", "page": 1, "id": "2503.03861_pg30_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1(X, \\mathscr O_X) = 0"}
{"pdf": "arxiv_math/2503.03861_pg30.pdf", "url": "https://arxiv.org/pdf/2503.03861", "page": 1, "id": "2503.03861_pg30_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1( X;\\mathbb C) = 0"}
{"pdf": "arxiv_math/2503.03861_pg30.pdf", "url": "https://arxiv.org/pdf/2503.03861", "page": 1, "id": "2503.03861_pg30_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "R := \\mathbb Z[1/2|G|]"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{n}(t)=[n_0(t),n_1(t),\\cdots,n_{M_r-1}(t)]^T"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{t,m}=\\sum_{i=0}^{m}d_{t,i},m\\in \\mathcal{M}_t\\triangleq[1,2,\\cdots,M_t-1]"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta\\in[-\\frac{\\pi}{2},\\frac{\\pi}{2}]"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\hat{\\tau},\\hat{v},\\theta)"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{m,q} \\in \\mathbb{K}"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{C}^{n\\times m}"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x}_{r}=[x_{r,0},x_{r,1},\\cdots,x_{r,M_r-1}]^T"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{a}(\\mathbf{x}_t,\\alpha)=[1,e^{j\\frac{2\\pi}{\\lambda}x_{t,1}\\sin\\alpha},\\cdots,e^{j\\frac{2\\pi}{\\lambda}x_{t,M_t-1}\\sin\\alpha}]^T"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{r,m}=\\frac{\\lambda}{2}m,m\\in\\mathcal{M}_r"}
{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{m,q}\\neq c_{m',q}, \\forall q, m\\neq m'"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_0 \\in C^0(\\overline\\Omega)"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_h^0 = \\pi^h \\varphi_0"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta_o = |\\max_o |\\phi_h^n| - 1 | > 0.5"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "h_c = \\frac{L_d}{N_c}"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "N_f = 8N_c = 2^{3+k} L_d"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega = \\prod_{i=1}^d (0,L_i) \\subset \\mathbb R^d"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "S^h = \\{ \\zeta \\in C^0(\\overline{\\Omega}) \\, : \\, \\zeta \\vert_{o} \\in P_1(o) \\quad \\forall o \\in \\mathcal{T}_h\\}"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\phi_h^{n+1}, \\mu_h^{n+1}) \\in S^h \\times S^h"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "h_f = \\frac{L_d}{N_f}"}
{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "d_0 : \\overline\\Omega \\to \\mathbb R"}
{"pdf": "arxiv_math/2503.09424_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09424", "page": 1, "id": "2503.09424_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\in \\{2,\\ldots,n-1\\}"}
{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi _{2k+1}(f)^2\\leq \\lambda _k(f)\\lambda _{k+1}(f)"}
{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\times f:X\\times X\\rightarrow X\\times X"}
{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\max _{j=0,\\ldots ,n}\\lambda _j(Fr_q^s\\circ f^t)"}
{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi _{2k}(f)=\\lambda _k(f)"}
{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda _k(Fr_q^s\\circ f^t)"}
{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "b_0,b_1,\\ldots ,b_{2n}"}
{"pdf": "arxiv_math/2503.07583_pg22.pdf", "url": "https://arxiv.org/pdf/2503.07583", "page": 1, "id": "2503.07583_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} A_{a,b,c} = 2(a+b+c) + 3 \\\\ C_{a,b|c,d} = 3 (2a+1)(2b+1) \\\\ G_{a} = (2a+1)(2a^2 + 2a + 1) \\end{aligned}"}
{"pdf": "arxiv_math/2503.07583_pg22.pdf", "url": "https://arxiv.org/pdf/2503.07583", "page": 1, "id": "2503.07583_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon_{[8,7]} = 8^3 + 7^3 = 855 = 9^3 + 5^3 + 1^3 = \\varepsilon_{[9,5,1]}"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_4\\|\\Z\\|_{F,1}^\\psi"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "r^0 = \\min\\{n_1, n_2\\}"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma(\\Z)=\\left\\lbrace j \\mid\\|\\Z(:,j,:)\\| \\neq 0, \\, j=1, \\ldots, r\\right\\rbrace"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varpi = \\max_{i\\in [6]}\\{\\rho_i+l_g\\}"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Z \\in \\mathbb{R}^{n_2 \\times r \\times b}"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma(\\Z^t)=\\Gamma(\\Z^{t+1})"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma(\\Z^t)\\neq\\Gamma(\\Z^{t+1})"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\Z^{t+1}-\\Z^t\\| < \\min\\{(2\\hat{\\lambda}_4(1-p))^{1/(2-p)}, \\nu\\}"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\|\\Z^{t+1}-\\Z^t\\right\\|\\ge\\left\\|\\Z^{t+1}(:,j,:)-\\Z^t(:,j,:)\\right\\|\\ge \\min\\{(2\\hat{\\lambda}_4(1-p))^{1/(2-p)}, \\nu\\}"}
{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{t\\to\\infty} \\|\\Z^{t+1}-\\Z^t\\| =0"}
{"pdf": "arxiv_math/2503.09021_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09021", "page": 1, "id": "2503.09021_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{supp}(m)\\subset B_\\rho:=\\{x\\in\\mathbb{R}^2:|x|<\\rho\\}"}
{"pdf": "arxiv_math/2503.09021_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09021", "page": 1, "id": "2503.09021_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "d\\in\\mathbb{S}^1:=\\{x\\in\\mathbb{R}^2:|x|=1\\}"}
{"pdf": "arxiv_math/2503.09021_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09021", "page": 1, "id": "2503.09021_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "u^i=u^i(x,d):=e^{ikx\\cdot d}"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x_n\\in \\partial (t_n \\widetilde{W} )"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(t_n+ t )\\widetilde w (e)e"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "x_n\\in t_n \\widetilde{W}"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "x_n/t_n\\in \\partial\\widetilde{W}"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_n \\to \\xi_\\infty \\in [0,1]^N,\\qquad x_n/t_n\\to \\zeta\\in \\partial\\widetilde{W}"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{R} (t_n) \\widetilde{W}"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "(t,x)\\in[0,+\\infty)\\times\\R^n"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_n := x_n - h_n \\in [0,1)^N"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "(t_n+t) \\widetilde{W}"}
{"pdf": "arxiv_math/2503.07128_pg28.pdf", "url": "https://arxiv.org/pdf/2503.07128", "page": 1, "id": "2503.07128_pg28_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\in\\partial\\big(t_n\\widetilde{W}-\\{x_n\\}\\big)"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_hF:=\\dfrac{1}{2}\\left(\\dfrac{\\partial F}{\\partial \\alpha_h}-\\mathcal{J}_h\\left(\\dfrac{\\partial F}{\\partial\\beta_h}\\right)\\right),\\qquad \\overline{\\partial}_hF:=\\dfrac{1}{2}\\left(\\dfrac{\\partial F}{\\partial \\alpha_h}+\\mathcal{J}_h\\left(\\dfrac{\\partial F}{\\partial\\beta_h}\\right)\\right)"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "F\\in Stem(D)\\cap\\mathcal{C}^1(D)"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "z=(z_1,\\dots,z_n)\\in\\mathbb{C}^n"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{e_K\\}_{K\\in\\mathcal{P}(n)}"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J}=\\left\\{\\mathcal{J}_h: \\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}\\rightarrow \\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}\\right\\}_{h=1}^n"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "F:D\\subset\\mathbb{C}^n\\to \\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\in D\\iff\\overline{z}^h\\in D"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "F:D\\to \\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}"}
{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_{J_1}\\times...\\times\\phi_{J_n}:\\mathbb{C}^n\\ni(z_1,...,z_n)\\mapsto\\left(\\phi_{J_1}(z_1),...,\\phi_{J_n}(z_n)\\right)\\in(\\mathbb{R}^{m+1})^n"}
{"pdf": "arxiv_math/2503.08360_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08360", "page": 1, "id": "2503.08360_pg10_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathcal{P}_\\ell(\\mathcal{T}_h, \\mathbb{R}^{m\\times n})"}
{"pdf": "arxiv_math/2503.08360_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08360", "page": 1, "id": "2503.08360_pg10_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{P}_\\ell(\\mathcal{F}_h, \\mathbb{R}^{m\\times n})"}
{"pdf": "arxiv_math/2503.04623_pg54.pdf", "url": "https://arxiv.org/pdf/2503.04623", "page": 1, "id": "2503.04623_pg54_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(0, \\mu^{\\sharp\\prime}, \\ldots, \\mu^{\\sharp\\prime})"}
{"pdf": "arxiv_math/2503.04623_pg54.pdf", "url": "https://arxiv.org/pdf/2503.04623", "page": 1, "id": "2503.04623_pg54_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu\\in X_\\bullet(G^*)"}
{"pdf": "arxiv_math/2503.04623_pg54.pdf", "url": "https://arxiv.org/pdf/2503.04623", "page": 1, "id": "2503.04623_pg54_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu^\\sharp\\in X_\\bullet(G^\\sharp)"}
{"pdf": "arxiv_math/2503.04623_pg54.pdf", "url": "https://arxiv.org/pdf/2503.04623", "page": 1, "id": "2503.04623_pg54_math_025", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_1^\\sharp, \\ldots, \\omega_{n-2}^\\sharp, \\omega_{n-1}^\\sharp+\\omega_{n-2}^\\sharp, 2\\omega_{n-1}^\\sharp, 2\\omega_{n-2}^\\sharp"}
{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\frac{1}{p}, \\frac{1}{q})"}
{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\frac{3}{8}, \\frac{1}{8})"}
{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "1 \\leq r \\leq \\frac{3}{2}"}
{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{s}_{C, \\epsilon}(p, q, r) = \\begin{cases} -\\frac{2}{q} + \\epsilon, & \\text{for } q \\geq 3p' \\text{ and } \\frac{1}{p} \\leq \\frac{1}{4}; \\\\ \\frac{2}{p} - \\frac{2}{q} - \\frac{1}{2} + \\epsilon, & \\text{for } q \\geq 3p' \\text{ and } \\frac{1}{4} < \\frac{1}{p} < \\frac{1}{2} - \\frac{1}{q}; \\\\ \\frac{1}{p} - \\frac{3}{q} + \\epsilon, & \\text{for } q \\geq 3p' \\text{ and } \\frac{1}{p} \\geq \\frac{1}{2} - \\frac{1}{q}; \\\\ \\frac{3}{2p} - \\frac{3}{2q} - \\frac{1}{2} + \\epsilon, & \\text{for } p' < q < 3p'; \\\\ \\frac{2}{p} - \\frac{1}{q} - 1 + \\epsilon, & \\text{for } q \\leq p'. \\end{cases}"}
{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\frac{1}{2}, \\frac{1}{6})"}
{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{s}_{C, 0}(p, q, r) < 0"}
{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "1 \\leq r < p \\leq q \\leq \\infty"}
{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "1 + (1 + \\omega)\\left(\\frac{1}{q} - \\frac{1}{p}\\right) > 0"}
{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{cases} \\frac{1}{q} > \\frac{2}{3p} - \\frac{1}{6}, & \\text{for } q \\geq 3p'; \\\\ \\frac{1}{q} > \\frac{1}{p} - \\frac{1}{3}, & \\text{for } p' < q < 3p'; \\\\ \\frac{1}{q} > \\frac{2}{p} - 1, & \\text{for } q \\leq p'. \\end{cases}"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "b_0 = 4(y_0 + y_1 - 2y_3), b_1 = 4(y_0 - y_3 + y_4 - y_5), b_2 = - 3y_0 - y_1 + 4y_3"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "b_3 = 4(y_0 - y_3 + y_4 - y_5), b_4 = 4(y_0 + y_2 - 2y_5), b_5 = -3y_0 - y_2 + 4y_5"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "a_0 = 4(x_0 + x_1 - 2x_3), a_1 = 4(x_0 - x_3 + x_4 - x_5), a_2 = - 3x_0 - x_1 + 4x_3"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_0 = \\arctan(\\sqrt{\\frac{\\phi_0}{\\phi_2}} \\phi_1)"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{0}^{1} (\\xi + \\phi_1)^p \\sqrt{\\phi_0(\\xi + \\phi_1 )^2 + \\phi_2}\\rm{d}\\xi = (\\frac{\\phi_2}{\\phi_0})^{\\frac{p+1}{2}} \\sqrt{\\phi_2} \\int_{\\theta_0}^{\\theta_1} tan^p \\theta sec^3 \\theta \\rm{d} \\theta"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} \\frac{\\partial x}{\\partial \\xi} = a_0\\xi + a_1\\eta + a_2, \\\\ \\frac{\\partial x}{\\partial \\eta} = a_3\\xi + a_4\\eta + a_5, \\\\ \\frac{\\partial y}{\\partial \\xi} = b_0\\xi + b_1\\eta + b_2, \\\\ \\frac{\\partial y}{\\partial \\eta} = b_3\\xi + b_4\\eta + b_5, \\\\ \\end{aligned}"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a_3 = 4(x_0 - x_3 + x_4 - x_5), a_4 = 4(x_0 + x_2 - 2x_5), a_5 = -3x_0 - x_2 + 4x_5"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "I_n = \\int \\sec^n \\theta \\rm{d} \\theta = \\frac{1}{n - 1} (tan\\theta \\sec^{n - 2} \\theta + (n - 2)I_{n-2}). \\nonumber"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int \\tan^{2k+1} \\theta \\sec^3 \\theta \\rm{d} \\theta = \\int (sec^2 \\theta - 1)^k sec^2 \\theta \\rm{d} \\sec \\theta"}
{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int \\tan^{2k} \\theta \\sec^3 \\theta \\rm{d} \\theta = \\int (sec^2 \\theta - 1)^k sec^2 \\theta \\rm{d} \\theta"}
{"pdf": "arxiv_math/2503.09254_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09254", "page": 1, "id": "2503.09254_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "M = \\{ m_1 , ..., m_r \\}"}
{"pdf": "arxiv_math/2503.09254_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09254", "page": 1, "id": "2503.09254_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "G := \\{ m_1 - \\overline{m_1}^{G_<} , ..., m_r - \\overline{m_r}^{G_<} \\}"}
{"pdf": "arxiv_math/2503.04917_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04917", "page": 1, "id": "2503.04917_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "D(-\\Delta_{g})=H^{2}(\\Omega)\\cap H_{0}^{1}(\\Omega)"}
{"pdf": "arxiv_math/2503.06838_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06838", "page": 1, "id": "2503.06838_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{X} \\in \\R^{N \\times n}"}
{"pdf": "arxiv_math/2503.06838_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06838", "page": 1, "id": "2503.06838_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T \\in \\R^{n \\times d}"}
{"pdf": "arxiv_math/2503.06838_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06838", "page": 1, "id": "2503.06838_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\min_{T} \\|\\mathbf{X} T - \\mathbf{Y}\\|^2"}
{"pdf": "arxiv_math/2503.06838_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06838", "page": 1, "id": "2503.06838_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{Y} \\in \\R^{N \\times d}"}
{"pdf": "arxiv_math/2503.04026_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04026", "page": 1, "id": "2503.04026_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{\\nu }=(\\nu _{1},\\nu _{2},\\nu _{3})"}
{"pdf": "arxiv_math/2503.04026_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04026", "page": 1, "id": "2503.04026_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "% \\mathfrak{e}=\\left( e_{1},e_{2},e_{3}\\right)"}
{"pdf": "arxiv_math/2503.07449_pg17.pdf", "url": "https://arxiv.org/pdf/2503.07449", "page": 1, "id": "2503.07449_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "q = 30 \\ {\\rm \\frac{J}{m^2}}"}
{"pdf": "arxiv_math/2503.06804_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06804", "page": 1, "id": "2503.06804_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "X^{\\text{work}}_n=S_n+I^{-}_n+R^{-}_n+R^{+}_n"}
{"pdf": "arxiv_math/2503.06804_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06804", "page": 1, "id": "2503.06804_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "X^{\\text{Test}}=I^{-}+R^{-}+S"}
{"pdf": "arxiv_math/2503.06804_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06804", "page": 1, "id": "2503.06804_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "C_T \\left(u^T_n X^{\\text{Test}}_n,\\overline{ \\text{x}}^{\\text{Test}}\\right)"}
{"pdf": "arxiv_math/2503.06804_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06804", "page": 1, "id": "2503.06804_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "X^{\\text{Test}}= N-(I^{+}+R^{+}+ H)"}
{"pdf": "arxiv_math/2503.06804_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06804", "page": 1, "id": "2503.06804_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "X^{\\text{work}}_n=N-I^+_n-H_n"}
{"pdf": "arxiv_math/2503.06804_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06804", "page": 1, "id": "2503.06804_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{ \\text{x}}^{\\text{Test}}>0"}
{"pdf": "arxiv_math/2503.06804_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06804", "page": 1, "id": "2503.06804_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "C_L \\left(u^L_n X^{ \\text{Work}}_n,0\\right)"}
{"pdf": "arxiv_math/2503.09076_pg12.pdf", "url": "https://arxiv.org/pdf/2503.09076", "page": 1, "id": "2503.09076_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\in \\{1, 2, \\ldots, t\\}"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x \\not\\in Z\\cap \\bigcap_{i=1}^nN(J_i)"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x \\in Z\\cap \\bigcap_{i=1}^nN(J)"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{sL}_\\mathbb{G}(x)"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in Y\\cap \\bigcap_{i=1}^n N(J_i)"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J}=\\{J_1,\\ldots,J_n\\}"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in \\bigcap_{J\\in \\mathcal{J}}N(J)"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J}= \\{J_1, \\ldots, J_n\\}"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "X^\\mathbb{G}:= X^\\odot \\cup \\bigcup\\limits_{x \\in X} \\mathrm{L}_\\mathbb{G}(x)"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in Y\\cap \\bigcap_{J\\in \\mathcal{J}}N(J)"}
{"pdf": "arxiv_math/2503.08411_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08411", "page": 1, "id": "2503.08411_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Cont}^\\triangle(X,\\mathbb{G})"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi(\\mathcal{G}_{n,d})"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{2}|\\mathcal{B}_{C}| \\leq (1+o(1))\\frac{n}{3}"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "s(\\mathcal{G}_{n,4}) \\leq 0.8327 n"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_005", "type": "math", "max_diffs": 0, "checked": "verified", "math": "h = 1, 2, \\dots, \\omega"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "s(\\mathcal{G}_{n,3})\\leq (1+o(1))\\frac{n}{3}"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi(\\mathcal{G}_{n,d}) = 4"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_008", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\alpha(\\mathcal{G}_{n,4}) \\leq 0.41635"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "s(\\mathcal{G}_{n,3}) = (1+o(1))\\frac{n}{4}"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "Z = \\sum_{i=0}^{i_{\\max}-1}\\sum_{h=1}^{\\omega}Z_{h}^{(i)}"}
{"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi(\\mathcal{G}_{n,d}) = (1+o(1))\\frac{n}{\\alpha(\\mathcal{G}_{n,d})} = (1+o(1))\\frac{d}{2\\log d}"}
{"pdf": "arxiv_math/2503.07741_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07741", "page": 1, "id": "2503.07741_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\rho(x_0,t)\\rangle"}
{"pdf": "arxiv_math/2503.05976_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05976", "page": 1, "id": "2503.05976_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "r \\in \\set{0} \\cup \\N \\cup \\set{\\infty}"}
{"pdf": "arxiv_math/2503.05976_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05976", "page": 1, "id": "2503.05976_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "z = (z_1, \\dots, z_n)"}
{"pdf": "arxiv_math/2503.05976_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05976", "page": 1, "id": "2503.05976_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta = (\\zeta_1, \\dots, \\zeta_n)"}
{"pdf": "arxiv_math/2503.07897_pg25.pdf", "url": "https://arxiv.org/pdf/2503.07897", "page": 1, "id": "2503.07897_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_r(0)=0 \\quad\\forall r\\geq 0"}
{"pdf": "arxiv_math/2503.07897_pg25.pdf", "url": "https://arxiv.org/pdf/2503.07897", "page": 1, "id": "2503.07897_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{\\Delta}(t) = \\mu_{r + 1}(t) - \\mu_r(t) = \\frac{\\gamma}{\\rho} \\; \\ln{(1 + \\rho \\; t)}"}
{"pdf": "arxiv_math/2503.07897_pg25.pdf", "url": "https://arxiv.org/pdf/2503.07897", "page": 1, "id": "2503.07897_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_r'(t)= \\lambda_r(t)"}
{"pdf": "arxiv_math/2503.07156_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07156", "page": 1, "id": "2503.07156_pg35_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "U\\in C([0,\\infty),X_p)"}
{"pdf": "arxiv_math/2503.07156_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07156", "page": 1, "id": "2503.07156_pg35_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_p\\!:\\!D(A_p)\\!\\subset \\!X_p\\to X_p"}
{"pdf": "arxiv_math/2503.07156_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07156", "page": 1, "id": "2503.07156_pg35_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "F^M(U(\\cdot))\\in L^1((0,t);X_p)"}
{"pdf": "arxiv_math/2503.07156_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07156", "page": 1, "id": "2503.07156_pg35_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "f_\\nu^M,\\psi^M,\\phi^M"}
{"pdf": "arxiv_math/2503.07156_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07156", "page": 1, "id": "2503.07156_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{t\\to0}e^{tA_p}U=U"}
{"pdf": "arxiv_math/2503.07156_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07156", "page": 1, "id": "2503.07156_pg35_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "C^1([0,\\infty);X_p)\\cap C([0,\\infty);D_p(A))"}
{"pdf": "arxiv_math/2503.07156_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07156", "page": 1, "id": "2503.07156_pg35_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "E:=\\{U\\in C([0,\\infty),X_p) : \\|U\\|_E=\\sup_{t\\ge0}e^{-(\\omega_p+\\theta)t}\\|U(t)\\|_{p}<\\infty\\}"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v_h = \\hat v_h \\circ \\bm F_K^{-1}"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{diam} T \\le Ch"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "B(\\bm z; r) = \\{\\bm x \\mid |\\bm x - \\bm z| \\le r\\}"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat v_h \\in \\mathbb P_k(\\hat T)"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\Delta g = \\eta \\quad\\text{in }\\; \\Omega, \\qquad g = 0 \\quad\\text{on }\\; \\Gamma"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "A(\\bm z; r, R) = \\{\\bm x \\mid r \\le |\\bm x - \\bm z|\\le R\\}"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{j=\\ell_1}^{\\ell_2} d_j^\\alpha"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{supp} \\eta \\subset \\Omega"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "v \\in W^{m, p}(\\tilde\\Omega)"}
{"pdf": "arxiv_math/2503.09190_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09190", "page": 1, "id": "2503.09190_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\emptyset \\neq T \\cap \\Gamma_h \\subset S"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "a_1(0,0)\\sharp b_2(0,0)"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "a_2(z_1,z_2)\\sharp b_1(z_1,z_2)"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\gamma}''_2(z_1,z_2)"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tilde{\\gamma}''_1(z_1,z_2), \\tilde{\\gamma}''_2(z_1,z_2))"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\gamma}_1''=\\tilde{\\gamma}''_1(0,0)"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde\\gamma_2''=\\tilde{\\gamma}''_2(0,0)"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(z_1,z_2)=(0,0) \\in [0,1]_{z_1} \\times [0,1]_{x_2}"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde\\gamma'_i(z_1,z_2)= a_i(z_1,z_2)\\sharp b_i(z_1,z_2)"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(z_1, 0) \\in [0,1] \\times [0,1]"}
{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tilde{\\gamma}'_1, \\tilde{\\gamma}'_2)=(\\tilde{\\gamma}'_1(0,0), \\tilde{\\gamma}'_2(0,0))"}
{"pdf": "arxiv_math/2503.05716_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05716", "page": 1, "id": "2503.05716_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "u(x_1,x_2,t) = t^4 + \\sin(x_1)\\cdot\\sin(x_2)\\cdot\\sin(t)"}
{"pdf": "arxiv_math/2503.05716_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05716", "page": 1, "id": "2503.05716_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega_1=[0,2\\pi]\\times[0,2\\pi], t\\in(0,2)"}
{"pdf": "arxiv_math/2503.05716_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05716", "page": 1, "id": "2503.05716_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega_2=[0,10\\pi]\\times[0,10\\pi], t\\in(0,10)"}
{"pdf": "arxiv_math/2503.05716_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05716", "page": 1, "id": "2503.05716_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{GELU}(x) = x \\cdot \\frac{1}{2}[1+erf(\\frac{x}{\\sqrt{2}})]"}
{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi\\in C^\\infty([0,\\infty))"}
{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi(r)\\sim cr^{\\frac32}"}
{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "r^{\\frac32}, r^{-\\frac12}"}
{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle L_1 \\phi , \\chi_b(r) r^{\\frac12}\\rho_1\\rangle = \\lambda_0 \\langle \\phi, \\chi_b(r) r^{\\frac12}\\rho_1\\rangle"}
{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "L_1(r^{\\frac12}\\rho_1)=0"}
{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "L_1\\phi =\\lambda_0\\phi"}
{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "q = (0.62,0.18,0.1,0.1)"}
{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "1-(1+\\frac{2}{3})^{-1} = 0.4"}
{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat q = (0.5,0.5,0,0)"}
{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta R^{2}(\\omega )"}
{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta R^{1}(\\omega )"}
{"pdf": "arxiv_math/2503.05594_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05594", "page": 1, "id": "2503.05594_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{split} \\big(D^X(s-)+\\tfrac12 \\gamma(s) \\Delta X(s)\\big)^\\top \\Delta X(s) & = D^X(s-)\\Delta X(s-) + \\tfrac12 (\\Delta X(s))^\\top \\gamma(s) \\Delta X(s) . \\end{split}"}
{"pdf": "arxiv_math/2503.05594_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05594", "page": 1, "id": "2503.05594_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "C(x,d,X) = \\int_{[0,T]} (D^X(s-))^\\top\\, dX(s) + \\frac12 \\int_{[0,T]} (\\Delta X(s))^\\top \\gamma(s)\\, dX(s)"}
{"pdf": "arxiv_math/2503.05594_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05594", "page": 1, "id": "2503.05594_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "D^X(s)=D^X(s-)+\\Delta D^X(s) = D^X(s-)+ \\gamma(s)\\Delta X(s)"}
{"pdf": "arxiv_math/2503.06167_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06167", "page": 1, "id": "2503.06167_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g_{u}(u) := \\rho \\left[ \\frac{u}{\\rho}\\right]"}
{"pdf": "arxiv_math/2503.06167_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06167", "page": 1, "id": "2503.06167_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "g_l: \\mathbb{R} \\mapsto \\mathbb{R}"}
{"pdf": "arxiv_math/2503.06167_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06167", "page": 1, "id": "2503.06167_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_{x_j} f_j(k)"}
{"pdf": "arxiv_math/2503.06167_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06167", "page": 1, "id": "2503.06167_pg12_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "g_l(\\partial_{x_j} f_j(k))"}
{"pdf": "arxiv_math/2503.09500_pg70.pdf", "url": "https://arxiv.org/pdf/2503.09500", "page": 1, "id": "2503.09500_pg70_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi=\\mathrm{BC}_E(\\pi)"}
{"pdf": "arxiv_math/2503.09500_pg70.pdf", "url": "https://arxiv.org/pdf/2503.09500", "page": 1, "id": "2503.09500_pg70_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi=\\bigotimes_v\\pi_v"}
{"pdf": "arxiv_math/2503.09500_pg70.pdf", "url": "https://arxiv.org/pdf/2503.09500", "page": 1, "id": "2503.09500_pg70_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi_{v_1}\\simeq \\pi_{v_1}\\otimes \\pi_{v_1}"}
{"pdf": "arxiv_math/2503.09500_pg70.pdf", "url": "https://arxiv.org/pdf/2503.09500", "page": 1, "id": "2503.09500_pg70_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f' = \\sum_\\tau p_{\\tau,!}(f^\\tau)"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "E_j(\\theta) = E_{\\Gamma_0}(\\theta) + \\mu_j"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\Gamma_0}(\\theta)=2\\sum_{i=1}^d \\cos 2\\pi\\theta_i"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\Gamma_0}(\\theta) = 2\\cos 2\\pi\\theta_1+2\\cos 2\\pi\\theta_2+2\\cos 2\\pi(\\theta_1+\\theta_2)"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "E_j(\\theta) = (1+\\mu_j) E_{\\Gamma_0}(\\theta)+\\mu_j"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_3=\\Gamma_0 \\boxtimes G_F"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_1 = \\Gamma_0 \\mathop\\square G_F"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle a\\rangle_p = \\sum_{q=1}^{\\nu_F} \\langle a(\\cdot+v_q)\\rangle \\sum_{s=1}^{\\nu'_F} |P_{\\mu_s}(v_p,v_q)|^2\\,"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_2 = \\Gamma_0 \\times G_F"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "E_j(\\theta) = \\mu_j E_{\\Gamma_0}(\\theta)"}
{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{\\mu_s}(v_p,v_q) = \\sum_{j\\,\\mu_j=\\mu_s} w_j(v_p)\\overline{w_j(v_q)}"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\langle L_I,h\\rangle| \\leq \\alpha \\int_0^T \\int_{\\mathbb{R}^N} |\\xi(t,x)|dx dt + 2\\beta \\|I\\|_{L^\\infty}\\|h\\|_{L^\\infty}T + \\gamma \\int_{\\mathbb{R}^N} |\\xi(T,x)|dx \\leq C_1 \\|h\\|_{L^\\infty(0,T)}"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I \\in \\mathcal{U}_{ad}"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi_I(\\varepsilon)= \\dfrac{J(I^\\varepsilon) - J(I)}{\\varepsilon}"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "h_1, h_2 \\in L^{\\infty}(0,T)"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "h = h_1 + \\lambda h_2"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\mathbb{R}"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "I^\\varepsilon = I + \\varepsilon h"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{\\varepsilon \\to 0} \\left\\|\\frac{G(I + \\varepsilon h) - G(I)}{\\varepsilon} - (\\xi, \\eta)\\right\\|_X = \\lim_{\\varepsilon \\to 0} \\left\\|(\\tilde{\\xi}, \\tilde{\\eta})\\right\\|_X = 0"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "L_I: L^{\\infty}(0,T) \\to \\mathbb{R}"}
{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "(p^\\varepsilon, d^\\varepsilon) = G(I^\\varepsilon)"}
{"pdf": "arxiv_math/2503.06055_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06055", "page": 1, "id": "2503.06055_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T_\\tau \\, v \\preceq T_\\sigma \\, v"}
{"pdf": "arxiv_math/2503.05685_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05685", "page": 1, "id": "2503.05685_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_m(j(E_1), j(E_2))"}
{"pdf": "arxiv_math/2503.05685_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05685", "page": 1, "id": "2503.05685_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "m \\gg_\\epsilon( \\# S)^{5+\\epsilon}"}
{"pdf": "arxiv_math/2503.04567_pg38.pdf", "url": "https://arxiv.org/pdf/2503.04567", "page": 1, "id": "2503.04567_pg38_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau_{ij} \\ne \\tau_{ji}"}
{"pdf": "arxiv_math/2503.03772_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03772", "page": 1, "id": "2503.03772_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g \\cdot (h \\cdot x) = (gh) \\cdot x"}
{"pdf": "arxiv_math/2503.03772_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03772", "page": 1, "id": "2503.03772_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdot : G \\times X \\to X"}
{"pdf": "arxiv_math/2503.03772_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03772", "page": 1, "id": "2503.03772_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau(g \\cdot x) = g \\cdot \\tau(x)"}
{"pdf": "arxiv_math/2502.15977_pg21.pdf", "url": "https://arxiv.org/pdf/2502.15977", "page": 1, "id": "2502.15977_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_1 + ... + \\theta_n"}
{"pdf": "arxiv_math/2503.06731_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06731", "page": 1, "id": "2503.06731_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "x.\\omega=\\omega(S(x).-),\\; \\forall\\omega\\in V^\\ast, x\\in A"}
{"pdf": "arxiv_math/2503.07447_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07447", "page": 1, "id": "2503.07447_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "n^{-1}\\log^2 n \\ll p \\ll n^{-1/2}\\log^{1/4} n"}
{"pdf": "arxiv_math/2503.07447_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07447", "page": 1, "id": "2503.07447_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "p \\gg n^{-2/3}\\log^{2/3} n"}
{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta_2 = \\partial_y^2-\\alpha^2"}
{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta\\phi_{\\pm\\alpha}(t,y)"}
{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\Phi(t, \\pm 1) = \\partial_y \\Phi(t,\\pm 1) = 0"}
{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_y\\delta\\phi_0"}
{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\vec{\\nabla}\\cdot \\vec{u} = 0"}
{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H \\coloneqq (\\delta_{m+i, j})_{i=1, \\dots, d-m; \\: j=1, \\dots, d}"}
{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\gamma^{(n)} - \\gamma\\|_2 \\to 0"}
{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lVert Z_n - Z \\rVert_2 \\to 0"}
{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "X_{m+1}(t),\\dots,X_d(t)"}
{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_013", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\gamma \\in L_t(X, d')"}
{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma\\in \\R^{d\\times d}"}
{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "Z_n \\in \\mathcal{E}(Y, t, d')"}
{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_018", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|I_t H - Z\\| = \\lim_{n \\to \\infty} \\|Z_n - Z\\| = 0"}
{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat\\Sigma(0,-1):=\\mathrm{Cov}(X(0))"}
{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\log\\left( \\frac{P_1^{\\rho_0+\\theta}(X_{(k-1)/n},X_{k/n};1/n)}{P_2^{\\rho_0}(X_{(k-1)/n},X_{k/n};1/n)}\\right) = \\log\\left(\\frac{\\beta}{\\alpha}\\right) -\\frac{(X_{k/n}-X_{(k-1)/n})^2}{2/n}\\left(\\frac{1}{\\alpha^2}-\\frac{1}{\\beta^2}\\right) + R_{k,\\theta}"}
{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "g_k := \\log\\left(\\frac{\\beta}{\\alpha}\\right) - \\frac{n}{2}\\left(\\frac{1}{\\alpha^2}-\\frac{1}{\\beta^2}\\right)(X_{k/n}-X_{(k-1)/n})^2"}
{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\mathcal{I}_4(\\theta)}"}
{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{k,\\theta} = \\log\\left( \\frac{1-\\frac{\\alpha-\\beta}{\\alpha+\\beta}\\exp\\left(-\\frac{2}{\\alpha^2/n}(X_{k/n}-\\rho_0-\\theta)(X_{(k-1)/n}-\\rho_0-\\theta)\\right)}{1+\\frac{\\alpha-\\beta}{\\alpha+\\beta}\\exp\\left(-\\frac{2}{\\beta^2/n}(X_{k/n}-\\rho_0)(X_{(k-1)/n}-\\rho_0)\\right)}\\right)"}
{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{I}_2(\\theta)"}
{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\mathcal{I}_5(\\theta)}"}
{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "V_j= B(z,(1+u)2^{-j})"}
{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak F_{W,r/100}"}
{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "B(z,2^{-j_0})\\subseteq D"}
{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{F}_{W,r/100}"}
{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak F_{W,r} = \\{(\\ell^1_i,\\ell^2_i,y_i)\\}_{i=1,\\ldots,M^*}"}
{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_016", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathfrak F_{W_{z,j},r}"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{C}_{m}(S,H)"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\varphi(i),\\varphi(j)) \\in E(S)"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "w_0, w_1,\\ldots,w_m,w_{m+1}"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau(H)=|E(H)|-|V(H)|"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E_A = \\{ (u,v) \\in E: u,v \\in A \\}"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathsf{Dist}_{H}(u,v)"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "H_{\\setminus A} = (V,E_{\\setminus A})"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_007", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathsf{Diam}(H)=\\max_{u,v \\in V(H)} \\mathsf{Dist}_H(u,v)"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "H,S \\subset \\mathcal{K}_n"}
{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "V(H)=\\{ u,v,w_1,\\ldots,w_m \\}"}
{"pdf": "arxiv_math/2503.09087_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09087", "page": 1, "id": "2503.09087_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "L_\\alpha^{\\red(w)}(G)"}
{"pdf": "arxiv_math/2503.06589_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06589", "page": 1, "id": "2503.06589_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{23},\\;B_{23},\\;R,\\;Y"}
{"pdf": "arxiv_math/2503.06589_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06589", "page": 1, "id": "2503.06589_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "[x_{\\rm min},x_{\\rm max}]"}
{"pdf": "arxiv_math/2503.06589_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06589", "page": 1, "id": "2503.06589_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mu-m)e^2+(a^2-n^2)\\mu<0"}
{"pdf": "arxiv_math/2503.06605_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06605", "page": 1, "id": "2503.06605_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "B_t=(b_{ij}^t)_{n\\times n}"}
{"pdf": "arxiv_math/2503.06605_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06605", "page": 1, "id": "2503.06605_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bf x}_t=(x_{1;t},\\ldots,x_{n;t})"}
{"pdf": "arxiv_math/2503.06605_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06605", "page": 1, "id": "2503.06605_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat y_{k;t}={\\bf x}_t^{B_t{\\bf e}_k}"}
{"pdf": "arxiv_math/2503.06605_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06605", "page": 1, "id": "2503.06605_pg1_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "F_u^t(y_1,\\ldots,y_n)\\in\\mathbb Z[y_1,\\ldots,y_n]"}
{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\det A_{n;3}(x_1,x_2)=0"}
{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{n;3}(0.97,\\sqrt{0.97})"}
{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{array}{rcl} p_n(x,\\sqrt{x})&=&-k_n(x)k_n(1/x)-2k_n(\\sqrt{x})k_n(1/\\sqrt{x})\\\\ \\ q_n(x,\\sqrt{x})&=&k_n(x)k_n(1/\\sqrt{x})^2+k_n(1/x)k_n(\\sqrt{x})^2. \\end{array}"}
{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{n;3}(0.098,\\sqrt{0.098})"}
{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "P=(0.098,\\sqrt{0.098})"}
{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_005", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\det A_{n;3}(x,\\sqrt{x})=\\delta_n^3+p_n(x,\\sqrt{x})\\delta_n+q_n(x,\\sqrt{x})=0"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "|N_{G}(U\\setminus B)|\\geq \\frac{s|U\\setminus B|}{2r}"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|B|\\leq \\frac{|U|}{10r}+\\frac{|U|}{10}\\le \\frac{|U|}{2}"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "N_{G,r}(U\\setminus B)\\subseteq X"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "|U'|\\leq \\frac{r|X \\cap L|}{t}\\le \\frac{r}{t}\\cdot |L| \\leq \\frac{r}{t}\\cdot \\frac{|U|\\cdot t}{10r}= \\frac{|U|}{10}"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "X\\subseteq V(G)\\setminus U"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "V(G)\\setminus (U\\cup L)"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|L| = |C|\\cdot t \\le \\frac{|U|}{10r}\\cdot t"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "X=X_k=V(H_k)\\setminus U"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "|C| \\le \\frac{|U|}{10r}"}
{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|U\\setminus B|\\geq \\frac{|U|}2"}
{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{GC}_\\text{id}(M)=\\mathcal{GC}(M)"}
{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{id}=(\\text{id},\\ldots,\\text{id})"}
{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(w,x_1),\\ldots,(w,x_s)"}
{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "c\\colon V\\rightarrow \\{1,\\ldots,m\\}\\times\\{1,\\ldots,m+1\\}\\times\\{1,\\ldots,n\\}"}
{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(v,x_1),\\ldots,(v,x_s)"}
{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "[b_2,d_2)\\lhd [b_1,d_1)"}
{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{GC}_\\text{id}(M)"}
{"pdf": "arxiv_math/2503.08910_pg26.pdf", "url": "https://arxiv.org/pdf/2503.08910", "page": 1, "id": "2503.08910_pg26_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Xi(\\{1\\})=\\frac{1}{3}"}
{"pdf": "arxiv_math/2503.08910_pg26.pdf", "url": "https://arxiv.org/pdf/2503.08910", "page": 1, "id": "2503.08910_pg26_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Xi(a) = \\Xi_0(g(\\sigma^0))"}
{"pdf": "arxiv_math/2503.08910_pg26.pdf", "url": "https://arxiv.org/pdf/2503.08910", "page": 1, "id": "2503.08910_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Xi_1(b_{i'}^{\\sigma(i')})= \\Xi_1({\\sim}b_{i'})=0"}
{"pdf": "arxiv_math/2503.08910_pg26.pdf", "url": "https://arxiv.org/pdf/2503.08910", "page": 1, "id": "2503.08910_pg26_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Xi\\left(g(\\sigma)\\wedge \\bigwedge_{i\\in J}b^{\\sigma(i)}_i\\right)=0"}
{"pdf": "arxiv_math/2503.08910_pg26.pdf", "url": "https://arxiv.org/pdf/2503.08910", "page": 1, "id": "2503.08910_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Xi(\\{0\\})=\\Xi(a\\cap \\{0,1\\}) = \\Xi_0(a)=\\frac{2}{3}"}
{"pdf": "arxiv_math/2503.05469_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05469", "page": 1, "id": "2503.05469_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta (r\\wedge s)^{-\\gamma} (r\\vee s)^{\\gamma-1}"}
{"pdf": "arxiv_math/2503.05469_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05469", "page": 1, "id": "2503.05469_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\sum_{k=r}^m \\frac1k < V(v) \\leq -\\sum_{k=r+1}^m \\frac1k"}
{"pdf": "arxiv_math/2503.05469_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05469", "page": 1, "id": "2503.05469_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal Y_1, \\ldots, \\mathcal Y_d"}
{"pdf": "arxiv_math/2503.05469_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05469", "page": 1, "id": "2503.05469_pg12_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{a}2u^{-\\rho_-}+1"}
{"pdf": "arxiv_math/2503.05469_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05469", "page": 1, "id": "2503.05469_pg12_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "V(v)=-\\sum_{k=r}^m \\frac1k"}
{"pdf": "arxiv_math/2503.09086_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09086", "page": 1, "id": "2503.09086_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\theta_{P},\\lambda_{P}) := \\text{arg} \\left( \\max_{\\lambda} \\min_{\\theta} L_{P,M}(\\theta,\\lambda) \\right) \\quad \\text{resp.} \\quad (\\theta_{R},\\lambda_{R}) := \\text{arg} \\left( \\max_{\\lambda} \\min_{\\theta} L_{R,M}(\\theta,\\lambda) \\right)"}
{"pdf": "arxiv_math/2503.09086_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09086", "page": 1, "id": "2503.09086_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "U({\\bf x};\\theta)-g({\\bf x})"}
{"pdf": "arxiv_math/2503.09086_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09086", "page": 1, "id": "2503.09086_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "L_{P,M}(\\theta,\\lambda) := J_{P,M}(\\theta)+\\frac{1}{X(\\partial \\Omega)} \\sum_{{\\bf x} \\in X(\\partial \\Omega)} (U({\\bf x};\\theta)-g({\\bf x}))\\lambda({\\bf x})"}
{"pdf": "arxiv_math/2503.09086_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09086", "page": 1, "id": "2503.09086_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "J_{R_I}(\\theta) := \\int_{\\Omega} \\left( \\frac{1}{2} |\\nabla \\widetilde{U}({\\bf x};\\theta)|^2- f({\\bf x}) \\widetilde{U}({\\bf x};\\theta)\\right) \\, d{\\bf x}"}
{"pdf": "arxiv_math/2503.09086_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09086", "page": 1, "id": "2503.09086_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{U}({\\bf x};\\theta)"}
{"pdf": "arxiv_math/2503.09086_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09086", "page": 1, "id": "2503.09086_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "U({\\bf x};\\theta_{R})"}
{"pdf": "arxiv_math/2503.09086_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09086", "page": 1, "id": "2503.09086_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "L_{R,M}(\\theta,\\lambda) := J_{R,M}(\\theta)+\\frac{1}{X(\\partial \\Omega)} \\sum_{{\\bf x} \\in X(\\partial \\Omega)} (U({\\bf x};\\theta)-g({\\bf x}))\\lambda({\\bf x})"}
{"pdf": "arxiv_math/2503.09086_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09086", "page": 1, "id": "2503.09086_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "U({\\bf x};\\theta_{P})"}
{"pdf": "arxiv_math/2503.09086_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09086", "page": 1, "id": "2503.09086_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda=\\lambda+ \\alpha \\nabla_{\\lambda} L_{P,M} \\quad \\text{or} \\quad \\lambda=\\lambda+\\alpha \\nabla_{\\lambda} L_{R,M}"}
{"pdf": "arxiv_math/2503.07355_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07355", "page": 1, "id": "2503.07355_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{C}(2k)\\simeq \\mathbb{C}(2^k) \\qquad \\text{and}\\qquad \\mathcal{C}(2k+1)\\simeq \\mathbb{C}(2^k)\\oplus\\mathbb{C}(2^k)"}
{"pdf": "arxiv_math/2503.07355_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07355", "page": 1, "id": "2503.07355_pg11_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "s_{r,s}=\\begin{cases} 2 \\text{ if } r-s=2,4 \\text{ mod 4},\\\\ 1 \\text{ if } r-s=1,3 \\text{ mod 4}. \\end{cases}"}
{"pdf": "arxiv_math/2503.08178_pg15.pdf", "url": "https://arxiv.org/pdf/2503.08178", "page": 1, "id": "2503.08178_pg15_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(v_{\\bar{c}_1},v_{\\bar{c}_2})"}
{"pdf": "arxiv_math/2503.08178_pg15.pdf", "url": "https://arxiv.org/pdf/2503.08178", "page": 1, "id": "2503.08178_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(v_{\\bar{c}_1},v_{\\bar{c}_r})"}
{"pdf": "arxiv_math/2503.08178_pg15.pdf", "url": "https://arxiv.org/pdf/2503.08178", "page": 1, "id": "2503.08178_pg15_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "P'~\\in~\\Set{(v_{c_1'},v_{c_2}), (v_{c_1'},v_{c_2'}), (v_{c_1'},v_{c_2''})}"}
{"pdf": "arxiv_math/2503.08178_pg15.pdf", "url": "https://arxiv.org/pdf/2503.08178", "page": 1, "id": "2503.08178_pg15_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "P'\\in\\Set{(v_{c_1},v_{c_2'}), (v_{c_1},v_{c_2''})}"}
{"pdf": "arxiv_math/2503.04471_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04471", "page": 1, "id": "2503.04471_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\kappa^{<\\kappa}=\\kappa"}
{"pdf": "arxiv_math/2503.04471_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04471", "page": 1, "id": "2503.04471_pg7_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\kappa^{<\\kappa}>\\kappa"}
{"pdf": "arxiv_math/2503.05736_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05736", "page": 1, "id": "2503.05736_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "U_0(x)=U_\\mathrm{b}(x)+U_\\mathrm{r}^\\omega(x)"}
{"pdf": "arxiv_math/2503.05736_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05736", "page": 1, "id": "2503.05736_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(b_k(\\omega))_{|k|\\leq K}"}
{"pdf": "arxiv_math/2503.05736_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05736", "page": 1, "id": "2503.05736_pg17_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "U_j^{-1}((-\\infty,c])"}
{"pdf": "arxiv_math/2503.05736_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05736", "page": 1, "id": "2503.05736_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "U_\\mathrm{r}(x)=\\sum_{|k|\\leq K}a_ke_k(x)"}
{"pdf": "arxiv_math/2503.05736_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05736", "page": 1, "id": "2503.05736_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "U_\\mathrm{b}+U_\\mathrm{r}^\\omega"}
{"pdf": "arxiv_math/2503.05736_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05736", "page": 1, "id": "2503.05736_pg17_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "U_\\mathrm{r}^\\omega(x):=\\sum_{|k|\\leq K}a_kb_k(\\omega)e_k(x)"}
{"pdf": "arxiv_math/2503.09552_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09552", "page": 1, "id": "2503.09552_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "[\\lambda]\\in\\mathrm{Mod}(S)"}
{"pdf": "arxiv_math/2503.09552_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09552", "page": 1, "id": "2503.09552_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_j+\\gamma_j=0\\in H_1(S)"}
{"pdf": "arxiv_math/2503.09552_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09552", "page": 1, "id": "2503.09552_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\beta_j=\\tau_*(\\beta_j)"}
{"pdf": "arxiv_math/2503.09552_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09552", "page": 1, "id": "2503.09552_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "cbabcb=(bcbabc)^{-1}=c^{-1}b^{-1}a^{-1}b^{-1}c^{-1}b^{-1}"}
{"pdf": "arxiv_math/2503.09552_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09552", "page": 1, "id": "2503.09552_pg18_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\alpha_j=\\gamma_j=\\tau_*(\\alpha_j)"}
{"pdf": "arxiv_math/2503.09552_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09552", "page": 1, "id": "2503.09552_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "cbabcb=cbacbc=cbcabc=bcbabc"}
{"pdf": "arxiv_math/2503.09552_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09552", "page": 1, "id": "2503.09552_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Psi(\\tau)=\\tau_*=-\\mathrm{Id}_{2n}\\in\\mathrm{Sp}_{2n}(\\mathbb{Z})"}
{"pdf": "arxiv_math/2503.09552_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09552", "page": 1, "id": "2503.09552_pg18_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "[\\lambda]=(cab)^2\\in\\Theta_n^1"}
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{"pdf": "arxiv_math/2503.06570_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06570", "page": 1, "id": "2503.06570_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu(\\phi_\\ell)=\\frac{1}{2} (\\deg \\phi_\\ell - \\dim X) \\phi_\\ell"}
{"pdf": "arxiv_math/2503.06570_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06570", "page": 1, "id": "2503.06570_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\tau , d \\rangle \\to -\\infty"}
{"pdf": "arxiv_math/2503.06570_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06570", "page": 1, "id": "2503.06570_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H^2(X) \\times \\mathbb{C}^\\times"}
{"pdf": "arxiv_math/2503.06570_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06570", "page": 1, "id": "2503.06570_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "d \\in \\mathsf{Eff}_{\\neq 0}"}
{"pdf": "arxiv_math/2503.06570_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06570", "page": 1, "id": "2503.06570_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathsf{Eff} \\subseteq H_2(X,\\mathbb{Z})"}
{"pdf": "arxiv_math/2503.06570_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06570", "page": 1, "id": "2503.06570_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tau,z)=(\\sum_{j=1}^{b^{2}(X)} t_j \\phi_j, z) \\in H^2(X) \\times \\mathbb{C}^\\times"}
{"pdf": "arxiv_math/2503.06570_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06570", "page": 1, "id": "2503.06570_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\alpha , \\beta , \\gamma \\rangle_{0,3,d} = 0"}
{"pdf": "arxiv_math/2503.06570_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06570", "page": 1, "id": "2503.06570_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\deg \\alpha + \\deg \\beta + \\deg \\gamma \\neq \\langle c_1,d \\rangle"}
{"pdf": "arxiv_math/2503.06570_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06570", "page": 1, "id": "2503.06570_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu:H^*(X) \\rightarrow H^*(X)"}
{"pdf": "arxiv_math/2503.04932_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04932", "page": 1, "id": "2503.04932_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "u(x,y,z,t) = \\sum\\limits_{k=1}^{2}{e^{-3dk^2t}\\sin{(k(x-t))}\\sin{(k(y-t))}\\sin{(k(z-t))}}"}
{"pdf": "arxiv_math/2503.04932_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04932", "page": 1, "id": "2503.04932_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "u_0(x,y,z)=\\text{exp}(-(x^2+9y^2+z^2))"}
{"pdf": "arxiv_math/2503.04932_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04932", "page": 1, "id": "2503.04932_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\mathbf{\\Omega}|=(2\\pi)^3"}
{"pdf": "arxiv_math/2503.04932_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04932", "page": 1, "id": "2503.04932_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "u(x,y,z,t)=\\text{exp}(-(x^2+2y^2+3z^2+3dt))"}
{"pdf": "arxiv_math/2503.06126_pg25.pdf", "url": "https://arxiv.org/pdf/2503.06126", "page": 1, "id": "2503.06126_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{ \\left\\Vert \\nabla u_{n}\\right\\Vert _{L^{m}}:p_{n}^{-}\\geqslant m\\right\\}"}
{"pdf": "arxiv_math/2503.07785_pg20.pdf", "url": "https://arxiv.org/pdf/2503.07785", "page": 1, "id": "2503.07785_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "N_x\\times N_v=49\\times97"}
{"pdf": "arxiv_math/2503.05880_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05880", "page": 1, "id": "2503.05880_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{0}\\in \\mathbf{R}^{d}"}
{"pdf": "arxiv_math/2503.05880_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05880", "page": 1, "id": "2503.05880_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( W\\left( x+x_{0}\\right) -W\\left( x_{0}\\right) \\right) _{x\\in \\mathbf{R}^{d}}"}
{"pdf": "arxiv_math/2503.05880_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05880", "page": 1, "id": "2503.05880_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\Vert x\\right\\Vert"}
{"pdf": "arxiv_math/2503.05880_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05880", "page": 1, "id": "2503.05880_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( W\\left( x\\right) \\right) _{x\\in \\mathbf{R}^{d}}"}
{"pdf": "arxiv_math/2503.04498_pg27.pdf", "url": "https://arxiv.org/pdf/2503.04498", "page": 1, "id": "2503.04498_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bmod p^{\\prime \\prime}"}
{"pdf": "arxiv_math/2503.03952_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03952", "page": 1, "id": "2503.03952_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} f(t,0;q)=\\frac{t-q}{1-q}. \\end{aligned}"}
{"pdf": "arxiv_math/2503.03952_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03952", "page": 1, "id": "2503.03952_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} f(t,q/t;q)=\\frac{t+q/t-2q}{1-q}. \\end{aligned}"}
{"pdf": "arxiv_math/2503.03952_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03952", "page": 1, "id": "2503.03952_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} (x;q)_\\infty=\\prod_{k=0}^\\infty (1-xq^k),\\qquad (x;q)_n=\\prod_{k=0}^{n-1}(1-xq^k),\\qquad (x;q)_0=1, \\end{aligned}"}
{"pdf": "arxiv_math/2503.03952_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03952", "page": 1, "id": "2503.03952_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "I_N(q^{1/2}, q^{1/2}; q)"}
{"pdf": "arxiv_math/2503.06701_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06701", "page": 1, "id": "2503.06701_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "1.262 \\cdot \\left(\\lvert e \\rvert\\right)^{\\frac{1}{5}} + 2"}
{"pdf": "arxiv_math/2503.06701_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06701", "page": 1, "id": "2503.06701_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(1-\\lvert e \\rvert )/70"}
{"pdf": "arxiv_math/2503.06701_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06701", "page": 1, "id": "2503.06701_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "-2 \\times 10^{-6} \\cdot i - 10^{-6} \\cdot c"}
{"pdf": "arxiv_math/2503.06701_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06701", "page": 1, "id": "2503.06701_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(1-\\lvert e \\rvert )/20"}
{"pdf": "arxiv_math/2503.05392_pg19.pdf", "url": "https://arxiv.org/pdf/2503.05392", "page": 1, "id": "2503.05392_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{1},K_{2}\\in\\mathcal{K}_o^n"}
{"pdf": "arxiv_math/2503.05392_pg19.pdf", "url": "https://arxiv.org/pdf/2503.05392", "page": 1, "id": "2503.05392_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "[S_{e_n}(K^\\circ)]^\\circ"}
{"pdf": "arxiv_math/2503.05392_pg19.pdf", "url": "https://arxiv.org/pdf/2503.05392", "page": 1, "id": "2503.05392_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "K_1|e_n^\\perp=K_2|e_n^\\perp"}
{"pdf": "arxiv_math/2503.05392_pg19.pdf", "url": "https://arxiv.org/pdf/2503.05392", "page": 1, "id": "2503.05392_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{1},K_{2}\\in\\mathcal{K}^n_o"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "t_1\\coloneqq\\sup\\{t\\ge0\\ |\\ d_X(o,\\gamma(t))\\le R\\}"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(Z_n,\\rho_n)\\xrightarrow{AI\\ conv.} (Z,\\rho_0)"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\xi|\\eta)_o\\ge(\\gamma_1(R)|\\gamma_2(R))_o= \\frac{1}{2}\\bigg(d_X(o,\\gamma_1(R))+d_x(o,\\gamma_2(R))-d_X(\\gamma_1(R),\\gamma_2(R))\\bigg)\\ge R-\\frac{\\hbox{diam}(U)}{2}"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma(t_0)\\in S_X(o,R)"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\gamma_1(t)|\\gamma_2(t))_o\\uparrow (\\xi|\\eta)_o"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_008", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\begin{split} d_X(x_n,p)&\\le d_X(x_n,p_n)+d_X(p_n,p)\\\\ &=d_X(o,p_n)-d_X(o,x_n)+ d_X(p_n,p)\\\\ &=d_X(o,p_n)-R+d_X(p_n,p)\\xrightarrow{n\\to \\infty} d_X(o,p)-R+0=0. \\end{split}"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\partial_P X_n,\\rho_{x_n})\\xrightarrow{AI conv.}(\\partial_P X,\\rho_x)"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_1(R),\\gamma_2(R)\\in U"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "(Z,\\rho_0)=(\\partial_P X,\\rho_x)"}
{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "2(x|y)_o=d_X(x,o)+d_X(y,o)-d_X(x,y)=2R-d_X(x,y)"}
{"pdf": "arxiv_math/2503.06334_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06334", "page": 1, "id": "2503.06334_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{C_{v_0},\\cdots,C_{v_{n-1}}\\}"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "L_{\\xi}=\\cup_{\\xi(\\pi(m_i))\\neq1}l_i"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "L_{\\xi}=l_{i_1}\\cup \\cdots l_{i_k}"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "m_1,...,m_d \\in \\pi_1(M-L;\\Z)"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_1(M-L) \\to \\pi_1(M-L)^{ab}"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ker \\tilde F_n\\subset \\ker \\tilde F_{n-1}"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|H_1(M_{\\pi};\\Z)|=\\frac{|G|}{\\prod_{\\xi \\in \\hat{G}^{(1)}}|1-\\xi(\\pi(m_{i(\\xi)}))|}\\prod_{\\xi \\in \\hat{G}}|\\Delta_{L_{\\xi}}(\\xi(\\pi(m_{i_1})),...,\\xi(\\pi(m_{i_k})))|"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varprojlim_nG_n\\cong\\Z_p^{\\,d}"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_{\\Gamma}:\\pi_1(M-L)\\to \\Z_p^d/\\Gamma"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{G}^{(1)}=\\{\\xi \\in \\hat{G}\\mid L_{\\xi} \\text{ has a single component}\\}"}
{"pdf": "arxiv_math/2503.06194_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06194", "page": 1, "id": "2503.06194_pg10_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Z^d \\hookrightarrow \\Z_p^d"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[\\|\\hat{x}_\\eta-x^*\\|^2]\\leq \\tfrac{2}{\\mu_f}\\left(\\|\\nabla f(x^*)\\| \\,\\sqrt[\\kappa]{\\tfrac{1}{\\alpha}\\left(\\texttt{Err}_{\\eta}+\\eta M\\right)} + \\tfrac{1}{\\eta}\\texttt{Err}_{\\eta}\\right)"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tfrac{\\mu_f}{2}\\mathbb{E}[\\|\\hat{x}_\\eta-x^*\\|^2]+\\nabla f(x^*)^\\top\\mathbb{E}\\left[\\left(\\hat{x}_\\eta- \\Pi_{X^*_h}[\\hat{x}_\\eta]\\right)\\right]\\leq \\mathbb{E}[f(\\hat{x}_\\eta)] - f^*"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi_{X^*_h}[\\hat{x}_\\eta] \\in X^*_h"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta\\le\\tfrac{\\alpha}{2\\|\\nabla f(x^*)\\|}"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\le \\mathbb{E}[h(\\hat{x}_\\eta)]-h^*\\le \\texttt{Err}_{\\eta}+\\eta M"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\le \\mathbb{E}[ h(\\hat{x}_\\eta)]-h^*\\le \\texttt{Err}_{\\eta}+\\eta M"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[h(\\hat{x}_\\eta)] -h^*\\geq 0"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[h(\\hat{x}_\\eta)]-h^*\\leq \\texttt{Err}_{\\eta}+\\eta M"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[h(\\hat{x}_\\eta) + \\eta f(\\hat{x}_\\eta)] -f^*_\\eta \\leq \\texttt{Err}_{\\eta}"}
{"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "f^*- \\mathbb{E}[ f(\\hat{x}_\\eta)]<M"}
{"pdf": "arxiv_math/2503.04881_pg25.pdf", "url": "https://arxiv.org/pdf/2503.04881", "page": 1, "id": "2503.04881_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "U_{\\rm EF}=U_{\\rm EF,0}"}
{"pdf": "arxiv_math/2503.04881_pg25.pdf", "url": "https://arxiv.org/pdf/2503.04881", "page": 1, "id": "2503.04881_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "V_{\\rm EF}=V_{\\rm EF,0}"}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi(\\omega_0) \\coloneqq \\log b(\\omega_0)"}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi(\\omega_0) \\coloneqq - \\log a(\\omega_0)"}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int \\left| \\log \\sin \\angle (\\mathbf{E}_1, \\mathbf{E}_2) \\right| \\, d\\mu"}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "F(\\omega) = \\begin{pmatrix} a(\\omega_0) & b(\\omega_0) \\\\ 0 & 1 \\end{pmatrix} \\, "}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int \\log |a| \\, d\\pi < 0"}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_1(\\nu) > \\lambda_2(\\nu)"}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_2 = \\int \\log |a| \\, d\\pi"}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega = (\\omega)_{n \\in \\Z} \\in \\Omega"}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "X(\\omega) = \\sum_{n=0}^\\infty a(\\omega_{-1}) a(\\omega_{-2}) \\cdots a(\\omega_{-n}) b(\\omega_{-n-1}) \\, "}
{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lbrace \\mathbf{E}_1(\\omega), \\mathbf{E}_2(\\omega)\\rbrace = S"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c_1x_1^\\alpha + c_2 x_2^\\beta + g(x_1,x_2)"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "v(D) = v(f_t^*D) = pq"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "-K_X\\cdot D = f_t^*\\left(\\sum_i C_i\\right)\\cdot D_t =\\left( (p+q)E_t + \\sum_i C_{i,t}\\right) \\cdot D_t = p+q"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(K_{Y_t} + D_t) \\cdot D_t = -2"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "-2\\leq (K_{Y_t} + D_t) \\cdot D_t = \\left(-\\sum_{i}C_{i,t} - E_t + D_t\\right) \\cdot D_t = D_t^2 - E_t\\cdot D_t < -E_t\\cdot D_t"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "-2\\leq D_t^2 - E_t\\cdot D_t"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "p+q = D\\cdot C \\geq (D\\cdot C)_x = \\alpha + \\beta \\geq p + q"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f_t^*D = D_t + pq E_t"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "f_t^*(\\sum_{i}C_i)\\cdot D_t = -K_X \\cdot D> 0"}
{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "B_t = \\sum_{i=1}^k a_iC_{i,t}"}
{"pdf": "arxiv_math/2503.07452_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07452", "page": 1, "id": "2503.07452_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} & \\underset{\\mathbf{b}}{\\text{minimize}} & & \\epsilon_d(\\mathbf{b}) = \\dfrac{1}{n_{MP}} \\sum_{i=1}^{n_{MP}} (\\mathbf{u}_{r_i} - \\mathbf{u}_{s_i}(\\mathbf{b}))^2\\\\ & \\text{subject to} & & \\mathbf{b}_l \\mathbf{\\leq b} \\leq \\mathbf{b}_u \\end{aligned}"}
{"pdf": "arxiv_math/2503.09042_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09042", "page": 1, "id": "2503.09042_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "Y_1 = Y_2 = \\cdots = Y_{k-1}"}
{"pdf": "arxiv_math/2503.09042_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09042", "page": 1, "id": "2503.09042_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pr{g(X) \\geq k \\text{ or } g(-X) \\geq k} \\leq 2\\epsilon"}
{"pdf": "arxiv_math/2503.09042_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09042", "page": 1, "id": "2503.09042_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pr{g(-X) \\geq k} = \\Pr{g(X) \\geq k} \\leq \\epsilon"}
{"pdf": "arxiv_math/2503.09042_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09042", "page": 1, "id": "2503.09042_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pr{X_{f(Y)}=X_{f(-Y)}}{Y=y} \\geq 1/2"}
{"pdf": "arxiv_math/2503.09042_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09042", "page": 1, "id": "2503.09042_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pr{B_Y} = 2^{-(k-2)}"}
{"pdf": "arxiv_math/2503.05138_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05138", "page": 1, "id": "2503.05138_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle Au,u-v\\rangle\\le\\Phi(v)-\\Phi(u)+I_\\Delta(\\psi^0(\\gamma_\\psi u;\\gamma_\\psi v-\\gamma_\\psi u))-\\langle f,v-u\\rangle"}
{"pdf": "arxiv_math/2503.05138_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05138", "page": 1, "id": "2503.05138_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle Au^h, u^h-v^h\\rangle \\le \\Phi(v^h)-\\Phi(u^h) + I_\\Delta(\\psi^0(\\gamma_\\psi u^h;\\gamma_\\psi v^h-\\gamma_\\psi u^h))-\\langle f,v^h-u^h\\rangle"}
{"pdf": "arxiv_math/2503.05138_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05138", "page": 1, "id": "2503.05138_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi^0(z;z_1+z_2)\\le \\psi^0(z;z_1)+\\psi^0(z;z_2)\\quad \\forall\\,z,z_1,z_2\\in \\mathbb{R}^m"}
{"pdf": "arxiv_math/2503.05348_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05348", "page": 1, "id": "2503.05348_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{-\\infty}^t F_X(x)dx\\geq \\int_{-\\infty}^t F_Y(x)dx\\ \\text{ for all }t"}
{"pdf": "arxiv_math/2503.05348_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05348", "page": 1, "id": "2503.05348_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{t\\to -\\infty} \\pi_X(t)-\\pi_Y(t)=0"}
{"pdf": "arxiv_math/2503.05348_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05348", "page": 1, "id": "2503.05348_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_t^{\\infty} \\bar F_X(x)dx\\leq \\int_t^{\\infty} \\bar F_Y(x)dx\\ \\text{ for all }t"}
{"pdf": "arxiv_math/2503.05348_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05348", "page": 1, "id": "2503.05348_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar F_X\\leq \\bar F_Y"}
{"pdf": "arxiv_math/2503.05348_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05348", "page": 1, "id": "2503.05348_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(t)=\\int_t^{\\infty} \\bar F(x)dx"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi(x)=e^{|x|}(1+|x|)-1"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T_e^{2t+1}=G_e^{2t+1}"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T_e^{2t+2}=G_e^{2t+2}"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E^{2t} \\cap \\{B_u = \\tilde{B}_u, \\forall u \\in \\partial G_e^{2t}\\}"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E^{2t+1} \\cap E^{2t} \\cap \\{B_u = \\tilde{B}_u, \\forall u \\in \\partial G_e^{2t}\\}"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in \\partial G_e^{2t}"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in \\partial G_e^{2t}\\equiv \\partial T_e^{2t}"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "[n]\\setminus V(G_e^{2t})"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\partial G_e^{2t}| \\le (2k\\lambda+2)^t \\log n"}
{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "(2k\\lambda +2)^t \\log n = n^{o(1)}"}
{"pdf": "arxiv_math/2503.07624_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07624", "page": 1, "id": "2503.07624_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x = a r\\cos\\theta, y = b r \\sin\\theta"}
{"pdf": "arxiv_math/2503.07624_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07624", "page": 1, "id": "2503.07624_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_1 = \\frac{\\cos^2\\theta}{a^2} + \\frac{\\sin^2\\theta}{b^2}, \\qquad \\omega_2 = \\frac{\\cos^2\\theta}{b^2} + \\frac{\\sin^2\\theta}{a^2}, \\qquad \\omega_3 = \\sin2\\theta(\\frac{1}{a^2}-\\frac{1}{b^2})"}
{"pdf": "arxiv_math/2503.07624_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07624", "page": 1, "id": "2503.07624_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega = \\bigl\\{ (x, y): \\tfrac{x^2}{a^2} + \\tfrac{y^2}{b^2} \\leq 1 \\bigr\\}"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial\\left(\\cdot\\right)"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{A} \\succeq \\mathbf{0}"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}^{M \\times N}"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}_{+}^{M \\times N}"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lVert \\mathbf{A} \\rVert_{\\rm F}"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_{\\mathbf{x}}f\\left(\\mathbf{x}\\right)"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lVert \\mathbf{a} \\rVert_2"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{CN}(\\mathbf{0}, \\sigma^2 \\mathbf{I})"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\left(\\mathbf{x}\\right)"}
{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_{\\mathbf{x}}^2f\\left(\\mathbf{x}\\right)"}
{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{matrix} & \\alpha'& \\beta'& \\gamma'&&& \\delta'\\\\ {\\rm angles:}& 67 & 68 & 71 & 74 & 5 & 14 & /78\\\\ \\times ~3: & 15& 16& 19& 22& 5 & 14 & /26 \\end{matrix}"}
{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\beta'=\\widehat\\gamma"}
{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha'=\\widehat{\\beta}"}
{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha'=\\widehat\\beta,\\quad \\beta'=\\widehat\\gamma,\\quad {\\rm and}\\quad \\delta'= \\widehat\\alpha~"}
{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat\\gamma=\\beta'"}
{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{matrix}& \\gamma & & & \\delta & \\alpha &\\beta\\\\ {\\rm angles:}& 16& 19 & 22& 31 & 40 & 41 & /78 \\\\ \\times ~3: & 16 & 19& 22& 5 & 14 & 15 & /26 \\end{matrix}"}
{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{x}_\\ell(\\omega)"}
{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "S_z(\\omega) = \\frac{1}{N} \\sum_{\\ell=1}^N S_{x_\\ell}(\\omega)"}
{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\langle e^{i(\\theta_m(t') - \\theta_n(t))} \\right\\rangle = \\delta_{mn} \\left\\langle e^{i(\\theta_m(t') - \\theta_m(t))} \\right\\rangle"}
{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{x_\\ell}(\\omega) = \\lim_{T \\to \\infty} \\frac{1}{T} \\left\\langle |\\tilde{x}_\\ell(\\omega)|^2 \\right\\rangle, \\quad \\tilde{x}_\\ell(\\omega) = \\int_0^T e^{i\\omega t} x_\\ell(t)\\,dt"}
{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "m \\ddot{\\theta}_\\ell(t) + \\dot{\\theta}_\\ell(t) = \\omega_\\ell + \\text{Im}\\left( e^{-i\\theta_\\ell(t)} \\zeta_\\ell(t) \\right)"}
{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "x_\\ell = e^{i\\theta_\\ell}"}
{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "S_z(\\omega) = \\frac{1}{N} \\sum_{\\ell=1}^{N} |\\tilde{x}_\\ell(\\omega)|^2"}
{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\omega}^1 = 0.001\\sqrt{\\Delta{t}_{\\text{Filter}}}\\tilde{\\varepsilon}^1"}
{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{\\gamma} = 1/\\alpha_{\\gamma}"}
{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\varepsilon}^i \\sim \\mathcal{N}(0, \\pmb{I}_l)"}
{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\omega}^2 = 0.1\\sqrt{\\Delta{t}_{\\text{Filter}}}\\tilde{\\varepsilon}^2"}
{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "Nt_{\\text{Filter}}=50"}
{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_{\\text{Darcy}} = 0.0001\\epsilon_D"}
{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_i = 1/d_i, i=1, 2"}
{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\epsilon_{\\text{Darcy}} \\sim N(0, \\pmb{I})"}
{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta{t}_{\\text{Filter}} = T/ Nt_{\\text{Filter}}"}
{"pdf": "arxiv_math/2503.05993_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05993", "page": 1, "id": "2503.05993_pg15_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_2=\\arctan \\frac{y_1-y_2}{x_1-x_2}"}
{"pdf": "arxiv_math/2503.05993_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05993", "page": 1, "id": "2503.05993_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi,\\dot{\\phi},\\cos \\phi"}
{"pdf": "arxiv_math/2503.05993_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05993", "page": 1, "id": "2503.05993_pg15_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_1=\\arctan \\frac{y_1}{x_1}"}
{"pdf": "arxiv_math/2503.05993_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05993", "page": 1, "id": "2503.05993_pg15_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi=\\omega + \\frac{3\\pi}{2}"}
{"pdf": "arxiv_math/2503.05993_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05993", "page": 1, "id": "2503.05993_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta_*=\\{\\phi,\\dot{\\phi}, \\cos \\phi,\\sin \\phi,\\phi^2,\\phi \\dot{\\phi},\\ldots \\}"}
{"pdf": "arxiv_math/2503.05654_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05654", "page": 1, "id": "2503.05654_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\tau, \\omega\\rangle =\\frac{2-\\|\\tau-\\omega\\|^2}{2}, \\quad \\forall \\tau, \\omega \\in \\mathbb{R}^d"}
{"pdf": "arxiv_math/2503.04430_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04430", "page": 1, "id": "2503.04430_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(b_1, \\ldots, b_n) \\in T(n)"}
{"pdf": "arxiv_math/2503.04430_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04430", "page": 1, "id": "2503.04430_pg9_math_001", "type": "math", "max_diffs": 0, "checked": "verified", "math": "Y_1 \\lor (\\cdots (Y_{n-1} \\lor Y_n)) = q^n_2(x, y_1[1], \\ldots, y_n[1])"}
{"pdf": "arxiv_math/2503.04430_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04430", "page": 1, "id": "2503.04430_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(q^n_m(e^n_i, z_1, \\ldots, z_n), q^n_m(e^n_j, z_1, \\ldots, z_n)) \\in \\theta"}
{"pdf": "arxiv_math/2503.04430_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04430", "page": 1, "id": "2503.04430_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(b_1, \\ldots, b_n, 0, \\ldots, 0) \\in T(m)"}
{"pdf": "arxiv_math/2503.04430_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04430", "page": 1, "id": "2503.04430_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{T} \\simeq \\mathbf{N}"}
{"pdf": "arxiv_math/2503.04430_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04430", "page": 1, "id": "2503.04430_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lnot c=\\bigvee\\{a_k : k \\neq i\\}"}
{"pdf": "arxiv_math/2503.04430_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04430", "page": 1, "id": "2503.04430_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(e^n_i, e^n_j) \\in \\theta"}
{"pdf": "arxiv_math/2503.04430_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04430", "page": 1, "id": "2503.04430_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "B \\simeq B / \\theta(c,1) \\times B/\\theta(c,0)"}
{"pdf": "arxiv_math/2503.04430_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04430", "page": 1, "id": "2503.04430_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "a:=(b_1, \\ldots, b_n) \\in T(n)"}
{"pdf": "arxiv_math/2503.08618_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08618", "page": 1, "id": "2503.08618_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_{n_j}=0,~~ j=0, 1, \\ldots, n"}
{"pdf": "arxiv_math/2503.08618_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08618", "page": 1, "id": "2503.08618_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_{n_j}, j=0, 1, 2,\\ldots,n"}
{"pdf": "arxiv_math/2503.08618_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08618", "page": 1, "id": "2503.08618_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{n_j}\\ne a_{n_{j+1}}, j=0, 1, 2, \\cdots, k"}
{"pdf": "arxiv_math/2503.08618_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08618", "page": 1, "id": "2503.08618_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "p(q)=\\sum_{\\nu=0}^{n}q^{n_j}a_{n_j}"}
{"pdf": "arxiv_math/2503.08618_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08618", "page": 1, "id": "2503.08618_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{n_j}=a_{n_j}, ~~j=0, 1, \\ldots, n"}
{"pdf": "arxiv_math/2503.08618_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08618", "page": 1, "id": "2503.08618_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{n_0}\\le \\alpha_{n_1}\\le \\cdots\\le \\alpha_{n_k}"}
{"pdf": "arxiv_math/2503.08618_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08618", "page": 1, "id": "2503.08618_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{n_l}=\\alpha_{n_l}+\\beta_{n_l}i"}
{"pdf": "arxiv_math/2503.08618_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08618", "page": 1, "id": "2503.08618_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "p(q)=\\sum_{\\nu=0}^{k} q^{n_\\nu} a_{n_\\nu}"}
{"pdf": "arxiv_math/2503.08618_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08618", "page": 1, "id": "2503.08618_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_{n_j},~ 0\\le j\\le n"}
{"pdf": "arxiv_math/2503.08982_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08982", "page": 1, "id": "2503.08982_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bm{b}=(b(1), b(2), \\dots, b(|\\cal S|))}"}
{"pdf": "arxiv_math/2503.08982_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08982", "page": 1, "id": "2503.08982_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{s \\in \\cal S} b(s) =1"}
{"pdf": "arxiv_math/2503.03899_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03899", "page": 1, "id": "2503.03899_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d(n)=e^{\\frac{2\\sqrt{n}}{A}+O(\\log n)}"}
{"pdf": "arxiv_math/2503.03899_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03899", "page": 1, "id": "2503.03899_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "a_n=\\alpha_n\\sqrt{n}, \\qquad b_n=\\beta_n\\sqrt{n}"}
{"pdf": "arxiv_math/2503.03899_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03899", "page": 1, "id": "2503.03899_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "x_n \\in \\{\\pm n^{-\\frac{1}{4}}\\}"}
{"pdf": "arxiv_math/2503.03899_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03899", "page": 1, "id": "2503.03899_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "a_n,b_n \\in \\mathbb{N}_0"}
{"pdf": "arxiv_math/2503.03899_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03899", "page": 1, "id": "2503.03899_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n \\to \\infty} P_n\\left(\\sup_{t \\geq 0}\\left|\\frac{1}{\\sqrt{n}}\\sum_{k \\leq t\\sqrt{n}}X_k- L(t)\\right|\\leq n^{-\\frac{1}{4}+\\delta} \\right)=1"}
{"pdf": "arxiv_math/2503.03899_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03899", "page": 1, "id": "2503.03899_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\limsup_{n \\to \\infty} n^{-\\delta}\\log P_n\\left(\\inf_{t \\geq 0}\\left|\\frac{1}{\\sqrt{n}}\\sum_{k \\leq t\\sqrt{n}}X_k- L(t)\\right|> n^{-\\frac{1}{4}+\\delta} \\right) <0"}
{"pdf": "arxiv_math/2503.03899_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03899", "page": 1, "id": "2503.03899_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_n,\\beta_n \\geq n^{-\\frac{1}{4}+\\delta}"}
{"pdf": "arxiv_math/2503.06780_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06780", "page": 1, "id": "2503.06780_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi(\\mu,E,\\hbar ,x) = \\sum_{k=0}^\\infty \\phi_k(\\mu ,E,x) \\hbar^k"}
{"pdf": "arxiv_math/2503.06780_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06780", "page": 1, "id": "2503.06780_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "x= \\dots ,-2\\pi ,-\\pi,0,\\pi , \\dots"}
{"pdf": "arxiv_math/2503.08964_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08964", "page": 1, "id": "2503.08964_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{t_1\\times 1,t_2\\times 2,\\dots}"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bigcup_{e \\in E} eRe = R"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi: S^{\\times} \\rightarrow G"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\phi}_g: \\mathcal T_c(E_{g^{-1}}) \\rightarrow \\mathcal T_c(E_g)"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_{\\mid S_1^{\\times}}"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "L_R(S_1) \\subseteq L_R(S_2)"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "V_x \\mapsto V_{\\phi_g(x)}"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "E_g = \\{x \\in E \\colon x \\leq ss^{\\ast} \\text{ for some } s \\in S \\text{ with } \\varphi(s) = g\\} \\subseteq E"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_g(x) = sxs^{\\ast}"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_g: E_{g^{-1}} \\rightarrow E_g"}
{"pdf": "arxiv_math/2503.06715_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06715", "page": 1, "id": "2503.06715_pg4_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "(\\mathcal T_c(E_1), \\Phi_1) \\subseteq (\\mathcal T_c(E_2), \\Phi_2)"}
{"pdf": "arxiv_math/2503.03948_pg3.pdf", "url": "https://arxiv.org/pdf/2503.03948", "page": 1, "id": "2503.03948_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{(x,y,E)} \\approx 0"}
{"pdf": "arxiv_math/2503.03948_pg3.pdf", "url": "https://arxiv.org/pdf/2503.03948", "page": 1, "id": "2503.03948_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{(\\mathbf{x},E)} = 0"}
{"pdf": "arxiv_math/2503.03948_pg3.pdf", "url": "https://arxiv.org/pdf/2503.03948", "page": 1, "id": "2503.03948_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Vert \\delta H \\Vert < \\mu_{(0,0,E)}(X,Y,H)"}
{"pdf": "arxiv_math/2503.03948_pg3.pdf", "url": "https://arxiv.org/pdf/2503.03948", "page": 1, "id": "2503.03948_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "H \\rightarrow H + \\delta H"}
{"pdf": "arxiv_math/2503.08898_pg56.pdf", "url": "https://arxiv.org/pdf/2503.08898", "page": 1, "id": "2503.08898_pg56_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ch_k^H(\\gamma), \\ch_k^H(e_j)"}
{"pdf": "arxiv_math/2503.08898_pg56.pdf", "url": "https://arxiv.org/pdf/2503.08898", "page": 1, "id": "2503.08898_pg56_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha=(r,d,f_\\bullet, c)\\in C(I)"}
{"pdf": "arxiv_math/2503.08898_pg56.pdf", "url": "https://arxiv.org/pdf/2503.08898", "page": 1, "id": "2503.08898_pg56_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "[M_\\alpha]\\in \\widecheck P_0"}
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{"pdf": "arxiv_math/2503.07533_pg5.pdf", "url": "https://arxiv.org/pdf/2503.07533", "page": 1, "id": "2503.07533_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma^2 \\rightarrow 0"}
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{"pdf": "arxiv_math/2503.03994_pg108.pdf", "url": "https://arxiv.org/pdf/2503.03994", "page": 1, "id": "2503.03994_pg108_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "v_p(\\Theta_0)=0=-1-v_p(x_1)"}
{"pdf": "arxiv_math/2503.03994_pg108.pdf", "url": "https://arxiv.org/pdf/2503.03994", "page": 1, "id": "2503.03994_pg108_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\alpha_0}{\\beta_0}=\\frac{\\alpha_1}{\\beta_1}=p\\Theta_0\\Theta_1"}
{"pdf": "arxiv_math/2503.03994_pg108.pdf", "url": "https://arxiv.org/pdf/2503.03994", "page": 1, "id": "2503.03994_pg108_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{\\Theta_0}=\\frac{1}{\\Theta_1}=0"}
{"pdf": "arxiv_math/2503.03994_pg108.pdf", "url": "https://arxiv.org/pdf/2503.03994", "page": 1, "id": "2503.03994_pg108_math_018", "type": "math", "max_diffs": 0, "checked": null, "math": "v_p(\\Theta_0)\\not\\in\\{-1,0,v_p(x_0),-1-v_p(x_1)\\}"}
{"pdf": "arxiv_math/2503.03994_pg108.pdf", "url": "https://arxiv.org/pdf/2503.03994", "page": 1, "id": "2503.03994_pg108_math_019", "type": "math", "max_diffs": 0, "checked": null, "math": "\\max\\{-1,-1-v_p(x_1)\\}\\leq v_p(\\Theta_0)=1-v_p(\\Theta_1)\\leq \\min\\{0,v_p(x_0)\\}"}
{"pdf": "arxiv_math/2503.03994_pg108.pdf", "url": "https://arxiv.org/pdf/2503.03994", "page": 1, "id": "2503.03994_pg108_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "v_p(\\Theta_0)\\not\\in\\{0,v_p(x_0)\\}"}
{"pdf": "arxiv_math/2503.03994_pg108.pdf", "url": "https://arxiv.org/pdf/2503.03994", "page": 1, "id": "2503.03994_pg108_math_027", "type": "math", "max_diffs": 0, "checked": null, "math": "A:= \\begin{bmatrix} -\\frac{x_1}{\\Theta_1}\\alpha_1 & \\alpha_1 \\\\ -\\beta_1 & 0 \\end{bmatrix} \\begin{bmatrix} -\\frac{x_0}{\\Theta_0}\\alpha_0 & \\alpha_0 \\\\ -\\beta_0 & 0 \\end{bmatrix} = \\begin{bmatrix} \\frac{x_0x_1}{\\Theta_0\\Theta_1}\\alpha_0\\alpha_1-\\beta_0\\alpha_1 & -\\frac{x_1}{\\Theta_1}\\alpha_0\\alpha_1 \\\\ \\frac{x_0}{\\Theta_0}\\alpha_0\\beta_1 & -\\alpha_0\\beta_1 \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.03994_pg108.pdf", "url": "https://arxiv.org/pdf/2503.03994", "page": 1, "id": "2503.03994_pg108_math_028", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_0\\alpha_1=\\lambda^2\\Theta_0\\Theta_1"}
{"pdf": "arxiv_math/2503.03994_pg108.pdf", "url": "https://arxiv.org/pdf/2503.03994", "page": 1, "id": "2503.03994_pg108_math_029", "type": "math", "max_diffs": 0, "checked": null, "math": "v_p(\\Theta_0)=v_p(x_0)=-1-v_p(x_1)\\not\\in\\{-1,0\\}"}
{"pdf": "arxiv_math/2503.07410_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07410", "page": 1, "id": "2503.07410_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "LV_{M_{Dir}, \\ell^\\infty}(\\lambda)"}
{"pdf": "arxiv_math/2503.07410_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07410", "page": 1, "id": "2503.07410_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\| M \\|_{p \\rightarrow q}"}
{"pdf": "arxiv_math/2503.07410_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07410", "page": 1, "id": "2503.07410_pg12_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "LV_{M, \\ell^p}(N^\\sigma)"}
{"pdf": "arxiv_math/2503.07410_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07410", "page": 1, "id": "2503.07410_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "LV_{M, \\ell^p}(\\lambda)"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(q(t),\\dot{q}(t),u(t)) \\in \\mathcal{E}"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dot{\\lambda}_v(t)^{\\top} = D_2 C(q(t),v(t),u(t)) - \\lambda_v(t)^{\\top} \\, D_2 f(q(t),v(t),u(t)) - \\lambda_q(t)^{\\top}"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dot{v}(t) = f(x(t),v(t),u(t))"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dot{\\kappa}(T)^{\\top} = D_1 \\phi(q(T),\\dot{q}(T))"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dot{\\lambda}_q(t)^{\\top} = D_1 C(q(t),v(t),u(t)) - \\lambda_v(t)^{\\top}\\, D_1 f(q(t),v(t),u(t))"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ddot{\\kappa}(t)^{\\top} = \\frac{d}{dt} \\left[ D_2 C(q(t),\\dot{q}(t),u(t)) - \\kappa(t)^{\\top} D_2 f(q(t),\\dot{q}(t),u(t)) \\right]"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ddot{\\kappa}(t)^{\\top}"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "T^{(2)} \\mathcal{Q} \\oplus_{T\\mathcal{Q}} \\mathcal{E}"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "0 = D_3 C(q(t),\\dot{q}(t),u(t)) - \\kappa(t)^{\\top} \\, D_3 f(q(t),\\dot{q}(t),u(t))"}
{"pdf": "arxiv_math/2503.04466_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04466", "page": 1, "id": "2503.04466_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(q,v,\\lambda_q,\\lambda_v)"}
{"pdf": "arxiv_math/2503.05521_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05521", "page": 1, "id": "2503.05521_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f:I\\rightarrow (0, \\infty)"}
{"pdf": "arxiv_math/2503.05521_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05521", "page": 1, "id": "2503.05521_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma=(\\alpha, \\beta)"}
{"pdf": "arxiv_math/2503.05521_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05521", "page": 1, "id": "2503.05521_pg22_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\frac{1}{2}(\\alpha')^2 + \\frac{1}{2f^2\\circ \\alpha}=E"}
{"pdf": "arxiv_math/2503.05521_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05521", "page": 1, "id": "2503.05521_pg22_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{c_\\gamma}{2f^2\\circ \\alpha}"}
{"pdf": "arxiv_math/2503.05521_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05521", "page": 1, "id": "2503.05521_pg22_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(p,x), (q,y), (r,z)\\in I\\times F"}
{"pdf": "arxiv_math/2503.04527_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04527", "page": 1, "id": "2503.04527_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}_{det}=0.95"}
{"pdf": "arxiv_math/2503.04527_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04527", "page": 1, "id": "2503.04527_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}_{det}= 1 - \\prod_{s=1}^{t} (1-\\phi_s)^{N_s}"}
{"pdf": "arxiv_math/2503.05068_pg47.pdf", "url": "https://arxiv.org/pdf/2503.05068", "page": 1, "id": "2503.05068_pg47_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{K}_{k\\Delta t, U(\\cdot, 0), \\theta}\\coloneqq \\max_{1\\leq i\\leq k}K_{i\\Delta t, U(\\cdot, 0), \\theta}"}
{"pdf": "arxiv_math/2503.05068_pg47.pdf", "url": "https://arxiv.org/pdf/2503.05068", "page": 1, "id": "2503.05068_pg47_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "U(\\bm{x}, 0)=\\hat{U}(\\bm{x}, 0)"}
{"pdf": "arxiv_math/2503.05068_pg47.pdf", "url": "https://arxiv.org/pdf/2503.05068", "page": 1, "id": "2503.05068_pg47_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} W_{2, \\delta}^{2, \\text{e}}\\big(\\hat{U}(\\cdot, k\\Delta t;\\hat{\\theta}), \\hat{U}_n(\\cdot, k\\Delta t;\\hat{\\theta} )\\big) &\\leq \\sup_{\\hat{\\theta}, U(\\bm{x}, 0)} K_{T, U(\\cdot, 0), \\theta}\\lambda_{N+1}^{-1}(\\hat{\\theta}). \\end{aligned}"}
{"pdf": "arxiv_math/2503.05068_pg47.pdf", "url": "https://arxiv.org/pdf/2503.05068", "page": 1, "id": "2503.05068_pg47_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nu_{U(\\cdot, 0)}(k\\Delta t)"}
{"pdf": "arxiv_math/2503.05068_pg47.pdf", "url": "https://arxiv.org/pdf/2503.05068", "page": 1, "id": "2503.05068_pg47_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "U_n(\\bm{x}, T;\\theta)"}
{"pdf": "arxiv_math/2503.05068_pg47.pdf", "url": "https://arxiv.org/pdf/2503.05068", "page": 1, "id": "2503.05068_pg47_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{k\\Delta t, U(\\cdot, 0), \\theta}"}
{"pdf": "arxiv_math/2503.05068_pg47.pdf", "url": "https://arxiv.org/pdf/2503.05068", "page": 1, "id": "2503.05068_pg47_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nu_{n, U_n(\\cdot, 0)}(k\\Delta t)"}
{"pdf": "arxiv_math/2503.05068_pg47.pdf", "url": "https://arxiv.org/pdf/2503.05068", "page": 1, "id": "2503.05068_pg47_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\big(U(\\bm{x}, k\\Delta t; \\theta), U_n(\\bm{x}, k\\Delta t;\\theta)\\big)"}
{"pdf": "arxiv_math/2503.05068_pg47.pdf", "url": "https://arxiv.org/pdf/2503.05068", "page": 1, "id": "2503.05068_pg47_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "U(\\bm{x}, k\\Delta t; \\theta)"}
{"pdf": "arxiv_math/2503.05062_pg35.pdf", "url": "https://arxiv.org/pdf/2503.05062", "page": 1, "id": "2503.05062_pg35_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "[i,i+r-2](\\bmod\\ {n+1})=\\begin{cases} [i,i+r-2],\\ &\\text{if}\\ i+r-2\\leq n;\\\\ [i,n]\\cup [1,(i+r-2\\bmod n+1)+1],\\ &\\text{if}\\ i+r-2> n. \\end{cases}"}
{"pdf": "arxiv_math/2503.09469_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09469", "page": 1, "id": "2503.09469_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "% \\lambda_{-1}(0) &= \\lambda_{-1}(\\B) \\\\ % \\lambda^{\\prime}_{-1}(0) &= -\\vect{v}_{-1}(\\B)^T \\C \\vect{v}_{-1}(\\B) =: - \\xi %"}
{"pdf": "arxiv_math/2503.09469_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09469", "page": 1, "id": "2503.09469_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_{-1}(\\omega) \\le \\lambda_{-1}(0) + \\lambda^{\\prime}_{-1}(0) \\omega"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\circ T = \\sigma \\circ f"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(-1)^m(d_m - x_{k+m}) < 0"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{\\beta}\\setminus P_{\\beta}^{\\Z}"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdots 211 211 \\cdots 211 \\bullet 2(1)^{\\infty}"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdots 210 210 \\cdots 210 \\bullet 2 (1)^{\\infty}"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{k+1} \\cdots x_{k+m-1} = d_1\\cdots d_{m-1}"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "d(l_{\\beta}, \\beta) = \\bullet2 (1)^{\\infty}"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdots 210 210 210 \\cdots 210 \\bullet (2)^{\\infty}"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\circ T = \\sigma\\circ f"}
{"pdf": "arxiv_math/2503.06597_pg18.pdf", "url": "https://arxiv.org/pdf/2503.06597", "page": 1, "id": "2503.06597_pg18_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "d_1d_2 \\cdots = 2 (1)^{\\infty}"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{H}_c \\subset \\mathbb{G}_\\mathbb{R}"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{G}_\\mathrm{R}"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{G}_{\\mathbf{comp}}"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\mathrm{G_\\theta}}{\\mathrm{H_\\theta}}"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{G}_\\mathrm{R}"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{\\mathcal{H}}_c"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{G_\\mathrm{R}/H_c}"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{H}_\\theta = \\mathbb{H}_c"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{G}_{\\mathrm{R}}"}
{"pdf": "arxiv_math/2503.07626_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07626", "page": 1, "id": "2503.07626_pg12_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{H}_c\\subset \\mathrm{G}_\\mathrm{R}"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_000", "type": "math", "max_diffs": 0, "checked": "verified", "math": "(Q,{\\bf d})=(Q',{\\bf d}|_{Q'})\\#_i(Q'',{\\bf d}|_{Q''})"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\iota:\\mathbb{Z}(Q_{\\bf d})_0\\rightarrow\\mathbb{Z}(Q_{\\bf e})_0"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\iota:Q_{\\bf d}\\rightarrow Q_{\\bf e},\\;\\;\\; (i,k)\\mapsto(i,k+e_i-d_i)"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{\\widehat{\\bf d}}^{\\rm flag}(Q_{\\bf d})"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\iota:W_{Q_{\\bf d}}\\rightarrow W_{Q_{\\bf e}}"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "1\\leftrightarrow 2\\leftrightarrow 3"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{\\widehat{\\bf d}}^{\\rm flag}(Q)"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bf e}\\leq\\widehat{\\bf d}"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "B_{\\bf d}\\subset G_{\\bf d}"}
{"pdf": "arxiv_math/2503.05407_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05407", "page": 1, "id": "2503.05407_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bf d}\\in\\mathbb{N}Q_0"}
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{"pdf": "arxiv_math/2503.08345_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08345", "page": 1, "id": "2503.08345_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal J_F(\\mathbf 0)"}
{"pdf": "arxiv_math/2503.08345_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08345", "page": 1, "id": "2503.08345_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{e_\\alpha\\}_{\\alpha\\in\\mathbb N^n}"}
{"pdf": "arxiv_math/2503.08345_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08345", "page": 1, "id": "2503.08345_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left(A_1 f\\right)_l = \\sum_{\\beta \\in \\mathbb N^n} (\\beta_l + 1) f_{(\\beta_1, \\ldots, \\beta_{l-1}, \\beta_l + 1, \\beta_{l+1},\\ldots, \\beta_n)} e_\\beta"}
{"pdf": "arxiv_math/2503.08345_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08345", "page": 1, "id": "2503.08345_pg8_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "f_\\beta = \\langle f, e_\\beta\\rangle"}
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{"pdf": "arxiv_math/2503.08345_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08345", "page": 1, "id": "2503.08345_pg8_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb H^2(\\mathbb{D}^n)"}
{"pdf": "arxiv_math/2503.08345_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08345", "page": 1, "id": "2503.08345_pg8_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "A_F f = \\sum_{\\alpha\\in \\mathbb N^n} \\left( \\sum_{\\beta\\in\\mathbb N^n} {A_F}_{\\alpha,\\beta} f_\\beta \\right)e_\\alpha\\\\"}
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{"pdf": "arxiv_math/2503.04189_pg11.pdf", "url": "https://arxiv.org/pdf/2503.04189", "page": 1, "id": "2503.04189_pg11_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "a_1 \\in \\widetilde{S}_s^{m-1, k}"}
{"pdf": "arxiv_math/2503.04189_pg11.pdf", "url": "https://arxiv.org/pdf/2503.04189", "page": 1, "id": "2503.04189_pg11_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "Q(a e^{i \\frac{\\varphi}{h}}) = Q'_\\varphi({a}) e^{i \\frac{\\varphi}{h}} + R(a), \\tag{3.12}"}
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{"pdf": "arxiv_math/2503.04189_pg11.pdf", "url": "https://arxiv.org/pdf/2503.04189", "page": 1, "id": "2503.04189_pg11_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi(x, \\xi') - \\varphi(y, \\xi') = \\Sigma(x, y', \\xi')(x' - y')"}
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{"pdf": "arxiv_math/2503.06267_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06267", "page": 1, "id": "2503.06267_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "V\\cong\\operatorname{Im}\\pi \\oplus \\ker\\pi=W\\oplus W'"}
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{"pdf": "arxiv_math/2503.09455_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09455", "page": 1, "id": "2503.09455_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "i_1^*b_J + \\delta \\phi"}
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{"pdf": "arxiv_math/2503.09455_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09455", "page": 1, "id": "2503.09455_pg21_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "p^*i_0^*[i_1^*b_J + \\delta \\phi]"}
{"pdf": "arxiv_math/2503.09440_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09440", "page": 1, "id": "2503.09440_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d_T(w_1,r)\\geq d_T(w_j,r)"}
{"pdf": "arxiv_math/2503.09440_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09440", "page": 1, "id": "2503.09440_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "w_1 \\in V(T_i)\\cap V(T_j)"}
{"pdf": "arxiv_math/2503.09440_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09440", "page": 1, "id": "2503.09440_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "V(T_i)\\cap V(T_k)\\neq \\emptyset"}
{"pdf": "arxiv_math/2503.09440_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09440", "page": 1, "id": "2503.09440_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "V(T_1)\\cap V(T_j)\\neq \\emptyset"}
{"pdf": "arxiv_math/2503.09440_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09440", "page": 1, "id": "2503.09440_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "N_G(v_j)\\subseteq N_G(v_i)"}
{"pdf": "arxiv_math/2503.09440_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09440", "page": 1, "id": "2503.09440_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "V(T_i)\\cap V(T_j)\\neq \\emptyset"}
{"pdf": "arxiv_math/2503.09440_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09440", "page": 1, "id": "2503.09440_pg14_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "d_T(r,w_k)\\leq d_T(r,w_1)"}
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{"pdf": "arxiv_math/2503.09549_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09549", "page": 1, "id": "2503.09549_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{x}\\in \\mathbb{R}^d"}
{"pdf": "arxiv_math/2503.09549_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09549", "page": 1, "id": "2503.09549_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "x_k(t_0) = x_k^0 \\in \\mathbb{R}^d"}
{"pdf": "arxiv_math/2503.09549_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09549", "page": 1, "id": "2503.09549_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\Omega, \\mathcal{F}, \\mathbb{P})"}
{"pdf": "arxiv_math/2503.09549_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09549", "page": 1, "id": "2503.09549_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{F} = (\\mathcal{F}_t)_{t \\geq t_0}"}
{"pdf": "arxiv_math/2503.09549_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09549", "page": 1, "id": "2503.09549_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "x_k(t) \\in \\mathbb{R}^d"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(k_r+k_r^{-1})^{-2}=\\vartheta r^2/2"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal G(x,y,S)\\equiv0"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{(x_1,x_2)\\in\\mathbb R^2: 2U(x_1)+x_2^2=r^2\\}"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{1,2,p}(r,\\varphi,S)\\equiv \\alpha_{2,2,p}(r,\\varphi,S)\\equiv 0"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "0<|r_0|<(2\\vartheta)^{-1/2}"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi(t)\\equiv \\nu(\\varrho_0)t+\\phi_0"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "0<|r|<(2\\vartheta)^{-1/2}"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu(t)\\equiv t^{-1/4}"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "0<|r|< (2\\vartheta)^{-1/2}"}
{"pdf": "arxiv_math/2503.07537_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07537", "page": 1, "id": "2503.07537_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varkappa\\in\\mathbb Z_+"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(g,h)=t_1^n\\otimes t_2^m\\in\\mathcal{A}"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "t_1^nh_1t_1^nh_1^{-1}\\in \\langle u\\rangle"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "t_1^{2n}\\otimes t_2^{2m}=(t_1^nh_1t_1^nh_1^{-1}\\otimes t_2^mh_2t_2^mh_2^{-1})(h_1t_1^{-n}h_1^{-1}t_1^n\\otimes h_2t_2^{-m}h_2^{-1}t_2^m)\\in\\mathcal{A}"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "h_1t_1^{-n}h_1^{-1}t_1^n\\otimes h_2t_2^{-m}h_2^{-1}t_2^m\\in\\mathcal{A}"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}_{t_1}\\otimes\\mathbb{E}_{t_2}(a)(h_1t_1^{2n}h_1^{-1}\\otimes h_2t_2^{2m}h_2^{-1})=\\sum_{k,l\\in\\mathbb{Z}}c_{t_1^k,t_2^l}t_1^kh_1t_1^{2n}h_1^{-1}\\otimes t_2^lh_2t_2^{2m}h_2^{-1}"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{(t_1^n,t_2^m)}\\ne0"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}_{t_1}\\otimes\\mathbb{E}_{t_2}(a)(h_1t_1^{2n}h_1^{-1}\\otimes h_2t_2^{2m}h_2^{-1})\\in\\mathcal{A}"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "t_2^mh_2t_2^{2m}h_2^{-1}\\in\\langle v'\\rangle"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "t_1^nh_1t_1^nh_1^{-1}\\otimes t_2^mh_2t_2^mh_2^{-1}\\in\\mathcal{A}"}
{"pdf": "arxiv_math/2503.09548_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09548", "page": 1, "id": "2503.09548_pg18_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "t_1^nh_1^2t_1^nh_1^{-2}\\otimes t_2^mh_2^2t_2^mh_2^{-2}\\in\\mathcal{A}"}
{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{V}:=\\{(v,q)\\in H^1_0(\\Omega)\\times L^2_0(\\Omega)\\colon -\\Delta v+\\nabla q \\in L^2(\\Omega), \\nabla\\cdot v\\in P\\}"}
{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sin\\lambda\\omega=\\pm\\lambda\\sin\\omega"}
{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Re}\\lambda\\in[\\frac12,4]"}
{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(y,p)\\in L^2(\\Omega)\\times P'"}
{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(u,p)\\in H^1(\\Omega)\\times L^2_0(\\Omega)"}
{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "U(\\omega)=-(1-\\lambda)c_3U^{(1)}(\\omega) -(1+\\lambda)c_4U^{(2)}(\\omega)+ c_3U^{(3)}(\\omega)+c_4U^{(4)}(\\omega)"}
{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\min(1,\\xi)<\\lambda\\le0"}
{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "u\\in H^s(\\Omega),\\quad p\\in H^{s-1}(\\Omega) \\quad\\forall s<1+\\lambda"}
{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "P=\\{v\\in H^1(\\Omega)\\cap L^2_0(\\Omega): r^{-1} v\\in L^2(\\Omega)\\}"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_{\\mathcal{A}^\\vee/X}"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\longrightarrow \\omega_{\\mathcal{A}^\\vee/X,\\tau_i}\\longrightarrow\\H(\\mathcal{A}/X)_{\\tau_i}\\longrightarrow \\textnormal{Lie}_{\\mathcal{A}/X,\\tau_i}\\longrightarrow 0"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "p\\mathcal{O}_E=\\mathfrak{q}\\mathfrak{q}^c"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle\\cdot,\\cdot\\rangle:\\H(\\mathcal{A}/X)_{\\tau_i}\\times \\H(\\mathcal{A}/X)_{\\tau_i^c}\\longrightarrow \\mathcal{O}_X"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(m_i,n_i)_{1\\le i\\le N}"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_{\\mathcal{A}^\\vee/X,\\tau_i^c}=\\omega_{\\mathcal{A}^\\vee/X,\\tau_i}^\\perp"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{L}_Y(\\lambda)"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\tau_1,\\dots,\\tau_N\\}"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mathcal{L}_Y(\\lambda)\\cdot C)"}
{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_{\\mathcal{A}^\\vee/X,\\tau_i}"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in\\Omega\\setminus\\tilde{\\Omega}"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(0,\\infty)\\ni s\\mapsto\\dfrac{\\Phi(x,s)}{s^{r-1}}"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "w_1,w_2\\in W^{1,p(x)}(\\Omega)"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\nabla w_1(x)|,|\\nabla w_2(x)|>0"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{\\Phi(x,|\\nabla w_1(x)|)}{|w_1(x)|^{r-1}}=\\dfrac{\\Phi(x,\\lambda|\\nabla w_1(x)|)}{\\big (\\lambda|w_1(x)|\\big )^{r-1}}"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{|\\nabla w_1(x)|}{w_1(x)}=\\dfrac{|\\nabla w_2(x)|}{w_2(x)}\\ \\Longrightarrow\\ \\dfrac{|\\nabla w_1(x)|}{w_1(x)}=\\dfrac{\\lambda(x)|\\nabla w_1(x)|}{w_2(x)}\\ \\Longrightarrow\\ \\lambda(x)=\\dfrac{w_2(x)}{w_1(x)}"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla w_2=\\lambda\\nabla w_1=0"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\nabla w_1(x)|\\cdot |\\nabla w_2(x)|=\\nabla w_1(x)\\cdot\\nabla w_2(x)"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{w_1}{w_2},\\ \\dfrac{w_2}{w_1}\\in L^{\\infty}(\\Omega)"}
{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{\\nabla w_1(x)}{w_1(x)}=\\dfrac{\\nabla w_2(x)}{w_2(x)}=0"}
{"pdf": "arxiv_math/2503.06581_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06581", "page": 1, "id": "2503.06581_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{I}_f,\\mathcal{I}_p,\\mathcal{I}_s,\\mathcal{I}_E"}
{"pdf": "arxiv_math/2503.06581_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06581", "page": 1, "id": "2503.06581_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho\\in L^2(\\mathbb R^3)"}
{"pdf": "arxiv_math/2503.06581_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06581", "page": 1, "id": "2503.06581_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "J\\in\\left(H^1(\\mathbb R^3)\\right)^3"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "H^* = \\operatorname{Span}\\{f_1, f_2, \\cdots, f_n\\}"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\beta_i\\}_{i\\leq n}"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{b_i\\}_{i\\leq n} \\subseteq X_{\\leq 1+\\sigma}"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_1 = r(y_1) >\\dfrac{1}{2}"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "r \\in X^{\\ast\\ast}\\backslash Q(X)"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "E_2 = \\operatorname{Span}\\{b_1, b_2\\}"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_n\\|\\sum_{i \\leq n}e_i\\| < \\infty"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "y_2 \\in X^*_{\\leq 1}\\cap E_1^{\\perp}"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\lambda_1, \\lambda_2, \\cdots, \\lambda_n)\\in\\mathbb{C}^n"}
{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_n = \\sup\\big\\{ r(y)\\,\\vert\\, y\\in X^{\\ast}_{= 1}\\cap E_n^{\\perp}\\big\\}"}
{"pdf": "arxiv_math/2503.06466_pg22.pdf", "url": "https://arxiv.org/pdf/2503.06466", "page": 1, "id": "2503.06466_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha(e^{-1}) = (\\alpha(e))^{-1}"}
{"pdf": "arxiv_math/2503.06466_pg22.pdf", "url": "https://arxiv.org/pdf/2503.06466", "page": 1, "id": "2503.06466_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha: D(\\Gamma) \\rightarrow G"}
{"pdf": "arxiv_math/2503.06466_pg22.pdf", "url": "https://arxiv.org/pdf/2503.06466", "page": 1, "id": "2503.06466_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "V(\\Gamma^{\\alpha}) = V(\\Gamma) \\times G"}
{"pdf": "arxiv_math/2503.06466_pg22.pdf", "url": "https://arxiv.org/pdf/2503.06466", "page": 1, "id": "2503.06466_pg22_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "e = (u,v) \\in D(\\Gamma)"}
{"pdf": "arxiv_math/2503.08808_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08808", "page": 1, "id": "2503.08808_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\frac{1}{2\\pi i} \\right)^2 \\int_{c - i\\infty}^{c + i\\infty} \\int_{c - i\\infty}^{c + i\\infty} \\frac{1}{\\left[ (z_1 + a)(z_2 + a) - b^2 \\right]^k} e^{z_1 x_1 + z_2 x_2} \\,dz_1 dz_2 \\notag \\\\ = \\frac{1}{\\Gamma(k) b^{k-1}} (x_1 x_2)^{\\frac{k-1}{2}} e^{-\\alpha (x_1 + x_2)} I_{k-1} \\left( 2b \\sqrt{x_1 x_2} \\right)"}
{"pdf": "arxiv_math/2503.08808_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08808", "page": 1, "id": "2503.08808_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "b = \\frac{\\sqrt{\\rho}}{\\sigma(1-\\rho)}"}
{"pdf": "arxiv_math/2503.08808_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08808", "page": 1, "id": "2503.08808_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{\\Gamma(k) b^{k-1}} \\int_0^{\\infty} \\int_0^{\\infty} (x_1 x_2)^{\\frac{k-1}{2}} e^{-\\alpha (x_1 + x_2)} I_{k-1} \\left( 2b \\sqrt{x_1 x_2} \\right) e^{-z_1 x_1 - z_2 x_2} \\,dx_1 dx_2 \\notag \\\\ = \\frac{1}{\\left[ (z_1 + a)(z_2 + a) - b^2 \\right]^k}"}
{"pdf": "arxiv_math/2503.08808_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08808", "page": 1, "id": "2503.08808_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha = \\frac{1}{\\sigma(1-\\rho)}"}
{"pdf": "arxiv_math/2503.07729_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07729", "page": 1, "id": "2503.07729_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T \\Delta E \\to \\infty"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "id^*\\nabla^\\mathrm{flat}=\\nabla^\\mathrm{flat}"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{\\nabla}\\in[\\overline{\\nabla}]"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "X, Y\\in\\mathfrak{L}(\\mathcal{F}^M)"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "id:\\mathbb{R}^3\\rightarrow\\mathbb{R}^3"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(F_*\\hat{\\nabla}^M)_XY=W"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(F^*\\hat{\\nabla}^N)_XY=Z"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "X, Y \\in\\mathfrak{X}(M)"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "Z(p):=d\\Phi_p^{-1}\\left(\\hat{\\nabla}^N_{\\Phi_*X}\\Phi_*Y\\right)_{\\Phi(p)}"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "W(q):=d\\Phi_p\\left(\\hat{\\nabla}_{\\Phi^*X}\\Phi^*Y\\right)_{p}"}
{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "V\\in\\mathfrak{X}(\\mathcal{F}^M)"}
{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_000", "type": "math", "max_diffs": 0, "checked": "verified", "math": "p(\\tau, \\tau \\sqcup \\{ij\\}) = \\left\\{ \\begin{array}{cl} \\displaystyle \\frac{\\alpha_{ij} w_{i, ij}}{\\sum_{k=1}^d \\alpha_{ik} w_{i, ik}} \\alpha_i , & \\text{if } i \\in \\partial \\tau, \\\\[15pt] \\alpha_{ij}, & \\text{if } i \\notin \\partial \\tau \\text{ and } ij \\notin \\tau, \\end{array} \\right."}
{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_i = \\alpha_{i1} + \\ldots + \\alpha_{id} \\in [0, 1]"}
{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{6} \\left(\\frac{w_1 + w_2}{2}\\right)^3 = \\frac{1}{12} \\left(\\frac{w_1 + w_2}{2}\\right)^3 + \\frac{1}{48} \\sum_{1 \\leq i, j \\leq 2} w_i w_j \\left(\\frac{w_1 + w_2}{2}\\right)"}
{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\prod_{i \\in \\overset{\\circ}{\\tau}} \\frac{\\alpha_i}{\\sum_{j=1}^d \\alpha_{ij} w_{i, ij}}}=\\left(\\frac{d}{\\sum_{j=1}^dw_j}\\right)^{|\\tau|- |\\partial \\tau|}"}
{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "j \\in \\{1, \\ldots, d\\}"}
{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\binom{n}{k_1,\\cdots,k_d}"}
{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma^{(j)} \\sqcup \\bigsqcup_{l\\neq j} \\sigma^{(l)}"}
{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal M_{\\varepsilon} \\rightarrow e(F,\\rho)"}
{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "I_i = \\frac 43 \\int \\rho_i^{\\frac 32} dt"}
{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal M_{\\varepsilon}"}
{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "e \\geq e_{\\gamma_i, \\rho_i}"}
{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal L_\\varepsilon"}
{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "e(0,0,F(t),t) = 0, \\forall t \\in [0,1]"}
{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho(t)= \\begin{cases} \\frac{1}{a^2(3-\\sqrt{8a})^2}, & \\text{if } t \\leq 2a \\\\ \\frac{2}{at(3-\\sqrt{8a})^2}, & \\text{if } 2a \\leq t \\leq 1. \\end{cases}"}
{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "e_{\\gamma, \\rho}(x_1,t_1,x_2,t_2) = - \\infty"}
{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "G(U)=\\begin{pmatrix}g_{1}(U) \\\\ g_{2}(U) \\\\ g_{3}(U) \\end{pmatrix}=\\begin{pmatrix} m P(H(M)-a_{1} \\lambda M ) \\\\ -m a_{2} \\lambda M S \\\\m_{s}S(1-M)-\\eta M P \\end{pmatrix},\\quad U_{R}=(0,S_{R},M_{R})\\quad \\text{and}\\quad U_{0}=(P_0,S_{0},M_{0})"}
{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{L^1([0,T]; X)} = \\int_0^T \\|f(t)\\|_X dt"}
{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{C(\\Omega_R)} = \\sup_{x \\in \\Omega_R} |f(x)| "}
{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "W^{2,1}(\\Omega_R) = \\{u \\in L^1(\\Omega_R) : D^\\alpha u \\in L^1(\\Omega_R) \\quad\\text{for all}\\quad |\\alpha| \\leq 2\\}"}
{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{C([0,T]; X)} = \\sup_{t \\in [0,T]} \\|f(t)\\|_X "}
{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{L^{\\infty}(\\Omega_R)} = \\operatorname{ess\\,sup}_{x \\in \\Omega_R} |f(x)|"}
{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "C\\left([0,T],C^{1}_{b}(\\R^d)\\right)"}
{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{C^1_b(\\Omega_R)} = \\|f\\|_{C(\\Omega_R)} + \\|\\nabla f\\|_{C(\\Omega_R)}"}
{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\displaystyle \\mathcal{A}\\left(U\\right)(t,x)=\\left[\\operatorname{div}\\left(\\vec{\\alpha_{p}}(t,x)\\left(\\int_{ \\Omega_{R}}\\gamma_{p}(x-y)P(t,y)d y \\right) P(t,x)\\right),\\operatorname{div}\\left(\\vec{\\alpha_{s}}(t,x)\\left(\\int_{ \\Omega_{R}}\\gamma_{s}(x-y)S(t,y)d y \\right)S(t,x)\\right),-D\\Delta M\\right]"}
{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}_A(\\phi)\\rvert_{H^{\\perp}} \\in L^1(H^{\\perp})"}
{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{\\perp} \\cong \\widehat{A/H}"}
{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}_{A/H}(\\phi^H) = \\mathcal{F}_A(\\phi)\\rvert_{H^{\\perp}}"}
{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "A= (\\Z/N\\Z)^{d_1} \\times \\R^{d_2}"}
{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_H \\phi(xh)dh = \\int_{H^{\\perp}} \\mathcal{F}_A(\\phi)(\\chi)\\chi(x)d\\chi"}
{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi^H(xH) = \\int_H \\phi(xh)dh"}
{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\wp_\\Upsilon(\\upsilon,t)d\\upsilon"}
{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Upsilon(t)\\in[\\upsilon,\\upsilon+d\\upsilon]"}
{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{\\sigma}^{-1}(\\kappa)"}
{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma = \\left[ \\frac{1}{N} \\sum_{k=1}^N \\, (z_k-\\nu)^2 \\right]^{\\frac{1}{2}} \\; , \\quad \\nu = \\frac{1}{N} \\sum_{k=1}^N z_k \\; "}
{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_010", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\wp_\\Upsilon(\\upsilon,t)"}
{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "h_{\\varphi }=\\pi /100"}
{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\alpha ,\\varepsilon ,\\lambda \\right)"}
{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "g_{0}^{\\xi _{0}}\\left( \\mathbf{x},\\mathbf{x}_{0}\\right)"}
{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x}\\in \\partial \\Omega"}
{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\alpha ,\\varepsilon \\right) "}
{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\alpha ,\\varepsilon \\right)"}
{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "% \\lambda \\in \\left[ 1,5\\right] "}
{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in C([0,T]; L^2(\\R^n))"}
{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "V \\in L^1((0,T); L^\\infty(\\R^n))"}
{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "f \\in L^2(\\R^n) \\mapsto u \\in C([0, T]; L^2(\\R^n))"}
{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ u(t, \\centerdot) : t \\in [0, T] \\}"}
{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_007", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathcal{U}_T : f \\in L^2(\\R^n) \\mapsto u(T, \\centerdot) \\in L^2(\\R^n)"}
{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "V \\in L^1((0, T); L^\\infty (\\R^n))"}
{"pdf": "arxiv_math/2503.04047_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04047", "page": 1, "id": "2503.04047_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "C_T = \\frac{\\partial \\mathbb{E}[E(T)] }{\\partial T}"}
{"pdf": "arxiv_math/2503.04047_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04047", "page": 1, "id": "2503.04047_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "t^* = \\underset{t \\geq M}{\\arg\\max} \\, \\hat{C}(t)"}
{"pdf": "arxiv_math/2503.04047_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04047", "page": 1, "id": "2503.04047_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma^2(\\{f(x_{t-M+1}), \\cdots, f(x_{t}) \\})"}
{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "1-c_0 n^{-\\tilde{\\epsilon}}"}
{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i \\neq j \\in [n] : |i-j|\\leq 2}\\frac{1}{(|\\lambda_{j}-\\lambda_{i}| + n^{-1.5-\\tilde{\\epsilon}})^2} \\leq 8c_0 n^{3.5 + 2\\tilde{\\epsilon}}"}
{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "1-n^{-\\tilde{\\epsilon}}"}
{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\epsilon} = \\epsilon/6"}
{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\min_{i \\in [n-1]} |\\lambda_{i+1}-\\lambda_i| \\geq n^{-1.5-\\tilde{\\epsilon}}"}
{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\min_{i \\in [n-3]} |\\lambda_{i+3}-\\lambda_i| \\leq n^{-6/5-\\tilde{\\epsilon}/5}"}
{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i, j \\in [n]: i \\neq j} \\frac{1}{(\\lambda_j-\\lambda_i)^2} \\leq 32c_0 n^{3.5 + 2\\tilde{\\epsilon}} + \\pi^2 n^{17/5+2\\tilde{\\epsilon}/5} \\leq n^{3.5 + 3\\tilde{\\epsilon}}"}
{"pdf": "arxiv_math/2503.07448_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07448", "page": 1, "id": "2503.07448_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "w\\colon E(H)\\to \\mathbb{R}^+"}
{"pdf": "arxiv_math/2503.07448_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07448", "page": 1, "id": "2503.07448_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "c \\colon V(G)\\to \\{1,\\dots,k\\}"}
{"pdf": "arxiv_math/2503.07448_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07448", "page": 1, "id": "2503.07448_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^n w(v_{i-1}v_i)"}
{"pdf": "arxiv_math/2503.07448_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07448", "page": 1, "id": "2503.07448_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "P=(v_0,v_1,\\dots,v_n)"}
{"pdf": "arxiv_math/2503.07448_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07448", "page": 1, "id": "2503.07448_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "w\\colon E(H) \\to \\mathbb{N}"}
{"pdf": "arxiv_math/2503.07448_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07448", "page": 1, "id": "2503.07448_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi\\colon V(G)\\to V(H)"}
{"pdf": "arxiv_math/2503.09299_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09299", "page": 1, "id": "2503.09299_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "W_1(x, y) = \\sqrt{|x - y|}"}
{"pdf": "arxiv_math/2503.08127_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08127", "page": 1, "id": "2503.08127_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla\\cdot(\\bm{C}\\otimes \\bm{u})"}
{"pdf": "arxiv_math/2503.08127_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08127", "page": 1, "id": "2503.08127_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\bm{u}\\cdot \\nabla)\\bm{C}"}
{"pdf": "arxiv_math/2503.08127_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08127", "page": 1, "id": "2503.08127_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla\\cdot(\\bm{u}\\otimes \\bm{u})=(\\bm{u}\\cdot \\nabla)\\bm{u}+(\\nabla\\cdot\\bm{u})\\bm{u}=(\\bm{u}\\cdot \\nabla)\\bm{u}"}
{"pdf": "arxiv_math/2503.08127_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08127", "page": 1, "id": "2503.08127_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\bm{u}\\otimes\\bm{v})_{ij}=u_i v_j, ~(\\bm{C}\\otimes\\bm{u})_{ijk}= C_{ij}u_k"}
{"pdf": "arxiv_math/2503.08127_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08127", "page": 1, "id": "2503.08127_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\bm{u}\\cdot \\nabla)\\bm{u}"}
{"pdf": "arxiv_math/2503.08127_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08127", "page": 1, "id": "2503.08127_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla\\cdot(\\bm{u}\\otimes \\bm{u})"}
{"pdf": "arxiv_math/2503.08127_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08127", "page": 1, "id": "2503.08127_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "F:=\\partial K^+\\bigcap \\partial K^-"}
{"pdf": "arxiv_math/2503.08127_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08127", "page": 1, "id": "2503.08127_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla\\cdot(\\bm{C}\\otimes \\bm{u})=(\\bm{u}\\cdot \\nabla)\\bm{C}"}
{"pdf": "arxiv_math/2503.04465_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04465", "page": 1, "id": "2503.04465_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "u\\in L^2((0,T)\\times G_0)"}
{"pdf": "arxiv_math/2503.04465_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04465", "page": 1, "id": "2503.04465_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi\\mapsto y(T,\\cdot;\\xi;y_0;u)"}
{"pdf": "arxiv_math/2503.04465_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04465", "page": 1, "id": "2503.04465_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}\\left[\\omega\\;:\\; y(T,\\cdot;\\alpha(\\omega);y_0;u)=0\\right]=0"}
{"pdf": "arxiv_math/2503.04465_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04465", "page": 1, "id": "2503.04465_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "y_0\\in L^2(G)\\setminus\\{0\\}"}
{"pdf": "arxiv_math/2503.04238_pg26.pdf", "url": "https://arxiv.org/pdf/2503.04238", "page": 1, "id": "2503.04238_pg26_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "S = \\begin{pmatrix} 1/4 & 0 & 0 \\\\ 0 & 1/2 & 0 \\\\ 0 & 0 & 1/4 \\end{pmatrix}, \\qquad \\mathcal Q = \\begin{pmatrix} -2 & 2 & 0 \\\\ 1 & -2 & 1 \\\\ 0 & 2 & -2 \\end{pmatrix}"}
{"pdf": "arxiv_math/2503.04238_pg26.pdf", "url": "https://arxiv.org/pdf/2503.04238", "page": 1, "id": "2503.04238_pg26_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "a_0=\\frac1{2+\\omega L}"}
{"pdf": "arxiv_math/2503.04238_pg26.pdf", "url": "https://arxiv.org/pdf/2503.04238", "page": 1, "id": "2503.04238_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left< \\cdot, \\cdot \\right>_S"}
{"pdf": "arxiv_math/2503.04238_pg26.pdf", "url": "https://arxiv.org/pdf/2503.04238", "page": 1, "id": "2503.04238_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nu = \\Omega\\left(\\frac{\\omega}{1 + (\\omega L)^2}\\right)"}
{"pdf": "arxiv_math/2503.04238_pg26.pdf", "url": "https://arxiv.org/pdf/2503.04238", "page": 1, "id": "2503.04238_pg26_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nu \\propto \\omega^{-1} L^{-2}"}
{"pdf": "arxiv_math/2503.04238_pg26.pdf", "url": "https://arxiv.org/pdf/2503.04238", "page": 1, "id": "2503.04238_pg26_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left<x, y\\right>_S = x^\\top S y"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{i-1} = (I_{i} \\setminus \\{i\\})\\cup\\{j\\}"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{i-1}\\setminus I_{i} = \\{j\\}"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{I}_f = (I_1, \\dots, I_n)"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{f}^{-1}_{\\mathcal{I}}(i) = j"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{a \\in [n] \\mid a <_{i} \\bar{f}(a)\\}"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "I = \\{i \\in [n] \\mid f(i) = i+n\\}"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{i-1} = I_{i} = (I_{i}\\setminus\\{i\\})\\cup\\{i\\}"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{i-1}\\setminus I_{i}"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "f \\mapsto \\mathcal{I}_{f}"}
{"pdf": "arxiv_math/2503.04923_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04923", "page": 1, "id": "2503.04923_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\{i \\in [n] : f(i) > n\\}| = k"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{v_0,+,j_1}<a_{v_0,+,j_2}"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{v_0,-,j_1}<a_{v_0,-,j_2}"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mathcal{S},D_{\\mathcal{S}})"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\epsilon}_{v_0,-,j_1}"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "0<{\\epsilon}_{v_0,-,j},{\\epsilon}_{v_0,+,j} <{\\epsilon}^{\\prime}<\\epsilon"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "F_{D_{\\mathcal{S}},1}"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "p_{j^{\\prime}} \\in S_{0,c_G}"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "F_{D_{\\mathcal{S}},2}"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\pi}_{2,1,1} {\\mid}_{S_{0,c_G}}"}
{"pdf": "arxiv_math/2503.09195_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09195", "page": 1, "id": "2503.09195_pg13_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "({\\pi}_{2,1,1}(v) \\pm {\\epsilon}_{v,\\pm,j_0},a_{v,\\pm,j_0})"}
{"pdf": "arxiv_math/2503.08169_pg16.pdf", "url": "https://arxiv.org/pdf/2503.08169", "page": 1, "id": "2503.08169_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} C^2(n_0) &\\ := \\ \\max_{x\\in [-\\frac{1}{n_0+2},\\frac{1}{n_0+2}]} (1-\\eta x) ^2+(\\sigma x)^2,\\\\ c^2(n_0) &\\ := \\ \\min_{x\\in [-\\frac{1}{n_0+2},\\frac{1}{n_0+2}]} (1-\\eta x) ^2+(\\sigma x)^2, \\end{aligned}"}
{"pdf": "arxiv_math/2503.08169_pg16.pdf", "url": "https://arxiv.org/pdf/2503.08169", "page": 1, "id": "2503.08169_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|(\\mathrm{I}_n+\\tfrac12 z \\mathrm{M}_{\\bm{n}})^{-1}|2"}
{"pdf": "arxiv_math/2503.08169_pg16.pdf", "url": "https://arxiv.org/pdf/2503.08169", "page": 1, "id": "2503.08169_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "f(x): = 1-2\\eta x +|z|^2 x^2"}
{"pdf": "arxiv_math/2503.08169_pg16.pdf", "url": "https://arxiv.org/pdf/2503.08169", "page": 1, "id": "2503.08169_pg16_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\left(\\frac{1}{n_0+2}\\right) = 1-\\frac{2\\eta}{n_0+2}+\\frac{\\sigma^2+\\eta^2}{(n_0+2)^2}= \\left(1-\\frac{\\eta}{ n_0+2}\\right)^2+\\frac{\\sigma^2}{(n_0+2)^2}"}
{"pdf": "arxiv_math/2503.08169_pg16.pdf", "url": "https://arxiv.org/pdf/2503.08169", "page": 1, "id": "2503.08169_pg16_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\argmin_{x\\in\\mathbb{R}} f(x) = \\frac{\\eta}{\\sigma^2+\\eta^2} = \\frac{\\eta}{|z|^2}=:x_0"}
{"pdf": "arxiv_math/2503.08169_pg16.pdf", "url": "https://arxiv.org/pdf/2503.08169", "page": 1, "id": "2503.08169_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "C^2(n_0)\\leq 1+\\frac{2|\\eta|}{n_0+2}+\\frac{|z|^2}{(n_0+2)^2}\\leq \\left(1+\\frac{|z|}{n_0+2}\\right)^2"}
{"pdf": "arxiv_math/2503.08169_pg16.pdf", "url": "https://arxiv.org/pdf/2503.08169", "page": 1, "id": "2503.08169_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "c^2(n_0)\\geq \\begin{cases} \\frac{\\sigma^2}{|z|^2}, & \\text{if }x_0\\leq \\frac{1}{n_0+2}, \\\\ f\\left(\\frac{1}{n_0+2}\\right), & \\text{if }x_0> \\frac{1}{n_0+2}. \\end{cases}"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "P \\left( \\sigma(W) \\geq \\frac{1}{k} \\right)>0"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_001", "type": "math", "max_diffs": 0, "checked": "verified", "math": "P(C^{\\leq k,w})=P(C^{k,w})"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "P(X_1 = X_2 = \\dots =X_{n_0} = 1) > 0"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "P \\left( N=n_0 \\text{ for the sequence generated by }\\{X_i\\}_{i> n_0}\\right) > 0"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{A(n)}{n} \\geq \\begin{cases} 1 & \\text{if } n \\leq n_0 \\\\ \\frac{1}{2} & \\text{if } n_0 < n \\leq 2n_0 \\\\ \\frac{1}{k} & \\text{if } n > 2n_0 \\end{cases}"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(s) = \\inf_{n}{\\frac{A(n)}{n}}"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{n=1}^{\\infty}{P(N=n)} = 1"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "d(\\{W_n\\}_{n \\in \\mathbf{N}}) > \\frac{1}{k}"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "A(n) = |\\{i: s_i \\leq n\\}|"}
{"pdf": "arxiv_math/2503.05177_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05177", "page": 1, "id": "2503.05177_pg10_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "N = \\min\\left\\{n:\\forall m \\geq n, \\quad \\frac{A(m)}{m} > \\frac{1}{k}\\right\\}"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "n-|\\mu|-\\mu_1-(|\\eta|-1)>i"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{maj}(\\sigma)=2+1+1+1=5"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_n(\\sigma)=\\sigma'"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma:\\nu\\rightarrow \\N"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "n\\geq |\\mu|+\\mu_1+|\\eta|+i"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_n:M^{(n)}\\rightarrow M^{(n+1)}"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{maj}(\\sigma):=\\sum_{u\\in \\text{Des}(\\sigma)}(\\text{leg}(u)+1)"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle [q^it^j]\\tilde{H}_{\\mu[n]},h_{\\eta[n]}\\rangle"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{st}(\\sigma)(u),\\text{st}(\\sigma)(w),\\text{st}(\\sigma)(v)"}
{"pdf": "arxiv_math/2503.04950_pg23.pdf", "url": "https://arxiv.org/pdf/2503.04950", "page": 1, "id": "2503.04950_pg23_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma:\\nu\\rightarrow \\{1,\\cdots,|\\nu|\\}"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma' \\colon X' \\longrightarrow X"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda'_{j}-\\lambda_{j}=\\nu(T'_{j},V)-\\nu(T_{j},V)=0"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{j} \\in \\{\\theta_{j}\\}"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta_{j}=(c \\|\\{\\theta_{j}\\}\\|+\\frac{\\epsilon}{2c_{V}})^{-1}\\frac{\\lambda'_{j}-\\lambda_{j}}{2}"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nu(T'_{j},V) = \\nu(R'_{j},V')=\\nu(R_{j},V') = \\nu(T_{j},V)"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathscr{S} \\cap \\mathscr{B}"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "T'_{1}, \\dots , T'_{m}"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{j} = (\\sigma')^{*}T_{j}"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "[V] \\wedge \\omega^{n-m} \\neq 0"}
{"pdf": "arxiv_math/2503.09233_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09233", "page": 1, "id": "2503.09233_pg16_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{1}', \\dots, R'_{m}"}
{"pdf": "arxiv_math/2503.04555_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04555", "page": 1, "id": "2503.04555_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdot \\ : M\\times R\\rightarrow M"}
{"pdf": "arxiv_math/2503.04555_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04555", "page": 1, "id": "2503.04555_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{split} a(b+c) & = ab + ac \\\\ (a+b)c & = ac + bc \\end{split}"}
{"pdf": "arxiv_math/2503.04041_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04041", "page": 1, "id": "2503.04041_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\epsilon=2.22\\times10^{-16}"}
{"pdf": "arxiv_math/2503.04041_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04041", "page": 1, "id": "2503.04041_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\kappa(A)=\\frac{\\sigma_{\\max}(A)}{\\sigma_{\\text{min}}(A)}"}
{"pdf": "arxiv_math/2503.04041_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04041", "page": 1, "id": "2503.04041_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "L=\\left(\\begin{array}{cccc} 3 & 1 & & \\\\ 1 & \\ddots & \\ddots & \\\\ & \\ddots & \\ddots & 1 \\\\ & & 1 & 3 \\end{array}\\right)"}
{"pdf": "arxiv_math/2503.07260_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07260", "page": 1, "id": "2503.07260_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0=3H^{2}-\\frac{3}{4}\\dot{\\beta}_{+}^{2}-\\frac{1}{4}\\dot{\\beta}_{-}^{2}% -3\\dot{\\phi}\\left( H+\\dot{\\beta}_{+}\\right) \\left( 2H-\\dot{\\beta}_{+}% -\\dot{\\beta}_{-}\\right) \\left( 2H-\\dot{\\beta}_{-}+\\dot{\\beta}_{+}\\right) -2\\Lambda, \\label{ai.08}%"}
{"pdf": "arxiv_math/2503.04382_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04382", "page": 1, "id": "2503.04382_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d^p= d^q \\Rightarrow p=q"}
{"pdf": "arxiv_math/2503.04382_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04382", "page": 1, "id": "2503.04382_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "d_p= d_q \\Rightarrow p=q"}
{"pdf": "arxiv_math/2503.04382_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04382", "page": 1, "id": "2503.04382_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I^+(p)=I^+(q)\\Rightarrow p=q"}
{"pdf": "arxiv_math/2503.04382_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04382", "page": 1, "id": "2503.04382_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ell(\\gamma\\vert_{[\\delta,1]})> m"}
{"pdf": "arxiv_math/2503.04382_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04382", "page": 1, "id": "2503.04382_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "d^r(p)= d^r(q) \\Rightarrow p=q"}
{"pdf": "arxiv_math/2503.04382_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04382", "page": 1, "id": "2503.04382_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "I^-(p)=I^-(q)\\Rightarrow p=q"}
{"pdf": "arxiv_math/2503.04382_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04382", "page": 1, "id": "2503.04382_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "d_p(r)\\ge \\ell(\\gamma\\vert_{[\\delta,1]})> m"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "I_1,I_2 \\in \\mathcal{C}^0([0,T], \\R)"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{C}^0([0,T],\\R)"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "2T\\|\\partial_Ir\\|_\\infty r_M<1"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned}\\|\\mu(n_1)-\\mu(n_2)\\|_X&\\leq 2T\\|\\partial_Ir\\|_\\infty \\|I_1-I_2\\|_\\infty. \\\\ &\\leq 2T\\|\\partial_Ir\\|_\\infty r_M\\|n_1-n_2\\|_X \\end{aligned}"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "r_1(t,x):=r(x,I_1(t))"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall t \\in [0,T],\\qquad \\|\\mu_1(t)-\\mu_2(t)\\|_1 \\leq \\|\\mu_1(0)-\\mu_2(0)\\|_1+2\\|\\partial_Ir\\|_\\infty\\int_0^t\\|I_1(\\tau)-I_2(\\tau)\\|_\\infty d\\tau"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_1(0)= \\mu_2(0) = n_\\text{ini}"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall n \\in X, \\qquad \\Phi(n,F(n)) = 0"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\mu_1-\\mu_2\\|_X\\leq 2T\\|\\partial_Ir\\|_\\infty \\|I_1-I_2\\|_\\infty"}
{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\partial_nF(n)\\|\\leq \\big\\|\\big(\\partial_I\\Phi(n,I)\\big)^{-1}\\big\\|\\:\\big\\|\\partial_n\\Phi(n,I)\\big\\|\\leq 1\\times r_M"}
{"pdf": "arxiv_math/2503.03765_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03765", "page": 1, "id": "2503.03765_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\boldsymbol{d}_l\\}_{l=1}^{L}"}
{"pdf": "arxiv_math/2503.03765_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03765", "page": 1, "id": "2503.03765_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{h}=[{h}_{1},\\cdots,{h}_{n}]^{T}"}
{"pdf": "arxiv_math/2503.03765_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03765", "page": 1, "id": "2503.03765_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{y}\\in \\mathbb{C}^n"}
{"pdf": "arxiv_math/2503.03765_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03765", "page": 1, "id": "2503.03765_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{h}, \\boldsymbol{x} \\in \\mathbb{C}^n"}
{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "|(D_p x)_i|_p = \\frac{1}{p^{\\text{ord}_p((D_p x)_i)}}"}
{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "D_p \\in \\mathbb{M}_{n\\times n}(\\mathbb{Q}_p)"}
{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|x|_p = p^{-k}, \\quad \\text{where } k \\in \\mathbb{N} \\text{ and } k \\geq 1"}
{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(D_p x)_i = \\sum_{j=1}^n d_{ij} x_j \\in \\mathbb{Q}_p"}
{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "x = (x_1, x_2, \\ldots, x_n)"}
{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{ord}_p((D_p x)_i)"}
{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta_p(x) = |D_p x|_p \\in \\mathbb{Q}_p^n"}
{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "x, \\delta_p(x), \\delta_p^2= \\delta_p(\\delta_p(x))= \\delta_p \\circ\\delta_p, \\dots, \\delta_p^n(x), \\dots"}
{"pdf": "arxiv_math/2503.08711_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08711", "page": 1, "id": "2503.08711_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{BSPA}_{0.1,30,30,90}"}
{"pdf": "arxiv_math/2503.06657_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06657", "page": 1, "id": "2503.06657_pg2_math_000", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\langle A,\\wedge,\\vee\\rangle"}
{"pdf": "arxiv_math/2503.06657_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06657", "page": 1, "id": "2503.06657_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{A}=\\langle A,\\wedge,\\vee, \\cdot,\\backslash,/,1\\rangle"}
{"pdf": "arxiv_math/2503.06657_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06657", "page": 1, "id": "2503.06657_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{K}=\\mathbf{S}_3"}
{"pdf": "arxiv_math/2503.06657_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06657", "page": 1, "id": "2503.06657_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle A,\\cdot,1\\rangle"}
{"pdf": "arxiv_math/2503.04486_pg21.pdf", "url": "https://arxiv.org/pdf/2503.04486", "page": 1, "id": "2503.04486_pg21_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\frac{\\|\\cdot\\|^2}{2}"}
{"pdf": "arxiv_math/2503.04486_pg21.pdf", "url": "https://arxiv.org/pdf/2503.04486", "page": 1, "id": "2503.04486_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "L_2 = \\mu_1 - \\mu_F \\geq -\\mu_F"}
{"pdf": "arxiv_math/2503.04486_pg21.pdf", "url": "https://arxiv.org/pdf/2503.04486", "page": 1, "id": "2503.04486_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_2 > -\\mu_1 = L_2 + \\mu_F"}
{"pdf": "arxiv_math/2503.05342_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05342", "page": 1, "id": "2503.05342_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{Z} \\times \\mathbb{Z}"}
{"pdf": "arxiv_math/2503.05342_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05342", "page": 1, "id": "2503.05342_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\vec{\u03bb}=(\u03bb_1,\\ldots ,\u03bb_c)"}
{"pdf": "arxiv_math/2503.07794_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07794", "page": 1, "id": "2503.07794_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ell\\in\\{500,600,\\ldots,1000\\}"}
{"pdf": "arxiv_math/2503.07794_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07794", "page": 1, "id": "2503.07794_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_t\\langle p^2\\rangle(x,t;\\ell)"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_{\\lambda_{i}} \\Phi = \\sum_{j=1}^n l_{ij} - 1"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi(L,\\lambda):=D(L,M) +\\sum_{i=1}^n \\lambda_i \\left(\\sum_{j=1}^n l_{ij} - 1\\right)"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{t\\to \\infty} \\frac{h(tx)}{t}=\\infty"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\argmin_{L\\in C_2} D(L,M) = \\left(\\frac{m_{ij}}{\\sum_{k=1}^n m_{kj}}\\right)_{1\\le i, j\\le n}"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{r\\to \\infty}f(r)=+\\infty"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\argmin_{L\\in C_1} D(L,M) = \\left(\\frac{m_{ij}}{\\sum_{k=1}^n m_{ik}}\\right)_{1\\le i, j\\le n}"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{k\\to\\infty} M^k=\\argmin_{L\\in C} D(L,M^0) =\\argmin_{L\\in C} H(L)"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{t\\to 0+} h'(x+t(y-x))(y-x)=-\\infty"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_{l_{ij}} \\Phi = \\log(l_{ij})-\\log(m_{ij}) +\\lambda_i"}
{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "f(r):=\\sum_{j=1}^n (j-1)m_{ij}r^{j-1}- x_i"}
{"pdf": "arxiv_math/2503.04323_pg24.pdf", "url": "https://arxiv.org/pdf/2503.04323", "page": 1, "id": "2503.04323_pg24_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "u,\\; v \\in \\mathcal{L}^{2}"}
{"pdf": "arxiv_math/2503.04323_pg24.pdf", "url": "https://arxiv.org/pdf/2503.04323", "page": 1, "id": "2503.04323_pg24_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "k \\equiv \\sum_{i=1}^{N}a_i=: A"}
{"pdf": "arxiv_math/2503.04323_pg24.pdf", "url": "https://arxiv.org/pdf/2503.04323", "page": 1, "id": "2503.04323_pg24_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_xG(t,s)=-bG(t,s)"}
{"pdf": "arxiv_math/2503.04323_pg24.pdf", "url": "https://arxiv.org/pdf/2503.04323", "page": 1, "id": "2503.04323_pg24_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_xG(t,s)=-\\sum_{i=1}^n b_iG_i(t,s),\\,0<s<t<T"}
{"pdf": "arxiv_math/2503.04323_pg24.pdf", "url": "https://arxiv.org/pdf/2503.04323", "page": 1, "id": "2503.04323_pg24_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "G(t,s)=\\sum_{i=1}^n G_i(t,s), \\; s,t \\in [0,T]"}
{"pdf": "arxiv_math/2503.04323_pg24.pdf", "url": "https://arxiv.org/pdf/2503.04323", "page": 1, "id": "2503.04323_pg24_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_{x}G(t,s)=-\\sum_{i=1}^n a_ib_i\\exp\\{-b_i(t-s)\\}, \\quad 0 \\le s < t \\le T"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "G'=G-\\{x_1,\\ldots,x_{k-1}\\}"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|S|\\le \\frac{2(n-4)}{5}"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|S|\\le \\frac{2(n-6)+2}{5}"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "D_1\\not\\in \\mathcal{B}"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "N_{G'}(x_4)\\cap S'\\neq \\emptyset"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "|S_2|=3\\le \\frac{2|V(D_2)|+2(k-1)}{5}"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "N_{G}(x_4)\\cap S'\\neq \\emptyset"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|S_1|\\le \\frac{2|V(D_1)|}{5}"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_h(G)\\le \\frac{2n}{5}"}
{"pdf": "arxiv_math/2503.04124_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04124", "page": 1, "id": "2503.04124_pg14_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "|S_1\\cup S_2|\\le \\frac{2|V(D_1)|+2|V(D_2)|+2(k-1)}{5}= \\frac{2n}{5} "}
{"pdf": "arxiv_math/2503.07373_pg21.pdf", "url": "https://arxiv.org/pdf/2503.07373", "page": 1, "id": "2503.07373_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "Q_0(e^\\mu_a)=-e^\\nu_a e_b^\\mu Q_0(e_\\nu^b)"}
{"pdf": "arxiv_math/2503.07373_pg21.pdf", "url": "https://arxiv.org/pdf/2503.07373", "page": 1, "id": "2503.07373_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "e_a^\\mu e_\\mu^b=\\delta^b_a"}
{"pdf": "arxiv_math/2503.07373_pg21.pdf", "url": "https://arxiv.org/pdf/2503.07373", "page": 1, "id": "2503.07373_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{2}\\iota_{[\\xi,\\xi]}\\delta_\\chi\\omega + \\iota_\\xi\\mathrm{L}^\\omega_\\xi \\delta_\\chi\\omega - \\frac{1}{2}\\iota_\\xi\\iota_\\xi d_\\omega \\delta_\\chi\\omega=0"}
{"pdf": "arxiv_math/2503.05606_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05606", "page": 1, "id": "2503.05606_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\in C^0([t_0, T]; \\R^d)"}
{"pdf": "arxiv_math/2503.05606_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05606", "page": 1, "id": "2503.05606_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dot{x}(t) = N_t(x(t)) + B(t)u(t)"}
{"pdf": "arxiv_math/2503.05175_pg7.pdf", "url": "https://arxiv.org/pdf/2503.05175", "page": 1, "id": "2503.05175_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(f_z^*-\\hat{f}_z)/f_z^*"}
{"pdf": "arxiv_math/2503.05175_pg7.pdf", "url": "https://arxiv.org/pdf/2503.05175", "page": 1, "id": "2503.05175_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\max_{j} \\left[ \\bar{g}_z^j\\left(h(z)\\right) \\right]"}
{"pdf": "arxiv_math/2503.05175_pg7.pdf", "url": "https://arxiv.org/pdf/2503.05175", "page": 1, "id": "2503.05175_pg7_math_005", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathcal{D} = \\{(r_i, c_i^0,d^0_i,Q_i,\\hat{u}_i)\\}_{i=1}^{N}"}
{"pdf": "arxiv_math/2503.04876_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04876", "page": 1, "id": "2503.04876_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "p_1(1-p_2)/(p_2(1-p_1))"}
{"pdf": "arxiv_math/2503.04247_pg22.pdf", "url": "https://arxiv.org/pdf/2503.04247", "page": 1, "id": "2503.04247_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\leq x, \\quad 0 \\leq y, \\quad m y < x"}
{"pdf": "arxiv_math/2503.04247_pg22.pdf", "url": "https://arxiv.org/pdf/2503.04247", "page": 1, "id": "2503.04247_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_{\\ell} = \\prod_{r=2}^{\\ell} \\alpha_r"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi_{j+q} = \\psi_j + 2\\pi p"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_j = \\delta \\{ \\varphi_j \\}/2 = (\\varphi_{j+1} - \\varphi_j)/2"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_{j+q} = \\theta_j"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\lambda_j = -\\partial_1 S(\\varphi_j,\\varphi_{j+1}) = \\partial_2 S(\\varphi_{j-1},\\varphi_j)"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "2q \\mu\\big \\{ h(\\psi_j) \\sin \\theta_j \\big \\} = A\\{ \\varphi_j\\} = L,\\quad \\mu \\{ h'(\\psi_j) \\sin \\theta_j \\} = 0, \\quad \\mu \\{\\theta_j\\} = \\pi p/q"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi_j = (\\varphi_{j+1} + \\varphi_j)/2"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi_j = \\sigma \\{ \\varphi_j \\}/2"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "S(\\varphi_j,\\varphi_{j+1}) = 2 h(\\psi_j) \\sin \\theta_j"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_{j+q} = \\varphi_j + 2\\pi p"}
{"pdf": "arxiv_math/2503.07488_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07488", "page": 1, "id": "2503.07488_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu \\{ S(\\varphi_{j_0+j},\\varphi_{j_0+j+1}) \\} = 2\\mu\\{ h(\\psi_{j_0+j}) \\sin \\theta_{j_0+j} \\} = A\\{ \\varphi_j \\}/q"}
{"pdf": "arxiv_math/2503.08578_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08578", "page": 1, "id": "2503.08578_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f \\in \\mathcal{C}^2(\\mathbb{R}^d)"}
{"pdf": "arxiv_math/2503.08578_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08578", "page": 1, "id": "2503.08578_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline f := \\min f >0"}
{"pdf": "arxiv_math/2503.08578_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08578", "page": 1, "id": "2503.08578_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "W^{k,p}_{loc}(\\Omega)"}
{"pdf": "arxiv_math/2503.08578_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08578", "page": 1, "id": "2503.08578_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varrho=\\frac{\\text{d}\\mu}{\\text{d}x}"}
{"pdf": "arxiv_math/2503.08578_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08578", "page": 1, "id": "2503.08578_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\nabla^2 f\\|_\\infty"}
{"pdf": "arxiv_math/2503.08578_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08578", "page": 1, "id": "2503.08578_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi f\\in W^{k,p}(\\Omega)"}
{"pdf": "arxiv_math/2503.06459_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06459", "page": 1, "id": "2503.06459_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\sqrt{n-1}}{\\sqrt{n-2}}\\delta"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "A=(\\theta_--\\theta_+)/2"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda(du)=\\lambda(\\delta_1(du)+ \\delta_{-1}(du))"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "B=(\\theta_-+\\theta_-)/2"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "k(x)=\\lambda e^{A x- B|x|}"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal L(G^k_1)\\sim \\Gamma\\left(\\lambda,\\frac{(k+c_+)(k+c_-)}{ 2 \\sigma^2}\\right)"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "k(x)=x^{-\\alpha} e^{A x- B|x|}"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "k(x)=e^{A x/\\sigma -B|x|/\\sigma}/(1-e^{-|x|/\\sigma})"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "H_a(z)= \\frac{1}{\\Gamma(-a)}\\int_0^\\infty e^{-x^2/2-xz}x^{-a-1}dx, \\qquad z>0"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau_+^{-A}=\\tau_-^{A}= \\lambda \\delta_B"}
{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal L(T_1) \\sim \\Gamma(\\lambda, \\theta_+\\theta_-/2)"}
{"pdf": "arxiv_math/2503.05829_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05829", "page": 1, "id": "2503.05829_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall i \\in [k]\\colon (v_i, v_{i+1}) \\in E \\lor (v_{i+1}, v_i) \\in E"}
{"pdf": "arxiv_math/2503.05829_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05829", "page": 1, "id": "2503.05829_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "E \\subset \\{(v,w)\\colon v,w \\in V\\}"}
{"pdf": "arxiv_math/2503.05829_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05829", "page": 1, "id": "2503.05829_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall i \\in [k]\\colon (v_i, v_{i+1}) \\in E"}
{"pdf": "arxiv_math/2503.05829_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05829", "page": 1, "id": "2503.05829_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(v_1, v_2,\\ldots,v_{k+1})"}
{"pdf": "arxiv_math/2503.05829_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05829", "page": 1, "id": "2503.05829_pg8_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\N \\coloneq \\{1,2,\\ldots\\}"}
{"pdf": "arxiv_math/2503.05829_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05829", "page": 1, "id": "2503.05829_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\N_0\\coloneq\\N\\cup\\{0\\}"}
{"pdf": "arxiv_math/2503.05829_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05829", "page": 1, "id": "2503.05829_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "[n] \\coloneq \\{1,\\ldots,n\\}"}
{"pdf": "arxiv_math/2503.09591_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09591", "page": 1, "id": "2503.09591_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "e\\leq e(n-1)+1\\leq e(n)"}
{"pdf": "arxiv_math/2503.09591_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09591", "page": 1, "id": "2503.09591_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "e\\leq 6(n-n_1)-4\\sqrt{6(n-n_1)-6}+6 \\leq 6n-\\sqrt{96n-63} \\text{ for }n_1=2,3 \\text{ if }n\\geq 6"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "N_j(G^{\\sigma}[H^{\\sigma}]) = \\{ x \\in V(G^{\\sigma}) \\setminus H^{\\sigma} \\mid d(x, G^{\\sigma}[H^{\\sigma}]) = j, \\quad j = 1, 2, \\dots, n \\}"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\lfloor \\frac{g_r}{2} \\right\\rfloor + k \\geq g_r"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\lceil \\frac{g_r}{2} \\right\\rceil \\leq k"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "y, y' \\in V(C^\\sigma)"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "y' \\sim y \\in N_1(C^{\\sigma})"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "p^+ (C^\\sigma) \\geq 2"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(V(P^\\sigma) \\setminus \\{y, y'\\}) \\cap V(C^\\sigma) = \\emptyset"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "y' \\in N_2(C^{\\sigma})"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "G^{\\sigma}[H^{\\sigma}]"}
{"pdf": "arxiv_math/2503.05135_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05135", "page": 1, "id": "2503.05135_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "p^{+}(P^\\sigma_n)=\\lfloor \\frac{n}{2} \\rfloor"}
{"pdf": "arxiv_math/2503.04494_pg21.pdf", "url": "https://arxiv.org/pdf/2503.04494", "page": 1, "id": "2503.04494_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta = (c_4^3-c_6^2)/1728"}
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{"pdf": "arxiv_math/2503.09571_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09571", "page": 1, "id": "2503.09571_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}^{n+1 \\choose 2}"}
{"pdf": "arxiv_math/2503.09571_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09571", "page": 1, "id": "2503.09571_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "p\\in \\mathbb{R}^{1+d}"}
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{"pdf": "arxiv_math/2503.09571_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09571", "page": 1, "id": "2503.09571_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "s_{ij} \\,\\,= \\,\\,\\lambda_i \\lambda_j \\! \\left( 1 - \\langle x^{(i)}, x^{(j)} \\rangle \\right)"}
{"pdf": "arxiv_math/2503.09571_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09571", "page": 1, "id": "2503.09571_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "D = {\\rm diag}(1,-1,-1,\\ldots,-1)"}
{"pdf": "arxiv_math/2503.09571_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09571", "page": 1, "id": "2503.09571_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "s_{ij} = p^{(i)} \\cdot p^{(j)}"}
{"pdf": "arxiv_math/2503.09571_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09571", "page": 1, "id": "2503.09571_pg3_math_010", "type": "math", "max_diffs": 0, "checked": "verified", "math": "{\\cal M}_{n,\\leq 1+d}"}
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{"pdf": "arxiv_math/2503.05922_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05922", "page": 1, "id": "2503.05922_pg8_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "A(t) = \\infty\\cdot\\chi_{(1, \\infty]}(t)"}
{"pdf": "arxiv_math/2503.05922_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05922", "page": 1, "id": "2503.05922_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "1\\leq q_1 < q_2\\leq \\infty"}
{"pdf": "arxiv_math/2503.08365_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08365", "page": 1, "id": "2503.08365_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{cr}(G)\\le 3.\\overline{3}n"}
{"pdf": "arxiv_math/2503.08365_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08365", "page": 1, "id": "2503.08365_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{cr}(G)\\le 6.6n"}
{"pdf": "arxiv_math/2503.08365_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08365", "page": 1, "id": "2503.08365_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{cr}(G)\\le n-2"}
{"pdf": "arxiv_math/2503.08949_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08949", "page": 1, "id": "2503.08949_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "F^\\circ F^j \\tilde u(x)"}
{"pdf": "arxiv_math/2503.08949_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08949", "page": 1, "id": "2503.08949_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ln \\lambda_{s,E_{2}} - \\ln \\lambda_{s, E_1}"}
{"pdf": "arxiv_math/2503.08949_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08949", "page": 1, "id": "2503.08949_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\| F^n_{s,E_2}\\|_1 - \\|F^n_{s,E_1}\\|_1 \\le 0"}
{"pdf": "arxiv_math/2503.08949_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08949", "page": 1, "id": "2503.08949_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{s,E_2}(-y-t^2/x) - \\rho_{s,E_1} (-y-t^2/x) < c(E_2-E_1)"}
{"pdf": "arxiv_math/2503.08949_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08949", "page": 1, "id": "2503.08949_pg8_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "F_{s,E_2}^{n-j -1} (F^\\circ) F_{s,E_1}^{j}"}
{"pdf": "arxiv_math/2503.07925_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07925", "page": 1, "id": "2503.07925_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "Q=\\{x\\in\\R^n:Mx\\leq b\\}"}
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{"pdf": "arxiv_math/2503.09298_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09298", "page": 1, "id": "2503.09298_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\llbracket U \\rrbracket"}
{"pdf": "arxiv_math/2503.09298_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09298", "page": 1, "id": "2503.09298_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "T = \\llbracket U \\rrbracket"}
{"pdf": "arxiv_math/2503.05896_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05896", "page": 1, "id": "2503.05896_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "B_{e^{-Kt_0}r}^{g_i(0)}(p_i)"}
{"pdf": "arxiv_math/2503.05896_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05896", "page": 1, "id": "2503.05896_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{g_i(0)}(x_i,y_i)\\geq e^{-Kt_0}d_{g_i(t_0)}(x_i,y_i)"}
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{"pdf": "arxiv_math/2503.05896_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05896", "page": 1, "id": "2503.05896_pg10_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{e_1,e_2,\\ldots,e_n\\}"}
{"pdf": "arxiv_math/2503.05896_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05896", "page": 1, "id": "2503.05896_pg10_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{vol}(g_i(0))\\Big(B_{e^{-Kt_0}r_0}^{g_i(0)}(x_i)\\Big)\\geq \\frac{A(n)}{n^n}r_0^ne^{-Knt_0}"}
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{"pdf": "arxiv_math/2503.05896_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05896", "page": 1, "id": "2503.05896_pg10_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "B_{e^{-Kt_0}r}^{g_i(0)}(p_i)\\subset B_r^{g_i(t_0)}(p_i)"}
{"pdf": "arxiv_math/2503.05896_pg10.pdf", "url": "https://arxiv.org/pdf/2503.05896", "page": 1, "id": "2503.05896_pg10_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "r> e^{Kt_0}d_{g_i(0)}(p_i,q_i)\\geq d_{g_i(t_0)}(p_i,q_i)"}
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{"pdf": "arxiv_math/2503.07508_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07508", "page": 1, "id": "2503.07508_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_0\\omega_1\\dotsb"}
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{"pdf": "arxiv_math/2503.07508_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07508", "page": 1, "id": "2503.07508_pg14_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_0^{l}=\\omega_0\\dotsb \\omega_l"}
{"pdf": "arxiv_math/2503.07508_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07508", "page": 1, "id": "2503.07508_pg14_math_016", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mu_{\\omega,n}\\sim D_n"}
{"pdf": "arxiv_math/2503.02004_pg9.pdf", "url": "https://arxiv.org/pdf/2503.02004", "page": 1, "id": "2503.02004_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "l\\leq N_{\\text{init}}"}
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{"pdf": "arxiv_math/2503.02004_pg9.pdf", "url": "https://arxiv.org/pdf/2503.02004", "page": 1, "id": "2503.02004_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "i=\\underset{i\\in T_{(l)}^{c}}{\\arg\\max}\\underset{k}{\\min}\\ \\|\\mathbf{h}_k\\|_2,\\ \\text{s.t.}\\ \\mathbf{h}_k\\in\\text{Col}(\\mathbf{H}_{T_{(l)}\\cup\\{i\\}})"}
{"pdf": "arxiv_math/2503.02004_pg9.pdf", "url": "https://arxiv.org/pdf/2503.02004", "page": 1, "id": "2503.02004_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{O}(2^M\\cdot K)"}
{"pdf": "arxiv_math/2503.02004_pg9.pdf", "url": "https://arxiv.org/pdf/2503.02004", "page": 1, "id": "2503.02004_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\mathcal{I}_s}"}
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{"pdf": "arxiv_math/2503.09472_pg15.pdf", "url": "https://arxiv.org/pdf/2503.09472", "page": 1, "id": "2503.09472_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{cases} \\frac{\\mathrm{d}x_1}{\\mathrm{d}t}=x_2+\\frac{b_{11}}{b_{10}}x_1x_2+\\frac{b_{20}}{b_{10}^2}x_2^2+\\frac{b_{12}}{b_{10}}x_1^2x_2+\\frac{b_{21}}{b_{10}^2}x_1x_2^2+\\frac{b_{30}}{b_{10}^3}x_1^3+O(\\|x\\|^4)\\\\ \\frac{\\mathrm{d}x_2}{\\mathrm{d}t}=a_{11}x_1x_2+\\frac{a_{20}}{b_{10}}x_2^2+\\frac{a_{21}}{b_{10}}x_1x_2^2+a_{12}x_1^2x_2+\\frac{a_{30}}{b_{10}^2}x_2^3+O(\\|x\\|^4) \\end{cases}"}
{"pdf": "arxiv_math/2503.09472_pg15.pdf", "url": "https://arxiv.org/pdf/2503.09472", "page": 1, "id": "2503.09472_pg15_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{cases} x_3=x_1 \\\\ x_4=x_2+\\frac{b_{11}}{b_{10}}x_1x_2+\\frac{b_{20}}{b_{10}^2}x_2^2+\\frac{b_{12}}{b_{10}}x_1^2x_2+\\frac{b_{21}}{b_{10}^2}x_1x_2^2+\\frac{b_{30}}{b_{10}^3}x_1^3+O(\\|x\\|^4) \\end{cases}"}
{"pdf": "arxiv_math/2503.09472_pg15.pdf", "url": "https://arxiv.org/pdf/2503.09472", "page": 1, "id": "2503.09472_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{cases} x_1=x_3 \\\\ x_2=x_4+v_{11}x_3x_4+v_{02}x_4^2+v_{21}x_3^2x_4+v_{12}x_3x_4^2+v_{03}x_4^3+O(\\|x\\|^4) \\end{cases}"}
{"pdf": "arxiv_math/2503.06185_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06185", "page": 1, "id": "2503.06185_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{k}^{RBB}\\in[\\alpha_{k}^{BB1}, \\ \\alpha_{k}^{BB2}]"}
{"pdf": "arxiv_math/2503.06185_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06185", "page": 1, "id": "2503.06185_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{k}^{RBB}=\\mathop{\\text{argmin}}_{\\alpha\\in\\mathbb{R}} \\Big\\{\\Vert\\alpha\\Delta y^{k-1}-\\Delta\\Psi^{k-1}\\Vert_{2}^{2} + \\tau_{k}\\Vert\\alpha \\sqrt{H_{k}}\\Delta y^{k-1}- \\sqrt{H_{k}}\\Delta\\Psi^{k-1}\\Vert_{2}^{2}\\Big\\}"}
{"pdf": "arxiv_math/2503.06185_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06185", "page": 1, "id": "2503.06185_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau_{k}\\in[0,\\infty)"}
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{"pdf": "arxiv_math/2503.06185_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06185", "page": 1, "id": "2503.06185_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|r^k\\|_{2}>\\|d^k\\|_{2}"}
{"pdf": "arxiv_math/2503.06185_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06185", "page": 1, "id": "2503.06185_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}^{m\\times n}"}
{"pdf": "arxiv_math/2503.08577_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08577", "page": 1, "id": "2503.08577_pg6_math_000", "type": "math", "max_diffs": 0, "checked": "verified", "math": "H_P^{(t)}(\\cdot, \\sigma)"}
{"pdf": "arxiv_math/2503.08577_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08577", "page": 1, "id": "2503.08577_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\lambda} \\coloneqq \\mathrm{dim} \\left(V_{\\pi_{\\lambda}}\\right)"}
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{"pdf": "arxiv_math/2503.08577_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08577", "page": 1, "id": "2503.08577_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{f}(x)=f(e^{i x})"}
{"pdf": "arxiv_math/2503.08577_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08577", "page": 1, "id": "2503.08577_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{U}(1) \\rightarrow \\mathrm{U}(1)"}
{"pdf": "arxiv_math/2503.08577_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08577", "page": 1, "id": "2503.08577_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "e^{i\\phi} \\mapsto e^{i \\lambda \\phi}"}
{"pdf": "arxiv_math/2503.08577_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08577", "page": 1, "id": "2503.08577_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "f: \\mathrm{U}(1) \\rightarrow \\mathbb{C}"}
{"pdf": "arxiv_math/2503.08577_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08577", "page": 1, "id": "2503.08577_pg6_math_007", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\hat{f}(\\lambda)=\\frac{1}{2 \\pi} \\int_{-\\pi}^{\\pi} e^{-i \\lambda \\phi} f(e^{i\\phi}) d \\phi \\mathrm{,}"}
{"pdf": "arxiv_math/2503.08577_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08577", "page": 1, "id": "2503.08577_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{End}(V_{\\pi_{\\lambda}})"}
{"pdf": "arxiv_math/2503.08577_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08577", "page": 1, "id": "2503.08577_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{U}(1) \\cong S^1"}
{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "N_i+\\left(\\bigcap_{j \\neq i} N_j\\right)=M"}
{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "M / \\bigcap_{i \\in I} N_i \\rightarrow \\oplus_{i \\in I} M / N_i"}
{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(P_{k})_{k=1}^{\\infty}"}
{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(N_{k})_{k=1}^{\\infty}"}
{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{N_1, N_2, \\ldots, N_n\\right\\}"}
{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "N_1 \\cap N_2 \\cap \\cdots \\cap N_n"}
{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{N_i \\mid i \\in I\\right\\}"}
{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "M=N_{k}\\oplus P_{k}\\oplus P_{k-1}\\oplus \\cdots \\oplus P_{1}\\quad \\text{with}\\quad N_{k}=N_{k+1}\\oplus P_{k+1}"}
{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "F^{-1}|S|+F(U)+F(\\mathcal{T})+\\|divC\\|_{HS(V)} \\leq K_0"}
{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "C(X,Y,Z):=C_{ijk}X^iY^jZ^k=\\frac{1}{4}\\frac{\\partial^3F^2(x,y)}{\\partial y^i\\partial y^j\\partial y^k}X^iY^jZ^k"}
{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "G^i=\\frac12\\Gamma^i_{jk}y^jy^k"}
{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "X,Y,Z\\in TM\\setminus\\{0\\}"}
{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "V(u)=\\exp(\\frac12F^{*2}(Du))"}
{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "N^i_j=\\frac{\\partial G^i}{\\partial y^j}"}
{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\delta}{\\delta x^i}=\\frac{\\partial}{\\partial x^i}-N^j_i\\frac{\\partial}{\\partial y^j}"}
{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "divC=FC^{i}_{\\,jk\\vert i}dx^j\\otimes dx^k"}
{"pdf": "arxiv_math/2503.08397_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08397", "page": 1, "id": "2503.08397_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{T}=\\sum_{k\\in \\{0\\}\\cup [K]} T_k"}
{"pdf": "arxiv_math/2503.08397_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08397", "page": 1, "id": "2503.08397_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon=o\\left({1}/{\\sqrt{N^{1-\\alpha_s}}}\\right)"}
{"pdf": "arxiv_math/2503.04578_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04578", "page": 1, "id": "2503.04578_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{t(n)\\}_{n \\in \\mathbb{N}}"}
{"pdf": "arxiv_math/2503.04578_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04578", "page": 1, "id": "2503.04578_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{t(n)\\}_{n\\in \\mathbb{N}}"}
{"pdf": "arxiv_math/2503.04578_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04578", "page": 1, "id": "2503.04578_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi:G\\rightarrow U(L^2(M,\\mu))"}
{"pdf": "arxiv_math/2503.04578_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04578", "page": 1, "id": "2503.04578_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "s\\in S\\setminus \\{e\\}"}
{"pdf": "arxiv_math/2503.04578_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04578", "page": 1, "id": "2503.04578_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bigsqcup M\\times \\{t(n)\\}"}
{"pdf": "arxiv_math/2503.04578_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04578", "page": 1, "id": "2503.04578_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_M \\xi(x) d\\mu(x)=0"}
{"pdf": "arxiv_math/2503.04433_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04433", "page": 1, "id": "2503.04433_pg10_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "C(f_k)=F(f_k)+jG(f_k)=\\dfrac{H^{(2)}_1(f_k)}{H^{(2)}_1(f_k)+jH^{(2)}_1(f_k)}"}
{"pdf": "arxiv_math/2503.04433_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04433", "page": 1, "id": "2503.04433_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "p_0 = \\delta \\pm jk_0"}
{"pdf": "arxiv_math/2503.04433_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04433", "page": 1, "id": "2503.04433_pg10_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "p_1 = \\delta \\pm jk_1"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d_2 = \\big[\\frac{t^2}{2!}\\big]d(t) = \\frac{d}{dt}^2d(t)\\bigr|_{t=0}"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "c(t) =\\sum_{i=0}^\\infty c_i\\frac{t^i}{i!}"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{n,k} = [t^n]d(t)h(t)^k,\\nonumber"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "d_{n,k}=\\Big[\\frac{t^n}{n!}\\Big]d(t)\\frac{h(t)^k}{k!}"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{k=0}^\\infty d_{n,k}c_k = \\sum_{k=0}^nd_{n,k}c_k = [t^n]d(t)c(h(t))\\nonumber"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{R}(d(t),h(t))*\\mathcal{R}(g(t),f(t)):= \\mathcal{R}(d(t)g(h(t)),f(h(t))\\nonumber"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(d_{n,k})_{n,k\\in\\mathbb{N}_0}"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{R}_e[d(t),h(t)]"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{R}(d(t),h(t))^{-1}:= \\mathcal{R}\\Bigg(\\frac{1}{d(\\bar{h}(t))},\\bar{h}(t)\\Bigg)\\nonumber"}
{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{R}(g(t),f(t))"}
{"pdf": "arxiv_math/2503.06856_pg9.pdf", "url": "https://arxiv.org/pdf/2503.06856", "page": 1, "id": "2503.06856_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c_i: [0, T] \\to [0, 1], \\, i = 0, 1"}
{"pdf": "arxiv_math/2503.06856_pg9.pdf", "url": "https://arxiv.org/pdf/2503.06856", "page": 1, "id": "2503.06856_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}(t) \\coloneqq \\overline{\\sigma}(B(s), 0 \\leq s \\le t)"}
{"pdf": "arxiv_math/2503.06856_pg9.pdf", "url": "https://arxiv.org/pdf/2503.06856", "page": 1, "id": "2503.06856_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{F} \\coloneqq \\{\\mathcal{F}(t)\\}_{t \\ge 0}"}
{"pdf": "arxiv_math/2503.06856_pg9.pdf", "url": "https://arxiv.org/pdf/2503.06856", "page": 1, "id": "2503.06856_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau \\in \\mathcal{T}_{T-t}"}
{"pdf": "arxiv_math/2503.06856_pg9.pdf", "url": "https://arxiv.org/pdf/2503.06856", "page": 1, "id": "2503.06856_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi^\\pi(\\cdot), \\, \\pi \\in [0,1]"}
{"pdf": "arxiv_math/2503.06856_pg9.pdf", "url": "https://arxiv.org/pdf/2503.06856", "page": 1, "id": "2503.06856_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "c_i(\\cdot), \\, i = 0, 1"}
{"pdf": "arxiv_math/2503.06856_pg9.pdf", "url": "https://arxiv.org/pdf/2503.06856", "page": 1, "id": "2503.06856_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "B = \\{B(t), \\, 0 \\le t < \\infty\\}"}
{"pdf": "arxiv_math/2503.06856_pg9.pdf", "url": "https://arxiv.org/pdf/2503.06856", "page": 1, "id": "2503.06856_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi^\\pi = \\{\\Pi^\\pi(t), 0 \\le t < \\infty \\}"}
{"pdf": "arxiv_math/2503.04583_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04583", "page": 1, "id": "2503.04583_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "E_s = (e_1, ..., e_s)"}
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{"pdf": "arxiv_math/2503.05396_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05396", "page": 1, "id": "2503.05396_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "m \\sim n \\sim \\sqrt{X/D_2}"}
{"pdf": "arxiv_math/2503.05396_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05396", "page": 1, "id": "2503.05396_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda^{\\sharp}_{1}(k,T)"}
{"pdf": "arxiv_math/2503.05396_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05396", "page": 1, "id": "2503.05396_pg5_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathcal{Q}_{n_1,n_2}(k)"}
{"pdf": "arxiv_math/2503.05396_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05396", "page": 1, "id": "2503.05396_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "L(1,(\\tfrac{4n_1n_2}{\\cdot}))^{-1}"}
{"pdf": "arxiv_math/2503.05396_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05396", "page": 1, "id": "2503.05396_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{x} f(\\tfrac{x}{A}) d(x^2+y^3)"}
{"pdf": "arxiv_math/2503.05396_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05396", "page": 1, "id": "2503.05396_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_{\\chi \\xi^{\\ell} } (k)"}
{"pdf": "arxiv_math/2503.05396_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05396", "page": 1, "id": "2503.05396_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda^\\sharp_{1}(k,T)=\\sum_{c \\leq T} (\\tfrac{4n_1n_2}{c})"}
{"pdf": "arxiv_math/2503.05396_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05396", "page": 1, "id": "2503.05396_pg5_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Upsilon_{n_1,n_2}(k)"}
{"pdf": "arxiv_math/2503.05396_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05396", "page": 1, "id": "2503.05396_pg5_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{n_1,n_2 \\sim N} \\frac{\\beta_{n_1} \\beta_{n_2}}{ \\sqrt{n_1n_2} L(1,(\\tfrac{4n_1n_2}{\\cdot}))} \\sum_{k \\asymp NX} \\Upsilon_{n_1,n_2}(k) \\lambda_{1}(k)"}
{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "[K(h), \\ldots, [K(h),\\widetilde{K}(h)]\\ldots]"}
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{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_1 = \\frac{1}{2} \\pm i \\frac{1}{2 \\sqrt{3}}"}
{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha = \\frac{1}{2} (1\\pm i )"}
{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Psi_{\\alpha_1 h}^{[2]}"}
{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Re(\\alpha) = \\frac{1}{2}, \\Re(\\alpha^2) = 0"}
{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "S_h = (\\alpha_1, \\alpha_2, \\ldots, \\alpha_r, \\overline{\\alpha}_1, \\overline{\\alpha}_2, \\ldots, \\overline{\\alpha}_r)"}
{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "S_h = \\Phi_{\\alpha h} \\, \\Phi_{\\overline{\\alpha} h}"}
{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{O}(h^{2p+1})"}
{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_3 \\in \\mathbb{R}"}
{"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04590", "page": 1, "id": "2503.04590_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "%\\label{time1} T(u(0)) \\leq \\dfrac{V(u(0))^{1-p}}{K(1-p)}"}
{"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04590", "page": 1, "id": "2503.04590_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T(u)\\equiv f(u)- P_{\\Phi(u)}(f(u)-\\alpha u)"}
{"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04590", "page": 1, "id": "2503.04590_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dot{V}(u) \\leq -K.\\big(V(u)\\big)^p \\quad \\forall u\\in U\\setminus \\{u^*\\}"}
{"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04590", "page": 1, "id": "2503.04590_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "f(u)- P_{\\Phi(u)}(f(u)-\\alpha u)=0"}
{"pdf": "arxiv_math/2503.07166_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07166", "page": 1, "id": "2503.07166_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(e^+)^2 - (e^-)^2 = e^+ + e^-"}
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{"pdf": "arxiv_math/2503.08475_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08475", "page": 1, "id": "2503.08475_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_1\\times\\ldots \\times \\pi_k"}
{"pdf": "arxiv_math/2503.08475_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08475", "page": 1, "id": "2503.08475_pg18_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "[a,b]_\\rho+\\ldots+[a+o(\\rho)-1,b+o(\\rho)-1]_\\rho"}
{"pdf": "arxiv_math/2503.08475_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08475", "page": 1, "id": "2503.08475_pg18_math_027", "type": "math", "max_diffs": 0, "checked": null, "math": "l([1,b']_\\rho)\\ge l([c,d+1]_\\rho)"}
{"pdf": "arxiv_math/2503.08475_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08475", "page": 1, "id": "2503.08475_pg18_math_028", "type": "math", "max_diffs": 0, "checked": null, "math": "l([1,b']_\\rho)\\le l([c,d+1]_\\rho)"}
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{"pdf": "arxiv_math/2503.08924_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08924", "page": 1, "id": "2503.08924_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hbox{\\bf detpol}(\\Delta)"}
{"pdf": "arxiv_math/2503.08924_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08924", "page": 1, "id": "2503.08924_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "Sres_0(A,B;x)= sres_0(A,B;x)"}
{"pdf": "arxiv_math/2503.08924_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08924", "page": 1, "id": "2503.08924_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "A(x)=\\sum_{i=0}^m a_{i}x^i\\qquad \\hbox{and} \\qquad B(x)=\\sum_{i=0}^n b_{i}x^i"}
{"pdf": "arxiv_math/2503.08924_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08924", "page": 1, "id": "2503.08924_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hbox{\\bf detpol}(\\Delta)=\\sum_{k=0}^{n-m}\\det(\\Delta_k)x^{n-m-k}"}
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{"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "h|_{T}=h_{T} :={\\rm diam}(T)"}
{"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{0}(\\omega)=L^{2}(\\omega)"}
{"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\|\\cdot\\|_{m,p,\\omega}"}
{"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H_{D}^{1}(\\Omega)=\\{v\\in H^{1}(\\Omega): v|_{\\Gamma_{D}}=0\\}"}
{"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "V_{h}\\subset H^{1}(\\Omega)"}
{"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\cdot,\\cdot)_{\\omega}"}
{"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "W^{m,2}(\\omega)=H^{m}(\\omega)"}
{"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\cdot\\|_{0,\\infty,\\omega}"}
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{"pdf": "arxiv_math/2503.07322_pg9.pdf", "url": "https://arxiv.org/pdf/2503.07322", "page": 1, "id": "2503.07322_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde C_2:=B_{d+1}\\otimes \\Lambda (V_1\\oplus \\overline{V}_1)\\otimes \\Lambda^+(V_2\\oplus \\overline{V}_2)"}
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{"pdf": "arxiv_math/2503.08109_pg12.pdf", "url": "https://arxiv.org/pdf/2503.08109", "page": 1, "id": "2503.08109_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "S=q~\\textrm{or}~S=q^\\dag"}
{"pdf": "arxiv_math/2503.08109_pg12.pdf", "url": "https://arxiv.org/pdf/2503.08109", "page": 1, "id": "2503.08109_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\mathcal Mat}_2(\\mathbb{C})"}
{"pdf": "arxiv_math/2503.07629_pg32.pdf", "url": "https://arxiv.org/pdf/2503.07629", "page": 1, "id": "2503.07629_pg32_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\underset {j \\epsilon S} \\oplus}{\\bm e}(\\tfrac{1}{n},j)({\\bf A}{\\bf w}(f,g))"}
{"pdf": "arxiv_math/2503.07629_pg32.pdf", "url": "https://arxiv.org/pdf/2503.07629", "page": 1, "id": "2503.07629_pg32_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\big\\{{\\bm e}(\\tfrac{1}{n},j), j=1,n\\big\\}"}
{"pdf": "arxiv_math/2503.07629_pg32.pdf", "url": "https://arxiv.org/pdf/2503.07629", "page": 1, "id": "2503.07629_pg32_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bm u}(\\tfrac{1}{n},j),j=1,n"}
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{"pdf": "arxiv_math/2503.07092_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07092", "page": 1, "id": "2503.07092_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{\\textnormal{true}}"}
{"pdf": "arxiv_math/2503.07092_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07092", "page": 1, "id": "2503.07092_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "S \\subseteq \\{1,2,\\dots, n\\} \\times \\{1,2,\\dots, f+g\\}"}
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{"pdf": "arxiv_math/2503.07092_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07092", "page": 1, "id": "2503.07092_pg4_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "Y \\in \\mathbb{R}^{m \\times p}[x]"}
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{"pdf": "arxiv_math/2503.07441_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07441", "page": 1, "id": "2503.07441_pg11_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "H: [0,1] \\times M \\to \\mathbb{R} ^{}"}
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{"pdf": "arxiv_math/2503.08630_pg15.pdf", "url": "https://arxiv.org/pdf/2503.08630", "page": 1, "id": "2503.08630_pg15_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "e \\in G(\\Lambda)^1, f_{i,j} \\in C_{n, k_2}^1"}
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{"pdf": "arxiv_math/2503.06226_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06226", "page": 1, "id": "2503.06226_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "W_y\\in \\mathbb{R}^{n \\times np}"}
{"pdf": "arxiv_math/2503.06226_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06226", "page": 1, "id": "2503.06226_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bar {\\mathcal{A}}}_y \\in \\mathbb{R}^{np \\times np}"}
{"pdf": "arxiv_math/2503.06226_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06226", "page": 1, "id": "2503.06226_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bar {\\mathcal{B}}}_u \\in \\mathbb{R}^{nm \\times m}"}
{"pdf": "arxiv_math/2503.06226_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06226", "page": 1, "id": "2503.06226_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{k \\rightarrow \\infty} \\hat x(k) = x(k)"}
{"pdf": "arxiv_math/2503.06226_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06226", "page": 1, "id": "2503.06226_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "({\\bar {\\mathcal{A}}}_u , ~{\\bar {\\mathcal{B}}}_u)"}
{"pdf": "arxiv_math/2503.06226_pg4.pdf", "url": "https://arxiv.org/pdf/2503.06226", "page": 1, "id": "2503.06226_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "({\\bar {\\mathcal{A}}}_y , ~{\\bar {\\mathcal{B}}}_y)"}
{"pdf": "arxiv_math/2503.09234_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09234", "page": 1, "id": "2503.09234_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g = U^{\\frac{4}{n-4}} \\overset{\\circ}{g}"}
{"pdf": "arxiv_math/2503.09234_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09234", "page": 1, "id": "2503.09234_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "g= u^{\\frac{4}{n-4}} g_{\\rm euc}"}
{"pdf": "arxiv_math/2503.09234_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09234", "page": 1, "id": "2503.09234_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "u:\\R^n \\backslash \\{ 0 \\} \\rightarrow (0,\\infty)"}
{"pdf": "arxiv_math/2503.08105_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08105", "page": 1, "id": "2503.08105_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f(z) = h(z) + \\overline{g(z)}"}
{"pdf": "arxiv_math/2503.08105_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08105", "page": 1, "id": "2503.08105_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f^2=(h + \\overline{g})^2 = h^2 +\\overline{g}^2+ 2h\\overline{g}"}
{"pdf": "arxiv_math/2503.08105_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08105", "page": 1, "id": "2503.08105_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\partial h}{\\partial z}= 0, ~~ ~~ \\frac{\\partial \\overline{g}}{\\partial\\overline{z} } =0"}
{"pdf": "arxiv_math/2503.08105_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08105", "page": 1, "id": "2503.08105_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "f = h + \\overline{g}"}
{"pdf": "arxiv_math/2503.08105_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08105", "page": 1, "id": "2503.08105_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\partial^2 f^2}{\\partial z\\overline{z}} = 0"}
{"pdf": "arxiv_math/2503.08105_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08105", "page": 1, "id": "2503.08105_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\partial^2 f^2}{\\partial z\\overline{z}} = 2\\frac{\\partial h}{\\partial z}\\frac{\\partial \\overline{g}}{\\partial\\overline{z} } =0"}
{"pdf": "arxiv_math/2503.04116_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04116", "page": 1, "id": "2503.04116_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_1(\\overrightarrow f_{j})= \\overleftarrow f_{j}; 1\\leq i \\leq 3 \\text{ and } 1 \\leq j \\leq 7"}
{"pdf": "arxiv_math/2503.04116_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04116", "page": 1, "id": "2503.04116_pg9_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\sigma_{0}=C_1 \\dots C_{5,}"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(P_1, \\ldots, P_5, b_{12345})"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_001", "type": "math", "max_diffs": 0, "checked": "verified", "math": "(P_{\\widehat{1}}, \\ldots, P_{\\widehat{5}})"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "q^{(0)}_{2}(u_i) = w^{(1)}_2(\\hat{e}_i)+ (f_i + b_i f_6) b_{12345} = P_if_6"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "H=\\lbrace i \\rbrace\\subset\\{1,\\cdots,5\\}"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "w^{(6)}_2(f_h)= b_1b_{12345}+2P_h \\quad \\textnormal{for}\\quad 1\\leq h \\leq 5\\quad \\textnormal{and}\\quad w^{(6)}_2(f_6)= -b_{12345}"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{\\widehat{1}}= x_{23}x_{45} - x_{24} x_{35} + x_{25} x_{34}"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "q^{(0)}_{2}(u_h) = b_{12345} \\cdot w^{(1)}_{1}(u_I)"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "w^{(0)}_{1,2}(\\theta_{ij}) = w^{(0)}_{1,2}(\\theta_{\\widehat{i}j}) = w^{(0)}_{1,2}(\\theta_{i \\widehat{j}}) = 0"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "w^{(0)}_{2}(u_i) = P_i"}
{"pdf": "arxiv_math/2503.08813_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08813", "page": 1, "id": "2503.08813_pg19_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "q^{(6)}_2(f_1)= b_1b_{12345}f_6 + b_{12345}(f_1 + b_1f_6) + P_1f_6 + w^{(1)}_2(\\hat{e}_1) = (b_1b_{12345}+2P_1)f_6"}
{"pdf": "arxiv_math/2503.03759_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03759", "page": 1, "id": "2503.03759_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "- i \\sum_{z} P(z) \\theta(P(z))"}
{"pdf": "arxiv_math/2503.03759_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03759", "page": 1, "id": "2503.03759_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H(Z) = \\mathbb{E}[I(Z)] = \\sum_{z} P(z) I(z)= - \\sum_{z} P(z) \\log P(z)"}
{"pdf": "arxiv_math/2503.03759_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03759", "page": 1, "id": "2503.03759_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "- \\sum_{z} P(z) \\log |P(z)|"}
{"pdf": "arxiv_math/2503.03759_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03759", "page": 1, "id": "2503.03759_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\log P(z) = \\log |P(z)| + i \\theta(P(z))"}
{"pdf": "arxiv_math/2503.08996_pg21.pdf", "url": "https://arxiv.org/pdf/2503.08996", "page": 1, "id": "2503.08996_pg21_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "G(k_1, \\gamma(k_1))=0"}
{"pdf": "arxiv_math/2503.08996_pg21.pdf", "url": "https://arxiv.org/pdf/2503.08996", "page": 1, "id": "2503.08996_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\mathcal{O}_3(k_1)|\\lesssim |k_1|^3"}
{"pdf": "arxiv_math/2503.08996_pg21.pdf", "url": "https://arxiv.org/pdf/2503.08996", "page": 1, "id": "2503.08996_pg21_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\partial_{k_2}\\tilde{\\varphi}(\\mathbf{k}; v, \\theta)=0"}
{"pdf": "arxiv_math/2503.08996_pg21.pdf", "url": "https://arxiv.org/pdf/2503.08996", "page": 1, "id": "2503.08996_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma(k_1)=\\frac{3b_\\theta}{2}k_1+\\frac{9a_\\theta b_\\theta}{2}k_1^2+\\mathcal{O}_3(k_1)"}
{"pdf": "arxiv_math/2503.08996_pg21.pdf", "url": "https://arxiv.org/pdf/2503.08996", "page": 1, "id": "2503.08996_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\varphi}(\\mathbf{k}; v, \\theta)"}
{"pdf": "arxiv_math/2503.08996_pg21.pdf", "url": "https://arxiv.org/pdf/2503.08996", "page": 1, "id": "2503.08996_pg21_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{k}=(k_1, \\gamma(k_1))"}
{"pdf": "arxiv_math/2503.08996_pg21.pdf", "url": "https://arxiv.org/pdf/2503.08996", "page": 1, "id": "2503.08996_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma: (-\\delta_0, \\delta_0)\\to\\mathbb{R}"}
{"pdf": "arxiv_math/2503.08996_pg21.pdf", "url": "https://arxiv.org/pdf/2503.08996", "page": 1, "id": "2503.08996_pg21_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{C}_{\\mathbf{k}\\approx\\mathbf{K}_\\star}(t; v, \\theta)"}
{"pdf": "arxiv_math/2503.07361_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07361", "page": 1, "id": "2503.07361_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta=n^2+\\frac{1}{2}"}
{"pdf": "arxiv_math/2503.07361_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07361", "page": 1, "id": "2503.07361_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{v_1},\\dots,S_{v_h}"}
{"pdf": "arxiv_math/2503.07361_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07361", "page": 1, "id": "2503.07361_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "y_k-y_j> k-1-j \\geq 2"}
{"pdf": "arxiv_math/2503.07361_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07361", "page": 1, "id": "2503.07361_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "G=(V,E_s \\cup E_\\ell)"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Im}(B_t^p(u,u)) \\leq \\frac{M}{\\nu}\\mathrm{Re}(B_t^p(u,u))"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "U_p : (s,\\infty) \\rightarrow L^2_\\omega(\\mathbb{R}^n)"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(B^p_t)_{{t \\in \\mathbb{R}}}"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{H}^\\star v=0"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "L^2((s,\\mathfrak{T});H^1_\\omega(\\mathbb{R}^n))"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Re}(B_t^p(\\cdot,\\cdot))"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_007", "type": "math", "max_diffs": 0, "checked": "verified", "math": "U_p(t):=\\Gamma_p(t,s)f"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_x U_p(t) \\in L^2_\\omega(\\mathbb{R}^n)"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "(x,y) \\in \\mathbb{R}^{2n}"}
{"pdf": "arxiv_math/2503.07569_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07569", "page": 1, "id": "2503.07569_pg19_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta \\in \\mathcal{D}(\\mathbb{R})"}
{"pdf": "arxiv_math/2503.06051_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06051", "page": 1, "id": "2503.06051_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_i=\\alpha_{i+1}"}
{"pdf": "arxiv_math/2503.06051_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06051", "page": 1, "id": "2503.06051_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "s_{i_1},s_{i_2},\\ldots,s_{i_{\\ell(\\sigma)}}"}
{"pdf": "arxiv_math/2503.06051_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06051", "page": 1, "id": "2503.06051_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\sigma \\pi)_i = \\sigma_{\\pi_i}"}
{"pdf": "arxiv_math/2503.06051_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06051", "page": 1, "id": "2503.06051_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "s_1, s_2, \\ldots, s_{n-1}"}
{"pdf": "arxiv_math/2503.06051_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06051", "page": 1, "id": "2503.06051_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "s_{i_1}s_{i_2}\\cdots s_{i_{\\ell(\\sigma)}}"}
{"pdf": "arxiv_math/2503.06051_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06051", "page": 1, "id": "2503.06051_pg2_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\alpha}^{\\sigma}(\\mathbf{x}; q, t) = E_{\\alpha}^{\\sigma s_i}(\\mathbf{x}; q, t)"}
{"pdf": "arxiv_math/2503.06051_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06051", "page": 1, "id": "2503.06051_pg2_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_{i+1} = \\sigma_i \\pm 1"}
{"pdf": "arxiv_math/2503.07337_pg40.pdf", "url": "https://arxiv.org/pdf/2503.07337", "page": 1, "id": "2503.07337_pg40_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall x\\in (u^{-1}(0))^{\\alpha_0}, \\qquad |\\nabla u(x)|\\ge \\frac{k}{2}"}
{"pdf": "arxiv_math/2503.07337_pg40.pdf", "url": "https://arxiv.org/pdf/2503.07337", "page": 1, "id": "2503.07337_pg40_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "(K)^t =\\Set{p\\in \\R^n\\;|\\; d(p,K)\\le t}"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dim N(S(\\lambda))=\\dim N(T(\\lambda))<\\infty"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i_{\\lambda}\\bigl(N(S(\\lambda))\\bigr)\\supseteq N(T(\\lambda))"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "L^{\\infty}[-\\tau,\\tau]"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma\\colon X_-\\oplus X_+\\to \\R^d"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "R(T(\\lambda))+ V= L^{\\infty}(\\R)"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "S^\\pm(\\lambda):X_{\\pm}\\to Y_{\\pm}"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "p(u):=u|_{[-\\tau,\\tau]}"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "[-\\tau,\\tau]=[-\\tau,0]\\cup [0,\\tau]"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_012", "type": "math", "max_diffs": 0, "checked": "verified", "math": "D(S(\\lambda)) := \\set{u\\in W^{1,\\infty}[-\\tau,\\tau]\\mid u(-\\tau)\\in N(P_\\lambda^-(-\\tau)), u(\\tau)\\in R(P_\\lambda^+(\\tau))}"}
{"pdf": "arxiv_math/2503.07221_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07221", "page": 1, "id": "2503.07221_pg16_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "Ju:=(u_-,u_+) \\text{ and } J_{\\lambda}v:=(v_-,v_+)"}
{"pdf": "arxiv_math/2503.08820_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08820", "page": 1, "id": "2503.08820_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha\\in\\mathbb{Z}^n_{\\ge 0}"}
{"pdf": "arxiv_math/2503.05688_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05688", "page": 1, "id": "2503.05688_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{\\mathcal{H}}"}
{"pdf": "arxiv_math/2503.08848_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08848", "page": 1, "id": "2503.08848_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{L}^{\\text {Airy }}"}
{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{bmatrix}\\varphi^{\\xi,+}_{n}\\\\\\varphi^{\\xi,-}_{n}\\end{bmatrix}=e^{2\\pi in\\theta}\\begin{bmatrix}\\check{\\phi}^{+}(\\xi+n\\Phi)\\\\\\check{\\phi}^{-}(\\xi+n\\Phi)\\end{bmatrix}=\\frac1{\\sqrt2}e^{2\\pi in\\theta}\\begin{bmatrix}\\check\\psi^+(\\xi+n\\Phi)+i\\check\\psi^-(\\xi+n\\Phi)\\\\i\\check\\psi^+(\\xi+n\\Phi)+\\check\\psi^-(\\xi+n\\Phi)\\end{bmatrix}"}
{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "W_{\\lambda_1,\\lambda_2,\\Phi,\\theta}"}
{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi=\\left[\\psi^{+},\\psi^{-}\\right]^\\top"}
{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "W_{\\lambda_{1},\\lambda_{2},\\Phi,\\theta}\\psi=z\\psi"}
{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi=\\varphi^\\xi=\\left[\\varphi^{\\xi,+},\\varphi^{\\xi,-}\\right]^\\top"}
{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\in\\Sigma_{\\lambda_1,\\lambda_2,\\Phi}"}
{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\sin2\\pi(\\theta+n\\Phi)|<\\exp(-|n|^{\\frac{1}{2\\tau}})"}
{"pdf": "arxiv_math/2503.07281_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07281", "page": 1, "id": "2503.07281_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "% H^1_{\\Theta}=K^1_{\\Theta}\\oplus \\Theta H^1"}
{"pdf": "arxiv_math/2503.07281_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07281", "page": 1, "id": "2503.07281_pg12_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1_\\Theta/(K^1_{\\Theta}\\oplus \\Theta H^1 )"}
{"pdf": "arxiv_math/2503.07281_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07281", "page": 1, "id": "2503.07281_pg12_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "K^1_{\\Theta}\\oplus \\Theta H^1 \\subsetneq H^1_{\\Theta}"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{u_\\varepsilon=(u_{1,\\varepsilon},\\cdots,u_{n,\\varepsilon})\\}_\\varepsilon"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^n \\sum_{t=1}^N e^{2H_{i,t}(p^*)}h_i(p_{t}^*) L_{i,t}=0"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{u_{\\varepsilon}:=(u_{1,\\varepsilon},\\cdots,u_{n,\\varepsilon})\\}_\\varepsilon"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{\\varepsilon}:=(\\rho_{1,\\varepsilon},\\cdots,\\rho_{n,\\varepsilon})"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Lambda_{I,N}(\\rho_\\varepsilon)=-\\sum_{i=1}^n\\frac{2}{N}\\Big(D_{i,t}+\\sum_{s\\neq t}\\frac{e^{2H_{i,s}(p^*)}h_i(p_{s}^*)}{e^{2H_{i,t}(p^*)}h_i(p_{t}^*)} D_{i,s}+o(1)\\Big)\\varepsilon_t^{2}"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i\\in\\hat{I}}\\sum_{t=1}^N e^{(\\hat{m}^*-2)H_{i,t}(p^*)}h_i(p_{t}^*)D_{i,t}\\neq 0"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon_1,\\cdots,\\varepsilon_N"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{i,\\varepsilon}\\to\\rho_i^*"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^N \\sum_{t=1}^N e^{2H_{i,t}(p^*)}h_i(p_{t}^*)D_{i,t}\\neq 0"}
{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^n \\sum_{t=1}^N e^{2H_{i,t}(p^*)}h_i(p_{t}^*)L_{i,t}\\neq 0"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{W}H= e^{-i\\alpha z}\\mathcal{H}(E)"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_001", "type": "math", "max_diffs": 0, "checked": "verified", "math": "(D-\\lambda I)\\mathcal{J},~\\lambda\\in \\mathbb{C}"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "R=(D-\\overline{\\lambda}I|\\mathcal{J})^{-1}"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_\\lambda\\notin \\mathcal{J}"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{\\lambda} \\notin \\sigma(D|\\mathcal{J})"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(D|\\mathcal{J})=\\emptyset"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J} \\in \\mathscr{J}"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_{\\lambda}(x) = \\varphi(V(I-\\overline{\\lambda}V)^{-1}x)=\\varphi(Rx) = 0"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J} = C^{\\infty}(D)"}
{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{W}\\varphi(\\lambda) = 0"}
{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x', x \\rangle \\ge 0"}
{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x, x' \\rangle\\geq 0"}
{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma=\\overline{\\mathrm{conv}(S')}"}
{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma \\subseteq V'_+"}
{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x'', x' \\rangle \\ge \\gamma > \\langle x'', x'_0 \\rangle"}
{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x, x' \\rangle \\ge \\gamma > \\langle x, x'_0 \\rangle"}
{"pdf": "arxiv_math/2503.04607_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04607", "page": 1, "id": "2503.04607_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "g(X_1 \\# X_2) = g(X_1) + g(X_2)"}
{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "LTR(3645712)=\\{ 3,6,7\\}"}
{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tau_1 +n)\\cdots (\\tau_m +n)"}
{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi =\\pi_1 \\cdots \\pi_n \\in S_n"}
{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi^{r}=\\pi_n \\cdots \\pi_1"}
{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau =\\tau_1 \\cdots \\tau_m \\in S_m"}
{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_i =\\max \\{ \\pi_j \\, |\\, j\\leq i\\}"}
{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "S=\\bigcup_{n\\in \\mathbb{N}}S_n"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v_p = r_p + \\beta n_p"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "g_{ij}=\\langle \\partial x_i, \\partial x_j\\rangle"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mu_+ =\\lambda_{\\max}\\left(\\frac{M_{+}+M_{+}^T}{2}\\right)"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma\\in\\mathbb{R}^{r\\times r}"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_{r_p}X\\in \\text{Range}(\\nabla X|^*_p)"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\nabla_{v_p} X, v_p\\rangle = \\langle w_p, (\\nabla X|_p)^{-1}w_p\\rangle \\leq \\|(\\nabla X|_p)^{-1}\\|\\,\\|w_p\\|^2 = \\|(\\nabla X|_p)^{-1}\\|\\, \\|\\nabla_{v_p} X|\\|^2"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "v_p^T g \\mathcal{A}_{X,p}v_p \\leq -\\alpha_p v_p^T \\mathcal{A}_{X,p}^T g \\mathcal{A}_{X,p} v_p,\\quad\\forall v_p\\in\\mathbb{R}^d"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{Range}(\\nabla X|_p)"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\nabla_{r_p} X, r_p\\rangle + \\beta\\langle \\nabla_{r_p} X, n_p\\rangle \\leq -\\alpha\\|\\nabla_{r_p}X\\|^2"}
{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\alpha_p = \\max_{1\\leq i \\leq d} \\lambda_i\\left(\\frac{M+M^T}{2}\\right)"}
{"pdf": "arxiv_math/2503.05399_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05399", "page": 1, "id": "2503.05399_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f_i \\in C^{1,\\frac{1}{2}}(L_i)"}
{"pdf": "arxiv_math/2503.05399_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05399", "page": 1, "id": "2503.05399_pg6_math_005", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\sigma : [0,\\ell_i] \\rightarrow \\R^2"}
{"pdf": "arxiv_math/2503.05399_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05399", "page": 1, "id": "2503.05399_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_i \\cup \\tilde{\\Gamma}_i"}
{"pdf": "arxiv_math/2503.08283_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08283", "page": 1, "id": "2503.08283_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "t_{p}, 1.5 t_{p}, 3 t_{p}, 4 t_{p}, 6 t_{p}"}
{"pdf": "arxiv_math/2503.05390_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05390", "page": 1, "id": "2503.05390_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|[b,T_\\Omega]\\vec f-[b,T_{\\Omega,\\delta}]\\vec f\\|_{L^p(W)}\\lesssim_{b,\\Omega}\\delta\\, \\|M_W(W^\\frac{1}{p}\\vec f)\\|_{L^p(\\R^d)}\\lesssim_{W} \\delta\\,\\|\\vec f\\|_{L^p(W)}"}
{"pdf": "arxiv_math/2503.05390_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05390", "page": 1, "id": "2503.05390_pg14_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "[b,T_{\\Omega,\\delta}]\\colon L^p(W)\\to L^p(W)"}
{"pdf": "arxiv_math/2503.09367_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09367", "page": 1, "id": "2503.09367_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "k^{\\log_23}\\leq n\\leq 36k^{\\log_23}"}
{"pdf": "arxiv_math/2503.09367_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09367", "page": 1, "id": "2503.09367_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "e(G)\\leq 3n-6\\leq 3n-6-\\frac{n}{36k^{1+\\log_23}}+\\frac{k^3}{4}"}
{"pdf": "arxiv_math/2503.09367_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09367", "page": 1, "id": "2503.09367_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "m(B_r)\\leq m(G)-1<\\frac{n-(t-1)}{3t-7}-1=\\frac{n-(3t-7)-(t-1)}{3t-7}<\\frac{|B_r|-(t-1)}{3t-7}"}
{"pdf": "arxiv_math/2503.09367_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09367", "page": 1, "id": "2503.09367_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "|B_i|\\geq k^{\\log_23}"}
{"pdf": "arxiv_math/2503.09367_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09367", "page": 1, "id": "2503.09367_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "e(G)=ex_{\\mathcal{P}}(n,2C_k)"}
{"pdf": "arxiv_math/2503.09367_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09367", "page": 1, "id": "2503.09367_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "e(G)\\leq 3n-6-\\frac{n}{4k^{\\log_23}}\\leq 3n-6-\\frac{n}{36k^{1+\\log_23}}+\\frac{k^2}{2}"}
{"pdf": "arxiv_math/2503.06622_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06622", "page": 1, "id": "2503.06622_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "I_t^{\\mathbf{Y}} = \\int_0^t (h,(D_x h f + D_y h)) (s, X^{\\mathbf{Y}}_s, Y_s) d \\mathbf{Y}_s - \\frac{1}{2} \\int_0^t | h (s, X^{\\mathbf{Y}}_s, Y_s) |^2 ds"}
{"pdf": "arxiv_math/2503.06622_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06622", "page": 1, "id": "2503.06622_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ Y_t : 0 \\leqslant t \\leqslant T \\}"}
{"pdf": "arxiv_math/2503.06622_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06622", "page": 1, "id": "2503.06622_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "Y_t = \\int_0^t h (X_s, Y_s) ds + B^{\\perp}_t"}
{"pdf": "arxiv_math/2503.03855_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03855", "page": 1, "id": "2503.03855_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "T_0\\coloneqq \\{t\\in T\\mid \\forall\\chi\\in X^*(T),\\, \\omega(\\chi(t))=0\\}"}
{"pdf": "arxiv_math/2503.03855_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03855", "page": 1, "id": "2503.03855_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "V\\coloneqq X_*(T)\\otimes_\\Z \\R"}
{"pdf": "arxiv_math/2503.03855_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03855", "page": 1, "id": "2503.03855_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_i\\coloneqq c_i v_i"}
{"pdf": "arxiv_math/2503.03855_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03855", "page": 1, "id": "2503.03855_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_0 = \\sum_{i=1}^d c_i \\alpha_i"}
{"pdf": "arxiv_math/2503.03855_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03855", "page": 1, "id": "2503.03855_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta_\\alpha\\left(\\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}\\right)"}
{"pdf": "arxiv_math/2503.03855_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03855", "page": 1, "id": "2503.03855_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_i(\\omega_j)=\\delta_{ij}"}
{"pdf": "arxiv_math/2503.03855_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03855", "page": 1, "id": "2503.03855_pg5_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "T_r\\coloneqq \\{t\\in T_0\\mid \\forall\\chi\\in X^*(T),\\omega(\\chi(t)-1)\\geq r\\}"}
{"pdf": "arxiv_math/2503.03855_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03855", "page": 1, "id": "2503.03855_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma\\to\\Phi:a\\mapsto \\hat{a}=\\alpha"}
{"pdf": "arxiv_math/2503.03855_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03855", "page": 1, "id": "2503.03855_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "C^+\\coloneqq \\R_{\\geq 0}\\cdot C"}
{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(st)^{+} = s^{+} \\quad \\text{and} \\quad (st)^{\\ast} = t^{\\ast}"}
{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{L}_{a_1a_5} \\times \\widetilde{R}_{a_1a_5}"}
{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "W_1 W_2 \\cdots W_{m_e}"}
{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_{LC_n}(X) \\neq 0"}
{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{L}_e \\times \\widetilde{R}_e"}
{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{array}{c|cccc} & a_1a_5 & a_1a_2a_5 & a_1a_5a_6 & a_1a_2a_5a_6 \\\\ \\hline a_1a_5 & a_1a_5 & a_1a_2a_5 & a_1a_5a_6 & a_1a_2a_5a_6 \\\\ a_1a_4a_5 & a_1a_4a_5 & \\cdot & \\cdot & \\cdot \\\\ a_5a_na_1 & a_5a_na_1 & \\cdot & a_na_1a_5a_6 & \\cdot \\\\ a_4a_5a_na_1 & a_4a_5a_na_1 & \\cdot & \\cdot & \\cdot \\end{array}"}
{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{L}_{a_1a_5} = \\{a_1a_5, a_1a_4a_5, a_na_1a_5, a_na_1a_4a_5\\}"}
{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{R}_{a_1a_5} = \\{a_1a_5, a_1a_2a_5, a_1a_5a_6, a_1a_2a_5a_6\\}"}
{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{array}{c|ccc} & a_1 & a_2 & 1 \\\\ \\hline a_1 & a_1 & \\cdot & a_1 \\\\ a_2 & \\cdot & a_2 & a_2 \\\\ 1 & a_1 & a_2 & 1 \\end{array}"}
{"pdf": "arxiv_math/2503.08488_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08488", "page": 1, "id": "2503.08488_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "S_k=0, \\ \\text{if}\\ k \\in \\partial\\mathcal{L}"}
{"pdf": "arxiv_math/2503.08488_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08488", "page": 1, "id": "2503.08488_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{L}=[-L,L]^3\\bigcap \\mathbb{Z}^3"}
{"pdf": "arxiv_math/2503.08488_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08488", "page": 1, "id": "2503.08488_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial \\mathcal{L}=\\{(z_1,z_2,z_3)\\ | \\ \\max\\{|z_1|,\\ |z_2|,\\ |z_3|\\}=L\\}"}
{"pdf": "arxiv_math/2503.08488_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08488", "page": 1, "id": "2503.08488_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "k\\in \\partial \\mathcal{L}"}
{"pdf": "arxiv_math/2503.08488_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08488", "page": 1, "id": "2503.08488_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{L}=\\mathcal{L}^{o}\\cup \\partial \\mathcal{L}"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\text{min}}^{\\mathcal{C}} = 14"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathcal{C}: [21,8,6]"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\text{min}}^{\\mathcal{C}} = 6"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "i \\in \\{1, 2, \\ldots, n\\}"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\text{min}}^\\mathcal{Q} \\le n+2"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\text{min}}^{\\mathcal{Q}} = 7"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{r}_i = \\begin{bmatrix} b_{i,1}, & b_{i,2}, & \\cdots, & b_{i,n} \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "N = 26 \\times (3^2 + 4^2)"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\text{min}}^\\mathcal{C}"}
{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_012", "type": "math", "max_diffs": 0, "checked": "verified", "math": "b_{i,j} - \\infty = \\infty - b_{i,j} = \\infty - \\infty := -\\infty"}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rm{diag} (\\mathbf{a})"}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "z_{k,c}^{'}=\\mathbf h_{k,c} \\sum\\limits_{l \\ne k}^K \\mathbf p_{l}^c x_{l,c} + z_{k,c}"}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{A} = \\rm{Bdiag}\\{\\mathbf{A}_1, \\cdots, \\mathbf{A}_N\\}"}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{A}_1, \\cdots, \\mathbf{A}_N"}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf p_{k}^c \\in \\mathbb{C}^{M \\times 1}"}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\mathcal{R}_{k,c}} = \\mathrm{log}(1+{\\Gamma}_{k,c}^{-1}\\mathbf{h}_{k,c}\\mathbf p_k^c(\\mathbf p_k^c)^H \\mathbf{h}_{k,c}^H) "}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\rm{tr}}(\\mathbf{A})"}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathbf H_k={\\rm Bdiag}\\{\\mathbf h_{k,1},\\cdots,\\mathbf h_{k,N_v}\\}\\in \\mathbb{C}^{N_v \\times MN_v}"}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{CN}(0, \\sigma_z^2 )"}
{"pdf": "arxiv_math/2503.07090_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07090", "page": 1, "id": "2503.07090_pg2_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf x_{k,c}= \\sum\\limits_{k=1}^K \\mathbf p_{k}^c x_{k,c}"}
{"pdf": "arxiv_math/2503.05661_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05661", "page": 1, "id": "2503.05661_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "P(u,v)\\cap D_G(v,\\lfloor\\frac{3}{2}K\\rfloor)"}
{"pdf": "arxiv_math/2503.05661_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05661", "page": 1, "id": "2503.05661_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "d_G(v,x)\\le d_G(v,z) + d_G(z,x)\\le K-1+(K-1)/2=3/2(K-1)<\\lfloor\\frac{3}{2}K\\rfloor"}
{"pdf": "arxiv_math/2503.05661_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05661", "page": 1, "id": "2503.05661_pg16_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "H_w:=G[D_G(w,\\lfloor\\frac{3}{2}K\\rfloor)]"}
{"pdf": "arxiv_math/2503.05661_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05661", "page": 1, "id": "2503.05661_pg16_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "d_G(V(H_v),V(H_u))< K"}
{"pdf": "arxiv_math/2503.05661_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05661", "page": 1, "id": "2503.05661_pg16_math_008", "type": "math", "max_diffs": 0, "checked": "verified", "math": "H_v:=G[D_G(v,\\lfloor\\frac{3}{2}K\\rfloor)]"}
{"pdf": "arxiv_math/2503.05661_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05661", "page": 1, "id": "2503.05661_pg16_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "d_G(V(H_v),P(u_w,w_u))< K"}
{"pdf": "arxiv_math/2503.05661_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05661", "page": 1, "id": "2503.05661_pg16_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "d_G(v,P(u,w))< \\lfloor\\frac{3}{2}K\\rfloor+K< 3K"}
{"pdf": "arxiv_math/2503.05661_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05661", "page": 1, "id": "2503.05661_pg16_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "d_G(v,u)< \\lfloor\\frac{3}{2}K\\rfloor+K+ \\lfloor\\frac{3}{2}K\\rfloor\\le 4K"}
{"pdf": "arxiv_math/2503.05661_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05661", "page": 1, "id": "2503.05661_pg16_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "H_u:=G[D_G(u,\\lfloor\\frac{3}{2}K\\rfloor)]"}
{"pdf": "arxiv_math/2503.08921_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08921", "page": 1, "id": "2503.08921_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\smash{\\mathcal{O}(\\epsilon^{-1})}"}
{"pdf": "arxiv_math/2503.08921_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08921", "page": 1, "id": "2503.08921_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{O}(\\epsilon^{-2})"}
{"pdf": "arxiv_math/2503.08921_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08921", "page": 1, "id": "2503.08921_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\smash{\\mathcal{O}(\\epsilon^{-2})}"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{P_2} \\times \\widetilde{P_3}"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{K} \\cap \\widetilde{P_1} = 1"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{K} = \\widetilde{P_1} \\times \\widetilde{P_3}"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\prod_{i=1}^{5}D_i\\unlhd X"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{K} \\cap \\widetilde{P_1} \\neq 1"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "D_1 \\times D_2 \\times D_3 \\leq KV"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{K} \\cap (\\widetilde{P_1} \\times \\widetilde{P_2})"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{K} \\cap (\\widetilde{P_1} \\times \\widetilde{P_2}) \\lhd \\widetilde{P_1} \\times \\widetilde{P_2}"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{P_1} \\times \\widetilde{P_2}"}
{"pdf": "arxiv_math/2503.06979_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06979", "page": 1, "id": "2503.06979_pg8_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "P_1\\times P_2\\times P_3"}
{"pdf": "arxiv_math/2503.08185_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08185", "page": 1, "id": "2503.08185_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_\\star=\\min_{x\\in\\Omega}\\pi(x)"}
{"pdf": "arxiv_math/2503.08185_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08185", "page": 1, "id": "2503.08185_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_\\star\\asymp \\frac{1}{2^{n^2}}"}
{"pdf": "arxiv_math/2503.08185_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08185", "page": 1, "id": "2503.08185_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega\\left(\\tfrac{n^2}{\\log n}\\right)"}
{"pdf": "arxiv_math/2503.08185_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08185", "page": 1, "id": "2503.08185_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ell_i+\\sum_{j\\neq i} a_j \\ell_j"}
{"pdf": "arxiv_math/2503.05983_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05983", "page": 1, "id": "2503.05983_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "A\\cong A^{sq}\\oplus A^{dot}"}
{"pdf": "arxiv_math/2503.08062_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08062", "page": 1, "id": "2503.08062_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(1-\\frac{N_{\\tau}-N_{cp}}{N})^2"}
{"pdf": "arxiv_math/2503.04701_pg26.pdf", "url": "https://arxiv.org/pdf/2503.04701", "page": 1, "id": "2503.04701_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "% \\bar{w}_{n,k}^{(i)} = [\\bar{w}_{n,-k}^{(i)}]^{*}, \\quad k \\in Z. %"}
{"pdf": "arxiv_math/2503.04701_pg26.pdf", "url": "https://arxiv.org/pdf/2503.04701", "page": 1, "id": "2503.04701_pg26_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\bar{v}_k^{(i)} = [\\bar{v}_{-k}^{(i)}]^{*}, \\quad \\text{for} \\quad i=1,2, \\quad k \\in \\mathbb{Z}"}
{"pdf": "arxiv_math/2503.04701_pg26.pdf", "url": "https://arxiv.org/pdf/2503.04701", "page": 1, "id": "2503.04701_pg26_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{\\sigma} = 0.927447198734628"}
{"pdf": "arxiv_math/2503.06285_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06285", "page": 1, "id": "2503.06285_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "g:\\mathcal{H}\\to\\mathbb{R}\\cup\\{+\\infty\\}"}
{"pdf": "arxiv_math/2503.06285_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06285", "page": 1, "id": "2503.06285_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "A:\\mathcal{H}\\to\\mathcal{H}"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "i(T_r T_{r-1} \\dots T_2 T_1) = i(T_1) i(T_2) \\dots i(T_{r-1}) i(T_r)"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(T_k \\rho)(x) = \\mathbf{1}_{ \\{ \\rho(x) \\leq k \\} }"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T_k : \\mathcal{S} \\rightarrow \\mathcal{S}"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{\\tau_i} T_\\omega = T_{\\tau_i \\omega}"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega(i) < \\omega(i+1)"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{P}(\\omega) = T_{\\omega}"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "W_t = \\prod_{i=1}^{r} T_i"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho \\in \\mathbb{Z}^{I}"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\pi_{t}(1), \\pi_{t}(2), \\dots \\pi_{t}(k)) \\overset{(d)}{=} (\\pi_{t}^{-1}(1), \\pi_{t}^{-1}(2), \\dots \\pi_{t}^{-1}(k))"}
{"pdf": "arxiv_math/2503.08341_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08341", "page": 1, "id": "2503.08341_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "i(T_{\\omega}) = T_{\\omega^{-1}}"}
{"pdf": "arxiv_math/2503.09174_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09174", "page": 1, "id": "2503.09174_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "A_T A_R/(\\lambda^2 r^2)"}
{"pdf": "arxiv_math/2503.05329_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05329", "page": 1, "id": "2503.05329_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "0<r' \\leq r \\leq \\infty"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "N_R(z)= \\frac{z}{6}(z^2+z+4)"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_001", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\begin{aligned} y \\cdot \\frac{1}{z^d} &=\\int \\frac{\\alpha}{z} \\cdot \\frac{1}{z^d} d z+\\beta=-\\frac{\\alpha}{d z^d } +\\beta, \\end{aligned}"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{z^{d+1} +(d-1)z}{d}"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "p(z)=\\beta z^d-\\frac{\\alpha}{d}"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "N_R(z)=\\frac{1}{3}z(z+2)"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta R(cz)= \\frac{z^d}{ z^d -1}"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "y'-\\frac{d}{z}y=\\frac{\\alpha}{z}"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "R(z)=\\frac{z^d}{ \\beta z^d -\\frac{\\alpha}{d}}"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "c^d = \\frac{\\alpha}{d \\beta}"}
{"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "N_R(z)=\\frac{1}{3}z(z^3+2)"}
{"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{split} T_{\\Lambda}: &\\left(c_{1,0}, c_{1,1}, \\ldots, c_{1, m_1-1} ; c_{2,0}, c_{2,1}, \\ldots, c_{2, m_2-1} ; \\ldots ; c_{\\ell, 0}, c_{\\ell, 1}, \\ldots, c_{\\ell, m_{\\ell}-1}\\right)\\mapsto\\\\ &\\left(\\lambda_1 c_{1, m_1-1},c_{1,0}, \\ldots, c_{1, m_1-2} ; \\lambda_2 c_{2, m_2-1}, c_{2,0}, \\ldots, c_{2, m_2-2} ; \\ldots ; \\lambda_{\\ell} c_{\\ell, m_{\\ell}-1}, c_{\\ell, 0}, \\ldots, c_{\\ell, m_{\\ell}-2}\\right). \\end{split}"}
{"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{\\mathbf{g}_1, \\mathbf{g}_2, \\ldots, \\mathbf{g}_\\rho\\right\\} \\subseteq V"}
{"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\neq \\lambda_i \\in \\mathbb{F}_q"}
{"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "n = m_1 + m_2 + \\cdots + m_\\ell"}
{"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Lambda = (\\lambda_1, \\lambda_2, \\ldots, \\lambda_\\ell)"}
{"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left(m_1, m_2, \\ldots, m_\\ell\\right)"}
{"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "(m_1, m_2, \\ldots, m_\\ell)"}
{"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left(\\lambda_1^{-1}, \\lambda_2^{-1}, \\ldots, \\lambda_\\ell^{-1}\\right)"}
{"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "V=\\bigoplus_{i=1}^\\ell\\frac{\\mathbb{F}_q[x]}{\\langle x^{m_i}-\\lambda_i\\rangle}=\\frac{\\mathbb{F}_q[x]}{\\langle x^{m_1}-\\lambda_1\\rangle}\\oplus \\frac{\\mathbb{F}_q[x]}{\\langle x^{m_2}-\\lambda_2\\rangle}\\oplus \\cdots \\oplus \\frac{\\mathbb{F}_q[x]}{\\langle x^{m_\\ell}-\\lambda_\\ell\\rangle}"}
{"pdf": "arxiv_math/2503.07028_pg20.pdf", "url": "https://arxiv.org/pdf/2503.07028", "page": 1, "id": "2503.07028_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega({\\color{red}{t}})"}
{"pdf": "arxiv_math/2503.06817_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06817", "page": 1, "id": "2503.06817_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c = \\Delta x/\\Delta t"}
{"pdf": "arxiv_math/2503.06817_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06817", "page": 1, "id": "2503.06817_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta t\\propto \\Delta x^2"}
{"pdf": "arxiv_math/2503.06817_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06817", "page": 1, "id": "2503.06817_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta t\\rightarrow 0"}
{"pdf": "arxiv_math/2503.06817_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06817", "page": 1, "id": "2503.06817_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta t=\\xi\\Delta x^2"}
{"pdf": "arxiv_math/2503.06817_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06817", "page": 1, "id": "2503.06817_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{L}=\\Delta x\\mathbb{Z}^d"}
{"pdf": "arxiv_math/2503.06817_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06817", "page": 1, "id": "2503.06817_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "f_k^{eq}(\\mathbf{x},t)"}
{"pdf": "arxiv_math/2503.04678_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04678", "page": 1, "id": "2503.04678_pg20_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{0,N}=2^{N-1}+1, d_{1,N}, \\ldots, d_{N-1,N}"}
{"pdf": "arxiv_math/2503.04678_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04678", "page": 1, "id": "2503.04678_pg20_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{i+1,N}\\leq 5 \\cdot 2^{N-i} \\cdot 2^i \\cdot d_{i,N}=5 \\cdot 2^N\\cdot d_{i,N}"}
{"pdf": "arxiv_math/2503.07532_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07532", "page": 1, "id": "2503.07532_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "e \\, c \\, d \\, \\bar c \\, \\bar d \\, \\bar e \\, a \\, b \\, \\bar a \\, \\bar b \\, e"}
{"pdf": "arxiv_math/2503.07532_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07532", "page": 1, "id": "2503.07532_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde\\alpha \\subset U"}
{"pdf": "arxiv_math/2503.07532_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07532", "page": 1, "id": "2503.07532_pg14_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta(\\alpha) \\subset T"}
{"pdf": "arxiv_math/2503.05311_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05311", "page": 1, "id": "2503.05311_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "p^{({r-1})/{2}} \\ll n^{-1} (\\log n)^{{1}/({r-2})}"}
{"pdf": "arxiv_math/2503.05311_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05311", "page": 1, "id": "2503.05311_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "n^{-1/2-o(1)} \\le p^{\\Delta/2} \\ll 1"}
{"pdf": "arxiv_math/2503.05311_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05311", "page": 1, "id": "2503.05311_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "p^{({r-1})/{2}} n^{-1} (\\log n)^{{1}/({r-2})} \\ll p^{({r-1})/{2}} \\ll 1"}
{"pdf": "arxiv_math/2503.05311_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05311", "page": 1, "id": "2503.05311_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "p^{(r-1)/2} \\gg n^{-1}"}
{"pdf": "arxiv_math/2503.05311_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05311", "page": 1, "id": "2503.05311_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\set{n}:=\\{1,2,\\ldots, n\\}"}
{"pdf": "arxiv_math/2503.05311_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05311", "page": 1, "id": "2503.05311_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "p^{({r-1})/{2}} \\asymp n^{-1} (\\log n)^{{1}/({r-2})}"}
{"pdf": "arxiv_math/2503.05311_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05311", "page": 1, "id": "2503.05311_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "n^{-1/\\Delta} \\ll p \\ll 1"}
{"pdf": "arxiv_math/2503.07692_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07692", "page": 1, "id": "2503.07692_pg35_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "C_{\\min} := \\min_{i,j}{(\\min_{i,j}n_{i,j}^{m},\\min_{i,j}p_{i,j}^{m})}"}
{"pdf": "arxiv_math/2503.07692_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07692", "page": 1, "id": "2503.07692_pg35_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi^{m+1,0}=2\\phi^{m}-\\phi^{m-1}"}
{"pdf": "arxiv_math/2503.07692_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07692", "page": 1, "id": "2503.07692_pg35_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|e_{\\zeta}\\|=\\|\\zeta_{h}-\\zeta_{h/2}\\|"}
{"pdf": "arxiv_math/2503.07692_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07692", "page": 1, "id": "2503.07692_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "n^{m+1,0}=\\max{\\left(2n^{m}-n^{m-1},\\epsilon\\right)}"}
{"pdf": "arxiv_math/2503.07692_pg35.pdf", "url": "https://arxiv.org/pdf/2503.07692", "page": 1, "id": "2503.07692_pg35_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "p^{m+1,0}=\\max{\\left(2p^{m}-p^{m-1},\\epsilon\\right)}"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "s_i v \\rightharpoonup s_i w"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{std}(w) \\in S_n"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{i+1}(v) < \\alpha_i(v)"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_005", "type": "math", "max_diffs": 0, "checked": "verified", "math": "D([4,7,8,2,5,9,3,10,1,11,6])"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\text{std}(44425533116)=[4,7,8,2,5,9,3,10,1,11,6]"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(\\text{conv}(w)) = \\pi(w)"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{conv}(v)=\\text{conv}(w)"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "D(\\text{std}(\\text{conv}(w)))"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{i+1}(w) < \\alpha_i(w)"}
{"pdf": "arxiv_math/2503.08911_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08911", "page": 1, "id": "2503.08911_pg10_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "D(\\text{std}(\\text{conv}(w))) \\subset [k] \\times [n]"}
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{"pdf": "arxiv_math/2503.06509_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06509", "page": 1, "id": "2503.06509_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "v^Td<b~\\forall d\\in Conv\\bigg\\{\\displaystyle \\bigcup_{j\\in \\Lambda}\\partial H_j(x^*)\\bigg\\}"}
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{"pdf": "arxiv_math/2503.06509_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06509", "page": 1, "id": "2503.06509_pg11_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "H'_{j^0}(x^*,d) \\geq 0, ~\\forall~ d\\in D-\\{x^*\\}"}
{"pdf": "arxiv_math/2503.06509_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06509", "page": 1, "id": "2503.06509_pg11_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "x^*=\\arg\\displaystyle \\min_{x\\in D}\\breve{H}(x)"}
{"pdf": "arxiv_math/2503.06509_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06509", "page": 1, "id": "2503.06509_pg11_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla h_j(x^*,\\xi_i)^Ts<0"}
{"pdf": "arxiv_math/2503.06509_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06509", "page": 1, "id": "2503.06509_pg11_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "Conv\\bigg\\{\\displaystyle \\bigcup_{j\\in \\Lambda}\\partial H_j(x^*)\\bigg\\}"}
{"pdf": "arxiv_math/2503.06509_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06509", "page": 1, "id": "2503.06509_pg11_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "h_{j^0}(x,\\xi_i)\\geq h_{j^0}(x^*,\\xi_i) + \\nabla h_{j^0}(x^*,\\xi_i)^T(x-x^*),~ \\forall ~i\\in I_{j^0}(x^*)"}
{"pdf": "arxiv_math/2503.06509_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06509", "page": 1, "id": "2503.06509_pg11_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "h_{j^0}(x,\\xi_i)\\geq h_{j^0}(x^*,\\xi_i),~ \\forall~ i\\in I_{j^0}(x^*)"}
{"pdf": "arxiv_math/2503.06509_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06509", "page": 1, "id": "2503.06509_pg11_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\not\\in Conv\\bigg\\{\\displaystyle \\bigcup_{j\\in \\Lambda}\\partial H_j(x^*)\\bigg\\}"}
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{"pdf": "arxiv_math/2503.05089_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05089", "page": 1, "id": "2503.05089_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "f, g : (0, 1] \\to (0, 1]"}
{"pdf": "arxiv_math/2503.05089_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05089", "page": 1, "id": "2503.05089_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "p \\gg n^{-r+1} \\log n"}
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{"pdf": "arxiv_math/2503.05692_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05692", "page": 1, "id": "2503.05692_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u\\|_{C_T\\mathbb{X}}:=\\sup_{s\\in [0,T]}\\|u(s)\\|_\\mathbb{X}"}
{"pdf": "arxiv_math/2503.05692_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05692", "page": 1, "id": "2503.05692_pg8_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "(-\\Delta)^{-1} f(t,x)"}
{"pdf": "arxiv_math/2503.05692_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05692", "page": 1, "id": "2503.05692_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta=(\\zeta_1,\\zeta_2,\\zeta_3)"}
{"pdf": "arxiv_math/2503.05692_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05692", "page": 1, "id": "2503.05692_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(-\\Delta)^{-1} \\bar f(x)"}
{"pdf": "arxiv_math/2503.05692_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05692", "page": 1, "id": "2503.05692_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|(-\\Delta)^{-1} \\div(w\\otimes w)\\|_{L^2_x} \\gg \\|w\\|_{L^2_x}"}
{"pdf": "arxiv_math/2503.05692_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05692", "page": 1, "id": "2503.05692_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u^{\\mathcal{A}}\\|_{L^2_x} \\ll \\|w\\|_{L^2_x}"}
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{"pdf": "arxiv_math/2503.04024_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04024", "page": 1, "id": "2503.04024_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\kappa_\\text{min} > 0"}
{"pdf": "arxiv_math/2503.04024_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04024", "page": 1, "id": "2503.04024_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bm{c} \\in H^1_{\\text{div}}(\\Omega) = \\{\\bm{c} \\in [L^2(\\Omega)]^2 \\ | \\ \\nabla \\cdot \\bm{c} \\in L^2(\\Omega)\\}"}
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{"pdf": "arxiv_math/2503.03903_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03903", "page": 1, "id": "2503.03903_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{S}_w = \\prod_{i}h^i_{L(i)}"}
{"pdf": "arxiv_math/2503.03903_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03903", "page": 1, "id": "2503.03903_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i, j, k = i, i+1, i+2"}
{"pdf": "arxiv_math/2503.03903_pg9.pdf", "url": "https://arxiv.org/pdf/2503.03903", "page": 1, "id": "2503.03903_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{S}_w = h_{L(w)}"}
{"pdf": "arxiv_math/2503.05442_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05442", "page": 1, "id": "2503.05442_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\kappa(BS_{n-1})=4k+1"}
{"pdf": "arxiv_math/2503.05442_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05442", "page": 1, "id": "2503.05442_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(b,M_1^+\\cup M_3\\cup\\{a^+\\})"}
{"pdf": "arxiv_math/2503.05442_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05442", "page": 1, "id": "2503.05442_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "BS_n\\backslash (BS_{n-1}^1\\oplus BS_{n-1}^2\\oplus BS_{n-1}^3)"}
{"pdf": "arxiv_math/2503.05442_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05442", "page": 1, "id": "2503.05442_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "M_1^+\\cup M_3\\cup\\{a^-,c^-\\}"}
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{"pdf": "arxiv_math/2503.05442_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05442", "page": 1, "id": "2503.05442_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "M_1\\cup M_2\\cup\\{c^+\\}"}
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{"pdf": "arxiv_math/2503.05442_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05442", "page": 1, "id": "2503.05442_pg14_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{a^+\\}\\subseteq V(BS_{n-1}^3)"}
{"pdf": "arxiv_math/2503.05442_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05442", "page": 1, "id": "2503.05442_pg14_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "M_2^+\\cup M_3^+\\cup\\{a^+\\}"}
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{"pdf": "arxiv_math/2503.07400_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07400", "page": 1, "id": "2503.07400_pg11_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|G| \\ge b_2(3, g - 1, s - 1) \\ge b_2(3, g - 1, g - 1)"}
{"pdf": "arxiv_math/2503.07400_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07400", "page": 1, "id": "2503.07400_pg11_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "b_2(k, g, s) \\ge \\frac12 M(k, g)"}
{"pdf": "arxiv_math/2503.07400_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07400", "page": 1, "id": "2503.07400_pg11_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "|G| \\ge b_2(k, g, (k - 2)g)"}
{"pdf": "arxiv_math/2503.07400_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07400", "page": 1, "id": "2503.07400_pg11_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "|G| \\ge b_2(k, g, (k - 2)g)+1"}
{"pdf": "arxiv_math/2503.07400_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07400", "page": 1, "id": "2503.07400_pg11_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "|G| > b_1(k, g, (k - 2)g)"}
{"pdf": "arxiv_math/2503.07400_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07400", "page": 1, "id": "2503.07400_pg11_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|G'| \\ge b_2(k, g, (k - 2)g))"}
{"pdf": "arxiv_math/2503.07400_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07400", "page": 1, "id": "2503.07400_pg11_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "|G'| \\ge b_2(k, g, s-(k-4))"}
{"pdf": "arxiv_math/2503.07400_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07400", "page": 1, "id": "2503.07400_pg11_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "M(3, 2d + 1) = 3\\cdot 2^d - 2 > 4d^2 + 6d + \\frac{11}{2}"}
{"pdf": "arxiv_math/2503.07400_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07400", "page": 1, "id": "2503.07400_pg11_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "s - 1 \\le (k - 2)(g - 1)"}
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{"pdf": "arxiv_math/2503.05874_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05874", "page": 1, "id": "2503.05874_pg8_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "S_{i j}^{\\prime}=\\left[L_{j}, U_{j}\\right]"}
{"pdf": "arxiv_math/2503.05874_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05874", "page": 1, "id": "2503.05874_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "A^{-}=\\left[\\begin{array}{llllllllll} 0.70 & 0.70 & 0.32 & 0.44 & 0.00 & 0.16 & 0.20 & 0.50 & 0.40 & 0.39 \\\\ 0.70 & 0.65 & 0.14 & 0.12 & 0.80 & 0.76 & 0.00 & 1.00 & 0.15 & 0.79 \\\\ 0.17 & 0.24 & 0.20 & 0.20 & 0.06 & 0.25 & 0.13 & 0.19 & 0.22 & 0.02 \\\\ 0.14 & 0.10 & 0.04 & 0.00 & 0.10 & 0.00 & 0.14 & 0.02 & 0.15 & 0.08 \\\\ 0.70 & 0.04 & 0.27 & 0.36 & 0.60 & 0.40 & 0.48 & 0.50 & 0.50 & 0.50 \\\\ 0.66 & 0.63 & 0.14 & 0.73 & 0.53 & 0.46 & 0.61 & 0.85 & 0.85 & 0.39 \\\\ 0.00 & 0.15 & 0.15 & 0.05 & 0.02 & 0.03 & 0.10 & 0.12 & 0.18 & 0.09 \\\\ 0.63 & 0.03 & 0.55 & 0.77 & 0.79 & 0.49 & 0.21 & 0.32 & 0.80 & 0.71 \\\\ 0.27 & 0.30 & 0.35 & 0.24 & 0.35 & 0.07 & 0.29 & 0.35 & 0.20 & 0.75 \\\\ 0.59 & 0.34 & 0.26 & 0.38 & 0.02 & 0.60 & 0.52 & 0.43 & 0.27 & 0.44 \\end{array}\\right]"}
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{"pdf": "arxiv_math/2503.04056_pg25.pdf", "url": "https://arxiv.org/pdf/2503.04056", "page": 1, "id": "2503.04056_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v(t) = {v}(x_{i},y_{i}, t)"}
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{"pdf": "arxiv_math/2503.08649_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08649", "page": 1, "id": "2503.08649_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_{B\\subset\\Omega}\\left(\\frac1{|B|}{\\int_Bw\\ dx }\\right)\\left(\\frac1{|B|}{\\int_Bw^{-1}}dx \\right)<+\\infty"}
{"pdf": "arxiv_math/2503.08649_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08649", "page": 1, "id": "2503.08649_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega \\subset\\mathbb R^N"}
{"pdf": "arxiv_math/2503.08649_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08649", "page": 1, "id": "2503.08649_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "w(x):=d^\\beta(x), \\quad{\\rm \\ \\ }"}
{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau^*\\frac{\\partial S_t}{\\partial t}"}
{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta(f, g)\\colon \\begin{array}{ccc} \\Omega^1_0(M) & \\to & \\Omega^1_0(M)\\\\ \\zeta &\\mapsto & \\zeta^*[f,\\tau^*\\zeta^*g] \\end{array}"}
{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f, g\\in J_\\xi^\\infty"}
{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "(J_\\xi^\\infty,\\triangleright)"}
{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "%\\abs{\\xi}=1,\\quad \\abs{\\eta(f, g)}=\\abs{f}+\\abs{g}. %"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{G}(tX,t^\\beta Y) = t^{d-e}\\widetilde{G}(X,Y)"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f_-(t) = \\sum_{k = 0}^m (-1)^{d-rk}a_k t^{sk}"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\leq m,n\\leq \\lfloor d/r\\rfloor"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "F(tX,t^\\beta Y) = t^d F(X,Y)= t^d X^e \\widetilde{F}(X,Y)"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "g_-(t) = \\sum_{k = 0}^n (-1)^{d-rk}b_k t^{sk}"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{F}(tX,t^\\beta Y) = t^{d-e}\\widetilde{F}(X,Y)"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a,b\\in\\R\\setminus\\{0\\}"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "g_+(t) = \\sum_{k = 0}^n b_k t^{sk}"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "F(tX,t^\\beta Y) = (tX)^e \\widetilde{F}(tX, t^\\beta Y) = t^e X^e \\widetilde{F}(tX,t^\\beta Y)"}
{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\deg g_+ = \\deg g_- = sn"}
{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "D:=\\{\\omega\\in\\Omega\\colon \\|z(\\omega)\\|>\\sigma\\}"}
{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "D\\setminus N=\\bigcup_{i=1}^\\infty E_i"}
{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_0:=(E\\setminus D)\\cup N\\cup C"}
{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\lambda a(\\omega)+\\nu b(\\omega)\\|=\\|\\lambda a(\\omega)\\|+\\|\\nu b(\\omega)\\|"}
{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "%\\label{eq: ||int_(E_i) z dmu|| > (1-eps) int_(E_i) ||z|| dmu} \\begin{aligned} \\biggl\\|\\int_{E_i} z\\,d\\mu\\biggr\\| &\\geq\\re\\zs_i\\biggl(\\int_{E_i} z\\,d\\mu\\biggr) =\\int_{E_i}\\bigl(\\re\\zs_i(z_i)+\\re\\zs_i(z-z_i)\\bigr)\\,d\\mu\\\\ &\\geq\\int_{E_i}(\\|z_i\\|-\\|z-z_i\\|)\\,d\\mu \\geq\\int_{E_i}(\\|z\\|-2\\|z-z_i\\|)\\,d\\mu\\\\ &>\\int_{E_i}(\\|z\\|-\\eps\\sigma)\\,d\\mu \\geq\\int_{E_i}(\\|z\\|-\\eps\\|z\\|)\\,d\\mu =(1-\\eps)\\int_{E_i}\\|z\\|\\,d\\mu. \\end{aligned}"}
{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "C:=\\bigcup_{i=n+1}^\\infty E_i"}
{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bigcup_{i=0}^n E_i=E"}
{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\alpha a(\\omega)+\\beta b(\\omega)\\|=\\alpha\\|a(\\omega)\\|+\\beta\\|b(\\omega)\\|"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\bar{\\textbf{H}}_{1,U}}"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textbf{H}_{1,\\text D}"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\delta} = \\frac{\\rho_1 e^{j\\omega_1}}{\\rho_2 e^{j\\omega_2}}"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textbf{H}_{2,\\text D}"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}^{'}_{2,D}} = \\frac{\\mathrm{\\textbf{H}_{2,D}}}{\\rho_2 e^{j\\omega_2}}"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}_\\text D}"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}_D}"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}} = [\\mathrm{\\textbf{H}^{'}_{1,U}},\\mathrm{\\textbf{H}^{'}_{2,D}} ]"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_008", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathrm{\\textbf{H}^{'}_{1,U}} = \\frac{\\mathrm{\\bar{\\textbf{H}}}_{1,U}}{\\delta}"}
{"pdf": "arxiv_math/2503.06651_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06651", "page": 1, "id": "2503.06651_pg13_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathrm{\\textbf{H}_{1,U}}"}
{"pdf": "arxiv_math/2503.04535_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04535", "page": 1, "id": "2503.04535_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "X=C_1\\times \\ldots \\times C_n"}
{"pdf": "arxiv_math/2503.04535_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04535", "page": 1, "id": "2503.04535_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "L=L_1\\boxtimes\\ldots\\boxtimes L_n"}
{"pdf": "arxiv_math/2503.04535_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04535", "page": 1, "id": "2503.04535_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "X=C_1\\times\\ldots\\times C_n"}
{"pdf": "arxiv_math/2503.04535_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04535", "page": 1, "id": "2503.04535_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "d_1\\geq d_2\\geq\\ldots\\geq d_n>0"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "w_\\perp \\in \\overline{E}_{\\leq\\rho\\Lambda}^\\perp"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "r(\\sigma)=W_{\\perp}\\gamma(\\sigma)\\in \\overline{E}_{\\leq\\rho \\Lambda}^\\perp"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "w_\\perp^T A(\\sigma)w_\\perp -tw_\\perp M w_\\perp\\geq \\alpha w_\\perp^T \\overline{A}w_\\perp -tw_\\perp M w_\\perp\\geq (\\alpha \\rho \\Lambda-t)\\|w_\\perp\\|^2_M>0"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{E}_{\\leq\\rho\\Lambda}"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "A:S\\rightarrow \\mathbb{S}^{n\\times n}_{++}"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "x(\\sigma)=\\overline{x}(\\sigma)+r(\\sigma)"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "W_{\\perp}^T M\\overline{x}(\\sigma)=0"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "-W_\\perp (W_\\perp^T (A(\\sigma)-\\lambda(\\sigma)M)W_\\perp)^{-1}W_\\perp^T \\delta A(\\sigma)\\overline{x}(\\sigma)=W_\\perp \\gamma(\\sigma)"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "Z(\\sigma, t ) : S \\times (0,\\Lambda) \\mapsto \\mathbb{R}^{n\\times n}"}
{"pdf": "arxiv_math/2503.08465_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08465", "page": 1, "id": "2503.08465_pg8_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\| x(\\sigma) \\|_M = 1"}
{"pdf": "arxiv_math/2503.08927_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08927", "page": 1, "id": "2503.08927_pg11_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "r_S=0.027\\, \\mathrm{day}^{-1}"}
{"pdf": "arxiv_math/2503.08927_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08927", "page": 1, "id": "2503.08927_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "[0.002,0.1]\\ni \\hat f_0"}
{"pdf": "arxiv_math/2503.08927_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08927", "page": 1, "id": "2503.08927_pg11_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{0.25, 0.50, 0.75\\}\\ni n_0"}
{"pdf": "arxiv_math/2503.08927_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08927", "page": 1, "id": "2503.08927_pg11_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat f_0\\coloneqq r(0)/(r(0)+s(0))"}
{"pdf": "arxiv_math/2503.08927_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08927", "page": 1, "id": "2503.08927_pg11_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta\\coloneqq (\\hat d_D, \\hat d_T, \\hat r_R, \\hat f_0)"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta: L(V,\\mathbb{R}) \\to L(V,\\mathbb{R})"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "A \\coloneqq (\\omega(x,y))_{x,y\\in V}\\in \\mathbb{R}^{|V| \\times |V|}"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "d_G \\coloneqq \\max_{x \\in V} d_x"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega(x,y) = \\omega(y,x)"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_005", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\Gamma(u) \\coloneqq \\Gamma(u,u)"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathcal{G} = (V, E, w)"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{f} (\\chi^{-1}) = \\overline{\\widehat{f}(\\chi)}"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega: V \\times V \\rightarrow \\mathbb{R}_{\\geq 0}"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "0 = \\lambda_1< \\lambda_2 \\le \\cdots \\le \\lambda_n"}
{"pdf": "arxiv_math/2503.06908_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06908", "page": 1, "id": "2503.06908_pg3_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "d_x \\coloneqq \\sum_{y \\in V,\\; y \\sim x} \\omega(x,y)"}
{"pdf": "arxiv_math/2503.08326_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08326", "page": 1, "id": "2503.08326_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho(v) \\subset \\sigma_3"}
{"pdf": "arxiv_math/2503.08326_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08326", "page": 1, "id": "2503.08326_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "u^{(i)}_{13} - u^{(i)}_{10}-u^{(i)}_8-u^{(i)}_7=0"}
{"pdf": "arxiv_math/2503.08326_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08326", "page": 1, "id": "2503.08326_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "x^{11}-x^8-2x^6-x^5+x^3+x+1 = (x-1)(x^4+x^3+x^2+x+1)(x^6-x^3-x-1)"}
{"pdf": "arxiv_math/2503.08326_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08326", "page": 1, "id": "2503.08326_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "j \\in \\{1, \\dotsc, 13\\}"}
{"pdf": "arxiv_math/2503.08326_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08326", "page": 1, "id": "2503.08326_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "i \\in \\{1, \\dotsc, 13\\}"}
{"pdf": "arxiv_math/2503.08326_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08326", "page": 1, "id": "2503.08326_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "v=(4, 16, \\{(L_2, L_3), (R_1, R_2)\\})"}
{"pdf": "arxiv_math/2503.08326_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08326", "page": 1, "id": "2503.08326_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "x^{11}-x^8-2x^6-x^5+x^3+x+1"}
{"pdf": "arxiv_math/2503.08326_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08326", "page": 1, "id": "2503.08326_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "u^{(i)}_{n} - u^{(i)}_{n-3}-u^{(i)}_{n-5}-u^{(i)}_{n-6}=0"}
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{"pdf": "arxiv_math/2503.08186_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08186", "page": 1, "id": "2503.08186_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "q'(t) \\ge \\frac{4}{d} \\, (C'' - q(t))"}
{"pdf": "arxiv_math/2503.08186_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08186", "page": 1, "id": "2503.08186_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "||E_1||_{H^1 \\to H^1} = 1"}
{"pdf": "arxiv_math/2503.08186_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08186", "page": 1, "id": "2503.08186_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi : \\R^{d-1} \\to \\R"}
{"pdf": "arxiv_math/2503.08186_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08186", "page": 1, "id": "2503.08186_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "||| E_2\\,E_1 ||| = ||| E_2 ||| \\le 2 \\sqrt{1+\\frac{1}{121d^2}}"}
{"pdf": "arxiv_math/2503.08186_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08186", "page": 1, "id": "2503.08186_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_{\\Omega}(t,x,y) = \\Gamma_{\\Omega}(t,y,x)"}
{"pdf": "arxiv_math/2503.08186_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08186", "page": 1, "id": "2503.08186_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial\\Omega_d = \\partial\\Omega \\setminus \\partial\\Omega_n"}
{"pdf": "arxiv_math/2503.08186_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08186", "page": 1, "id": "2503.08186_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega_+\\setminus \\Omega"}
{"pdf": "arxiv_math/2503.08186_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08186", "page": 1, "id": "2503.08186_pg7_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega := B_d(0,R) \\cap \\{ (x',x_d) \\in \\R^d : x_d > \\phi(x') \\}"}
{"pdf": "arxiv_math/2503.08186_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08186", "page": 1, "id": "2503.08186_pg7_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "C'' := C_{\\mathrm{Nash},d}^{-2}\\, |||E |||^{-2 - \\frac4d}"}
{"pdf": "arxiv_math/2503.04077_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04077", "page": 1, "id": "2503.04077_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_1(\\mathbb{R})\\cup\\gamma_2(\\mathbb{R})"}
{"pdf": "arxiv_math/2503.04077_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04077", "page": 1, "id": "2503.04077_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}\\times S^1\\times\\mathbb{C}"}
{"pdf": "arxiv_math/2503.04077_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04077", "page": 1, "id": "2503.04077_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_i(t+1) = \\gamma_i(t) + \\sqrt{-1}"}
{"pdf": "arxiv_math/2503.04077_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04077", "page": 1, "id": "2503.04077_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "p_1,p_2\\in\\gamma_1(\\mathbb{R})"}
{"pdf": "arxiv_math/2503.04077_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04077", "page": 1, "id": "2503.04077_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_1, \\gamma_2\\colon \\mathbb{R}\\to \\mathbb{C}"}
{"pdf": "arxiv_math/2503.04077_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04077", "page": 1, "id": "2503.04077_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\overline{p_1p_2}| \\leq |\\overline{p_3p_4}|"}
{"pdf": "arxiv_math/2503.04077_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04077", "page": 1, "id": "2503.04077_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta\\in (0,\\frac{\\pi}{2}]"}
{"pdf": "arxiv_math/2503.04077_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04077", "page": 1, "id": "2503.04077_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "p_3,p_4\\in\\gamma_2(\\mathbb{R})"}
{"pdf": "arxiv_math/2503.09230_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09230", "page": 1, "id": "2503.09230_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma: \\widehat{\\Pi^+}\\to\\widehat{\\Pi^+}"}
{"pdf": "arxiv_math/2503.09230_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09230", "page": 1, "id": "2503.09230_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H \\cap \\Pi = G \\cap \\Pi"}
{"pdf": "arxiv_math/2503.09230_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09230", "page": 1, "id": "2503.09230_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi(f) := \\sigma^{-1}(f)"}
{"pdf": "arxiv_math/2503.09230_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09230", "page": 1, "id": "2503.09230_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi(f_i) = \\sigma^{-1}(f_i)"}
{"pdf": "arxiv_math/2503.09230_pg13.pdf", "url": "https://arxiv.org/pdf/2503.09230", "page": 1, "id": "2503.09230_pg13_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(G\\cap\\Pi^+) - V(G\\cap\\Pi)"}
{"pdf": "arxiv_math/2503.06826_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06826", "page": 1, "id": "2503.06826_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "q = \\Theta(\\sqrt{n t / \\log n})"}
{"pdf": "arxiv_math/2503.06826_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06826", "page": 1, "id": "2503.06826_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta = \\alpha n / (8N)"}
{"pdf": "arxiv_math/2503.06826_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06826", "page": 1, "id": "2503.06826_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "k \\ge \\frac{\\alpha N}{64 \\ell} =: q"}
{"pdf": "arxiv_math/2503.06826_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06826", "page": 1, "id": "2503.06826_pg10_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "V(\\Gamma) = R \\cup A \\cup B"}
{"pdf": "arxiv_math/2503.06826_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06826", "page": 1, "id": "2503.06826_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "|W| = \\left| \\bigcup_{i = 1}^k W_i \\right| < \\alpha N / 64"}
{"pdf": "arxiv_math/2503.06826_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06826", "page": 1, "id": "2503.06826_pg10_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "|W_i| = \\sqrt{\\frac{2 K N \\log N}{d}} =: \\ell"}
{"pdf": "arxiv_math/2503.06826_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06826", "page": 1, "id": "2503.06826_pg10_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "|N_\\Gamma(W_i) \\cap U| \\ge d|W_i|/2"}
{"pdf": "arxiv_math/2503.06826_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06826", "page": 1, "id": "2503.06826_pg10_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|N_\\Gamma(D) \\cap U| < d |D| / 2"}
{"pdf": "arxiv_math/2503.06826_pg10.pdf", "url": "https://arxiv.org/pdf/2503.06826", "page": 1, "id": "2503.06826_pg10_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "U_i = N_\\Gamma(W_i) \\cap U"}
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{"pdf": "arxiv_math/2503.04251_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04251", "page": 1, "id": "2503.04251_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "F(f)(s)\\otimes t = s\\otimes G(f)(t)"}
{"pdf": "arxiv_math/2503.04251_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04251", "page": 1, "id": "2503.04251_pg10_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_{A,B,x}(s\\otimes t)=\\llbracket s\\otimes t\\rrbracket"}
{"pdf": "arxiv_math/2503.04251_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04251", "page": 1, "id": "2503.04251_pg10_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\llbracket s\\otimes f\\rrbracket"}
{"pdf": "arxiv_math/2503.04251_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04251", "page": 1, "id": "2503.04251_pg10_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "k[\\Upsilon_{\\mathrm{add}}]\\circ \\Theta_{A,B}"}
{"pdf": "arxiv_math/2503.04251_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04251", "page": 1, "id": "2503.04251_pg10_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "k \\otimes_{\\mathbb{Z}}T_i =0 =\\mathrm{Tor}_1^{\\mathbb{Z}}(k,T_{j})=0\\;, \\text{ for $0<i<e$ and $0<j<e-1$.}"}
{"pdf": "arxiv_math/2503.04251_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04251", "page": 1, "id": "2503.04251_pg10_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Upsilon_{\\mathrm{add}}"}
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{"pdf": "arxiv_math/2503.04251_pg10.pdf", "url": "https://arxiv.org/pdf/2503.04251", "page": 1, "id": "2503.04251_pg10_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "s\\otimes t\\in F(x)\\otimes_K G(x)"}
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{"pdf": "arxiv_math/2503.06293_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06293", "page": 1, "id": "2503.06293_pg24_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\chi_{-}^{-T}-\\chi_{-}^{-1})=\\chi^+_{2,5} = \\chi^-_{2,5}"}
{"pdf": "arxiv_math/2503.06293_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06293", "page": 1, "id": "2503.06293_pg24_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi^+_{2,6} = \\chi^-_{2,6} = \\left( \\begin{array}{ccccccccc} 0 & 6 & 15 & 70 & 105 & 420 & 490 & 1764 & 1764 \\\\ -6 & 0 & 6 & 36 & 70 & 315 & 420 & 1624 & 1764 \\\\ -15 & -6 & 0 & 6 & 15 & 70 & 105 & 420 & 490 \\\\ -70 & -36 & -6 & 0 & 6 & 36 & 70 & 315 & 420 \\\\ -105 & -70 & -15 & -6 & 0 & 6 & 15 & 70 & 105 \\\\ -420 & -315 & -70 & -36 & -6 & 0 & 6 & 36 & 70 \\\\ -490 & -420 & -105 & -70 & -15 & -6 & 0 & 6 & 15 \\\\ -1764 & -1624 & -420 & -315 & -70 & -36 & -6 & 0 & 6 \\\\ -1764 & -1764 & -490 & -420 & -105 & -70 & -15 & -6 & 0 \\\\ \\end{array} \\right)"}
{"pdf": "arxiv_math/2503.06293_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06293", "page": 1, "id": "2503.06293_pg24_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi^+_{2,4} = \\chi^-_{2,4} = \\left( \\begin{array}{cccccc} 0 & 4 & 6 & 10 & 20 & 20 \\\\ -4 & 0 & 4 & 4 & 16 & 20 \\\\ -6 & -4 & 0 & 0 & 4 & 6 \\\\ -10 & -4 & 0 & 0 & 4 & 10 \\\\ -20 & -16 & -4 & -4 & 0 & 4 \\\\ -20 & -20 & -6 & -10 & -4 & 0 \\\\ \\end{array} \\right)"}
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{"pdf": "arxiv_math/2503.06293_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06293", "page": 1, "id": "2503.06293_pg24_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi^{+}_{k,N}=\\chi^{-}_{k,N}"}
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{"pdf": "arxiv_math/2503.05960_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05960", "page": 1, "id": "2503.05960_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "a \\star b = \\mu (a, b)"}
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{"pdf": "arxiv_math/2503.05960_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05960", "page": 1, "id": "2503.05960_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "A = A \\star A = A \\star A \\star A' = A \\star A'"}
{"pdf": "arxiv_math/2503.05960_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05960", "page": 1, "id": "2503.05960_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "b \\in \\mathrm{Hom} (C, D)"}
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{"pdf": "arxiv_math/2503.05960_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05960", "page": 1, "id": "2503.05960_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "g^{- 1} \\star g = g^{- 1} \\star g \\star A' = A'"}
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{"pdf": "arxiv_math/2503.05960_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05960", "page": 1, "id": "2503.05960_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "a \\in \\mathrm{Hom} (A, B)"}
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{"pdf": "arxiv_math/2503.09421_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09421", "page": 1, "id": "2503.09421_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_1^\\infty t^{s-2} \\, dt < \\infty"}
{"pdf": "arxiv_math/2503.09421_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09421", "page": 1, "id": "2503.09421_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_n^{n+1} \\| (A + x)^{-1} \\|^s \\, dx \\leq C(s)"}
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{"pdf": "arxiv_math/2503.09421_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09421", "page": 1, "id": "2503.09421_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "U_\\omega^{(L)} = U_\\omega^{\\Lambda_L} \\oplus U_\\omega^{\\Lambda_L^C}"}
{"pdf": "arxiv_math/2503.09421_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09421", "page": 1, "id": "2503.09421_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "U_\\omega^{\\Lambda_L^C}"}
{"pdf": "arxiv_math/2503.07868_pg30.pdf", "url": "https://arxiv.org/pdf/2503.07868", "page": 1, "id": "2503.07868_pg30_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{R_i\\}_{i\\in\\mathbb{N}}"}
{"pdf": "arxiv_math/2503.08500_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08500", "page": 1, "id": "2503.08500_pg11_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "q(x)=x^{\\intercal} A x"}
{"pdf": "arxiv_math/2503.08500_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08500", "page": 1, "id": "2503.08500_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(K)=\\text{sign}(G_F(K))-(l+2m+n)"}
{"pdf": "arxiv_math/2503.08500_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08500", "page": 1, "id": "2503.08500_pg11_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "C(K) \\le 2 \\cdot g(K)+1"}
{"pdf": "arxiv_math/2503.08500_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08500", "page": 1, "id": "2503.08500_pg11_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "C(7_4)=3=2 \\cdot g(7_4)+1"}
{"pdf": "arxiv_math/2503.08500_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08500", "page": 1, "id": "2503.08500_pg11_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{Z}/2\\mathbb{Z} \\to \\mathbb{Z}/8 \\mathbb{Z}"}
{"pdf": "arxiv_math/2503.08500_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08500", "page": 1, "id": "2503.08500_pg11_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "q:H_1(F;\\mathbb{Z}/2\\mathbb{Z}) \\to \\mathbb{Z}/2\\mathbb{Z}"}
{"pdf": "arxiv_math/2503.08500_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08500", "page": 1, "id": "2503.08500_pg11_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "H_1(F;\\mathbb{Z}/2\\mathbb{Z})"}
{"pdf": "arxiv_math/2503.08500_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08500", "page": 1, "id": "2503.08500_pg11_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "G_F(K)=\\begin{bmatrix} l & m \\\\ m & n \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.05232_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05232", "page": 1, "id": "2503.05232_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\lambda^{irr}, N^{irr}, \\phi^{irr})"}
{"pdf": "arxiv_math/2503.05232_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05232", "page": 1, "id": "2503.05232_pg22_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\frac{b}{2}, +\\infty)"}
{"pdf": "arxiv_math/2503.07247_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07247", "page": 1, "id": "2503.07247_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "t_g = \\inf_{z \\in \\mathbb{H}} d(z, gz)"}
{"pdf": "arxiv_math/2503.07247_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07247", "page": 1, "id": "2503.07247_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\operatorname{Tr} g| < 2"}
{"pdf": "arxiv_math/2503.07247_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07247", "page": 1, "id": "2503.07247_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{Tr} g \\neq 0"}
{"pdf": "arxiv_math/2503.07247_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07247", "page": 1, "id": "2503.07247_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "g \\in \\operatorname{PSL}^{\\pm}_2(\\mathbb{R})"}
{"pdf": "arxiv_math/2503.07247_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07247", "page": 1, "id": "2503.07247_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{2}|\\!\\operatorname{Tr} gh| = \\left|\\sinh\\left(\\frac{t_g}{2}\\right)\\sinh\\left(\\frac{t_h}{2}\\right) + \\cosh\\left(\\frac{t_g}{2}\\right)\\cosh\\left(\\frac{t_h}{2}\\right)\\cos\\theta_P\\right|"}
{"pdf": "arxiv_math/2503.07247_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07247", "page": 1, "id": "2503.07247_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{PSL}^{\\pm}_2(\\mathbb{R}) = \\left\\{ \\pm \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\middle| a, b, c, d \\in \\mathbb{R}, ad - bc = \\pm 1 \\right\\}"}
{"pdf": "arxiv_math/2503.07247_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07247", "page": 1, "id": "2503.07247_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{Tr} g = 0"}
{"pdf": "arxiv_math/2503.07247_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07247", "page": 1, "id": "2503.07247_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\operatorname{Tr} g| = 2"}
{"pdf": "arxiv_math/2503.07247_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07247", "page": 1, "id": "2503.07247_pg2_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\operatorname{Tr} g| > 2"}
{"pdf": "arxiv_math/2503.08986_pg11.pdf", "url": "https://arxiv.org/pdf/2503.08986", "page": 1, "id": "2503.08986_pg11_math_000", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\overline{\\mathcal{C}}_{\\mathrm{sum},u}=\\overline{\\mathcal{C}}_{\\mathrm{c},u}+\\overline{\\mathcal{C}}_{\\mathrm{p},u}"}
{"pdf": "arxiv_math/2503.06722_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06722", "page": 1, "id": "2503.06722_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "Q \\colon \\mathbf{2} \\to \\mathbf{Fin}"}
{"pdf": "arxiv_math/2503.06722_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06722", "page": 1, "id": "2503.06722_pg17_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{Fun}(\\mathbf{2},\\mathbf{Fin})"}
{"pdf": "arxiv_math/2503.05476_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05476", "page": 1, "id": "2503.05476_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi_{n}(t)\\le\\psi(t)"}
{"pdf": "arxiv_math/2503.05476_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05476", "page": 1, "id": "2503.05476_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi_{n+1}(t) \\leq \\psi(t)"}
{"pdf": "arxiv_math/2503.05476_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05476", "page": 1, "id": "2503.05476_pg14_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi_n(t) \\leq \\psi(t)"}
{"pdf": "arxiv_math/2503.05476_pg14.pdf", "url": "https://arxiv.org/pdf/2503.05476", "page": 1, "id": "2503.05476_pg14_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi(t) = 1 \\geq \\psi_{n+1}(t)"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_000", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathcal{C}_{k, n, w}"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i \\in \\{i+1, i+2, \\dots, n\\}"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^{i_0-1}(k-1)k^{k-1}"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "I(k, n, w) \\leq \\frac{k^n(k-1)}{4} \\sum_{i=1}^{i_0-1}(k^{n-i}-k^{n-i-(n-i)}) = \\frac{k(k-1)}{4} \\sum_{i=1}^{i_0-1}k^{n-1}(k^{n-i}-1) = \\binom{k}{2} \\sum_{i=1}^{i_0-1}k^{i-1}\\binom{k^{n-i}}{2}"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "I(k, n, w) = \\binom{k}{2} \\sum_{i=1}^{i_0-1}k^{i-1}\\binom{k^{(c_w)_{i}} }{2}k^{2n-2i-2(c_w)_i} = \\frac{k-1}{4}\\sum_{i=1}^{i_0-1}(k^{2n-i} - k^{2n-i-(c_w)_i})"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^{i_0-1} (k-1)k^{i-1}"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "I(k, n, w) = \\frac{k^n(k-1)}{4}\\sum_{i=1}^n (k^{n-i} - k^{n-i-(c_w)_i}) = \\frac{k^n(k-1)}{4}\\sum_{i=1}^{i_0-1}(k^{n-i} - k^{n-i-w_i+1})"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\binom{k}{2}\\sum_{i=1}^{i_0-1} k^{i-1}\\binom{k^{n-i}}{2}"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "i \\in \\{i_0, i_0+1, \\dots, n \\}"}
{"pdf": "arxiv_math/2503.09577_pg20.pdf", "url": "https://arxiv.org/pdf/2503.09577", "page": 1, "id": "2503.09577_pg20_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\binom{k}{2} \\sum_{i=1}^{i_0-1}k^{i-1}\\binom{k^{n-i}}{2}"}
{"pdf": "arxiv_math/2503.05642_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05642", "page": 1, "id": "2503.05642_pg4_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "D_s(G):=|\\{(u,v)~|~u,v\\in V,~d_{u,v}=s\\}|"}
{"pdf": "arxiv_math/2503.05642_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05642", "page": 1, "id": "2503.05642_pg4_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\begin{aligned} k(e_{u_1,v_1},e_{u_2,v_2})=k_v(l_{u_1},l_{u_2})\\cdot k_e(d_{u_1,v_1},d_{u_2,v_2})\\cdot k_v(l_{v_1},l_{v_2}) \\end{aligned}"}
{"pdf": "arxiv_math/2503.05642_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05642", "page": 1, "id": "2503.05642_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} k_{\\mathit{SP}}(G^1, G^2)=\\sum\\limits_{u_1,v_1\\in V^1,u_2,v_2\\in V^2}k(e_{u_1,v_1},e_{u_2,v_2}) \\end{aligned}"}
{"pdf": "arxiv_math/2503.04523_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04523", "page": 1, "id": "2503.04523_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}^{n_1\\times n_2\\times\\cdots\\times n_d}"}
{"pdf": "arxiv_math/2503.08088_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08088", "page": 1, "id": "2503.08088_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(D\\setminus \\{u\\})\\cup \\{v\\}"}
{"pdf": "arxiv_math/2503.08088_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08088", "page": 1, "id": "2503.08088_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{e\\in E(G) \\, \\colon e"}
{"pdf": "arxiv_math/2503.08088_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08088", "page": 1, "id": "2503.08088_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "N_G(v), N_G[v], N_G(X)"}
{"pdf": "arxiv_math/2503.08088_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08088", "page": 1, "id": "2503.08088_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{v\\in V(G)\\setminus X \\, \\colon v \\text{ is adjacent to a vertex in } X\\}"}
{"pdf": "arxiv_math/2503.08088_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08088", "page": 1, "id": "2503.08088_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "N(v)\\cap D\\subseteq A_D"}
{"pdf": "arxiv_math/2503.08088_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08088", "page": 1, "id": "2503.08088_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{v \\in V(G) \\setminus D \\, \\colon N(v) \\cap D = \\{u\\}\\}"}
{"pdf": "arxiv_math/2503.05886_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05886", "page": 1, "id": "2503.05886_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i,j}\\tilde p_{ij}\\log\\frac{\\tilde p_{ij}}{p_{ij}}"}
{"pdf": "arxiv_math/2503.05886_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05886", "page": 1, "id": "2503.05886_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal L= \\sum_{i,j}\\left(\\tilde p_{ij}\\log\\frac{\\tilde p_{ij}}{p_{ij}}+\\lambda_i(\\tilde p_{ij}-\\tilde \\alpha_i)+\\gamma_j(\\tilde p_{ij}-\\tilde \\beta_j)\\right)"}
{"pdf": "arxiv_math/2503.05886_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05886", "page": 1, "id": "2503.05886_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde\\rho_1\\neq \\sum_{i,k,j}L_{ikj}(1,0)\\tilde\\rho_0L_{ikj}(1,0)^\\dagger"}
{"pdf": "arxiv_math/2503.04086_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04086", "page": 1, "id": "2503.04086_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in (\\Z/n)^{\\times}"}
{"pdf": "arxiv_math/2503.04086_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04086", "page": 1, "id": "2503.04086_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gcd(a,n) = \\gcd(b,n)"}
{"pdf": "arxiv_math/2503.03909_pg14.pdf", "url": "https://arxiv.org/pdf/2503.03909", "page": 1, "id": "2503.03909_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "u_{xx} + u_{yy} + \\lambda e^{u} = 0, \\ \\ (x,y) \\in [0,1] \\times [0,1]"}
{"pdf": "arxiv_math/2503.03909_pg14.pdf", "url": "https://arxiv.org/pdf/2503.03909", "page": 1, "id": "2503.03909_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "X^{k+1}(i,j) = G(i,j;X^{k},\\alpha) \\equiv X^k(i,j) + \\alpha M(G_{\\rm B}(i,j;X^{k}))"}
{"pdf": "arxiv_math/2503.03909_pg14.pdf", "url": "https://arxiv.org/pdf/2503.03909", "page": 1, "id": "2503.03909_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "X(i,j) \\approx u(x_i,y_j)"}
{"pdf": "arxiv_math/2503.03909_pg14.pdf", "url": "https://arxiv.org/pdf/2503.03909", "page": 1, "id": "2503.03909_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\rm TOL}=10^{-6}, \\hat{m}=5, \\theta=0.9, \\alpha = 0.125h_x^2"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\star\\frac{x}{y}=t\\in T\\setminus\\{0\\}"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|E_i|=2^{2m-t}+2^m-2^{m-t}"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "y_1\\star x + y_2\\star x=z"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(1-2^{-t})(2^{2m}-2^m)"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\star\\frac{x}{y} \\in T"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "|E_1|=|f_1^{-1}(f_1(0,0))| = 2\\cdot 2^m-1+(2^{m-t}-1)(2^m-1)"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left(1-\\frac{1}{2^m}\\right)\\big(2^n-2^{n/2}\\big)"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{y_1}(x)\\neq \\alpha_{y_2}(x)"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma'(y_1)\\neq \\sigma'(y_2)"}
{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "W_f(a,b)=\\begin{cases} \\pm 2^{n/2} & \\text{if $b\\neq 0$,} \\\\ 0 & \\text{if $a\\neq 0$, $b=0$,} \\\\ 2^n & \\text{if $a=0$, $b=0$.} \\end{cases}"}
{"pdf": "arxiv_math/2503.05503_pg13.pdf", "url": "https://arxiv.org/pdf/2503.05503", "page": 1, "id": "2503.05503_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\cdot, \\cdot \\rangle"}
{"pdf": "arxiv_math/2503.05503_pg13.pdf", "url": "https://arxiv.org/pdf/2503.05503", "page": 1, "id": "2503.05503_pg13_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{l,m}= \\sqrt{\\frac{(2 l+1)}{4 \\pi} \\frac{(l-m)!}{(l+m)!}}"}
{"pdf": "arxiv_math/2503.05503_pg13.pdf", "url": "https://arxiv.org/pdf/2503.05503", "page": 1, "id": "2503.05503_pg13_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ Y_l^m : l\\in \\N_0, \\, -l \\le m \\le l \\}"}
{"pdf": "arxiv_math/2503.07286_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07286", "page": 1, "id": "2503.07286_pg6_math_001", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\lambda \\in [\\underline{H},\\overline{H}]"}
{"pdf": "arxiv_math/2503.07286_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07286", "page": 1, "id": "2503.07286_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\mathbb E}X^2(t)= \\sum_{j=0}^{+\\infty} \\sum_{k=0}^{2^{j}-1} \\left|\\int_{0}^{1} (t-s)_{+}^{H_{j}(k/{2^j})-{1}/{2}} h_{j,k}(s)ds\\right|^2 < +\\infty"}
{"pdf": "arxiv_math/2503.07286_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07286", "page": 1, "id": "2503.07286_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\lambda,x) \\in [\\underline{H},\\overline{H}] \\times \\mathbb{R}"}
{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial H^h/\\partial l_i"}
{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "l_{\\sigma}=(l_{12}(l_{23}, l_{24}, l_{34}), l_{13}(l_{23}, l_{24}, l_{34}), l_{14}(l_{23}, l_{24}, l_{34}), l_{23}, l_{24}, l_{34})"}
{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H^h(l^h)=H(l^*,l^h)=H(l)= cov(l) - 2\\pi\\sum_{i\\in E} l_i"}
{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\partial H^h}{\\partial l_i} = - K_i, \\quad e_i\\in E^h"}
{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(l_{23}, l_{24}, l_{34})=l^h_{\\sigma} \\in \\R^3"}
{"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_k = \\|\\boldsymbol{x}_k-\\boldsymbol{x}_*\\|,\\quad f(k) = -\\ln \\xi_k"}
{"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{k\\to\\infty}\\frac{\\xi_{k+1}}{\\xi_k^q}=Q_q, \\quad q>1,\\quad 0<Q_q<\\infty"}
{"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{j=k}^\\infty\\frac{|d(j)|}{q^{\\,j+1}} \\le M\\sum_{j=k}^\\infty\\frac{1}{q^{\\,j+1}}"}
{"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "f(k)=q^{\\,k-1}f(1)+\\sum_{j=1}^{k-1}q^{\\,k-1-j}d(j)"}
{"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{k\\to\\infty} d(k) = -\\ln Q_q \\coloneqq d"}
{"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{f(k)}{q^k}=\\frac{f(1)}{q}+\\sum_{j=1}^{k-1}\\frac{d(j)}{q^{\\,j+1}}"}
{"pdf": "arxiv_math/2503.08504_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08504", "page": 1, "id": "2503.08504_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac1p=\\frac {d-1}2(\\frac12-\\frac1q)"}
{"pdf": "arxiv_math/2503.08504_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08504", "page": 1, "id": "2503.08504_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{2(d+1)}{d-1}< q< \\frac{2d}{d-2}"}
{"pdf": "arxiv_math/2503.08504_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08504", "page": 1, "id": "2503.08504_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "2\\le q<\\frac{2(d+1)}{d-1}"}
{"pdf": "arxiv_math/2503.08504_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08504", "page": 1, "id": "2503.08504_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{2(d+1)}{d-1}< q<\\infty"}
{"pdf": "arxiv_math/2503.08504_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08504", "page": 1, "id": "2503.08504_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac1p=\\frac d2(\\frac12-\\frac1q)"}
{"pdf": "arxiv_math/2503.09589_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09589", "page": 1, "id": "2503.09589_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta < \\min(\\alpha;2-\\alpha)"}
{"pdf": "arxiv_math/2503.09589_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09589", "page": 1, "id": "2503.09589_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma:=\\frac{\\alpha-\\beta}{1-\\beta}"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta^{(n)} \\coloneq \\{b_i^{(n)}\\}_{i=1}^\\infty \\in S"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\beta^{(n_n)}\\}_{n=1}^\\infty"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^\\infty a_i <\\infty"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "S \\coloneq \\left\\{ \\{b_i\\}_{i=1}^\\infty \\left\\vert 0\\leq b_i\\leq a_i, \\sum_{i=1}^\\infty b_i = 1\\right.\\right\\}"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\leq b^{(n_n)}_i\\leq a_i"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{b_1^{(n)}\\}_{n=1}^\\infty"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_n(W((1-\\theta)p + \\theta q))>0"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{b^{(1_n)}_1\\}_{n=1}^\\infty"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\leq b^{(1_n)}_2 \\leq a_2"}
{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n\\to\\infty} \\sum_{i=1}^\\infty |b^{(n_n)}_i - c_i| = 0"}
{"pdf": "arxiv_math/2503.08206_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08206", "page": 1, "id": "2503.08206_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f(x) = \\begin{dcases} 1/q & \\text{ if } x=p/q\\text{ with } (p,q)=1;\\\\ 0 & \\text{ if } x \\text{ is irrational;} \\end{dcases}"}
{"pdf": "arxiv_math/2503.08206_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08206", "page": 1, "id": "2503.08206_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta^+(x) := B^+(x)-2B_0^+(x)\\quad \\text{ and }\\quad \\Delta^-(x) := W^-(x)-2B_0^-(x)"}
{"pdf": "arxiv_math/2503.05844_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05844", "page": 1, "id": "2503.05844_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "Q=\\begin{bmatrix} 10& 0\\\\ 0& 50\\\\ \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.05844_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05844", "page": 1, "id": "2503.05844_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0= [0.2; 0; 0; 0.1; 0]^T"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in {\\Sigma_\\vartheta}"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta: {\\Sigma_\\vartheta} \\times \\mathbb{T} \\to {\\Sigma_\\vartheta} \\times \\mathbb{T}"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(x,y) \\mapsto (T_\\omega (x) , T_\\omega(y))"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta^{(2)} : {\\Sigma_\\vartheta} \\times \\mathbb{T}^2 \\to {\\Sigma_\\vartheta} \\times \\mathbb{T}^2"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "T^n_\\omega (x) \\coloneqq T_{\\omega_{n-1}} \\circ \\cdots \\circ T_{\\omega_0} (x)"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in \\Omega_\\vartheta"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(x,y) \\mapsto T^{(2)}_\\omega (x,y)"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in \\Sigma_\\vartheta"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\Sigma_\\vartheta} \\mu^{(2)} \\left( \\left( T^{(2)}_\\omega\\right)^{-1} (A) \\right) \\, d\\mathbb{P} (\\omega) = \\mu^{(2)}(A)"}
{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma \\omega \\coloneqq (\\omega_{i+1})_{i\\in\\mathbb{N}}"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{3}^{\\circ}(3,1,2)"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{3}^{*}(3,1,2)"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(k_{1},k_{2})\\in \\mathcal{A}_{2}^{*}(3,1,2)"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(k_{1},\\ldots,k_{m-1})\\in \\mathcal{A}_{m-1}^{*}(3,1,2)"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{K}_{m}(3,2^{m-3},1,k) = 2(k-m)+3,\\quad \\mathbb{K}_{m-1}(2^{m-3},1,k) = k-m+2"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{r}^{*}(3,1,2)"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{r}^{\\circ}(3,1,2)"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_008", "type": "math", "max_diffs": 0, "checked": "verified", "math": "3m(k-1)-2k+5\\not\\equiv 2\\pmod{3}"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "3\\leqslant r\\leqslant m-1"}
{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "3(k-m)+5\\not\\equiv 2\\pmod{3}"}
{"pdf": "arxiv_math/2503.07696_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07696", "page": 1, "id": "2503.07696_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_1<\\gamma_2<\\ldots"}
{"pdf": "arxiv_math/2503.07696_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07696", "page": 1, "id": "2503.07696_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\arg(\\eta(s))\\ne\\pm\\pi/2"}
{"pdf": "arxiv_math/2503.07696_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07696", "page": 1, "id": "2503.07696_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda^\\prime\\ne\\rho^\\prime"}
{"pdf": "arxiv_math/2503.07696_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07696", "page": 1, "id": "2503.07696_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta^{\\prime\\prime}(s)"}
{"pdf": "arxiv_math/2503.05513_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05513", "page": 1, "id": "2503.05513_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "u\\colon X \\to \\R \\cup \\{-\\infty\\}"}
{"pdf": "arxiv_math/2503.05513_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05513", "page": 1, "id": "2503.05513_pg15_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "u|_{\\partial U} \\leq h|_{\\partial U}"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c\\wedge x_k\\not\\leqslant b"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\neq(a_1 \\wedge a_2) \\wedge n_i = (a_1 \\wedge a_2) \\wedge (n_i^1 \\wedge n_i^2) =(a_1 \\wedge n_i^1) \\wedge (a_2 \\wedge n_i^2)"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "c\\wedge x_1\\wedge x_2\\not\\leqslant b"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "((b\\vee c)\\wedge x_k)>b"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a_1 \\wedge n_i^1 \\neq 0"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "c\\wedge a\\wedge x_1\\wedge x_2\\neq 0"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "a_1 \\wedge \\left( \\bigwedge_{i=1}^2 n_i^1 \\right) \\neq 0\\quad \\text{and}\\quad a_2 \\wedge \\left( \\bigwedge_{i=1}^2 n_i^2 \\right) \\neq 0"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{1,3\\}^\\uparrow={\\{1\\}\\vee\\{3\\}}^\\uparrow"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "a\\wedge x_1\\wedge x_2\\neq 0"}
{"pdf": "arxiv_math/2503.06739_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06739", "page": 1, "id": "2503.06739_pg8_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "(a_1 \\wedge a_2) \\wedge \\left( \\bigwedge_{i=1}^2 n_i \\right) = (a_1 \\wedge a_2) \\wedge \\left( \\bigwedge_{i=1}^2 (n_i^1 \\wedge n_i^2) \\right) \\neq 0"}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{ \\left( 0_{X},0_{Y},-1\\right) \\right\\}"}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "C\\cap \\mathbb{A}% ^{-1}\\left( D\\right)"}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "C\\cap \\mathbb{A}^{-1}\\left( D\\right) "}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{U}\\cap V=U\\cap V"}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "% \\left\\{ \\left( 0_{X^{\\prime }},0\\right) \\right\\}"}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "f:X\\longrightarrow \\overline{\\mathbb{R}}"}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mathcal{A})\\Longleftrightarrow (\\mathcal{B}% )"}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{ x\\in X:f\\left( x\\right) \\geq 0\\right\\} "}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\notin \\overline{U}"}
{"pdf": "arxiv_math/2503.04226_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04226", "page": 1, "id": "2503.04226_pg3_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mathcal{A}% )\\Longleftrightarrow (\\mathcal{B})"}
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{"pdf": "arxiv_math/2503.04897_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04897", "page": 1, "id": "2503.04897_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho(m)=\\sum_{i} {m_{(0)}}_i\\otimes {m_{(1)}}_i\\in M\\otimes C"}
{"pdf": "arxiv_math/2503.04897_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04897", "page": 1, "id": "2503.04897_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(H, \\Delta, \\varepsilon)"}
{"pdf": "arxiv_math/2503.04897_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04897", "page": 1, "id": "2503.04897_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta\\colon H\\to H\\otimes H"}
{"pdf": "arxiv_math/2503.04897_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04897", "page": 1, "id": "2503.04897_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{(m)}m_{(0)}\\otimes m_{(1)}"}
{"pdf": "arxiv_math/2503.04897_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04897", "page": 1, "id": "2503.04897_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\colon M\\rightarrow M'"}
{"pdf": "arxiv_math/2503.04897_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04897", "page": 1, "id": "2503.04897_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu\\colon H\\otimes H\\to H"}
{"pdf": "arxiv_math/2503.04897_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04897", "page": 1, "id": "2503.04897_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(C,\\Delta, \\varepsilon)"}
{"pdf": "arxiv_math/2503.04897_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04897", "page": 1, "id": "2503.04897_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\colon (M,\\rho)\\rightarrow (M', \\rho')"}
{"pdf": "arxiv_math/2503.04897_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04897", "page": 1, "id": "2503.04897_pg4_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau(c\\otimes c')=c'\\otimes c"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "X\\subseteq V(H)\\setminus V (P)"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "S'_{A}=\\{w_1,w_2,\\dots,w_{k-\\ell}\\}"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "b:=(4k-2\\ell-1)(k-\\ell)+r"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "i_{\\mathcal{P}}(T_i)=(m-1,k-\\ell-m+1)"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "[V_{1}^{A},V_{2}^{A},\\dots,V_{k}^{A}]"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "t_1:=|S\\cap V_1|\\le a"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "R_1\\subseteq V(H)\\setminus V(P')"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|X\\cup R_1|\\le4(k-\\ell)p+p \\le(k-\\ell)\\beta^{2k-2\\ell+4}n"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "i_{\\mathcal{P}}(S_i)=(m,k-\\ell-m)"}
{"pdf": "arxiv_math/2503.05275_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05275", "page": 1, "id": "2503.05275_pg9_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "A\\setminus\\{w_1,w_2,\\dots,w_{k-\\ell}\\}"}
{"pdf": "arxiv_math/2503.07814_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07814", "page": 1, "id": "2503.07814_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bm{\\lambda}\\in\\Lambda"}
{"pdf": "arxiv_math/2503.04649_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04649", "page": 1, "id": "2503.04649_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\theta, \\phi) \\in \\left[0, 2 \\pi\\right) \\times [0, \\pi]"}
{"pdf": "arxiv_math/2503.04649_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04649", "page": 1, "id": "2503.04649_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_\\phi, \\partial_\\theta"}
{"pdf": "arxiv_math/2503.04649_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04649", "page": 1, "id": "2503.04649_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "L = 3, 6, 8, 10, 12, 15, 18, 22"}
{"pdf": "arxiv_math/2503.04649_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04649", "page": 1, "id": "2503.04649_pg14_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi \\in [\\tfrac{\\pi}{5}, \\tfrac{4\\pi}{5}]"}
{"pdf": "arxiv_math/2503.04649_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04649", "page": 1, "id": "2503.04649_pg14_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "a(u, v) = a, b(u, v) = b"}
{"pdf": "arxiv_math/2503.04649_pg14.pdf", "url": "https://arxiv.org/pdf/2503.04649", "page": 1, "id": "2503.04649_pg14_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "0.7 \\leq r(\\theta, \\phi) \\leq 1.3"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "E\\in \\big| \\omega_X^{\\otimes k} \\otimes \\alpha \\big|"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{ D}^b (\\mathrm{ Coh}(M))"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E'=R_k(E) \\in \\big| \\omega_Y^{\\otimes k} \\otimes \\beta \\big|"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{ D}^b(\\mathrm{ Unip}(X)) \\simeq \\mathrm{ D}^b ( \\mathrm{ Coh}_{\\{ \\hat{0} \\}}(Y) )"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "N:= \\omega_X^{\\otimes k}|_Z"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{ D}_{ \\mathrm{ Supp}(E) }^b(\\mathrm{ Coh}(X))\\simeq \\mathrm{ D}_{\\mathrm{ Supp}(E')}^b(\\mathrm{ Coh}(Y))"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "H^0(X,\\omega_X^{\\otimes k_0}\\otimes \\alpha) \\neq 0"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "R \\colon \\mathrm{ Aut}^0(X) \\times \\mathrm{ Pic}^0(X) \\to \\mathrm{ Aut}^0(Y) \\times \\mathrm{ Pic}^0(Y)"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{ D}_Z^b(\\mathrm{ Coh}(X))"}
{"pdf": "arxiv_math/2503.08419_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08419", "page": 1, "id": "2503.08419_pg6_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "S_Z(P) \\simeq P[ \\dim X ]"}
{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_n^{\\frac{m}{m+1}}"}
{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi,|\\nabla \\psi|^2\\in L^1(\\R^n,\\gamma_n^\\frac{m}{m+1})"}
{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "D^m\\psi,|\\nabla D^m\\psi|^2\\in L^1(\\R^{nm},\\gamma_{n,m})"}
{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla D^m\\psi(x_1,\\dots,x_m)=\\left\\{\\nabla \\psi(x_i) - \\nabla \\psi\\left(-\\sum_{i=1}^m x_i\\right) \\right\\}_{i=1}^m"}
{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "V_\\epsilon^\\star = \\frac{|x|^2}{2} - \\epsilon \\psi(x) + \\frac{\\epsilon^2}{2}|\\nabla \\psi|^2 + o(\\epsilon^3)"}
{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\nabla D^m\\psi(x_1,\\dots,x_m)|^2=\\sum_{i=1}^m\\left|\\nabla \\psi(x_i) - \\nabla \\psi\\left(-\\sum_{i=1}^m x_i\\right)\\right|^2"}
{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "V(x)=V_\\epsilon(x)=\\frac{|x|^2}{2}+\\epsilon \\psi(x)"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_i, g_i, q_i, \\omega_i, \\pi_{ij}"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta_i\\in C([\\tau,\\infty);\\mathbb R_{>0})"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{\\omega_i,T}:=\\max\\limits_{t\\in [t_0,T]}\\omega_i(t)\\|f(t)\\|"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_i(s)=\\|\\Omega_i(p,s)\\|"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_i, \\beta_i, \\gamma_i, \\delta_i"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0\\in B_{\\eta}\\cap K\\implies \\lim_{t\\to\\infty}\\|x(t;t_0,x_0)\\|=0"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\in C([t_0,T];L(\\mathcal H_i))"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb R_+K\\subset K"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "O\\in L(\\mathcal H_j;\\mathcal H_i)"}
{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0\\in B_{\\delta}\\cap K\\implies \\|x(t;t_0,x_0)\\|<\\varepsilon"}
{"pdf": "arxiv_math/2503.05270_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05270", "page": 1, "id": "2503.05270_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "b = \\beta_+^{-{2}/{3}}"}
{"pdf": "arxiv_math/2503.05270_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05270", "page": 1, "id": "2503.05270_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left({n^{1/3}}\\middle/{\\log^{{1}/{3}}}\\right)\\exp\\left\\{-g_n\\right\\}\\to\\infty"}
{"pdf": "arxiv_math/2503.05270_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05270", "page": 1, "id": "2503.05270_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{n}{\\log n}\\exp\\left\\{-g_n\\right\\}\\to\\infty"}
{"pdf": "arxiv_math/2503.05270_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05270", "page": 1, "id": "2503.05270_pg21_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{n^{1/3}}{\\log^{1/3}n}\\exp\\left\\{- \\frac{\\beta_+^2b^3\\log n }{6}\\right\\} = \\frac{n^{1/3}}{\\log^{1/3}n} \\exp\\left\\{-\\frac{1}{6}\\log n\\right\\} = \\frac{n^{1/6}}{\\log^{1/3}n}\\to \\infty \\text{, for } n\\to \\infty"}
{"pdf": "arxiv_math/2503.05270_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05270", "page": 1, "id": "2503.05270_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\exp\\left\\{-\\frac{\\beta_+^2b^2\\log^{2/3}n}{4n^{2/3}}-\\frac{\\beta_+^2b\\log^{{1}/{3}}n}{12n^{4/3}}\\right\\}\\to 1 \\text{, for } n\\to\\infty"}
{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "meg(G)=\\vert Man(G)\\vert"}
{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "d(x'_1,y)=d(x'_1,v_{i_l})"}
{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "d_P(x'_1,v'_1)= d(v'_1,v_{i_l})+d(x'_1,v_{i_l})"}
{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "d(x'_1,y)\\geq d(x'_1,v_{i_l})"}
{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "d(v'_1,y)\\geq d(v'_1,v_{i_l})"}
{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "Man(G_k)\\setminus \\{v_{i_l}\\}"}
{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "d(v'_1,y)=d(v'_1,v_{i_l})"}
{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "d_P(x'_1,v'_1)=d(x'_1,y)+d(y,v'_1)\\geq d(v'_1,v_{i_l})+d(x'_1,v_{i_l})"}
{"pdf": "arxiv_math/2503.04297_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04297", "page": 1, "id": "2503.04297_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda' = \\lambda^{( k-1)} + \\lambda^{( k)} + \\cdots"}
{"pdf": "arxiv_math/2503.04297_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04297", "page": 1, "id": "2503.04297_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{exp} : ((\\mathfrak{g}_A)_0, \\mathrm{BCH}, 0) \\cong (\\Gamma_A, \\circledcirc, 1) : \\mathrm{log}\\ "}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "1 \\le \\ell \\le \\lfloor \\frac{p-k-1}{2k-1} \\rfloor"}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R \\subseteq A \\times A"}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R \\subseteq A \\times B"}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| + |B| > (1+\\delta)p - O(1)"}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| = 6 \\lfloor \\frac{p}{11} \\rfloor - 3"}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| = p - (k-1) \\lfloor \\frac{p-k-1}{2k-1} \\rfloor - k + 1"}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| + |B| \\ge p-k+\\lfloor \\frac{p-k-1}{2k-1} \\rfloor + 4 > \\frac{2k}{2k-1} p - k + 2"}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| + |B| \\ge (1+\\delta)p"}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "|B| = k \\lfloor \\frac{p-k-1}{2k-1} \\rfloor + 2"}
{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| = p - (k-1) \\ell - k + 1"}
{"pdf": "arxiv_math/2503.09088_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09088", "page": 1, "id": "2503.09088_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{D}'(\\mathbb{R})"}
{"pdf": "arxiv_math/2503.09088_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09088", "page": 1, "id": "2503.09088_pg2_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "s<\\max\\{0,\\frac{1}{2}-\\frac{\\alpha}{k}\\}"}
{"pdf": "arxiv_math/2503.09088_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09088", "page": 1, "id": "2503.09088_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{Lu}(\\xi)=|\\xi|^{\\alpha + 1}\\widehat{\\omega}(\\xi)"}
{"pdf": "arxiv_math/2503.09088_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09088", "page": 1, "id": "2503.09088_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "C([0, T];H^s(\\mathbb{R}))"}
{"pdf": "arxiv_math/2503.09088_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09088", "page": 1, "id": "2503.09088_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha=\\frac{A}{h_{0}}\\ll 1, \\qquad \\beta=\\frac{h_{0}}{l^{2}}\\ll 1, \\qquad S=\\frac{\\alpha}{\\beta}=\\frac{Al^{2}}{h_{0}^{3}}\\approx 1"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "u, v \\in C^{1+\\beta, \\frac{1+\\beta}{2}}(\\Omega_{T_m-\\varepsilon})"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "|h'(t)| + |g'(t)| \\leq C_2"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u(t, \\cdot)\\|_{C^{2+\\beta}([g(t), h(t)])} \\leq Q^*"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "f(t,x,u)\\leq \\bar f(u)<0"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "h, g \\in C^{1+\\frac{\\beta}{2}}((0, T_m)), \\quad u \\in C^{1+\\frac{\\beta}{2}, 2+\\beta}(\\Omega_{T_m})"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u\\|_{C^{2+\\beta,1+\\frac{\\beta}{2} }(\\Omega_{T_m - \\varepsilon} \\setminus \\Omega_{T_0})} \\leq Q^*"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{u}(t, x) := \\max\\{q_0(ct - x - L), q_0(ct + x - L)\\}"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{(t,x): t>0, x\\in [g(t), h(t)]\\}"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_010", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\underline{u}(0, x) \\leq u_0(x) \\quad \\text{for } x \\in [-h_0, h_0]"}
{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega_{T_m} := \\{(t, x) : t \\in (0, T_m), x \\in [g(t), h(t)]\\}"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textsf{Sfas}(D_{\\sigma})"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_B(T_{\\sigma}, P_{\\sigma})"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textsf{Sfas}(D_{\\sigma},S)"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textsf{Sfas}(T_{\\sigma}) = \\textsf{Sfas}(T_{\\sigma} [A_I(T_{\\sigma}, P_{\\sigma})])+\\textsf{Sfas}(T_{\\sigma} [A_B(T_{\\sigma}, P_{\\sigma})])"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "S \\subseteq V_{\\sigma}"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{\\sigma} = (V_1, \\ldots, V_{\\ell})"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "A_I(T_{\\sigma}, P_{\\sigma})"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textsf{Sfas}(T_{\\sigma}) \\leq \\textsf{Sfas}\\big(T_{\\sigma}[A_I(T_{\\sigma}, P_{\\sigma})]\\big) + \\textsf{Sfas}\\big(T_{\\sigma}[A_B(T_{\\sigma}, P_{\\sigma})]\\big)"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{\\sigma}[A_I(T_{\\sigma}, P_{\\sigma})]"}
{"pdf": "arxiv_math/2503.07208_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07208", "page": 1, "id": "2503.07208_pg11_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textsf{Sfas}(T_{\\sigma}) \\geq \\textsf{Sfas}(T_{\\sigma}[A_1]) + \\textsf{Sfas}(T_{\\sigma}[A_2])"}
{"pdf": "arxiv_math/2503.06384_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06384", "page": 1, "id": "2503.06384_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\to\\xi(x,t)=x/\\rho(t)"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{bmatrix} 1 & 0 \\\\ 0 & 0 \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A(t)=t^\\gamma c B+o(t^\\gamma)"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "B_\\infty=\\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{PSL}_2\\mathbb{C}=\\mathbb{C}P^3"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde A(t)=t^\\alpha B+o(t^\\alpha) \\in V(\\mathbb{K})\\cap PSL_2(\\mathbb{K})"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\exp(-\\beta\\gamma^{-1} \\sigma)t^{\\alpha\\beta\\gamma^{-1}}"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{K}P^1\\ni[z:w]\\longmapsto Z(t)=\\begin{bmatrix} tw & z \\\\ 0 & t^{-1}w \\end{bmatrix}\\in\\overline{\\operatorname{PSL_2}\\mathbb{K}}=\\mathbb{K}P^3"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "[z:1]\\in\\mathbb{K}P^1"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{bmatrix} 0 & ct^\\gamma \\\\ 0 & 0 \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.09133_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09133", "page": 1, "id": "2503.09133_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{bmatrix} t & ct\\\\ 0 & 0 \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "S^{\\lambda^{+3}}_\\lambda"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle f_\\alpha : \\alpha < \\omega_{n+1} \\rangle"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\delta^i_\\eta : \\eta < \\lambda^{+k}\\}"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma = \\sup\\{\\beta_\\eta : \\eta < \\lambda^{+k}\\}"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "h_\\eta <^* f_{\\beta_\\eta}"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{cf}(h(i)) > \\lambda"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{cf}(h(i)) = \\lambda^{+k}"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma < \\lambda^{+3}"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle f_{\\alpha} : \\alpha < \\omega_{1} \\rangle"}
{"pdf": "arxiv_math/2503.06758_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06758", "page": 1, "id": "2503.06758_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_\\eta < \\lambda^{+3}"}
{"pdf": "arxiv_math/2503.04255_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04255", "page": 1, "id": "2503.04255_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "L^2(\\mathbb{R}^d, \\mathbb{R})"}
{"pdf": "arxiv_math/2503.04255_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04255", "page": 1, "id": "2503.04255_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "L^2(\\mathbb{R}^d, \\mathbb{R}^m)"}
{"pdf": "arxiv_math/2503.04255_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04255", "page": 1, "id": "2503.04255_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "L^2(\\mathbb{R}^2, \\mathbb{R}^2)"}
{"pdf": "arxiv_math/2503.04255_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04255", "page": 1, "id": "2503.04255_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "L^2(\\mathbb{R}, \\mathbb{R}^2)"}
{"pdf": "arxiv_math/2503.04255_pg1.pdf", "url": "https://arxiv.org/pdf/2503.04255", "page": 1, "id": "2503.04255_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "L^2(\\mathbb{R}^2, \\mathbb{R})"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{s\\downarrow\\cdot,s\\in\\mathbb{Q}_{+}}\\mathbb{P}-\\lim_{n\\to\\infty}\\int_{0}^{s}Yd\\widetilde{X}^{n}=\\int_{0}^{\\cdot}Yd\\widetilde{X}"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{t\\in[0,\\infty):f\\textrm{ is discontinuous at }t\\}=\\{f\\neq f_{-}\\}\\cup\\{f\\neq f_{+}\\}"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{k\\to\\infty}\\int_{0}^{1}fdg_{n_{k}}=\\int_{0}^{1}fdg"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{X}=\\lim_{s\\downarrow\\cdot,s\\in\\mathbb{Q}_{+}}\\lim_{n\\to\\infty}\\widetilde{X}^{n}_{s}"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(X^{n}_{0})_{n=1}^{\\infty}"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(n_{k})_{k=1}^{\\infty}"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(X^{n})_{n=1}^{\\infty}"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{X}^{n}\\in\\mathrm{co}\\{X^{m}:m\\geq n\\}"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{co}\\left\\{\\left\\vert{\\int_{0}^{t}\\xi dX^{n}}\\right\\vert:n\\in\\mathbb{N},\\xi\\textrm{ is predictable and }\\vert{\\xi}\\vert\\leq1\\right\\}"}
{"pdf": "arxiv_math/2503.06350_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06350", "page": 1, "id": "2503.06350_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(M^{n})_{n=1}^{\\infty}"}
{"pdf": "arxiv_math/2503.04467_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04467", "page": 1, "id": "2503.04467_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{ \\begin{aligned} \\frac{d}{dt}u + vAu &= B(u_{2})-B(u_{1})\\\\ u(0) &= 0 \\end{aligned} \\right. ,\\quad u \\in L^{\\infty}(0,T;V) \\cap L^{2}(0,T,D(A))"}
{"pdf": "arxiv_math/2503.04467_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04467", "page": 1, "id": "2503.04467_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\vert (B(u,u_{2}),u) \\vert \\leq c \\Vert u \\Vert_{L^{2}}^{3/2}\\Vert u_{2} \\Vert_{V}\\Vert u \\Vert_{V}^{\\frac{1}{2}}"}
{"pdf": "arxiv_math/2503.04467_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04467", "page": 1, "id": "2503.04467_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\vert (B(u,u_{2}),u) \\vert \\leq \\frac{v}{2}\\Vert u \\Vert_{V}^{2} + \\frac{c^{2}}{2v}\\Vert u \\Vert_{H}^{2} \\Vert u_{2} \\Vert_{V}^{2}"}
{"pdf": "arxiv_math/2503.04467_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04467", "page": 1, "id": "2503.04467_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{2}\\frac{d}{dt}\\Vert u_{m} \\Vert_{L^{2}}^{2} + va(u,u) = (B(u_{2})-B(u_{1}),u_{2}-u_{1})"}
{"pdf": "arxiv_math/2503.04467_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04467", "page": 1, "id": "2503.04467_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{d}{dt}\\Vert u \\Vert_{H}^{2} \\leq \\frac{c^{2}}{v}\\Vert u \\Vert_{H}^{2}\\Vert u_{2} \\Vert_{V}^{2}"}
{"pdf": "arxiv_math/2503.04467_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04467", "page": 1, "id": "2503.04467_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{ \\begin{aligned} (B(u_{1},u),u) &= 0\\\\ \\vert (B(u,u_{2}),u) \\vert &= \\vert \\int_{\\Omega} u \\cdot \\nabla u_{2} \\cdot u dxdy\\vert \\leq \\Vert u \\Vert_{L^{\\infty}}\\Vert \\nabla u_{2} \\Vert_{V} \\Vert u \\Vert_{L^{2}} \\end{aligned} \\right."}
{"pdf": "arxiv_math/2503.04467_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04467", "page": 1, "id": "2503.04467_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Vert u(t) \\Vert_{H}^{2} \\leq \\Vert u(0) \\Vert_{H}^{2} e^{\\frac{c^{2}}{v}\\int_{0}^{t}\\Vert u_{2}(s) \\Vert_{V}^{2}ds} = 0"}
{"pdf": "arxiv_math/2503.04467_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04467", "page": 1, "id": "2503.04467_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}(\\Omega) \\hookrightarrow L^{\\infty}(\\Omega)"}
{"pdf": "arxiv_math/2503.04467_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04467", "page": 1, "id": "2503.04467_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "B(u_{1})-B(u_{2}) = B(u_{1},u)+B(u,u_{2})"}
{"pdf": "arxiv_math/2503.09488_pg11.pdf", "url": "https://arxiv.org/pdf/2503.09488", "page": 1, "id": "2503.09488_pg11_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "I_1,\\dots, I_k \\sub [n]"}
{"pdf": "arxiv_math/2503.09488_pg11.pdf", "url": "https://arxiv.org/pdf/2503.09488", "page": 1, "id": "2503.09488_pg11_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "t_{ij}^k +t_{jl}^k-t_{il}^k"}
{"pdf": "arxiv_math/2503.09488_pg11.pdf", "url": "https://arxiv.org/pdf/2503.09488", "page": 1, "id": "2503.09488_pg11_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\langle \\{t_{ij}^k\\}_{i,j \\in I}^{1\\leq k\\leq d}\\right \\rangle"}
{"pdf": "arxiv_math/2503.09488_pg11.pdf", "url": "https://arxiv.org/pdf/2503.09488", "page": 1, "id": "2503.09488_pg11_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{d,n}(I_1,\\dots, I_k)"}
{"pdf": "arxiv_math/2503.08151_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08151", "page": 1, "id": "2503.08151_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ket{\\Psi_{t+1}}\\in\\mathcal{H}_p\\otimes\\mathcal{H}_c"}
{"pdf": "arxiv_math/2503.08151_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08151", "page": 1, "id": "2503.08151_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{Z}=\\left\\{0,\\pm 1,\\pm 2,\\ldots\\right\\}"}
{"pdf": "arxiv_math/2503.08151_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08151", "page": 1, "id": "2503.08151_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ket{0}=\\begin{bmatrix} 1\\\\0 \\end{bmatrix},\\quad \\ket{1}=\\begin{bmatrix} 0\\\\1 \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.08151_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08151", "page": 1, "id": "2503.08151_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{\\ket{x} : x\\in\\mathbb{Z}\\right\\}"}
{"pdf": "arxiv_math/2503.08151_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08151", "page": 1, "id": "2503.08151_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{\\ket{0},\\ket{1}\\right\\}"}
{"pdf": "arxiv_math/2503.08151_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08151", "page": 1, "id": "2503.08151_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{H}_p\\otimes\\mathcal{H}_c"}
{"pdf": "arxiv_math/2503.08151_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08151", "page": 1, "id": "2503.08151_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ket{\\Psi_t}\\in\\mathcal{H}_p\\otimes\\mathcal{H}_c"}
{"pdf": "arxiv_math/2503.05989_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05989", "page": 1, "id": "2503.05989_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega(\\cdot, \\cdot): \\mathbb{R}\\times\\mathbb{R}\\rightarrow \\mathbb{R}"}
{"pdf": "arxiv_math/2503.05989_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05989", "page": 1, "id": "2503.05989_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f(\\mathbf{0}) = \\mathbf{0}"}
{"pdf": "arxiv_math/2503.05989_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05989", "page": 1, "id": "2503.05989_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "f(\\cdot ) : \\mathbb{R}^n\\rightarrow \\mathbb{R}^n"}
{"pdf": "arxiv_math/2503.05989_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05989", "page": 1, "id": "2503.05989_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "h(\\cdot ) : \\mathbb{R}^n\\rightarrow \\mathbb{R}"}
{"pdf": "arxiv_math/2503.05989_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05989", "page": 1, "id": "2503.05989_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x}\\in\\mathbb{R}^n"}
{"pdf": "arxiv_math/2503.05989_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05989", "page": 1, "id": "2503.05989_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "S(\\cdot) : \\mathbb{R}^n\\rightarrow \\mathbb{R}"}
{"pdf": "arxiv_math/2503.05989_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05989", "page": 1, "id": "2503.05989_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "g(\\cdot ) : \\mathbb{R}^n\\rightarrow \\mathbb{R}^n"}
{"pdf": "arxiv_math/2503.08451_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08451", "page": 1, "id": "2503.08451_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{r}_t = \\sigma(\\mathbf{W}_r \\mathbf{x}_t + \\mathbf{U}_r \\mathbf{h}_{t-1} + \\mathbf{b}_r)"}
{"pdf": "arxiv_math/2503.08451_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08451", "page": 1, "id": "2503.08451_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Linear}_{d_h}"}
{"pdf": "arxiv_math/2503.08451_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08451", "page": 1, "id": "2503.08451_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{W}_z, \\mathbf{U}_z, \\mathbf{b}_z"}
{"pdf": "arxiv_math/2503.08451_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08451", "page": 1, "id": "2503.08451_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{h}_t = (1 - \\mathbf{z}_t) \\odot \\mathbf{h}_{t-1} + \\mathbf{z}_t \\odot \\tilde{\\mathbf{h}}_t"}
{"pdf": "arxiv_math/2503.08451_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08451", "page": 1, "id": "2503.08451_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathbf{h}}_t = \\tanh(\\mathbf{W}_h \\mathbf{x}_t + \\mathbf{U}_h (\\mathbf{r}_t \\odot \\mathbf{h}_{t-1}) + \\mathbf{b}_h)"}
{"pdf": "arxiv_math/2503.08451_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08451", "page": 1, "id": "2503.08451_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{h}_t = f(\\mathbf{h}_{t-1}, \\mathbf{x}_t; \\theta)"}
{"pdf": "arxiv_math/2503.08451_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08451", "page": 1, "id": "2503.08451_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{z}_t = \\sigma(\\mathbf{W}_z \\mathbf{x}_t + \\mathbf{U}_z \\mathbf{h}_{t-1} + \\mathbf{b}_z)"}
{"pdf": "arxiv_math/2503.08451_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08451", "page": 1, "id": "2503.08451_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ \\mathbf{x}_t \\}_{t=1}^T"}
{"pdf": "arxiv_math/2503.07611_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07611", "page": 1, "id": "2503.07611_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\emptyset, \\square\\}"}
{"pdf": "arxiv_math/2503.05542_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05542", "page": 1, "id": "2503.05542_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "X \\coloneqq (x_1,\\dots,x_n)^\\top"}
{"pdf": "arxiv_math/2503.05542_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05542", "page": 1, "id": "2503.05542_pg2_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\varepsilon \\coloneqq (\\varepsilon_1,\\dots,\\varepsilon_n)^{\\top}"}
{"pdf": "arxiv_math/2503.05542_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05542", "page": 1, "id": "2503.05542_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "y_i = x_i^{\\top} \\beta_0 + \\varepsilon_i, \\quad i=1,\\dots,n"}
{"pdf": "arxiv_math/2503.05542_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05542", "page": 1, "id": "2503.05542_pg2_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "y \\coloneqq (y_1,\\dots,y_n)^\\top"}
{"pdf": "arxiv_math/2503.09155_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09155", "page": 1, "id": "2503.09155_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "J:\\R^n_{\\geq 0}\\to\\R^{n\\times n}"}
{"pdf": "arxiv_math/2503.09155_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09155", "page": 1, "id": "2503.09155_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "J(x)=\\begin{bmatrix} -\\alpha_1 & 0 &0&\\dots& 0& -\\frac{m x_n^{m-1}}{(1+x_n^m)^2}\\\\ 1&-\\alpha_2&0&\\dots&0&0\\\\ 0 & 1&-\\alpha_3&\\dots&0&0\\\\ \\vdots&\\vdots&\\vdots&\\ddots&\\vdots&\\vdots\\\\ 0&0&0& \\dots&-\\alpha_{n-1}&0\\\\ 0&0&0& \\dots&1&-\\alpha_n \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.09155_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09155", "page": 1, "id": "2503.09155_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "a\\in\\R^n_{\\geq 0}\\setminus\\{e\\}"}
{"pdf": "arxiv_math/2503.09155_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09155", "page": 1, "id": "2503.09155_pg16_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "e\\in\\operatorname{int}\\left(\\mathcal{B}_G\\right)"}
{"pdf": "arxiv_math/2503.09155_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09155", "page": 1, "id": "2503.09155_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "e\\in \\operatorname{int}\\left(\\mathcal{B}_{G} \\right )"}
{"pdf": "arxiv_math/2503.09155_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09155", "page": 1, "id": "2503.09155_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in\\mathbb{R}^n_{\\geq 0}"}
{"pdf": "arxiv_math/2503.09374_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09374", "page": 1, "id": "2503.09374_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(x, y)=\\pi(y\\mid x)\\pi_0(x)"}
{"pdf": "arxiv_math/2503.09374_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09374", "page": 1, "id": "2503.09374_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_0(x)\\propto \\exp\\{-\\frac12\\|x\\|^2_{C}\\} =\\exp\\{-\\frac12 x^{\\top}C^{-1}x\\}"}
{"pdf": "arxiv_math/2503.09374_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09374", "page": 1, "id": "2503.09374_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}: \\mathbb{R}^d \\to \\mathbb{R}^{n}"}
{"pdf": "arxiv_math/2503.09374_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09374", "page": 1, "id": "2503.09374_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(x\\mid y)=N(\\mu_{post},C_{post})"}
{"pdf": "arxiv_math/2503.09374_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09374", "page": 1, "id": "2503.09374_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(x \\mid y)=\\frac{\\pi(y| x)\\pi_0(x)}{Z(y)}\\propto \\exp(-\\Phi(x))\\pi_0(x)"}
{"pdf": "arxiv_math/2503.09374_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09374", "page": 1, "id": "2503.09374_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\mathcal{F}(x)"}
{"pdf": "arxiv_math/2503.09374_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09374", "page": 1, "id": "2503.09374_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta\\thicksim N(0,\\Sigma)"}
{"pdf": "arxiv_math/2503.08494_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08494", "page": 1, "id": "2503.08494_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Vert J_i(\\mathbf{u})-J_i(\\mathbf{v})\\Vert \\leq l_J\\|\\mathbf{u}-\\mathbf{v}\\|"}
{"pdf": "arxiv_math/2503.08494_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08494", "page": 1, "id": "2503.08494_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "J_i(x_i,h(\\mathbf{x}))"}
{"pdf": "arxiv_math/2503.08494_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08494", "page": 1, "id": "2503.08494_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{-i}^*=\\mathrm{col}(x_1^*,\\dots, x_{i-1}^*, x_{i+1}^*, \\dots, x_N^*)"}
{"pdf": "arxiv_math/2503.08494_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08494", "page": 1, "id": "2503.08494_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega_i \\subseteq \\mathbb{R}^d"}
{"pdf": "arxiv_math/2503.08494_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08494", "page": 1, "id": "2503.08494_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "J_i(\\mathbf{u}), u \\in \\Omega"}
{"pdf": "arxiv_math/2503.08494_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08494", "page": 1, "id": "2503.08494_pg2_math_005", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\mathbf{x}^*=\\mathrm{col}(x_1^*,x_2^*,...,x_N^*) \\in \\Omega \\cap \\Upsilon"}
{"pdf": "arxiv_math/2503.08494_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08494", "page": 1, "id": "2503.08494_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "J_i(x_i^*,h(\\mathbf{x}^*)) \\leq J_i(x_i,\\frac{1}{N}x_i+\\sum_{j \\neq i} \\frac{1}{N}x_j^*), \\nonumber"}
{"pdf": "arxiv_math/2503.08494_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08494", "page": 1, "id": "2503.08494_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{u}, \\mathbf{v} \\in \\Omega"}
{"pdf": "arxiv_math/2503.08494_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08494", "page": 1, "id": "2503.08494_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_i f(\\mathbf{x})"}
{"pdf": "arxiv_math/2503.04068_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04068", "page": 1, "id": "2503.04068_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^mA_i\\Sigma(W_ix+b_i)"}
{"pdf": "arxiv_math/2503.09337_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09337", "page": 1, "id": "2503.09337_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla G(\\mathcal{X})=\\mathcal{D}^{T}\\ast_{t}(\\mathcal{D}\\ast_{t}\\mathcal{X}-\\mathcal{B})"}
{"pdf": "arxiv_math/2503.09337_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09337", "page": 1, "id": "2503.09337_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{\\alpha}(\\mathcal{X})_{i_{1}i_{2}\\dots i_{N}}=(\\vert \\mathcal{X}_{i_{1}i_{2}\\dots i_{N}}\\vert -\\alpha)_{+}sgn(\\mathcal{X}_{{i_{1}i_{2}\\dots i_{N}}})"}
{"pdf": "arxiv_math/2503.09337_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09337", "page": 1, "id": "2503.09337_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "G(\\mathcal{X})=\\frac{1}{2}\\Vert \\mathcal{D}\\ast_{t}\\mathcal{X}-\\mathcal{B} \\Vert_{F}^{2}"}
{"pdf": "arxiv_math/2503.09337_pg14.pdf", "url": "https://arxiv.org/pdf/2503.09337", "page": 1, "id": "2503.09337_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{T}_{\\alpha}: \\mathbb{R}^{M_{1}\\times M_{2}\\times\\dots\\times M_{N}}\\rightarrow \\mathbb{R}^{M_{1}\\times M_{2}\\times\\dots\\times M_{N}}"}
{"pdf": "arxiv_math/2503.07632_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07632", "page": 1, "id": "2503.07632_pg11_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{n=1}^N \\sum_{m=1}^N \\int_0^T a_n a_m w(nf,t) w(mf,t)\\,dt"}
{"pdf": "arxiv_math/2503.07632_pg11.pdf", "url": "https://arxiv.org/pdf/2503.07632", "page": 1, "id": "2503.07632_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left|\\int_0^T w_c(f,N,A,t)^2\\, dt - \\frac{NT}{2}\\right| \\le \\frac{N^2}{2\\pi f}"}
{"pdf": "arxiv_math/2503.08355_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08355", "page": 1, "id": "2503.08355_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{y}_{X_{i}}(t_{j}) = y_{X_{i}}(t_{j}) + \\varepsilon_{i,j}"}
{"pdf": "arxiv_math/2503.08355_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08355", "page": 1, "id": "2503.08355_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "f: \\mathbb{R}^d \\to \\mathbb{R}^d"}
{"pdf": "arxiv_math/2503.08355_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08355", "page": 1, "id": "2503.08355_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "f:\\mathbb{R}^{D}\\rightarrow \\mathbb{R}^{D}"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{i_k\\}_{k \\in \\mathbb{N}} \\subset I"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{I} = (I, \\mathcal{E}^I, \\mathcal{P}(I))"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{dom}_1(C)"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "|I| \\leq |J| \\times \\aleph_0"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{\\kappa}(\\mathcal{X})"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{\\aleph_0}(\\mathcal{X})"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "I_C = \\{i \\in I \\mid A_i \\cap \\operatorname{dom}_1(C) \\neq \\emptyset\\} \\subseteq I"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "I = E[I_C] = \\bigcup_{i \\in I_C} E[\\{i\\}]"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "|I| = |I| \\times \\aleph_0 = |J| \\times \\aleph_0 = |J|"}
{"pdf": "arxiv_math/2503.07398_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07398", "page": 1, "id": "2503.07398_pg15_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|I_C| \\times \\sup_{i \\in I_C} |E[\\{i\\}]|"}
{"pdf": "arxiv_math/2503.06549_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06549", "page": 1, "id": "2503.06549_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_i^{(N)},\\lambda_i^{(N-1)}, \\lambda_i^{(N-2)}\\ldots"}
{"pdf": "arxiv_math/2503.06549_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06549", "page": 1, "id": "2503.06549_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\langle {\\bm w}_i^{(N)},{\\bm w}_j^{(N-k)}\\rangle|\\ll 1"}
{"pdf": "arxiv_math/2503.06549_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06549", "page": 1, "id": "2503.06549_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_i^{(n)}-\\lambda_{i+1}^{(n)}"}
{"pdf": "arxiv_math/2503.06549_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06549", "page": 1, "id": "2503.06549_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_i^{(1)}(t), \\mu_i^{(2)}(t)"}
{"pdf": "arxiv_math/2503.06549_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06549", "page": 1, "id": "2503.06549_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "k(a)\\sim N^{2(1-a)/3}"}
{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{I}\\left|f_n\\left(t_0\\right)(x)-f\\left(t_0\\right)(x)\\right| d x \\geq \\int_{\\tilde{I}}\\left|f_n\\left(t_0\\right)(x)-f\\left(t_0\\right)(x)\\right| d x>\\epsilon_0 \\cdot m(\\tilde{I})"}
{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\cdot\\|_{\\mathcal{Z}}"}
{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall n \\in \\mathbb{N}"}
{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "C\\left([0, T] ; L^2(I ; \\mathbb{R} \\otimes \\mathbb{R})\\right)"}
{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(f_n)_{n=1}^{\\infty} \\in \\mathcal{Z}"}
{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f(t_0)(x) > \\bar{A} + \\epsilon_0"}
{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "f(t_0)(x) < -\\bar{A} -\\epsilon_0"}
{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{F}_1 = 0.0099s_{xx}+0.9955u"}
{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\bm{l}/(\\bm{k}^2+\\bm{l}^2)\\tilde{s}"}
{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\bm{k}/(\\bm{k}^2+\\bm{l}^2)\\tilde{s}"}
{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{F}_1 = 0.0098s_{xx} + u - 0.0083s + 0.0046"}
{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "u(x,t) = (\\pi x)/5+u_1(t)"}
{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\gamma} = -1/(\\bm{k}^2+\\bm{l}^2)\\tilde{s}"}
{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma = -\\Delta^{-1} s"}
{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "u(x,y,t) = u_3(t)\\times \\{0.1\\sin[2\\pi (x + y)] + \\cos[2\\pi (x + y)]\\}"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{H}_{r}^{\\text{NLoS}} \\in \\mathbb{C}^{M_r \\times N}"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "e^{j 2 \\pi d_{\\text{R}} \\sin(\\gamma) \\left[ \\lfloor \\frac{n-1}{N_x} \\rfloor \\sin(\\zeta) + ((n-1)-\\lfloor \\frac{n-1}{N_x} \\rfloor N_x) \\cos(\\zeta) \\right]/\\lambda }, n = 1, 2, \\cdots, N"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{n}_{\\text{BS},i} \\sim \\mathcal{CN} \\left( \\mathbf{0}_{M_t}, \\sigma_{\\text{BS}}^2 \\mathbf{I}_{M_t} \\right)"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\varphi} = [\\varphi_1, \\varphi_2, \\cdots, \\varphi_N]^T"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{H}_t \\in \\mathbb{C}^{N \\times M_t}, \\mathbf{H}_r \\in \\mathbb{C}^{M_r \\times N}"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta \\in \\left[0, \\pi\\right)"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{H}_d \\in \\mathbb{C}^{M_r \\times M_t}"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_n = e^{j \\theta_n}"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta_d^{\\text{AoA}}"}
{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "M_s \\leq \\min\\left\\{M_t, M_r\\right\\}"}
{"pdf": "arxiv_math/2503.08947_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08947", "page": 1, "id": "2503.08947_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{ab}{}^I = - u^\\mu_a u^\\nu_b \\nabla_{(\\mu} n^I_{\\nu)} =n^I_\\mu D_a u^\\mu_b + \\frac{1}{2} \\hat \\upsilon^I \\tau_{ab}"}
{"pdf": "arxiv_math/2503.08947_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08947", "page": 1, "id": "2503.08947_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{R}^a{}_{bcd}= \\partial_c \\gamma^a_{d b} - \\partial_d \\gamma^a_{c b} + \\gamma^a_{cf} \\gamma^f_{d b}- \\gamma^a_{df} \\gamma^f_{c b}"}
{"pdf": "arxiv_math/2503.08947_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08947", "page": 1, "id": "2503.08947_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{D}_a T^I = D_a T^I - \\omega_a{}^I{}_J T^J"}
{"pdf": "arxiv_math/2503.08947_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08947", "page": 1, "id": "2503.08947_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_a{}^I{}_J = n^\\mu_J D_a n^I_\\mu \\, "}
{"pdf": "arxiv_math/2503.08947_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08947", "page": 1, "id": "2503.08947_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "2 \\gamma^c_{[ab]} = -\\hat \\upsilon^c F_{ab}"}
{"pdf": "arxiv_math/2503.08947_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08947", "page": 1, "id": "2503.08947_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta K_{ab}{}^I = \\frac{1}{2} F_{I a} \\partial_b \\sigma + \\frac{1}{2} F_{Ib} \\partial_a \\sigma "}
{"pdf": "arxiv_math/2503.08947_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08947", "page": 1, "id": "2503.08947_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{R}^a{}_{b cd}"}
{"pdf": "arxiv_math/2503.05572_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05572", "page": 1, "id": "2503.05572_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "b_k = \\Theta(e^{k^\\beta})"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "b_{i,l,j,l'}=D_{j}(\\frac{(E_{d,j})^{l'}.a_{i,l}}{l'!})"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "s=\\sum_{i_0=0}^\\infty\\sum_{i_1=0}^\\infty\\cdots\\sum_{i_{d-1}=0}^\\infty c_{i_0...i_{d-1}}(x_{0}-x_{0,n})^{i_0}\\cdots(x_{d-1}-x_{d-1,n})^{i_{d-1}}"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{d,i}(x_i-x_{i,n})=1"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "Z^{-1}E_{d,i}Z=E_{d,i}"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "Z=\\left( \\begin{matrix} I_{d\\times d}&0\\\\ (z_0,z_1,...,z_{d-1})&1 \\end{matrix} \\right)"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{i,l}=D_i(\\frac{(E_{d,i})^l.s}{l!})"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "s=\\sum_{l=0}^\\infty a_{i,l}(x_{i}-x_{i,n})^l"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{d,i}c_{i_0...i_{d-1}}=0"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_023", "type": "math", "max_diffs": 0, "checked": "verified", "math": "a_{i,l}=\\sum_{k=0}^\\infty b_{i,l,j,l'}(x_j-x_{j,n})^{l'}"}
{"pdf": "arxiv_math/2503.09352_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09352", "page": 1, "id": "2503.09352_pg5_math_025", "type": "math", "max_diffs": 0, "checked": null, "math": "z=[z_0:z_1:...:z_{d-1}:1]"}
{"pdf": "arxiv_math/2503.04667_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04667", "page": 1, "id": "2503.04667_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} \\mathcal{L}_{\\text{SIMax}} = \\mathbb{E}_{z \\sim {p_{\\theta}(z|x)}} \\{ & \\mathbb{E}_{t \\sim T} \\left[-\\log q_{\\phi_t}(y_{t}|z)\\right] \\\\ & - \\alpha \\log \\frac{\\exp(\\text{sim}(z, z^{+}) / \\tau)}{\\sum_{z' \\in \\mathcal{B^{+}}} \\exp(\\text{sim}(z, z') / \\tau)} \\}, \\end{aligned}"}
{"pdf": "arxiv_math/2503.04667_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04667", "page": 1, "id": "2503.04667_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I(X; Z) \\geq I(Z; Z^\\prime)"}
{"pdf": "arxiv_math/2503.04667_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04667", "page": 1, "id": "2503.04667_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "Y_t \\rightarrow X \\rightarrow Z \\rightarrow Z_t \\rightarrow \\hat{Y}_t"}
{"pdf": "arxiv_math/2503.04667_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04667", "page": 1, "id": "2503.04667_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "q_{\\phi_t}(y_{t} | z)"}
{"pdf": "arxiv_math/2503.04667_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04667", "page": 1, "id": "2503.04667_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{t=1}^{|T|}[I(Y_t;Z)]"}
{"pdf": "arxiv_math/2503.04667_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04667", "page": 1, "id": "2503.04667_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "Z \\leftarrow X \\rightarrow Z^\\prime"}
{"pdf": "arxiv_math/2503.07924_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07924", "page": 1, "id": "2503.07924_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "J_{ij} = -\\frac{Q_{ij}}{4}"}
{"pdf": "arxiv_math/2503.07924_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07924", "page": 1, "id": "2503.07924_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H(x) = \\sum_{i \\neq j} Q_{ij} x_i x_j + \\sum_i q_i x_i"}
{"pdf": "arxiv_math/2503.07924_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07924", "page": 1, "id": "2503.07924_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} H(\\sigma) &= \\sum_{i \\neq j} Q_{ij} \\cdot \\frac{(\\sigma_i + 1)(\\sigma_j + 1)}{4} + \\sum_i q_i \\cdot \\frac{\\sigma_i + 1}{2} \\\\ &= \\sum_{i \\neq j} \\frac{Q_{ij}}{4} \\sigma_i \\sigma_j + \\sum_i \\left( \\frac{q_i}{2} + \\frac{\\sum_{i \\neq j} (Q_{ij} + Q_{ji})}{4} \\right) \\sigma_i \\\\ &\\quad + \\left( \\frac{\\sum_i q_i}{2} + \\frac{\\sum_{i \\neq j} Q_{ij}}{4} \\right) \\end{aligned}"}
{"pdf": "arxiv_math/2503.07924_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07924", "page": 1, "id": "2503.07924_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} Q_{ij} = & \\, P_1 \\sum_{i \\neq j} x_{S,i} x_{S,j} + P_2 \\sum_{i \\neq j} x_{i,D} x_{j,D} \\\\ & + P_3 \\sum_{i \\not\\in \\{S,D\\}} \\left( \\sum_{(i,j) \\in E} \\sum_{\\substack{(i,k) \\in E \\\\ k \\neq j}} x_{i,j} x_{i,k} \\right. \\\\ & \\quad + \\left. \\sum_{(j,i) \\in E} \\sum_{\\substack{(k,i) \\in E \\\\ k \\neq j}} x_{j,i} x_{k,i} - 2 \\sum_{(i,j) \\in E} \\sum_{(k,i) \\in E} x_{i,j} x_{k,i} \\right) \\end{aligned}"}
{"pdf": "arxiv_math/2503.07924_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07924", "page": 1, "id": "2503.07924_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "H(\\sigma) = -\\sum_{i,j} J_{ij} \\sigma_i \\sigma_j - \\sum_i h_i \\sigma_i"}
{"pdf": "arxiv_math/2503.07924_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07924", "page": 1, "id": "2503.07924_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "h_i = -\\frac{q_i}{2} - \\frac{\\sum_{i \\neq j} (Q_{ij} + Q_{ji})}{4}"}
{"pdf": "arxiv_math/2503.07924_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07924", "page": 1, "id": "2503.07924_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_i \\in \\{-1,1\\}"}
{"pdf": "arxiv_math/2503.07924_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07924", "page": 1, "id": "2503.07924_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} q_i = & -P_1 \\sum_i x_{S,i} - P_2 \\sum_i x_{i,D} \\\\ & + P_3 \\sum_{i \\not\\in \\{S,D\\}} \\left( \\sum_{(i,j) \\in E} x_{i,j} + \\sum_{(k,i) \\in E} x_{k,i} \\right) + f(x) \\end{aligned}"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "L'=Z_n(L')\\leq Z_i(L)"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "[x_i+y,_{n_i}x_i]=[y,_{n_i}x_i]=0"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "A=Z_{i+1}(L')/Z_i(L')"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "ad^r_{x}(x+y)=ad^r_{x}(y)=0"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "[x+y,_{n}x]=[y,_{n}x]=0"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "Z_{i+1}(L')\\leq Z_{k+m}(L)"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "C_A(L)=C_A(x_1,..,x_n)"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "m=1+\\sum_{i=1}^{k} (n_i-1)"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "[y,_m L]\\leq Z_i(L')\\leq Z_k(L)"}
{"pdf": "arxiv_math/2503.06230_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06230", "page": 1, "id": "2503.06230_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "H_0=H\\lhd H_1 \\lhd...\\lhd H_n=L"}
{"pdf": "arxiv_math/2503.07705_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07705", "page": 1, "id": "2503.07705_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sqrt{g(r,\\theta)}drd\\theta"}
{"pdf": "arxiv_math/2503.07705_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07705", "page": 1, "id": "2503.07705_pg1_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\int_{\\tau_\\mathrm{min}}^\\infty p(x,y;t)dt<\\infty"}
{"pdf": "arxiv_math/2503.07705_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07705", "page": 1, "id": "2503.07705_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "ds^2=dr^2+f(r)^2\\,d\\theta^2"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_000", "type": "math", "max_diffs": 0, "checked": "verified", "math": "E_7: f(x,y,z)=x^2+y^3+yz^3"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "v_3\\in (\\mathbb{R}_{\\geq 0})^3"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "C_{v_3}(f)=\\langle (1,0,0),(0,1,0),(n+1,0,1),(0,n+1,1) \\rangle"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_6: f(x,y,z)=z^2+y^3+x^4"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "In_{{v_3}}(f)=-z^{n+1}"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "C_{v_1}(f)=\\langle (n+1,0,1),(0,n+1,1) \\rangle"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "A_n: f(x,y,z)=xy-z^{n+1}, \\ \\ n\\in \\mathbb{N}"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "D_n: f(x,y,z)=z^2-x(y^2+x^{n-2}), \\ \\ n\\in \\mathbb{N}, n\\geq 4"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "-s\\in \\mathbb{Z}_{\\leq 0}"}
{"pdf": "arxiv_math/2503.08118_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08118", "page": 1, "id": "2503.08118_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "X=V(f)\\subset \\mathbb{C}^3"}
{"pdf": "arxiv_math/2503.03949_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03949", "page": 1, "id": "2503.03949_pg1_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "s(D) \\in \\{0,\\cdots,N-2\\}"}
{"pdf": "arxiv_math/2503.03949_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03949", "page": 1, "id": "2503.03949_pg1_math_003", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\delta(D) \\in [1,\\frac{1}{2}(N-s(D))]"}
{"pdf": "arxiv_math/2503.08955_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08955", "page": 1, "id": "2503.08955_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x \\mapsto {}^{\\natural}x^{-1}"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "a \\leq \\|f\\|_{L^2(\\Omega)} \\leq b"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{f}_s \\in U_{ad} \\subset L^2(\\Omega)"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "f_s \\rightharpoonup f"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f_s\\|_{H^{-s}(\\Omega)}"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "N,\\Omega \\text{and} s"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "s\\in(0,1),\\Omega \\subset \\R^N"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "J_s(\\bar{f}_s):=\\min_{f_s \\in U_{ad}} J_s(f_s)"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}_s=\\left\\{f_s\\right\\}_{0<s<1} \\subset H^{-s}(\\Omega)"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{cases} (-\\Delta)u = f , & x\\in \\Omega, \\\\ u=0, & x\\in \\partial \\Omega, \\end{cases}"}
{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "H_0^{1-\\delta}(\\Omega)"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{P_{\\Delta_n}}"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x_i \\leq c_{i, \\Delta_n}^{P_{\\Delta_n}}"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{\\Delta_n} = (a_{i, j})"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "X_{\\Delta_n}(\\N^{2n})"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "x_i \\leq b_{i, \\Delta_n}"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{\\Delta_n}^{-1} = (a^{-1}_{i,j})"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\prod_{j = 1}^{P_{\\Delta_n}} F_{i, j} \\leq c_{i, \\Delta_n}^{P_{\\Delta_n}}"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\prod_{j = 1}^{P_{\\Delta_n}} F_{i, j} \\leq b_{i, \\Delta_n}^{P_{\\Delta_n}}"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "x_i \\leq b_{i, \\Delta_n}^{P_{\\Delta_n}}"}
{"pdf": "arxiv_math/2503.08800_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08800", "page": 1, "id": "2503.08800_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "F'_{i, 1} \\leq F'_{i, j}"}
{"pdf": "arxiv_math/2503.09519_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09519", "page": 1, "id": "2503.09519_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta_{12}(t;1/2,2)< 10^{-25}"}
{"pdf": "arxiv_math/2503.09135_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09135", "page": 1, "id": "2503.09135_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{v}(t_1)({{\\rho}}(t_1), \\dot{{\\rho}}(t_1), {\\bar{Az}}(t_1), {\\bar{El}}(t_1), \\dot{\\bar{Az}}(t_1), \\dot{\\bar{El}}(t_1))"}
{"pdf": "arxiv_math/2503.09135_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09135", "page": 1, "id": "2503.09135_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bm{x} = (\\rho(t_1), \\bar{Az}(t_1), \\bar{El}(t_1), \\rho(t_3), \\bar{Az}(t_3), \\bar{El}(t_3))"}
{"pdf": "arxiv_math/2503.09135_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09135", "page": 1, "id": "2503.09135_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{v}(t_2)({{\\rho}}(t_1), \\dot{{\\rho}}(t_1), {\\bar{Az}}(t_1), {\\bar{El}}(t_1), \\dot{\\bar{Az}}(t_1), \\dot{\\bar{El}}(t_1))"}
{"pdf": "arxiv_math/2503.09135_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09135", "page": 1, "id": "2503.09135_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\Delta}({{\\rho}}(t_1), \\dot{{\\rho}}(t_1), {\\bar{Az}}(t_1), {\\bar{El}}(t_1), \\dot{\\bar{Az}}(t_1, \\dot{\\bar{El}}(t_1))) = \\begin{pmatrix} \\rho^c(t_2) - {\\rho}(t_2) \\\\ \\dot{\\rho}^c(t_2) - \\dot{{\\rho}}(t_2) \\end{pmatrix}"}
{"pdf": "arxiv_math/2503.09135_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09135", "page": 1, "id": "2503.09135_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bm{\\Delta}^T\\bm{\\Delta}"}
{"pdf": "arxiv_math/2503.09135_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09135", "page": 1, "id": "2503.09135_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{r}(t_1)({{\\rho}}(t_1), {{\\rho}}(t_1), {\\bar{Az}}(t_1), {\\bar{El}}(t_1), \\dot{\\bar{Az}}(t_1), \\dot{\\bar{El}}(t_1))"}
{"pdf": "arxiv_math/2503.09135_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09135", "page": 1, "id": "2503.09135_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\Delta} = \\boldsymbol{0}"}
{"pdf": "arxiv_math/2503.09135_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09135", "page": 1, "id": "2503.09135_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bm{\\Delta}^T\\bm{\\Delta}=\\bigcup\\limits_{i=1}^{N_s}\\mathcal{T}_{\\bm{\\Delta}^T \\bm{\\Delta}}^i(\\bm{x})"}
{"pdf": "arxiv_math/2503.09135_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09135", "page": 1, "id": "2503.09135_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{r}(t_2)({{\\rho}}(t_1), \\dot{{\\rho}}(t_1), {\\bar{Az}}(t_1), {\\bar{El}}(t_1), \\dot{\\bar{Az}}(t_1), \\dot{\\bar{El}}(t_1))"}
{"pdf": "arxiv_math/2503.08509_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08509", "page": 1, "id": "2503.08509_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\epsilon_j \\propto N(0, C_d)"}
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{"pdf": "arxiv_math/2503.08240_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08240", "page": 1, "id": "2503.08240_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "a \\sim \\mathcal{N}(0,\\sigma^2)"}
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{"pdf": "arxiv_math/2503.06110_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06110", "page": 1, "id": "2503.06110_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\liminf_{l \\to \\infty} \\frac{\\log(b_1 b_2 \\dots b_l)}{l \\log N}"}
{"pdf": "arxiv_math/2503.06110_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06110", "page": 1, "id": "2503.06110_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{Z}_{\\mathcal{O}}^n"}
{"pdf": "arxiv_math/2503.06110_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06110", "page": 1, "id": "2503.06110_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "g_t u_x = u_y g_t u_{x_0}"}
{"pdf": "arxiv_math/2503.06110_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06110", "page": 1, "id": "2503.06110_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{F}_q((X^{-1}))^n"}
{"pdf": "arxiv_math/2503.06110_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06110", "page": 1, "id": "2503.06110_pg13_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_{t,C}(e_1 - y_1 e_2 - \\dots - y_n e_{n+1}) = 0"}
{"pdf": "arxiv_math/2503.06110_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06110", "page": 1, "id": "2503.06110_pg13_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "d([e_1], g_t u_x H_C) = d([e_1], u_y H_{t,C}) \\asymp d([u_{-y} e_1], H_{t,C}) \\asymp \\phi_{t,C}(u_{-y} e_1)"}
{"pdf": "arxiv_math/2503.06110_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06110", "page": 1, "id": "2503.06110_pg13_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu(C) = \\frac{1}{b_1 b_2 \\dots b_l}"}
{"pdf": "arxiv_math/2503.06110_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06110", "page": 1, "id": "2503.06110_pg13_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\phi_{t,C}(u_{-y} e_1) \\lesssim e^{-M}"}
{"pdf": "arxiv_math/2503.06110_pg13.pdf", "url": "https://arxiv.org/pdf/2503.06110", "page": 1, "id": "2503.06110_pg13_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "F_\\infty\\subset K_\\infty"}
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{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{W} \\subset \\mathcal{L}"}
{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{V}(\\cdot,\\cdot)"}
{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\hat{V}}_{\\mathcal{L}}(\\underline{Y}_{\\mathcal{L}})"}
{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall \\epsilon_e > 0"}
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{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left(\\mathcal{V},p_V\\right)"}
{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{V}_{\\mathcal{L}}"}
{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{V} \\in \\{0,1\\}"}
{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{\\underline{V}}_{\\mathcal{L}}(\\underline{Y}_{\\mathcal{L}})"}
{"pdf": "arxiv_math/2503.07310_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07310", "page": 1, "id": "2503.07310_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in \\mathcal{X}\\subseteq \\mathbb{R}^{n_x}"}
{"pdf": "arxiv_math/2503.04646_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04646", "page": 1, "id": "2503.04646_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K \\subset \\{\\mu \\in P(X \\times Y): \\mu_X^* \\in P(X) \\text{ is the } X\\text{-marginal of } \\mu\\}"}
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{"pdf": "arxiv_math/2503.04604_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04604", "page": 1, "id": "2503.04604_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dot{F}^{\\alpha,q}_p(\\R^n)"}
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{"pdf": "arxiv_math/2503.04604_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04604", "page": 1, "id": "2503.04604_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\R^n} Ta(x)x^{\\alpha}dx=0, \\quad \\forall f\\in L^2_{N_p}(\\R^n) \\text{ and } |\\alpha|\\leq N_p"}
{"pdf": "arxiv_math/2503.04604_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04604", "page": 1, "id": "2503.04604_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "|K(x,y)-K(x,z)|\\leq C|y-z|^{\\delta}|x-y|^{-n-\\delta}"}
{"pdf": "arxiv_math/2503.04604_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04604", "page": 1, "id": "2503.04604_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "T^*((\\cdot-c)^{\\alpha})"}
{"pdf": "arxiv_math/2503.09536_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09536", "page": 1, "id": "2503.09536_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A(X,e) \\cong\\text{\\AE}(X)"}
{"pdf": "arxiv_math/2503.09536_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09536", "page": 1, "id": "2503.09536_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_k \\lvert a_k - a\\rvert < \\infty"}
{"pdf": "arxiv_math/2503.09536_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09536", "page": 1, "id": "2503.09536_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\cdot, \\cdot\\rangle_A"}
{"pdf": "arxiv_math/2503.09536_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09536", "page": 1, "id": "2503.09536_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "U \\subset \\mathbb R^n"}
{"pdf": "arxiv_math/2503.09536_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09536", "page": 1, "id": "2503.09536_pg7_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "A(X,e) \\cong \\text{\\AE}(X)"}
{"pdf": "arxiv_math/2503.09536_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09536", "page": 1, "id": "2503.09536_pg7_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi \\mapsto (m \\mapsto \\langle m,\\phi\\rangle_{\\text{\\AE}})"}
{"pdf": "arxiv_math/2503.09536_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09536", "page": 1, "id": "2503.09536_pg7_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "m \\in \\text{\\AE}_0(X,e)"}
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{"pdf": "arxiv_math/2503.09536_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09536", "page": 1, "id": "2503.09536_pg7_math_025", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\cdot,\\cdot \\rangle_{\\text{\\AE}}"}
{"pdf": "arxiv_math/2503.09536_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09536", "page": 1, "id": "2503.09536_pg7_math_026", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal M(X) \\subset \\text{\\AE}(X)"}
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{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\varepsilon} \\cdot \\underline{x} = \\sum_{i=1}^{5} \\varepsilon_i x_i \\leq 1, \\quad \\underline{x}\\in \\mathbb{H}_5"}
{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Isom}(\\mathbb{H}_5)"}
{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "h_x = % (1-\\norm{x}^2)^{-1} g_x"}
{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\varepsilon}"}
{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\prod \\varepsilon_i=1"}
{"pdf": "arxiv_math/2503.07801_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07801", "page": 1, "id": "2503.07801_pg15_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "p+1 \\mid \\delta(p+1) \\mid (p-1)\\ell(p)"}
{"pdf": "arxiv_math/2503.07801_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07801", "page": 1, "id": "2503.07801_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ell(p) \\ge \\frac{p+1}{2}"}
{"pdf": "arxiv_math/2503.07801_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07801", "page": 1, "id": "2503.07801_pg15_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(p-1)\\ell(p) = (p+1)\\ell(p) - 2\\ell(p)"}
{"pdf": "arxiv_math/2503.07801_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07801", "page": 1, "id": "2503.07801_pg15_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "g_i\\ge\\prod_{\\substack{p~\\text{inert in~}K\\\\(p+1)\\mid M_{x_i}}}2>\\exp\\left((\\log2)\\exp\\left(C \\frac{\\log x_i}{\\log\\log x_i}\\right)\\right)>i"}
{"pdf": "arxiv_math/2503.07801_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07801", "page": 1, "id": "2503.07801_pg15_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "L(g_i)\\le M_{x_i}\\le x_i^2=(\\log i)^{(4/C)\\log\\log\\log i}<(\\log g_i)^{c_0\\log\\log\\log g_i}"}
{"pdf": "arxiv_math/2503.07573_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07573", "page": 1, "id": "2503.07573_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{P_* T} \\alpha = \\int_T P^* \\alpha"}
{"pdf": "arxiv_math/2503.07573_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07573", "page": 1, "id": "2503.07573_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\alpha(x)| \\lesssim_N \\langle x\\rangle^{-N}"}
{"pdf": "arxiv_math/2503.07573_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07573", "page": 1, "id": "2503.07573_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha \\in \\mathscr S"}
{"pdf": "arxiv_math/2503.07573_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07573", "page": 1, "id": "2503.07573_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x\\rangle := \\sqrt{1 + |x|^2}"}
{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{l} \\in \\mathbf{R}^{m}"}
{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "P \\in \\mathbb{R}^{n \\times n}"}
{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{l}_i= \\mathbf{u}_i"}
{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x} \\in \\mathbb{R}^{n}"}
{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "A \\in \\mathbb{R}^{m \\times n}"}
{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{u} \\in \\mathbb{R}^{m}"}
{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{q} \\in \\mathbb{R}^{n }"}
{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_000", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\bar{L}(n)\\mathbb{P}^x(\\tau_{B_{r_0}(x_0)}>\\tau_n)\\le \\bar{L}(|x-x_0|),\\qquad x\\in B^c_{r_0}(x_0)"}
{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "O\\subseteq B_{r_0}(x_0)"}
{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "B\\in\\mathcal{B}(\\R^d)"}
{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "B\\in\\mathcal{B}(B_{r_0}(x_0))"}
{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(0,\\infty)\\times \\bar{B}_{r_0}(x_0)\\times\\bar{B}_{r_0}(x_0)"}
{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}^x(\\tau_{B_{r_0}(x_0)}=\\infty)\\le \\frac{\\bar{L}(|x-x_0|)}{\\bar L(\\infty)}<1,\\qquad x\\in B^c_{r_0}(x_0)"}
{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{D}=\\mathcal{B}(B_{r_0}(x_0))"}
{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}^x(\\tau_{B_{r_0}(x_0)}<\\infty)>0"}
{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "B\\in \\mathcal{B}(\\R^d)"}
{"pdf": "arxiv_math/2503.08227_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08227", "page": 1, "id": "2503.08227_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial f / \\partial \\vec{n}"}
{"pdf": "arxiv_math/2503.04674_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04674", "page": 1, "id": "2503.04674_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "t\\in [-\\frac{1}{2},0]"}
{"pdf": "arxiv_math/2503.04674_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04674", "page": 1, "id": "2503.04674_pg19_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\phi(t,x,y)=\\mathrm{e}^{-t}x(1-x)y(1-y)"}
{"pdf": "arxiv_math/2503.06958_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06958", "page": 1, "id": "2503.06958_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta(\\varphi)=\\sup_{\\mu\\in \\mathcal{M}_T(X)}\\int \\varphi\\ d\\mu"}
{"pdf": "arxiv_math/2503.06958_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06958", "page": 1, "id": "2503.06958_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}^{\\mathbb{Z}^d}"}
{"pdf": "arxiv_math/2503.06958_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06958", "page": 1, "id": "2503.06958_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{M}_{{\\rm max}}(\\varphi)"}
{"pdf": "arxiv_math/2503.06958_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06958", "page": 1, "id": "2503.06958_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi:X\\rightarrow \\mathbb{R}"}
{"pdf": "arxiv_math/2503.05104_pg18.pdf", "url": "https://arxiv.org/pdf/2503.05104", "page": 1, "id": "2503.05104_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha = 0.3, 0.6, 0.9"}
{"pdf": "arxiv_math/2503.05885_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05885", "page": 1, "id": "2503.05885_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{B}^\\nu_{h,\\alpha} := \\big\\{ r \\in [1,\\infty) : m^\\nu([r, r+h]) \\geq \\frac{\\alpha h}{ r}\\big\\}"}
{"pdf": "arxiv_math/2503.05885_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05885", "page": 1, "id": "2503.05885_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{1,R}(\\overline{B}^\\nu_{h,\\alpha}) \\leq C\\alpha^{-1}"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}_{\\pm}(K_{n, p}, E[p^{\\infty}])^{\\vee} \\simeq \\Lambda_{n}^{2}/\\omega_{n}^{\\pm}\\Lambda_{n}^{2}"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}_{\\pm}(K_{\\infty, w}, E[p^{\\infty}]) \\coloneq \\varinjlim H^{1}_{\\pm}(K_{n, p}, E[p^{\\infty}])"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "E[p^{\\infty}]^{G_{K_{\\infty, p}}}=0"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}_{\\pm}(K_{\\infty, w}, E[p^{\\infty}])^{\\vee} \\simeq \\Lambda^{2}"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "B_{v_{\\infty}} \\coloneq E[p^{\\infty}]^{G_{K_{\\infty, v_{\\infty}}}}"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{\\infty, v_{\\infty}}/K_{n,v_{n}}"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "v \\in \\Sigma \\setminus \\{ p \\}"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{\\infty, v_{\\infty}}/K_{n, v_{n}}"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_019", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}_{\\pm}(K_{n, p}, E[p^{\\infty}])"}
{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "E[p^{\\infty}]^{G_{K_{\\infty}}} = 0"}
{"pdf": "arxiv_math/2503.07088_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07088", "page": 1, "id": "2503.07088_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nu=\\nu(q)=\\frac{1}{\\sqrt{1-q}}"}
{"pdf": "arxiv_math/2503.07088_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07088", "page": 1, "id": "2503.07088_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(a-x)_q^n=(-1)^nq^{\\frac{n(n-1)}{2}}\\left(x-q^{1-n}a\\right)_q^n"}
{"pdf": "arxiv_math/2503.07088_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07088", "page": 1, "id": "2503.07088_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_q^{x}= e_{\\frac{1}{q}}^x = \\sum_{k=0}^{+\\infty} q^{\\frac{k(k-1)}{2}}\\frac{x^k}{[k]_q!}"}
{"pdf": "arxiv_math/2503.07088_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07088", "page": 1, "id": "2503.07088_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "e_q^x = \\sum_{k=0}^{+\\infty}\\frac{x^k}{[k]_q!}"}
{"pdf": "arxiv_math/2503.07088_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07088", "page": 1, "id": "2503.07088_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{q^2}^{-\\frac{q^2x^2}{[2]_q}} = \\sum_{k=0}^{+\\infty}\\frac{q^{k(k+1)}(q-1)^k}{(1-q^2)_{q^2}^k} x^{2k}"}
{"pdf": "arxiv_math/2503.07351_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07351", "page": 1, "id": "2503.07351_pg7_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\|a \\wedge b\\|=\\min\\{\\|a\\|, \\|b\\|\\}"}
{"pdf": "arxiv_math/2503.07351_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07351", "page": 1, "id": "2503.07351_pg7_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\|a\\& b\\|=T(\\|a\\|, \\|b\\|)=\\|a\\|\\ast\\|b\\|"}
{"pdf": "arxiv_math/2503.07351_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07351", "page": 1, "id": "2503.07351_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|a \\wedge b\\|=T(\\|a\\|, \\|b\\|)=\\|a\\|\\ast\\|b\\|"}
{"pdf": "arxiv_math/2503.05181_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05181", "page": 1, "id": "2503.05181_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\leq x \\perp y \\geq 0"}
{"pdf": "arxiv_math/2503.05181_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05181", "page": 1, "id": "2503.05181_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f: \\mathbb{R}^n \\rightarrow \\mathbb{R}"}
{"pdf": "arxiv_math/2503.05181_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05181", "page": 1, "id": "2503.05181_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^m"}
{"pdf": "arxiv_math/2503.05181_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05181", "page": 1, "id": "2503.05181_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_{xx} f \\in \\mathbb{R}^{n \\times n}"}
{"pdf": "arxiv_math/2503.05181_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05181", "page": 1, "id": "2503.05181_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_x f \\in \\mathbb{R}^{m \\times n}"}
{"pdf": "arxiv_math/2503.05181_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05181", "page": 1, "id": "2503.05181_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "x, y \\in \\mathbb{R}^n"}
{"pdf": "arxiv_math/2503.08187_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08187", "page": 1, "id": "2503.08187_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma=0.5\\cdot\\mathbf{1}\\in\\mathbb{R}^n"}
{"pdf": "arxiv_math/2503.08187_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08187", "page": 1, "id": "2503.08187_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "z(\\theta)=\\min(\\max(\\theta,-\\frac{\\pi}{2}),\\frac{\\pi}{2})"}
{"pdf": "arxiv_math/2503.08187_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08187", "page": 1, "id": "2503.08187_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "|[g_{x'}]_i| \\ll |[g_{z'}]_i|"}
{"pdf": "arxiv_math/2503.08187_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08187", "page": 1, "id": "2503.08187_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "[\\sigma]_i^2[g_{x'}]_i^2 + (1-[\\sigma]_i)^2[g_{z'}]_i^2"}
{"pdf": "arxiv_math/2503.08187_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08187", "page": 1, "id": "2503.08187_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left([g_{x'}]_i, [g_{z'}]_i\\right)^T = \\bold{R}([{\\theta}]_i) [\\nabla \\bold{m}]_i"}
{"pdf": "arxiv_math/2503.08187_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08187", "page": 1, "id": "2503.08187_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "[g_{x'}]_i^2 + [g_{z'}]_i^2"}
{"pdf": "arxiv_math/2503.05644_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05644", "page": 1, "id": "2503.05644_pg3_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi = \\pi_0 + \\pi_1 + \\pi^\\prime"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d((x,y),L_i)<d((x,y),K)"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "d((x, y), L) < y=d((x,y), K)"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0<y_n=G(x_n)< f(x_n) < f(x)+\\varepsilon"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "d((x,y),L)=d((x,y),\\cup_{i\\in I\\ }L_i)\\leq d((x,y),L_i)<d((x,y),K)"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "y>\\min \\left\\{G_i(x)\\mid i\\in I\\right\\}"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "G_{\\min}(x)=\\min \\left\\{G_i(x)\\mid i\\in I\\right\\}"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "d((x, y_2), L)-d((x, y_1), L) < d((x,y_2), (x,y_1))=y_2 - y_1\\quad \\Rightarrow \\quad"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{G_i\\mid i\\in I\\right\\}"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "y<\\min \\left\\{G_i(x)\\mid i\\in I\\right\\}"}
{"pdf": "arxiv_math/2503.05901_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05901", "page": 1, "id": "2503.05901_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "d((x,y),K)<\\min \\left\\{d((x,y), L_i)\\mid i\\in I\\right\\}=d((x,y),\\cup_{i\\in I\\ } L_i)=d((x,y), L)"}
{"pdf": "arxiv_math/2503.08975_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08975", "page": 1, "id": "2503.08975_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "N\\in\\{1-45,47-58,61,63,64,65,67,68,72,73,75,80,81,91,109,121,125\\}"}
{"pdf": "arxiv_math/2503.08975_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08975", "page": 1, "id": "2503.08975_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "N\\in\\{1-29,31,32,34,36,37,43,45,49,50,54,64,81\\}"}
{"pdf": "arxiv_math/2503.08975_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08975", "page": 1, "id": "2503.08975_pg3_math_006", "type": "math", "max_diffs": 0, "checked": "verified", "math": "N\\in\\{1-33,35-37,39-41,43,46-50,53,59,61,65,71,79,83,89,101,131\\}"}
{"pdf": "arxiv_math/2503.06377_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06377", "page": 1, "id": "2503.06377_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma = (\\gamma_1, \\gamma_2, \\gamma_3) \\in X"}
{"pdf": "arxiv_math/2503.06377_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06377", "page": 1, "id": "2503.06377_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "X_{\\pm 3} = \\emptyset"}
{"pdf": "arxiv_math/2503.06377_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06377", "page": 1, "id": "2503.06377_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "I,J \\in \\binom{[10]}{5}"}
{"pdf": "arxiv_math/2503.06377_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06377", "page": 1, "id": "2503.06377_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in X_1 \\cup X_{-1}"}
{"pdf": "arxiv_math/2503.09115_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09115", "page": 1, "id": "2503.09115_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v_1 < v_2 < \\ldots < v_k"}
{"pdf": "arxiv_math/2503.09115_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09115", "page": 1, "id": "2503.09115_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "v_1 <_x v_2 <_x \\ldots <_x v_k"}
{"pdf": "arxiv_math/2503.09123_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09123", "page": 1, "id": "2503.09123_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\in\\mathbb{I}_{P^{(k)}}"}
{"pdf": "arxiv_math/2503.09123_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09123", "page": 1, "id": "2503.09123_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{t_0}^{t_f} \\dot{u}^{\\top}(t)R\\dot{u}(t)\\,dt"}
{"pdf": "arxiv_math/2503.09123_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09123", "page": 1, "id": "2503.09123_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\in\\mathbb{I}_{N^{(k)}}"}
{"pdf": "arxiv_math/2503.08773_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08773", "page": 1, "id": "2503.08773_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta^{i\\, \\tau}=e^{-i K \\tau}"}
{"pdf": "arxiv_math/2503.08773_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08773", "page": 1, "id": "2503.08773_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A=\\cup_{i=1}^N (l_i,r_i)"}
{"pdf": "arxiv_math/2503.05468_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05468", "page": 1, "id": "2503.05468_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varnothing \\rightarrow u_1 \\rightarrow u_1u_2 \\rightarrow ... \\rightarrow u_1...u_n=u"}
{"pdf": "arxiv_math/2503.07765_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07765", "page": 1, "id": "2503.07765_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pr[A\\cup B]\\le \\Pr[A]+\\Pr[B]"}
{"pdf": "arxiv_math/2503.07765_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07765", "page": 1, "id": "2503.07765_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "[-\\rho_{\\max},+\\rho_{\\max} ]"}
{"pdf": "arxiv_math/2503.07875_pg21.pdf", "url": "https://arxiv.org/pdf/2503.07875", "page": 1, "id": "2503.07875_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma_2^* = \\left(1,0,0,0,\\frac{1}{c_{1v}},0,0\\right)"}
{"pdf": "arxiv_math/2503.08235_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08235", "page": 1, "id": "2503.08235_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_\\mu=\\frac{\\partial}{\\partial \\lambda_\\mu}"}
{"pdf": "arxiv_math/2503.08235_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08235", "page": 1, "id": "2503.08235_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf V(\\hat{\\bm \\lambda})"}
{"pdf": "arxiv_math/2503.08235_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08235", "page": 1, "id": "2503.08235_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "F_{\\mu\\nu}=\\sum_k p_{\\bm \\lambda}(k)\\,\\partial_\\mu \\log p_{\\bm \\lambda}(k)\\, \\partial_\\nu \\log p_{\\bm \\lambda}(k) = \\sum_k \\frac{\\partial_\\mu p_{\\bm \\lambda}(k)\\, \\partial_\\nu p_{\\bm \\lambda}(k)}{p_{\\bm \\lambda}(k)} \\,"}
{"pdf": "arxiv_math/2503.08235_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08235", "page": 1, "id": "2503.08235_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{k} \\Pi_k=\\mathbb{I}"}
{"pdf": "arxiv_math/2503.08235_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08235", "page": 1, "id": "2503.08235_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "E_k(\\hat \\lambda_\\mu)"}
{"pdf": "arxiv_math/2503.08235_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08235", "page": 1, "id": "2503.08235_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bm E_\\nu(\\hat {\\bm \\lambda})=\\hat{\\bm \\lambda}\\, \\qquad \\partial_\\mu E_k(\\hat \\lambda_\\nu)=\\delta_\\mu^\\nu"}
{"pdf": "arxiv_math/2503.08235_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08235", "page": 1, "id": "2503.08235_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf V(\\hat{\\bm \\lambda})\\geq \\frac{1}{M \\mathbf F}\\,"}
{"pdf": "arxiv_math/2503.08235_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08235", "page": 1, "id": "2503.08235_pg3_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\bm\\lambda=(\\lambda_0, \\lambda_1,\\ldots, \\lambda_d)^T"}
{"pdf": "arxiv_math/2503.06540_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06540", "page": 1, "id": "2503.06540_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x}_\\mathrm{n}(t)"}
{"pdf": "arxiv_math/2503.06540_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06540", "page": 1, "id": "2503.06540_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "% \\mathbf{a}(\\theta)=\\dfrac{1}{\\sqrt M} \\Big[1, \\ e^{j % 2\\pi\\bar{d} \\sin\\theta}, \\ \\cdots, \\ e^{j % 2\\pi (M-1)\\bar{d}\\sin\\theta}\\Big]^\\mathrm{T}\\!\\!,\\raisetag{15pt} %"}
{"pdf": "arxiv_math/2503.06540_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06540", "page": 1, "id": "2503.06540_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{a}(\\theta)=\\frac{1}{\\sqrt M} \\Big[1, \\ e^{j 2\\pi\\bar{d} \\sin\\theta}, \\ \\cdots, \\ e^{j 2\\pi (M-1)\\bar{d}\\sin\\theta}\\Big]^\\mathrm{T}"}
{"pdf": "arxiv_math/2503.06540_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06540", "page": 1, "id": "2503.06540_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x}(t)= \\mathbf{x}_\\mathrm{s}(t)+\\mathbf{x}_\\mathrm{i}(t)+\\mathbf{x}_\\mathrm{n}(t)"}
{"pdf": "arxiv_math/2503.08457_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08457", "page": 1, "id": "2503.08457_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathbb{H}}^k\\mathbb{E}^0 = (\\mathbb{E}^0\\otimes 1)\\tilde{\\mathbb{H}}^k"}
{"pdf": "arxiv_math/2503.08457_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08457", "page": 1, "id": "2503.08457_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma(X, \\underline{F})"}
{"pdf": "arxiv_math/2503.08457_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08457", "page": 1, "id": "2503.08457_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\mathcal{S}}_X(X)"}
{"pdf": "arxiv_math/2503.08457_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08457", "page": 1, "id": "2503.08457_pg18_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\mathcal{S}}_X"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "X(\\theta_{i}): \\begin{bmatrix}x_{i} \\\\ p_{i} \\end{bmatrix} \\mapsto \\begin{bmatrix} \\cos(\\theta_{i}) & \\sin(\\theta_{i}) \\\\ -\\sin(\\theta_{i}) & \\cos(\\theta_i) \\end{bmatrix}\\begin{bmatrix}x_{i} \\\\ p_{i} \\end{bmatrix}"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "F_{SL(2n)} \\coloneqq \\frac{E_{SL(2n)}}{E[f_{0}]}"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{Sp(2)} = E_{SL(2)}"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "f_{0}\\circ \\phi^{-1} = \\frac{1}{|B(R)|} \\Theta(R^2-\\epsilon^{-1/2}(x^2+p_{x}^2+p_{y}^2)-\\epsilon^{3/2}y^2)"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_{i}^{H} \\lambda_{n+1-i}^{V} \\equiv \\text{const}"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "f_{0}(\\mathbf{z},R) = \\frac{6}{R^2 |B(R)|} \\chi_{B(R)} = \\frac{6}{R^2|B(R)|} \\Theta(R^2-|\\mathbf{z}|^2)"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "V(\\epsilon) = \\text{diag}(1,\\epsilon^2,1,1)"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{V} = \\text{diag}( 1, \\epsilon^{-1/2}, 1, \\epsilon^{1/2})"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}(\\mathbf{z},\\epsilon) = x^2 + \\epsilon^2 y^2 + p_{x}^2 + p_{y}^2"}
{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "E[f_{0}] = (3 + \\epsilon)"}
{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_2= \\frac{D_{\\rm W}}{4}"}
{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta = \\frac{c^2}{16\\pi^2 f_c^2 }"}
{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\psi}_m= (x_m,y_m,0)"}
{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{h}_{mk}=\\alpha_{mk}h_{mk}"}
{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\boldsymbol \\psi}_m^{\\rm Pin}=(x_m,\\beta_m,d)"}
{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_1=-\\frac{D_{\\rm W}}{4}"}
{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_m=-\\frac{D_{\\rm W}}{2}+(m-1)\\frac{D_{\\rm W}}{M}+\\frac{D_{\\rm W}}{2M}"}
{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "h_{mk}=\\frac{\\sqrt{\\eta} e^{-2\\pi j \\left(\\frac{ 1}{\\lambda}\\left| {\\boldsymbol \\psi}_m - {\\boldsymbol \\psi}_k^{\\rm Pin}\\right| +\\frac{1}{\\lambda_g}\\left| {\\boldsymbol \\psi}_0^{\\rm Pin} - {\\boldsymbol \\psi}_k^{\\rm Pin}\\right| \\right)}}{ \\left| {\\boldsymbol \\psi} _m - {\\boldsymbol \\psi}_k^{\\rm Pin}\\right|}"}
{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_0 = \\mathbb{C}\\,I_{\\mathcal{H}}"}
{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{B}(\\mathcal{H})"}
{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_s\\,\\mathcal{E}_t \\subseteq \\mathcal{E}_{s+t}"}
{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_0 = \\mathbb{C}\\,I_{\\mathcal{H}}, \\quad\\text{and for each integer } t \\ge 1"}
{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\mathcal{E}_t\\}_{t \\ge 0} \\;\\subseteq\\; \\mathcal{B}(\\mathcal{H})"}
{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_t^* = \\mathcal{E}_t"}
{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E} \\;\\subseteq\\; \\mathcal{B}(\\mathcal{H})"}
{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0\\in\\{\\frac{1}{3},\\frac{2}{3}\\}"}
{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}_{\\bm{y}}[F(\\bm{y})]"}
{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "F(\\bm{y})=G(u^s(\\cdot,\\bm{y}))=u^s(x_0,\\bm{y})"}
{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "y_1,\\ldots,y_s\\overset{i.i.d.}{\\sim}N(0,1)"}
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{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_j(\\frac{n}{N})"}
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{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "u^s(0,\\bm{y})=u^s(1,\\bm{y})=0"}
{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\frac{d}{dx}(a^s(x,\\bm{y})\\frac{du^s(x,\\bm{y})}{dx})=1"}
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{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "R=\\{(\\alpha,\\beta,\\gamma)\\in[0,1]^3:\\alpha\\beta+\\gamma>1,\\alpha\\gamma+\\beta>1,\\beta\\gamma+\\alpha>1\\}"}
{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "E(H)\\subset \\binom{[n]}{r}"}
{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\alpha,\\beta,\\gamma)\\in R_1"}
{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_004", "type": "math", "max_diffs": 0, "checked": "verified", "math": "T_{min}(\\alpha,\\beta,\\gamma)=\\alpha+\\beta+\\gamma-2"}
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{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x \\in \\complement \\overline{V_i} \\cap U_i"}
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{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_008", "type": "math", "max_diffs": 0, "checked": "verified", "math": "L_i(u,t) = \\begin{cases} p_i(H(u,2t)) & \\text{if } 0 \\leq t \\leq \\frac12 \\\\ G(h_i(u), 2t-1) & \\text{if } \\frac12 \\leq t \\leq 1. \\end{cases}"}
{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi\\colon X\\to T^n(\\mu)"}
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{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c \\in \\mathbb{F}_p^{\\times}"}
{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "I(C_n^{\\boxtimes d}, x)"}
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{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "I(C_n^{\\boxtimes d}, x) \\equiv c \\cdot (x+1)^m \\pmod{p}"}
{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{Frob}_p: \\, a \\mapsto a^p"}
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{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "I(C_n^{\\boxtimes d}, x) \\equiv (x+1)^m \\pmod{p}"}
{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(x+1)^{p^d} \\equiv x^{p^d} + 1 \\pmod{p}"}
{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "F(x) = \\sum_{S \\in \\mathcal{F}_p} x^{|S|}"}
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{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "0<\\varepsilon<\\varepsilon_{\\mathbf{I}}"}
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{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_009", "type": "math", "max_diffs": 0, "checked": "verified", "math": "L\\coloneqq 2e^{\\delta^{-1}k_\\mathbf{I}(e_\\mathbf{I}(\\varphi) + 10^{-1}\\varepsilon)}"}
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{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta' = \\{W', \\mathbf{b}'\\}"}
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{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{u} \\in [0, 1]^{d'}"}
{"pdf": "arxiv_math/2503.04397_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04397", "page": 1, "id": "2503.04397_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\left\\| {\\cdot} \\right\\|_2}"}
{"pdf": "arxiv_math/2503.04397_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04397", "page": 1, "id": "2503.04397_pg2_math_001", "type": "math", "max_diffs": 0, "checked": "verified", "math": "{\\cal CN}(0,\\sigma^2)"}
{"pdf": "arxiv_math/2503.04397_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04397", "page": 1, "id": "2503.04397_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{diag}\\left({\\mathbf a} \\right)"}
{"pdf": "arxiv_math/2503.04397_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04397", "page": 1, "id": "2503.04397_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\left( {\\cdot} \\right)^{\\rm{H}}}"}
{"pdf": "arxiv_math/2503.04397_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04397", "page": 1, "id": "2503.04397_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left| {\\cdot} \\right|"}
{"pdf": "arxiv_math/2503.04397_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04397", "page": 1, "id": "2503.04397_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}\\left( {\\cdot} \\right)"}
{"pdf": "arxiv_math/2503.09181_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09181", "page": 1, "id": "2503.09181_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Lambda^{(n)}=\\Lambda"}
{"pdf": "arxiv_math/2503.09181_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09181", "page": 1, "id": "2503.09181_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x_s=\\{x_i : i \\in s \\subset [d]\\}"}
{"pdf": "arxiv_math/2503.09181_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09181", "page": 1, "id": "2503.09181_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat y^{(n)} = f(x_s^{(n)},z^{(n)})"}
{"pdf": "arxiv_math/2503.09181_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09181", "page": 1, "id": "2503.09181_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\min_{\\theta,\\phi} \\mathbb{E}_{\\mathbf{y}, \\mathbf{x}} l(f_\\theta(\\mathbf{x}_s,z),\\mathbf{y})"}
{"pdf": "arxiv_math/2503.09181_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09181", "page": 1, "id": "2503.09181_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "s \\leftarrow s \\cup \\pi(x_s,z)"}
{"pdf": "arxiv_math/2503.09181_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09181", "page": 1, "id": "2503.09181_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(x^{(n)}_s, z^{(n)}) \\in \\Lambda^{(n)}"}
{"pdf": "arxiv_math/2503.06825_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06825", "page": 1, "id": "2503.06825_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v, w \\in \\mathbb{R}^n"}
{"pdf": "arxiv_math/2503.06825_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06825", "page": 1, "id": "2503.06825_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "B \\in \\mathbb{R}^{n \\times l}"}
{"pdf": "arxiv_math/2503.06825_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06825", "page": 1, "id": "2503.06825_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "A \\in \\mathbb{R}^{n \\times n}"}
{"pdf": "arxiv_math/2503.06825_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06825", "page": 1, "id": "2503.06825_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "C \\in \\mathbb{R}^{m \\times n}"}
{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{GoE}_m(t) = \\frac{u_m(t)}{\\Delta_m(t)}, \\forall m"}
{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{B}(\\cdot, \\cdot)"}
{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "v_{\\rm cpt}(x) = \\begin{cases} v^+_{\\rm cpt}(x) = (x - x_{\\rm ref})^{\\alpha_{\\rm cpt}}, ~x \\geq x_{\\rm ref};\\\\ v^-_{\\rm cpt}(x) = -\\lambda_{\\rm cpt} (x_{\\rm ref} - x)^{\\beta_{\\rm cpt}}, ~x < x_{\\rm ref}, \\end{cases}"}
{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{\\mathcal{P}}"}
{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_m, \\beta_m>0, \\forall m"}
{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle s(t), a(t), r(t), s(t+1) \\rangle_{t=0}^{T_{\\rm e}}"}
{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_008", "type": "math", "max_diffs": 0, "checked": "verified", "math": "w_{\\rm cpt}(x)=x, \\forall x"}
{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "g_m(t; y_m(t)) = \\operatorname{min}\\left\\{ 1, \\frac{y_m^{\\alpha_m-1}(t) (1 - y_m(t))^{\\beta_m-1}}{\\operatorname{B}(\\alpha_m, \\beta_m)} \\right\\}"}
{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{\\rm cpt}=\\beta_{\\rm cpt}=0.5"}
{"pdf": "arxiv_math/2503.07083_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07083", "page": 1, "id": "2503.07083_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{P}(T\\to \\infty)"}
{"pdf": "arxiv_math/2503.07083_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07083", "page": 1, "id": "2503.07083_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta(2)\\equiv \\langle T\\rangle =2"}
{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "h^{\\mathrm{BdG}}_8(\\mathbf{k})=\\frac{1}{2}\\left[\\sin k_x\\sigma^x+\\sin k_y \\sigma^z +\\sin k_z\\tau^z\\sigma^y\\right]"}
{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat S_0(\\textbf{k}) = \\tau^z \\qquad \\hat{Q}_0=\\sum_\\mathbf{k} d_{\\mathbf{k}}^\\dag \\hat S_0(\\mathbf{k})d_{\\mathbf{k}} \\\\ \\hat S_1(\\textbf{k}) = \\cos k_z \\tau^z + \\sin k_z \\tau^x \\qquad \\hat{Q}_1=\\sum_\\mathbf{k} d_{\\mathbf{k}}^\\dag \\hat S_1(\\mathbf{k})d_{\\mathbf{k}}"}
{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_002", "type": "math", "max_diffs": 0, "checked": "verified", "math": "d^\\dag_\\mathbf{k}\\equiv(c^\\dag_{\\mathbf{k}\\uparrow},c^\\dag_{\\mathbf{k}\\downarrow},c_{-\\mathbf{k}\\uparrow},c_{-\\mathbf{k}\\downarrow})"}
{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "m_j(\\mathbf{k})\\eta^z\\sigma^j"}
{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "e_{\\mathbf{k}}\\equiv(c_{\\mathbf{k}-\\mathbf{K}},c^\\dag_{-\\mathbf{k}+\\mathbf{K}},c_{\\mathbf{k}+\\mathbf{K}},c^\\dag_{-\\mathbf{k}-\\mathbf{K}})^T"}
{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textbf{k} = \\pm \\mathbf{K}"}
{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta = i\\sigma^y \\mathcal{K}"}
{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{k}+2\\mathbf{K}"}
{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat S_{1,K}(\\textbf{k}) = \\cos K\\ \\tau^z + \\sin K\\ \\tau^x"}
{"pdf": "arxiv_math/2503.09398_pg1.pdf", "url": "https://arxiv.org/pdf/2503.09398", "page": 1, "id": "2503.09398_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{V}=[ \\mathbf{v}_1, \\dots, \\mathbf{v}_K] \\in \\mathbb{C}^{N\\times K}"}
{"pdf": "arxiv_math/2503.09398_pg1.pdf", "url": "https://arxiv.org/pdf/2503.09398", "page": 1, "id": "2503.09398_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{h}_k\\in \\mathbb{C} ^{N\\times 1}"}
{"pdf": "arxiv_math/2503.09398_pg1.pdf", "url": "https://arxiv.org/pdf/2503.09398", "page": 1, "id": "2503.09398_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{H}=[\\mathbf{h}_1, \\dots, \\mathbf{h}_K] \\in \\mathbb{C}^{N\\times K}"}
{"pdf": "arxiv_math/2503.09398_pg1.pdf", "url": "https://arxiv.org/pdf/2503.09398", "page": 1, "id": "2503.09398_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "( \\cdot ) ^{\\mathrm{H}}"}
{"pdf": "arxiv_math/2503.09398_pg1.pdf", "url": "https://arxiv.org/pdf/2503.09398", "page": 1, "id": "2503.09398_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overrightarrow{( \\cdot ) }"}
{"pdf": "arxiv_math/2503.09398_pg1.pdf", "url": "https://arxiv.org/pdf/2503.09398", "page": 1, "id": "2503.09398_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{v}_k\\in \\mathbb{C} ^{N\\times 1}"}
{"pdf": "arxiv_math/2503.09398_pg1.pdf", "url": "https://arxiv.org/pdf/2503.09398", "page": 1, "id": "2503.09398_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "( \\cdot ) ^{\\mathrm{T}}"}
{"pdf": "arxiv_math/2503.09398_pg1.pdf", "url": "https://arxiv.org/pdf/2503.09398", "page": 1, "id": "2503.09398_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Tr}\\left( \\cdot \\right)"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi = (\\varphi_1,\\varphi_2, \\varphi_3) \\colon \\tilde{U}(B) \\rightarrow [X]"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "[x,y]\\alpha=[x\\alpha, y]-x[f,y]"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "f \\cdot g=(f \\ast (f' \\ast -), (- \\ast f) \\ast f'), f \\ast [f',-] + [f,-] \\ast f')"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "[x,yz]=[x,y]z+y[x,z], \\quad \\forall x,y,z \\in X"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha [x,y]=[\\alpha x, y]-[f,y]x"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_008", "type": "math", "max_diffs": 0, "checked": "verified", "math": "\\cdots\\vert \\;f \\ast [x,y]=[f \\ast x,y]-[f,y]x,  [x,y]*f=[x \\ast f,y]-x[f,y],  [f,xy]=[f,x]y+x[f,y] \\rbrace"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "(X^{2}, \\cdot)=(X,\\cdot)"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "[f,g]=(f \\ast [f',-] - [f',f \\ast -], [f',-] \\ast f - [f',- \\ast f], [f,[f',-]]-[f',[f,-]])"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdot \\colon X \\times X\\rightarrow X \\quad \\text{ and } \\quad [-,-] \\colon X \\times X \\rightarrow X"}
{"pdf": "arxiv_math/2503.04488_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04488", "page": 1, "id": "2503.04488_pg9_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "(b,x) \\cdot (b',x') = (bb', xx'+ \\varphi(b)x'+x\\varphi(b'))"}
{"pdf": "arxiv_math/2503.06877_pg16.pdf", "url": "https://arxiv.org/pdf/2503.06877", "page": 1, "id": "2503.06877_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(U_\\ast,\\lambda_\\ast)"}
{"pdf": "arxiv_math/2503.06877_pg16.pdf", "url": "https://arxiv.org/pdf/2503.06877", "page": 1, "id": "2503.06877_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi: \\mathrm{M} \\to \\mathbb{R}^n"}
{"pdf": "arxiv_math/2503.06877_pg16.pdf", "url": "https://arxiv.org/pdf/2503.06877", "page": 1, "id": "2503.06877_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i'=2}^k n_{i'} \\geq n_1 +k-1,\\quad \\sum_{i'=2}^k n_{i'} \\ge n_{i''} + k,\\quad 2 \\le i'' \\le k"}
{"pdf": "arxiv_math/2503.06877_pg16.pdf", "url": "https://arxiv.org/pdf/2503.06877", "page": 1, "id": "2503.06877_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ (U_{[p]},\\lambda_{[p]})\\}"}
{"pdf": "arxiv_math/2503.06877_pg16.pdf", "url": "https://arxiv.org/pdf/2503.06877", "page": 1, "id": "2503.06877_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal A\\in\\mathbb R^{n_1}\\otimes\\dots\\otimes\\mathbb R^{n_k}"}
{"pdf": "arxiv_math/2503.06877_pg16.pdf", "url": "https://arxiv.org/pdf/2503.06877", "page": 1, "id": "2503.06877_pg16_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "n_1 = \\cdots =n_k \\ge 2"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "D(\\widetilde{M})=\\Omega\\times Y"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{M}=\\widehat{\\Omega}\\times Y"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi:X_{\\kappa}\\times Y\\to Y"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "D\\gamma =\\rho(\\gamma)D"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega\\subset X_{\\kappa}"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma^{-1}(y)=\\widehat{\\Omega}"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{\\Omega}=D^{-1}\\left(\\Omega\\times\\left\\{y\\right\\}\\right)"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "D:\\widetilde{M}\\to X_{\\kappa}\\times Y"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{x_0\\right\\}\\times Y"}
{"pdf": "arxiv_math/2503.07843_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07843", "page": 1, "id": "2503.07843_pg3_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{\\Omega}\\times Y"}
{"pdf": "arxiv_math/2503.07802_pg90.pdf", "url": "https://arxiv.org/pdf/2503.07802", "page": 1, "id": "2503.07802_pg90_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{Q} \\subset B_1"}
{"pdf": "arxiv_math/2503.03879_pg4.pdf", "url": "https://arxiv.org/pdf/2503.03879", "page": 1, "id": "2503.03879_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "h_{l-1} - h_{l} = \\Delta h_{l}^+ - \\Delta h_{l}^-"}
{"pdf": "arxiv_math/2503.03879_pg4.pdf", "url": "https://arxiv.org/pdf/2503.03879", "page": 1, "id": "2503.03879_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "G(x) \\geq 0, \\; H(x) \\geq 0, \\; G_i(x)H_i(x) = 0, \\quad \\forall i = 1 \\dots n_c"}
{"pdf": "arxiv_math/2503.03879_pg4.pdf", "url": "https://arxiv.org/pdf/2503.03879", "page": 1, "id": "2503.03879_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "2N \\{\\hat \\lambda, \\hat \\nu\\} + (2N-2) \\{\\pi^{\\lambda},\\pi^{\\nu}\\} + (N-1) \\{\\tau\\} + (N-1) \\{\\eta\\}"}
{"pdf": "arxiv_math/2503.03879_pg4.pdf", "url": "https://arxiv.org/pdf/2503.03879", "page": 1, "id": "2503.03879_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{\\lambda_l} = \\sum^K_{k=1} \\lambda_{l,k}, \\quad \\hat{\\nu_l} = \\sum^K_{k=1} \\nu_{l,k}"}
{"pdf": "arxiv_math/2503.03879_pg4.pdf", "url": "https://arxiv.org/pdf/2503.03879", "page": 1, "id": "2503.03879_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{l=1}^N h_{l} = t_f"}
{"pdf": "arxiv_math/2503.03879_pg4.pdf", "url": "https://arxiv.org/pdf/2503.03879", "page": 1, "id": "2503.03879_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\leq (\\Delta h_{l}^+ + \\Delta h_{l}^-) \\perp \\eta_l \\geq 0"}
{"pdf": "arxiv_math/2503.03879_pg4.pdf", "url": "https://arxiv.org/pdf/2503.03879", "page": 1, "id": "2503.03879_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta h_{l}^+, \\Delta h_{l}^- \\geq 0"}
{"pdf": "arxiv_math/2503.03879_pg4.pdf", "url": "https://arxiv.org/pdf/2503.03879", "page": 1, "id": "2503.03879_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi^{\\lambda}_l = \\hat{\\lambda}_{l-1} \\odot \\hat{\\lambda_l}, \\quad \\pi^{\\nu}_l = \\hat{\\nu}_{l-1} \\odot \\hat{\\nu_l}"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_000", "type": "math", "max_diffs": 0, "checked": "verified", "math": "E_{\\infty}^{p,q}\\in \\mathscr{S}"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\leqslant i \\leqslant n"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "H^i(K(J), M)\\in \\mathscr{S}"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(1)\\Leftrightarrow (2)"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\leqslant q \\leqslant n"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\to N_{j-1}\\to N_j\\to N_i/N_{j-1}\\to 0"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "H^i(K(I), M)\\in \\mathscr{S}"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_018", "type": "math", "max_diffs": 0, "checked": null, "math": "E_2^{p,q}\\in \\mathscr{S}"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_020", "type": "math", "max_diffs": 0, "checked": null, "math": "0=N_0\\subset N_1\\subset \\cdots \\subset N_k=N"}
{"pdf": "arxiv_math/2503.06354_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06354", "page": 1, "id": "2503.06354_pg6_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "I=(x_1,x_2,\\cdots,x_n)"}
{"pdf": "arxiv_math/2503.04577_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04577", "page": 1, "id": "2503.04577_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tfrac{1}{p} \\|x-y\\|^p"}
{"pdf": "arxiv_math/2503.07453_pg68.pdf", "url": "https://arxiv.org/pdf/2503.07453", "page": 1, "id": "2503.07453_pg68_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_\\ell(x_\\ell,a_\\ell)\\in \\Gamma"}
{"pdf": "arxiv_math/2503.07453_pg68.pdf", "url": "https://arxiv.org/pdf/2503.07453", "page": 1, "id": "2503.07453_pg68_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\hat{x}_h,\\hat{a}_{h})"}
{"pdf": "arxiv_math/2503.07453_pg68.pdf", "url": "https://arxiv.org/pdf/2503.07453", "page": 1, "id": "2503.07453_pg68_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_{\\min}(\\Sigma_h)\\geq \\lambda"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "X(x) := X \\otimes_x O"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{v}^{(p)} \\triangleleft I_v"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "C^{\\bullet}(I_v , V )"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "V^{I_{v}^{(p)}} \\otimes_R S \\rightarrow (V \\otimes_R S)^{I_{v}^{(p)}}"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "e := \\frac{1}{\\# \\varphi(I_v^{(p)})}\\sum_{g \\in \\varphi(I_v^{(p)})}g \\in O[\\varphi(I_v^{(p)})]"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(V/V^{I_v^{(p)}} \\otimes_{R} S)^{I_v^{(p)}} = e(V/V^{I_v^{(p)}} \\otimes_{R} S) = e(V/V^{I_v^{(p)}}) \\otimes_{R} S = 0\\,"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdots \\rightarrow 0 \\rightarrow V^{I_{v}^{(p)}} \\xrightarrow{t-1} V^{I_{v}^{(p)}} \\rightarrow 0 \\rightarrow \\cdots"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1(I_{v},V)^{\\mathrm{Fr}_{v}=1}_{R-\\mathrm{tors}}"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\longrightarrow V^{I_v^{(p)}}\\otimes_{R} S \\longrightarrow V \\otimes_{R} S \\longrightarrow V/V^{I_v^{(p)}} \\otimes_{R} S \\longrightarrow 0"}
{"pdf": "arxiv_math/2503.07542_pg14.pdf", "url": "https://arxiv.org/pdf/2503.07542", "page": 1, "id": "2503.07542_pg14_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1(I_v, V) \\cong V^{I_v^{(p)}}"}