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\begin{align*}\tilde v:=\hat v-(\hat v)_{B_{R/2}(x_0)}\end{align*} |
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\begin{align*}P_{rT,\Omega }f=\sum a_{k}\lambda _{k}\phi _{k} \end{align*} |
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\Psi _ { r } ^ { B } = U _ { B } { \psi } _ { r } ^ { B } = n _ { B } ^ { 1 / 2 } U _ { B } \stackrel { \sim } { \psi } _ { r } ^ { B } |
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\begin{align*}{\cal L}_b(\frac{1}{2},\alpha,g)= \bar\psi [ \sigma_\mu (\partial_\mu + i a_\mu+ i C_\mu ) + M + ib ] \psi - \frac{i}{8 \pi \alpha} \epsilon_{\mu \nu \lambda} a_\mu \partial_\nu a_\lambda+ \frac{1}{g}b^2\;.\end{align*} |
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\begin{align*}\chi_{\infty ,N} (x) = ((N - 1)!)^{- 1} \int_{0}^{x^{- 1}}t^{N - 1} e^{- t}dt\end{align*} |
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\begin{align*}R(a)R(b) = R\left(R(a) b + a S(b)\right) , \\S(a)S(b) = S\left(R(a) b + a S(b)\right) . \end{align*} |
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\begin{align*}A&=M\begin{bmatrix}x_1 & 1\\ 0& x_1\end{bmatrix}M^{-1}.\end{align*} |
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\delta S = 2 \varepsilon \, g _ { \mu \nu } \Theta ^ { \mu \nu } = 0 . |
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\begin{align*} \varphi(s)=1, {\rm \ for \ } s \in [-\frac 34, \frac 34], {\rm \ and \ } \varphi (s)=0, {\rm \ for \ } \vert s \vert \geq 1.\end{align*} |
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\begin{align*}\hat{\chi}^*_{[a_0a_1)} (y|x) = \chi_{[a_0a_1)} (y_2,y_1 x) = \chi_{(a_1a_0]}(y) \ ,\end{align*} |
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\begin{align*}\sum_{n=0}^{\infty} \frac{q^{n}}{(zq^{n+1};q)_{n+2} (zq^{2n+4};q^2)_{\infty}}=\sum_{n=0}^{\infty}\frac{z^nq^{2n^2+2n}}{(q;q^2)_{n+1}(zq;q^2)_{n+1}}.\end{align*} |
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\begin{align*}E(T^*_{m,N})=(N-1)\sum_{j\leq m} \hat{\nu}(j) \sum_{k=j}^{N-1}{1\over N-k}.\end{align*} |
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\begin{align*}G^{d\,MN}(P_M-qA_M)(P_N-qA_N)+m^2e^{-2{{D-4}\over{D-3}}\phi}=0,\end{align*} |
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\begin{align*}0\rightarrow \Omega^{2}(2)\rightarrow {\cal O}^{\oplus 10}\rightarrow{\cal O}^{\oplus 5}(1)\rightarrow{\cal O}(2)\rightarrow 0\end{align*} |
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\begin{align*}\textrm{N}_{\vec d, I}= \bigoplus_{i\in I,\,\,j\in [N],\,\, i\ne j}\pi_*\left(\mathcal{K}_i^{\vee}\right)-\bigoplus_{i,j\in I,\,\,i\ne j}\pi_*\left(\mathcal{K}^{\vee}_i \otimes\mathcal{K}_j\right) .\end{align*} |
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\begin{align*}2(q-1)=i(n-t+2)+j,\quad\ 0\le j\le n-t+1.\end{align*} |
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\begin{align*}f(x)=\int_{\mathcal{Y}^{N}}g(y_{[1,N]})dP^{\mu}(y_{[1,N]}|x_{1}=x)\end{align*} |
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\begin{align*}var[Y_k] & = k\sum_{j=1}^k \frac{(k-j+1)(k-j)}{j^2}\\& =k \sum_{j=1}^k \frac{k^2-2kj+j^2+k-j}{j^2}\\& = (k^3+k^2)\sum_{j=1}^k \frac{1}{j^2} -(2k^2 +k)\sum_{j=1}^k \frac{1}{j} +k^2\\& \sim \frac{\pi^2}{6} k^3.\end{align*} |
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\begin{align*}&\big( S^{\wedge, (l)} \:V\: S^{\wedge, (r)} \big)(x,y) \\&= \sum_{n=0}^\infty \frac{1}{n!} \int_0^1 \alpha^{l} \:(1-\alpha)^r\:(\alpha - \alpha^2)^n \: (\Box^n V) \big|_{\alpha y + (1-\alpha) x} \:d\alpha\;S^{\wedge, (n+l+r+1)}(x,y) \:.\end{align*} |
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\begin{align*}F_N(a,b;t)=\frac{(1-tq^N)(1-b)}{(1-bq^N)(1-t)}\sum_{n=0}^{N}\frac{(q^{-N})_n(atq/b)_n(q)_nq^n}{(tq)_n(q^{1-N}/b)_n(q)_n}.\end{align*} |
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\begin{align*}\Phi(\{\phi_i,\phi_j\}) = [\Phi(\phi_i),\Phi(\phi_j)].\end{align*} |
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\begin{align*}(A+F)y=(b+f).\end{align*} |
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\begin{align*}2\partial_{\bar z}(\mathcal P_1\Theta^{-1}{\mathcal P_2^*})+A_1(\mathcal P_1\Theta^{-1}{\mathcal P_2^*})-(\mathcal P_1\Theta^{-1}{\mathcal P_2^*})A_2=0\quad\mbox{in}\,\,\Omega \setminus \mathcal X.\end{align*} |
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\begin{align*}\sum_{k=0}^\infty\binom{2k+e}{k}x^k=\frac{1}{(1-2\beta)(1-\beta)^e},\end{align*} |
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\begin{align*}\mathcal{D}_{K3}:=\left\{ [\omega]\in\mathbb{P}((U(2)^{\oplus2}\oplus A_{1}^{\oplus2})\otimes\mathbb{C})\mid\omega.\omega=0,\omega.\bar{\omega}>0\right\} ^{+},\end{align*} |
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\begin{align*}-3F(2\pi/3)+\sum_{i=1}^{3}F(\theta_{i})\leq-\frac{1}{\pi^{2}}\frac{1}{2}\lambda_{1,n}^{r,s}.\leq-\frac{9}{4\pi^{2}}\lambda_{1,n}^{r,s}\sum_{i=1}^{3}\Big(\frac{\theta_{i}}{2\pi}-1/3\Big)^{2}.\end{align*} |
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\begin{align*}\lim_{k \to \infty} \frac{| \alpha(q_k)-\alpha(p_k)|}{|q_k-p_k|} = L_g(\alpha)\end{align*} |
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\begin{align*}G_{S}(X)-G_{S}(\overline{X})=G'_{S}(\overline{X})H+O(\|H\|^{2})=G_S'(\overline{X})H+O(\|H\|^{2})\,,\end{align*} |
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\begin{align*}Tr(T)=\sum_{\widetilde s\in G/H}\varphi_T(s)=\sum_{\widetilde s\in G/H}(\delta_s,T\delta_s) T\in \big<N,B\big>\end{align*} |
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\begin{align*}I(U_*)= I(u_0)+k\left[A-\frac{B_1k^{N-2}}{r^{N-2}\lambda^{N-2}}+\frac{B_2u_0(r)}{\lambda^{\frac{N-2}{2}}}+O(\frac{1}{\lambda^{\frac{N-2}{2}(1+\delta)}})\right]\end{align*} |
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\begin{align*}\xi(s,u) = \begin{cases} \frac{y - b(s,0) - u}{t-t_1}, &\| u \| < R, \\0, &,\end{cases}\end{align*} |
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{ \cal L } ^ { c t } ( \phi , \lambda ) = { \frac { 1 } { 2 } } A ( \lambda ) \partial _ { \mu } \phi \partial ^ { \mu } \phi - { \frac { 1 } { 2 } } B ( \lambda ) m ^ { 2 } \phi ^ { 2 } - { \frac { 1 } { 4 ! } } C ( \lambda ) \phi ^ { 4 } . |
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\begin{align*} \sum_{n\le x} (-1)^{n-1} \frac1{\beta(n)} = K_3\log x+ K_4+O(x^{-u})\end{align*} |
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\begin{align*}\theta(v)x=\theta(\beta(\theta^{-1}(x)))\theta(v),\ \ \ x\in M,\end{align*} |
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\begin{align*}(\lambda^{s},\lambda^{t})=(\lambda^{2n-1-s},\lambda^{2n-1-t}),\end{align*} |
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\begin{align*}0=\Delta_f\phi=\frac{\partial^2 \phi}{\partial t^2}-k_g\frac{\partial \phi}{\partial t}+\frac{\partial^2 \phi }{\partial x}-\frac{\partial f}{\partial t}\frac{\partial \phi}{\partial t}-\frac{\partial f}{\partial x}\frac{\partial \phi}{\partial x}.\end{align*} |
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\begin{align*}Q_{3,k}&=|I_{k}|^{2(1-H)}\int_{0}^{1}\left|(a_{k}+v_1|I_{k}|)^{H-\frac{1}{2}} \cdot\int_{0}^{v_1}\frac{\varphi_k(v_1)-\varphi_k(v_2)}{\left(v_1-v_2\right)^{H+\frac{1}{2}}}dv_2\right|^{2}dv_1,\end{align*} |
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\exp ( i \Gamma ^ { ( 1 ) } ) = \int D v \exp ( \frac { i } { 2 } v \Delta v ) ; \, G a m m a ^ { ( 1 ) } = \frac { i } { 2 } T r \log \Delta |
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\begin{gather*}B_{\alpha}^{(i,j) }-B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) } =\widehat{\mu}_{\alpha},B_{\alpha-\alpha_{j}(\varepsilon_{j}-\varepsilon_{i}) }^{(i,j) } =0,\end{gather*} |
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\begin{align*}J_{\varepsilon}'\left(W_{\delta_{\varepsilon}(t),\eta}+\Phi_{\delta_{\varepsilon}(t),\eta}\right)\left[\frac{\partial}{\partial t}\Phi_{\delta_{\varepsilon}(t),\eta}\right]=o(|\varepsilon|).\end{align*} |
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\begin{align*}\begin{array}{lllllllllll}0=\omega(C)-\omega(C)&= |E_{g'=(0,0)}\cap E(C)|+|E(C)-E_{g'=(0,0)}|-\omega(C)\\&\le \frac{\omega(C)}{4}+\Omega+1+1+\Omega(7-1-2)-\omega(C)\\&=-2<0,\end{array}\end{align*} |
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\begin{align*}A = b^2 - ac, B = ad - bc, C = c^2 - bd.\end{align*} |
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\begin{align*}p_{N}^{\ell_1-}(x)=\dfrac{\sum\limits_{j=0}^N\dfrac{\Omega_j}{x-x_j}\left(f(x_j)+\sum\limits_{\ell=0}^Nc_{\ell}\tilde{\Phi}_{\ell}(x_j)\right)}{\sum\limits_{j=0}^N\dfrac{\Omega_j}{x-x_j}}.\end{align*} |
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\begin{align*}Tr K^{(1)}_{\bf a}\simeq -{1\over 2}f(p)=-{1\over \ln (tk^2_0)}+{\sqrt{t}k_0\over tk_0^2-1}~~.\end{align*} |
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\begin{gather*}C(z)-D(z)=\mathrm{i} g\sum_{\substack{j,k=1\\ (j\neq k)}}^n\frac{-1}{\lambda_j-\lambda_k}\prod_{\substack{\ell=1\\(\ell\neq k)}}^n(z-\lambda_\ell)=\mathrm{i} g\sum_{\substack{j,k=1\\ (j<k)}}^n\prod_{\substack{\ell=1\\ (\ell\neq j,k)}}^n(z-\lambda_\ell).\end{gather*} |
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{ \cal D } = \{ [ \omega ] \in { \bf P } ( M \otimes { \bf C } ) \mid \omega ^ { 2 } = 0 \, , \quad \omega \cdot \bar { \omega } > 0 \} \, , |
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\begin{align*}f_e(x|\eta)= \eta e^{-x\eta}.\end{align*} |
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\begin{align*} a \in \mathfrak A(\mathsf V_a) \cap \mathfrak A(\mathsf Y\setminus \{ y\}) = \mathfrak A(\mathsf V_a \setminus \{y\})\end{align*} |
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\begin{align*}J(u,v) = \lim_n\frac{1}{n}\log G(u_n,v_n). \end{align*} |
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\begin{align*}\begin{aligned}\bar{\boldsymbol{\psi}}(t)\!=\!\bar{\boldsymbol{\psi}}(t_k)\!+\!\frac{(t\!-\!t_k)}{b_{S,k+1}}\left[\bar{\boldsymbol{\psi}}(t_{k+1})\!-\!\bar{\boldsymbol{\psi}}(t_k)\right], t\!\in\![t_k, t_{k+1}].\end{aligned}\end{align*} |
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\int \left[ { \cal D } \varphi \right] \left[ { \cal D } \bar { \varphi } \right] e ^ { i \int d x \left( \bar { \varphi } { \cal P } \varphi + { \cal J } \varphi \bar { { \cal J } } \bar { \varphi } \right) } = |
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\begin{align*} g(u)= \left ( \begin{array}{cc} a & b \\ c & d \end{array}\right )\,.\end{align*} |
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\begin{align*} C\Delta_{{\bf T}_1}+D_{n_1}(\widehat\Delta_{{\bf T}_2}) {\bf T}_1^* = \widehat\Delta_{{\bf T}_2}, \end{align*} |
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\begin{align*} (i+1)g = (x)(\sigma_1\sigma_2)^i,\end{align*} |
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\begin{align*}L_m^G=\sum_{n=-\infty}^{\infty}(m-n)\;b_{m+n}\; c_{-n}- a\; \delta_{m}\end{align*} |
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\begin{align*} \left|\int_0^M h_d'(x)e^{2\pi\textnormal{i}h_d(x)\beta}dx\right|=\left|\int_0^{h_d(M)}e^{2\pi \i y \beta} \d y\right| \ll \min \{L, |\beta|^{-1} \} \end{align*} |
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\begin{align*} \bar{D}^{n} := \big((\partial_{t}, \nabla)^{\alpha} \,|\, 0 \leq |\alpha| \leq n\big) H^{0}_{0}(\Omega) \equiv H^{0}(\Omega) := L^{2}(\Omega). \end{align*} |
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\begin{align*}{\rm sgn}(r_1) = {\rm sgn}(r_2) = -{\rm sgn}(\tilde{a}) = -{\rm sgn}(a) \;.\end{align*} |
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\begin{align*}\begin{aligned} & \xi*\ddot{u}_{i}=u_{i}-v_{i}^{0}t-u_{i}^{0},\\ & l*\dot{\theta}=\theta-\theta^{0}l*\dot{\eta}=\eta-\eta^{0},\\ & \zeta*(\theta+\beta\dot{\theta})=\theta-\theta^{0}e^{-t/\beta},\\ & \chi*(\theta+\beta'\dot{\theta})=\theta-\theta^{0}e^{-t/\beta'},\end{aligned}\end{align*} |
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\begin{align*}\left(D_i(Q^{\ast})\right)_{mn} := \delta_{mn}\ \partial_i +g \epsilon_{mkn}\ Q^{\ast}_{ki}.\end{align*} |
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\begin{align*}\Phi^1_n:=\sqrt{q^2+p^2}\,T_n\left(q(q^2+p^2)^{-\frac{1}{2}}\right)-1\quad,\quad \Phi^2_n:=p\,U_{n-1}\left(q(q^2+p^2)^{-\frac{1}{2}}\right)\end{align*} |
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\begin{align*} \sum\limits_{k=0}^{\infty}a_{k}\bar{b}_{k}r^{k}= \mathbf{S}(f_{r}). \end{align*} |
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a _ { 0 } ^ { 2 } = \frac { \sqrt { 1 + \frac { { | \Lambda c | } } { 2 } } - 1 } { | { \Lambda | } } . |
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