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[
"Math in Canada versus Other Counties"
] |
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"math"
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"vc6jg4"
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Some places do not distinguish between real analysis and calculus and they mean the same thing; some places do not have the US and Canadian equivalent of calculus and only do straight up hard real analysis. It varies by country and region. In the US and Canada, calculus is applied real analysis with an eye towards problem solving with real world applications to physics and engineering; whereas real analysis is the rigorous theory, mostly devoid of direct application (since it was covered previously in calculus). It's easier in theory, but some of the students I have had who came from a place where they do "real analysis" bemoan actually going through derivations and theory in calculus (while claiming the material is beneath them) and struggle with what the concepts actually mean, so.. . My personal opinion is that having calculus before analysis helps to give a really strong intuitive foundation for the material. Not everyone needs that before taking analysis (I definitely did). It also helps one cross over to other subjects like physics if that is of interest. Pure analysis is great (see flair), but it can be bolstered by looking at applications and other disciplines.
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The University of Toronto has their first-year math specialists (what would be called math majors elsewhere) take MAT 157, which is an intro to real analysis course. Also, I'm entering the University of Waterloo this fall (graduating high school in a few weeks), and one of the courses that I'll be taking, MATH 145, typically covers number theory and introductory rings and groups. From what I've heard, though, most universities besides these two don't have quite the same caliber of math departments.
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Is it easier in Canada? Easier than what? You need to specify the thing to which you'd like it compared. Let's use the US for example. Below is a replication of a comment I've posted somewhere else : The US is a big place, so it's difficult to make these sorts of comparisons just anecdotally. To throw gasoline on that fire though, I've known a few US students who have come to Canada, were considered "advanced" in their state-level education system, and were woefully underprepared for the Canadian education system. That being said, the Ontario secondary curriculum is quite bad, and we as instructors are commonly astonished at just how bad the system has gotten. I can offer an interesting comparison at a broader level. For example, we find that most US based calculus textbooks are simply too low level to be used in Canadian universities. This speaks to the general level of preparedness of US students versus Canada. If we want objective data, one of the best metrics is the Program for International Student Assessment . It's run every three years, and has a rotating focus (so one year they focus more on math, then on science, then on literacy, etc). You can see that the United States ranks well below Canada in every domain. I should qualify that I have also had exceptional American students.
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Math "in Canada" is not a thing. A math undergrad at Waterloo or Toronto is completely different from a math undergrad at Guelph or Windsor, which is completely different from a math undergrad at a small liberal arts university. In general, a math undergrad in Canada will be very comparable to a math undergrad in the US at a school with a similar focus. So don't compare an undergrad at MIT to an undergrad at Windsor -- they don't have the same goal. Similarly, don't compare an undergrad at U of T to an undergrad at a tiny liberal arts college in the US.
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Waterloo also has MATH147 which would be comparable to MAT157 at UofT.
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[
"Old Maths Textbooks. We’re cleaning out the office in the school I work in and we’re coming across books that are 60, 70 years old and some that were originally published over 100 years ago. Would people here be interested in them? It’d be a shame to bin them."
] |
[
"math"
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"vc424k"
] |
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12
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""
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0.94
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I don’t have a complete set of the Bostock, Chandler and Rourke books that were big in the 90s. And I’d love them. Thanks u/pukeupmyring
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I might be interested. Which ones do you have?
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I'm very interested as well
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YAASSSSS
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I'm always in search of new math texts! I have a bunch of pdf copies of books that I want to eventually get but can't afford!
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[
"Can I really get a good grasp on Algebraic Geometry, Differential Geometry etc. by self studying?"
] |
[
"math"
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"vcmvmo"
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30
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I have started self studying and I am doing really slow progress. I am having very low self confidence because of that. I want to get into a good grad school but having a hard time trying to study these. How difficult are these to self study?
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I self-studied DiffGeo and seem to be doing fine. However, I do have gaps in my understanding (to the point where it’s somewhat affecting my research progress) so I am intending to take a course next year. I think anything is possible if you put your heart, mind, and soul to it.
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First and foremost, if you want to start understanding DiffGeo, you need a solid foundation of real analysis and linear algebra. Assuming you have that already, Jack Lee’s Introduction to Smooth Manifolds is great, and it’s what I used to self-study. I also learned a lot just from reading papers, picking up useful theorems, proofs, and techniques along the way.
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Self-studying Algebraic Geometry is much harder, IMO, but maybe it depends on the individual. For DG, you just need to know some analysis, calculus and basic point set topology. In contrast, for AG, you may also need to pick up some commutative algebra and category theory. Vakil's book walks you through this but the abstraction level is higher and it will take more work building intuitions for the subject.
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Honestly nothing can compare to a professor giving you weekly homework assignments and 2 exams and risking a bad grade if you don’t do well haha. I haven’t done many exercises, I usually just read the theorems and proofs that I feel are necessary to my research and move on. But there’s so much stuff I miss, and I know it. The little tiny details and machinery for proofs. For instance, I know there’s all these tricks to submanifolds and quotient manifolds that I lack. At this point in time, there are very few things I can prove from the ground up in DiffGeo. I usually just rely on theorems I have memorized and hope those are sufficient to move forward. With that being said, I’m not putting so much effort in my self-study. Anyone can self-study it without taking any courses, they just need sufficient effort hah.
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That crucially depends on your prior knowledge, your previous experience with self-study and your overall pure mathematics skills. Both topics are very large and go way above what a single person can learn in a life time. That being said, if you self study either choose an online lecture and not really self study (maybe Richard Borcherds yt for advanced undergrad/early graduate level) or follow a book that gives a lot of motivation and examples (maybe miles Reids undergraduate algebraic geometry). Both topics have very vast formalism but also some down to earth motivating examples and problems in the beginning where geometric intuition and computations in coordinates are as important as formalism. If you are not already Grad level maybe start there. I am not one for prereqs as in you need to have done commutative algebra for years before starting with geometry, but you should have at least done the equivalent of 2-3 analysis and 2-3 (linear) Algebra courses before this can make any sense to you.
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[
"Question about the Collatz Conjecture"
] |
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"math"
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"vcqt5l"
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Suppose that there exist a positive integer that does not satisfy the Collatz Conjecture. How can we verify that this is true? I mean, of course we can just run the Collatz Transformation a million times, but this won't proof it even if the result does not end in the 1-4-2 cycle.
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There are broadly two options:
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The fact we might not know if a sequence is a counterexample or not is why there is a possibility that the conjecture is independent from ZFC. It is known that generalized version of Collatz conjecture are undecidable. Therefore, there exist some variants of the conjecture that is independent from ZFC. Whether it's the Collatz conjecture itself that is independent, we don't know.
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This is a rabbit hole I would avoid anyday
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I don't have a real example, obviously, but there are a few possible ways this could be done: As a (trivial) example, consider "if the number is odd then add 2, if the number is even but larger than 0 then subtract 2, if it's 0 then don't change it". If you start with 1 it's immediately obvious that you will never reach 0 with it. You can formally prove this by showing that 1 is odd, 0 is even, and the transformation never changes an odd number to an even number.
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Independent just means that it can't be proved or refuted. Sometimes independence will tell you something about truth, but not in general.
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[
"Automated Theorem Proving"
] |
[
"math"
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[
"vcu0o5"
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145
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""
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[
true
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[
false
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0.94
] | null |
Proving a theorem is essentially like playing a game. There is a desired outcome, but many possible moves at every step. The number of possible logical paths is huge. It takes intelligence to find the right one. We've seen computers do this with chess and go for games, but it might take time for theorems. The space is so much more complex.
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I’ve used it a few times on both my masters and phd thesis. For my masters I used prover (prov3r or provr? It’s been a while) to check if combinatorial configurations satisfied the elliptic curve group law. Calculations which are pretty rote, but tedious. For my phd thesis, I used the magma language to verify if reflection groups satisfied specific properties I was after. Once again, tedious and rote work I didn’t want to waste time doing.
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It is amazing to me that this technology did not surpass humans yet. It's weird that humans can do something that is essentially symbolic manipulation better than computers. Remember that the size of the space to be explored is exponentially large, ridiculously so. Following intuition is much better than scrambling around in the dark. Great strides have been made with automated theorem proving, and I'm sure much more will be done in the future. But I think it's amazing that automated theorem can sometimes match humans, not the other way around.
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While theorem-proving is a bit hard to nail down as a specific problem, given that SAT is NP-complete, the total general case for being able to prove any theorem in any axiom system Edit: Is certainly NP-hard. (Thanks to u/sfurbo for catching this mistake!) It's also not co-NP in general, so the only thing polynomial time is me making mistakes. (Thanks /u/wrightm for catching this one!)
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magma language I thought this said “manga language” and I was like… you did math in Japanese?
|
[
"How do you actually retain math conceptual understanding?"
] |
[
"math"
] |
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"vclbr7"
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I’m just afraid the math I’m learning right now will just be…lost. I’m not too worried about the whole “use it or lose it” thing, since I am re-studying discrete math right now and I’m a CS major, those go hand in hand. I just feel so afraid of losing knowledge that I had once learned, I know I’ll lose the nuance of techniques applied to solve certain problems, but I’m more concerned about overall topics becoming fuzzy. Answers I’ve gotten before include making sure your conceptual understanding is really solid by 1) Teaching others/putting material into your own words 2) Having a part of your notes just be a list of conceptual questions and going back every few days to answer them Any other recommendations?
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Repetition. Once you stop using it it slowly oozes out of your brain.
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My dude there is no stopping it. You WILL lose it if you dont use it. It definitely WILL be fuzzy unless you keep repeating it over and over and over. Anyone who says otherwise is high on copium.
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I forget lots of things. But at this point, I know where I need to start looking if I need to (re-)learn or review something, and I am confident that I have the skills to re-learn anything I've forgotten. It's totally normal to have to go back sometimes. To add to your list, I would say that doing problems, lots and lots of problems, is a great way to really understand and learn the concepts you are working with. Try to make sure you understand all the proofs and fill in any details for yourself that textbook authors have left out. Read different textbooks and watch videos about the subject to get different ways of looking at things. Talk about it with your friends/classmates, professors, or strangers on the internet. Also think about why these concepts were ever defined in the first place, understand the motivation behind it.
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An advice I keep giving people but I rarely follow myself is to write down things in your own terms. Like really let loose and dont try to be elegant or concise or even too organized. Explain to yourself the concept you just got many times be redundant draw very clear pictures with lots of comments as if you were talking to yourself. I cannot count the number of times I go back to my notes and I find some cryptic one line that at the moment I thought was enough to summarize many hours of struggle. I think you inevitably forget things when you dont use them often and thats fine, its just you need to figure out the key points to connect it all once again.
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Yes! How do you retain muscle from your gym sessions? You keep working out consistently.
|
[
"Exponential Functions"
] |
[
"math"
] |
[
"lydk8m"
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1
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[
"Removed - post in the Simple Questions thread"
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[
0.6
] | null |
Well I mean you can. For example f(x) = e^x -1. But, I presume you mean f(x) = e^x, which does not. They have horizontal asymptotes (which may or may not actually be the x-axis) depending on what they are and how they're shifted.
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I believe you are looking for the concept of limits. For e : as x approaches negative infinity, e approaches 0. It never actually equals zero since infinity isn't a real number.
|
Look into the concept of limits. You don't seem to quite understand what's going on. Not sure what you mean by it being similar to pi, and I just explained why it can't be equal to 0. If it could be equal to zero, the "benefits" to math would be that calculus and things based on it would be incorrect... which doesn't seem feasible; it would be like the "benefits" of disproving addition.
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Yes they have, it’s called the extend real line and you get it by including the limit point at the positive and negative ends of the real line. ie you include positive and negative infinity as points. You might also be interested in the protectively extended real number line. where positive and negative negative infinity are made equal.
|
You mean 0 on the y axis?
|
[
"Best way to “understand” and do well in Linear Algebra?"
] |
[
"math"
] |
[
"lyenrh"
] |
[
7
] |
[
"Removed - post in the Simple Questions thread"
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[
true
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[
false
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[
0.82
] | null |
Not sure about proofs, but for intuition, 3Blue1Brown’s “Essence of Linear Algebra” series is : https://www.3blue1brown.com/essence-of-linear-algebra-page/ .
|
Many people benefit from a first course dedicated exclusively to reading, writing, and understanding mathematical proofs. You may look into some of the texts from these courses to get a feel for the intuition and methods of proof which are essentially the same across all branches. ‘How to prove it’ by Velleman is popular. There are several other transition to advanced math type texts which all offer similar content. ‘How to Solve It’ by Polya is the classic standard. Hope it helps, good luck!
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Easily 3blue1browns best series. The calculations of linear algebra are usually very easy, if grindy, but actual intuition of what they're doing can be a little hard to grasp without visualizations, and 3bl1br has the best mathematics visualizations I've found.
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Getting good at proofs is mainly just practice practice practice. In linear algebra I found that a combination of abstract algebra background and intuition goes a long way. For intuition the 3blue1brown series is excellent but for abstract algebra of recommend just some basic group theory book or something.
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e math. I guess what you actually meant is that the computations are easy but the actual This will be your greatest resource by far.
|
[
"Got about a free 50-60 days before joining my PhD. Any advice or suggestions on what to do in this free time ?"
] |
[
"math"
] |
[
"vcl2rq"
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[
301
] |
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[
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[
0.95
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Edit: thank you for all the suggestions. I was planning on meeting my old friends and have fun. You guys suggested something similar.
|
Go do something completely not math related, you’ll probably be doing math for the rest of your life so take this time to do something fun and unrelated. Go backpack through the wilderness or get in really good physical shape or start learning an instrument or something. You’ll have tons of time to do math so take now to have some unrelated fun.
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There's no point in starting early. Most people burn the first year of their PhD doing coursework, satisfying teaching requirements, and learning about a topic to even just come up with good questions. You can't really study to get ahead. It's a marathon of continuous work to understand the current state of a problem and then to come up with a novel solution. You need to go in rested (take a vacation) and ready to go (get your housing settled, move in, set up new insurance, licenses, etc...) So that you can hit the ground running and keep running. If I were you, if you have the money to survive the first few months until the stipend kicks in, I would take a nice, super memorable vacation. If not, I would find a way to take a cheap trip with friends.
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Should be enough time to prove the collatz conjecture
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Party, go to the gym and have a few adventures.
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I’m going on a three week backpacking trip before I start my economics PhD in August 👌
|
[
"What would you change about math?"
] |
[
"math"
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[
"vc449a"
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I'm going back in time to a decade or two before Euclid published his Elements. With me I'm bringing all my knowledge of modern math. Once I've stabilized my life as a wizard from the future, I will start writing down and teaching away everything I know. However, I will be in a very unique position to change things. What changes should I do to make the lives of future mathematicians as simple as possible? And let's just get τ=6.28 out of the way. That's a no-brainer, of course I'm doing that, and it is also discussed to death. I want all the other pieces of friction in modern math that not everyone has tired of yet.
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I'd keep the term imaginary from ever being applied to complex numbers. You'd be surprised by how many people I've met who believe "complex numbers don't exist" just because when they learned them in school, the word imaginary was used.
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The word "complex" itself also carries connotations of being difficult, when I believe the word is meant to denote "put together from parts" and nothing more.
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It would be a really fun idea to figure out how would the greeks and ancient civlizations would react to set theory as a foundation for mathematics. The greek in particular with their geometric bias I find it would be a bit hard to convince in light of the apparent contradictions that show up. But I wonder if there would be a point in between where you could try to formalize a geometric algebra ( like in Artin's book ) and convince them that this makes sense and see if maybe that kickstarts some giant development more algebra focused than geometry/analysis focused as we had in history. On the other hand Id be too woried to introduce my own biases , I think I would be much more curiou to learn about how they think about things than me trying to set them on my 21st century ways.
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Borderline unpopular opinions, mostly math education stuff: Just teach the damn Lebesgue integral from the get go (in analysis classes). Teach freshman calculus as a combined class with physics. Integrate learning some computer programming as part of the undergraduate curriculum. τ is a way less attractive symbol than 𝜋, has anyone learned to make a nice τ by hand? Mine are garbage.
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Make 51 prime
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[
"How do I find the solution of a slope-intercept, and a standard line?"
] |
[
"math"
] |
[
"ly0kax"
] |
[
0
] |
[
"Removed - post in the Simple Questions thread"
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[
true
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[
false
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[
0.17
] | null |
Substitution. Sub 7y for 7(3x+15). Gotta solve it twice tho
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Oh, yah... since y is 3x+15 that would make sense. Thank you!
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Thanks (:
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Let me ask you this: x + y = 5 What numbers add up to be 5? Is there only one answer? Did you consider negative numbers? How about fractions? There are infinitely many answers to an equation with 2 variables We graph lines when we can’t LIST the answers; when we have infinitely solutions We can’t TELL YOU all the answers, but we can draw a picture of it since the line goes on forever, and is made of “points”- pairs of numbers that follow a rule, such as numbers that add to be 5! You need to study(google): “Graphing lines in slope intercept form” And “Graphing lines in standard form”
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Thanks!
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[
"Can someone make a math puzzle for me?"
] |
[
"math"
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[
"lxtsi6"
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0
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[
"Removed - post in the Simple Questions thread"
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[
true
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[
false
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[
0.43
] | null |
> she struggled in high school math > I want to give her [a math] problem to solve It's up to you, but this sounds like a dangerous game.
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You could always go the Facebook route of replacing numbers with fruit emoji. 🍎 🍊 🍑 🍎 🍎+🍎=🍎•🍑 (🍎+🍑)/🍑=🍊 🍎 (🍎- 🍊) = 🍎
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(10 + 3 ) * 2 = 4324
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It was her idea; she likes a challenge.
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My biggest pep peeve is those things. They'll have a clock showing 3:00 that equals 3, but then the "solve" equation has a clock that shows 4:00 and you're supposed to assume it equals 4. It makes sense but it's mathematically wrong. They're two different variables and you can't make assumptions like that. Just because x=4 does not mean y=3 because you cut a leg off. Creative suggestion though, actually wish I had thought to use something like that.
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[
"Reliving integration after many years. Could use some guidance!"
] |
[
"math"
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[
"lxpqkz"
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1
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[
"Removed - post in the Simple Questions thread"
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[
true
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[
false
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[
0.67
] | null |
Hi thank for the answer but can you explain the steps?
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Hi thank for the answer but can you explain the steps?
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I don't see why dQ is disappearing from the equation. I'm familiar with the power rule but because I have dQ/dt which has a set value for a given time interval and 1/QT which also has a set value for a given time interval why is QT kept and dQ thrown?
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Since integration is like the reverse derivative. If you have a function f(x) = ln(x) and you take the derivative you get f’(x) = 1/x. So integrating 1/x you would get ln|x| + c where c is a constant.
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Thanks, so the heat flow term dQ/dt is.just dropped?
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[
"A question regarding the mathematical philosophy of Leibniz"
] |
[
"math"
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[
"lydjep"
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[
6
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""
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0.76
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Leibniz once said "The great foundation of mathematics, is the principle of contradiction or identity, that is, that a proposition cannot be true and false at the same time; and that therefore A is A, and cannot be not A. This single principle is sufficient to demonstrate every part of arithmetic and geometry, that is, all mathematical principles." I'm in agreement with the first sentence, but I'm not sure that a modern study of mathematics could produce agreement with the second. I know from past readings that the term "arithmetic" often referred to what we would now call "algebra," though I'm no historian and his scope may have been more narrow in his statement. What are your thoughts on this?
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Liebniz existed long before the philosophical foundations of mathematics as we know them today were developed. A lot of what we now consider theorems would've been considered "obvious", and a lot of definitions were just assumed to be present and not added explicitly as axioms or constructions in the theory. To get an idea, you can compare Hilbert's axioms of Euclidean geometry to the axioms that Euclid himself used to see the changes to our treatment of mathematical foundations over time.
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Gödel was a Platonist and fully agreed with Leibniz's point of view. Gödel showed that a given axiom system may not be able to the truth or falsity of a given proposition; nevertheless, that proposition has a definite truth value. He felt that the Continuum hypothesis, for example, was false.
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A paradox is not a valid proposition
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That's actually an interesting note since Godel was a bit of a Leibniz fan.
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It's not a well formed formula in propositional logic
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[
"Inspired by another post I wanted to share another small YouTube channel that has some excellent content, Professor Robert Ghrist. The production quality he has put into his video series is immense and I strongly urge you to check out his multivariable calculus videos."
] |
[
"math"
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"lxkb2i"
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629
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""
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true
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0.98
] | null |
His video series are great and I link my students to them all the time. Also check out his book and, hell, check out some of his papers. His papers have some of the greatest readability/advancedness ratios I have ever seen. His papers on sensor networks in particular are really cool and really readable.
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I also really liked his Single Variable Calculus Coursera course (which is broken up into 5 somewhat confusingly named parts on Coursera). I think it fills a nice niche of being more advanced and better explained than the Stewart-type textbooks, but less rigorous than the Spivak/Apostol/Courant/"Honors Calculus" group. He uses Taylor series as the basis for most of his lectures, which I thought provided pretty clean explanations for a lot of the concepts. I especially liked the video on "Discrete Calculus" which, among other things, shows how to use the "discrete anti-derivative" to find closed-form solutions to $\sum_{n=1} n $ for any k (with a little extrapolation). The relevant section starts around 4:59, but probably requires watching the preceding part of the video for context. I did find his speaking to be maddeningly slow. I usually watched the videos at between 1.75x and 2x speed. If you don't want to deal with Coursera (which has some good practice problems), all of the video lectures are on Youtube in this playlist .
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If anyone is interested in applications of topology, his book "Elementary Applied Topology" is great for a beginners introduction and building intuition (and comes with plenty of pictures)
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I personally don’t think there’s anything elementary or beginner friendly about that book, and I’m in an algebraic topology via differential forms course right now. It’s good as a sampler to know what’s out there.
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Took Calc and Dynamical Systems with him, super chill guy. Starts his classes by reading a passage from classic literature, I think it was the "consolation of philosophy" in my calc class.
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[
"I Feel Like My Maths Knowledge Is Getting Worse"
] |
[
"math"
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"ly7hd5"
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[
83
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""
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true
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Hello people! Junior physics major here. I will get to the point immediately. I have passed every maths course in my curriculum at this point, but I feel like I know absolutely nothing! I have noticed that I passed all of those maths courses by "memorizing" the course. I can still look up things on the internet and figure out how to solve a problem but I feel the lack of knowledge immensely. Have any of you experience something like this before? If so, what would be your suggestion? Should I just go over the course materials? I am curious to hear what you all think! I am open to all suggestions, resources and comments! Cheers!
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I think this is fairly common for a lot of people. Everything you have learned in your maths courses will come up again in more advanced courses and it will slowly solidify overtime. Think of your courses as an intro to a life long journey of deepening your understanding of abstraction. Linear algebra sticks out the most to me as I was able to cruise through grade-wise while having zero clue what I was actually computing. Fast forward a dozen graduate courses later and linear algebra is crisp and beautiful. Now new math subjects are the issue...
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I think a big part of that is that the for learning algebra gets ingrained in you. The first time around it's all new, but taking field theory after taking linear, group, and ring theory? "Here's an object. Here are it's sub-objects. Here are how we handle transformations. These are the conditions required for them to transform nicely. Oh, look! Half way through the term, here's something unique about them! Anyway, here are how they behave as direct/semi-direct/internal products, annnnnnnnd outta time, gg"
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That's perfectly normal. I feel that whenever I passed an exam, I had no idea what was going on. Only going back to it years later with more knowledge and experience you really get what you were doing. For me this was extremely apparent with linear algebra. When I did the course I had no idea whatsoever what I was doing. When about four years later I tutored for the same course, and hence had to review the material, it all became nice, clear, beautiful, intuitive. It was an amazing feeling finally grasping linear algebra xD So what I would suggest is this: if you feel confident enough, try tutoring someone on your previous math courses. This will give you the chance to review things, and explaining them to someone really forces you to grasp it!
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If you want to do higher level physics, my understanding is that you NEED this abstract stuff. I was in another field but my phys prof was old guy that just wanted to talk about symmetry :) This professor is awesome whenever I remember the basics of topology, I hear "And this is the discrete topology! It is utterly useless!" in his voice hahaha. If you are haveing trouble going from math to physics, he should help. For me the abstract made a lot more sense when I started to understand how to use proofs as tools. I went through 4-5 books designed for "transition" to higher math. 2 books really "unlocked" the whole thing for me. The 2nd one is free. My favorite is this one by Chartrand and friends. This one is not quite as good, but it definitely gets the job done. Enjoy :)
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Time is the best teacher. Reread your notes and you'll see it is much easier to understanding topics conceptually than when you initially learned them. Like for me, I didn't understand epsilon delta proofs as a concept when I first learned them in class. I would just do them methodically by memorizing how linear epsilon delta proofs are generally done since that was what we mainly focused on when we learned them, but now that I'm more knowledgeable and experienced, I can understand the concept of the proof. It came w time and doing some reading :)
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[
"Richard Borcherds Q&A"
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Classic Borcherds wit with a lot of these: "Why did you decide to become a mathematician?" - "I didn't decide to become a mathematician any more than a dolphin decides to become a dolphin." "What do you wish you knew when you were younger?" - "Wish I'd known to buy Bitcoin 10 years ago." "How does mathematics shape your view of the world around you?" - "Well, it makes car number plates more interesting because you can practice factorising them."
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There are many hierarchies of mathematics. The hierarchy of importance, order of understanding, distance from the axioms within some axiomatic framework, and so on. What you consider foundations depends on how you think about the structure of the field of mathematics. I suspect that a professor not working in logic or set theory will use order of understanding or something similar.
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He's basically taking his own interpretation of what "foundations" mean. When we usually say "foundations" we mean systems that, amongst other things, build up to us having a rigorous foundation for working with mathematical objects - set theory, logic, etc. However, I suspect Prof. Borcherds is trying to point out that the way in which you construct a rigorous foundation for the natural numbers (of which there are many) are not as important as finding out about the natural numbers. This number system which provides the starting point for so many different avenues of enquiry is what he considers the real foundation; the way in which you construct and systematise them is largely irrelevant (as long as they work, of course). He's basically hijacking the common interpretation and recasting it as "what are the fundamental questions in maths", the answer being "questions about number", so just take his response from this angle. After all, before set theory existed, we could still reason about numbers, but if we have nothing to use set theory with, it won't work the other way.
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Don't know if I should take solace in the fact that he says he can't understand some things or should I feel terror...
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I really liked, but also a bit confused about the foundation of math. I thought it was logic and not the natural numbers
|
[
"What are some examples of the natural logarithm and/or Euler's number in nature?"
] |
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I hear a lot about how Euler's number and the natural logarithm are incredibly important things that describe the way the natural world behaves, yet I am falling short on fining any concrete examples of it outside of pure mathematics and obviously compound interest which is most people's introduction to e.
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If you rather think of e as being defined as a function whose derivative is equal to itself, you get many more answers to your question in the field of differential equations (which I will add is likely a highly "visual" physical property we can match to mathematics).
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Population growth and radioactive decay are the two that leap immediately to mind.
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If we understand “describe the way the natural world behaves” as including physics, we gain a host of usages of e. Generally speaking, linear constant coefficient ordinary differential equations have a solutions which can be expressed in terms of exponentials. These ODEs model basic physical phenomena, such as oscillations and signals. The complex exponential enables us to simplify notation involving sine and cosine, and hence compactly represent wave phenomena. Speaking of wave phenomena, in a physical system where we expect waves, e will probably be lurking somewhere. Euler’s number is used when defining the Fourier transform, which is immensely important in quantum mechanics as well as signal processing. In general, the exponential enables us to describe systems with oscillatory or periodic characters. You can express solutions to various equations in electromagnetism or the heat equation or the wave equation using e.
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Natural logarithms appear in entropy and the Weber-Fechner laws. Functions expressed as powers of e occur in Planck's radiation formula in quantum mechanics. Do you consider probability part of pure math or an important use of math for the real world? I ask because the defining formula for bell curves in probability involves powers of e. Exponential functions, usually expressed as powers of e, show up when describing charge flow in an electric circuit. Something most students do not appreciate about logarithmic funtions is that logarithms (for different bases b > 1) are the same function up to an overall positive scaling factor. Therefore the choice of a specific logarithmic base could be regarded as a convention. The reason for the preference to use natural logarithms is their role in calculus (simplest derivative among all logarithmic functions), much like the preference for radians over degrees in some scientific fields is due to its better properties in calculus. Fourier series and the Fourier transform show up in many places in physics (solving wave equations, a mathematical formulation of the uncertainty principle, etc.). In Fourier series, functions of the form e appear. In Fourier transforms, functions of the form e appear.
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Imagine you are looking at a linear regression/line of best fit. You'll get some parameters for that line: the slope, the y-intercept. A general line looks like y = ax + b. But the "natural line," the line on which all other lines are modeled is y = x. But the line y = x doesn't usually appear in nature unless you're graphing one quantity versus itself. It's the same picture with the exponential function. y = e is the solution to the question: which function equals its own rate of growth. So here we are looking at some quantity whose rate of growth is proportional to itself. For instance, the number of people becoming infected by a virus is proportional to the amount of people with the virus. The growth rate of a population (without predators and with unlimited resources) is proportional to the size of the population. Radioactive decay is proportional to the amount of of radioactive isotope. Just like y = x is the model for a line, y = e is the model for "growth proportional to size." And just like with lines, there are parameters. y = e is the solution to "y(0) = 1 and the rate of growth is to the size." If we stick with y(0) = 1 and now say that the rate of growth is k times the size, the solution is now y = e (examples are y = 2 , y = 1/2 ). If instead of y(0) = 1, we had y(0) = A (the initial population, quantity, mass, whatever), you now get y = Ae . This is the general family of exponential functions, each of which is modeled after y = e . And like the function y = x, you don't typically see this platonic exponential function appear in nature, you get parameters: for the initial population, and for the rate of growth. The point: y = e is the "natural exponential" within the family of all exponential functions (e.g. y = 10 , y = 3 * 2 , y = 0.7 ) in the same way that y = x is the "natural line."
|
[
"Is the application of continued fractions to approximation theory still a relevant topic?"
] |
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"math"
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I have been reading a book called “The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory” by Alexey Khovanskii, published in 1963. This book is one of the craziest things I have ever read; I imagine it is what the average person thinks math looks like if you keep going past high school. There are just pages upon pages of formula derivations, with up to 10 or more variables using both Greek and Latin letters. There are special cases, setting some of those variables equal to 0 or 1, or to each other. On top of all that, there are precious few words describing what is going on, or why the author does what he does. Certainly, some of this my fault because, for example, differential equations aren’t my thing and I had to look up the Riccati equation on Wikipedia. With all that being said, I can’t look away from this book. I’m thinking of going through with pen and paper and working through all of the derivations. The key, though, is that I’ll have a computer at my side. The book presents 8 different ways to approximate sqrt(2). The approximations have differing levels of precision because they converge slowly and at different rates, and the author only shows six or so iterations for each, presumably because computer time was so dear and doing the calculations by hand would be arduous. I assume approximation theory is still quite relevant in today’s world, and I’m sure algorithms have been studied and optimized significantly over the last 60 years. Is anyone out there familiar with this application of continued fractions? Is any of this still relevant? I wouldn’t be surprised to find out that the topic has been completely studied and described, and then superseded by some other type of algorithm. Heck, for all I know, continued fractions themselves are considered passé.
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Continued fractions are an important step in Shor's algorithm to factor an integer with a quantum computer.
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I think continued fractions are relevant, and relevant to approximating. You might enjoy https://www.youtube.com/watch?v=FECj9A2e0OI
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The Kolmogorov Arnold Moser theorem, kam theory, which proves the stability of certain dynamical systems relies on continued fractions in its analysis. Check out https://galileo-unbound.blog/tag/kam-theory/
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Independently of what looks to you like crazy content in that book, continued fractions themselves are passé. See https://math.stackexchange.com/questions/585675/what-are-the-applications-of-continued-fractions
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You may be interested in the Ramanujan Machine .
|
[
"Automated Theorem Provers?"
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Mathematicians of , would you accept proofs that have been verified by a computer-assisted theorem prover (e.g. )? For example, imagine you are reviewing submissions to a mathematics journal and somebody submitted a paper that claims to have proven (or disproven) Goldbach's conjecture. Pretty amazing!!! However, instead of a human-readable proof, the author presents a 500000000-page long .lean file that no human could ever read and understand in any reasonable amount of time, but the automated Lean prover could verify as valid in a matter of seconds. Would you vote to accept that paper for publication? I use Golbach's conjecture as an example because it is a well-known (and very difficult) problem, but the same general question applies for any purported proof.
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I would accept it as a proof, but additionally demand the author to explain how they managed to get that 500000000-page long .lean file. The process required to get the lean file is humanly understandable and probably contains interesting insights.
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You're confusing automated theorem provers and proof assistants. These are not the same thing. An automated theorem prover uses AI or other similar, relatively opaque algorithms, to come up with the steps to the proof. Depending on how transparent the software is, and especially whether or not it is open-source, it may or may not be possible to figure out how the AI obtained the proof. A proof assistant doesn't have any AI. All the "work" is done by the person writing the proof. The proof assistant simply provides assistance in the form of keeping track of what you need to prove and what you have proved, and occasionally by providing an API for automation (essentially, you can write functions and code to help implement your proof). If someone presented a proof of Goldbach's conjecture in Lean then it would surely be combed over line by line by humans until we were able to extract the essential ideas of the proof. It is possible to publish a proof in a form that deliberately makes this process harder than usual, for example by expanding out all proof terms and submitting it as one monolithic blob instead of the normal presentation consisting of well-designed lemmas that feed into a main result, but doing so would be obviously pathological and I think the community would insist on publication of the original "source code" in that case.
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As others have pointed out, you've confused proof assistant, and automated provers. However, I would accept in either case. We believe the computer when it tells us 2 -1 is prime. Why would we not trust it with our theorems?
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Nobody could ever read the proof of the Poincaré Conjecture (Theorem) if they inlined all the definitions, all the proofs of lemmas, and restricted themselves to the use of the terminology of Set theory instead of Topology. But this is what your hypothetical Goldbach proof is analogous to. You're sort-of creating a straw-man argument (though I'm sure it's not your intent). Software is just as amenable to clarification via definitions, lemmas, and proper theoretical framing as are hand-written proofs. Your Goldbach example would proceed just like anything else in Math, with lots of people obtaining incremental results by building on top of the incremental results obtained by others. Edit: Grammar.
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I assume OP knows this, but for anyone else reading - most proof assistants do provide some form of automated theorem proving, which end up generating the complete proof terms, but the "source code" can be as simple as the following: (Example in Coq, as taken from the SoftwareFoundations book ) Lemma solving_by_apply : ∀ ( P Q : nat → Prop), ( ∀ n , Q n → P n ) → ( ∀ n , Q n ) → P 2. Proof. auto. Qed. What I mean is that it possible for a proof of the Golbach's conjecture to unintentionally be really hard to follow because a huge part of the work was done via proof search by the engine.
|
[
"Why doesn't the 6n+-1 formula for primes prove the twin prime conjecture?"
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The 6n±1 formula just says that any prime can be written in the form 6n±1 but it doesn't say that there exist infinitely many n such that 6n-1 and 6n+1 are primes
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Typically things that can be proved in one to two lines aren't called anything special like "Theorem" or even "Lemma"
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Typically things that can be proved in one to two lines aren't called anything special like "Theorem" or even "Lemma"
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Take any prime number p greater than 6. Divide it by 6 and consider the remainder r. r can’t be 0, or the prime is a multiple of 6. r can’t be 2 nor 4, or the prime would be even. r can’t be 3, or the prime would be divisible by 3. So r is 1 or 5, in the first case p = 6n+1 (where n is the quotient), in the second case p = 6n-1 (where n-1 is the quotient)
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It's a straightforward result in number theory that can be proved with the division algorithm.
|
[
"Hey math people of reddit Why do so many mathematics graduates end up being high school teacher?"
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"math"
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Hello every one i always had respect for my math teachers and any one studying math as its a pretty difficult subject i am cs student and the classes iam taking involve bunch of math one of my friends studying math agreed to help me when i asked her about her future plans she said she would become a high school mathematics teacher and later on try for assistant professor. My main question is this considering how hard math is and how in demand skill it is plus very few people have the brains to really master it why would any one want to become a math teacher when you could easily be a data scientist or in some other postion with high salary
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Because life is complex and not all about money. Some people may find working in finance or as a data scientist etc. unfulfilling, stressful, or even unethical. They may feel that the salary of a high school teacher is enough to live comfortably the way they want. They may want to devote less time to their career and more time to their hobbies or family.
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Thought I would reply to this post with my own example which might help the op. I'm not really a math person but I have a CS degree and teach secondary school (ages 11-18) computer science so my answer is probably relevant. My wife is also a teacher. We earn enough money to live comfortably in a nice part of a nice town. I have a 12 minute walk from my front door to my classroom (when it opens back up again!), and I have 13 weeks holiday in the year. My wife teaches in a private school and has 16! This means that I can collect my children from nursery / school and I have plenty of time with them, especially in the summer. I also value teaching highly. I had good opportunities as a child because my own parents spent their time on me, which led to me accessing education and qualifications that my peers did not. If I am able to provide similar opportunities to the children I teach then I feel that is time well spent.
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There are more math graduates than you think. Lots become teachers, but lots do data science and finance too
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Your wife has 20922789888000 weeks of holiday in the year? Nice!
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I am the head of the math department in a small high school. Between my experiences as a student and as a professional, I can say that most of the teachers I’ve met either didn’t or shouldn’t have received a degree in math.
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[
"Career and Education Questions"
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"math"
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This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered. Please consider including a brief introduction about your background and the context of your question. Helpful subreddits include , , , and . If you wish to discuss the math you've been thinking about, you should post in the most recent thread.
|
What's the best use of my time if I'm graduating (Masters) this year but won't be applying for a PhD until next year? Anything I could do to help further my chances of getting an offer whilst also being 'out of the system'? I'm considering using the time to create some sort of educational material, either in the form of videos or a 'textbook'. I think the field I'm focusing on really lacks a good, undergraduate/early graduate level textbook. But I'm not sure if that's a bit too ambitious for my current level?
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It’s not common to just offer a course on limits and series. Unless they don’t have real analysis at this university, this course sounds redundant. The standard is proof-based linear algebra-> 2 semesters of real analysis + algebra -> upper-division electives. If they expect you to take intro to proof and won’t cover that in other courses, it might be a good idea to take it (mine covered it in linear algebra). In general it would be helpful to finish the analysis or algebra sequence as soon as possible so you will have sufficient mathematical maturity for advanced classes. Algebraic geometry is graduate-level course and you want to build a strong foundation before even thinking about it.
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Both Warwick and Durham are excellent unis for maths, and I think you will accrue the benefits of having gone to a top uni for maths graduating from either. Tell me about what you think of each. Which course do you prefer? Where do you want to live? Can you meet the entry requirements for Warwick? I'm probably projecting, but you remind me of myself at your (presumed) age. I made a lot of mistakes planning the next phase of my life when I was in my last year of school, and my mum instincts have kicked in to try and avoid you making the same mistakes 😅
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Both Warwick and Durham are excellent unis for maths, and I think you will accrue the benefits of having gone to a top uni for maths graduating from either. Tell me about what you think of each. Which course do you prefer? Where do you want to live? Can you meet the entry requirements for Warwick? I'm probably projecting, but you remind me of myself at your (presumed) age. I made a lot of mistakes planning the next phase of my life when I was in my last year of school, and my mum instincts have kicked in to try and avoid you making the same mistakes 😅
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Thanks! That link looks like it’ll definitely help with reviewing the concepts
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"argument with a friend"
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It's just a matter of counting possibilities but we need more details. How many digits in your ID? Does a pair or trio have to be consecutive (xxx44xxx vs. xx4xxx4x)?
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Just wrote it more clear.👍
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I think you do not have understood the question. If we are playing poker and I ask "which served hand is more rare, a double pair or a poker?", do you answer "they are both as rare as any other combination of 5 cards" ? Clearly in this case OP was asking something along the lines: if I'm assigned a random ID, it is more probable this or that?"
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Yeah exactly thanks,
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To compute the exact probabilities (assuming the IDs are given at random and can be any combination of a certain number of digits) we still need the total number of digits in the ID and if the pairs and triple need to be consecutive. Just for example, let's assume that there are 6 digits and that repeats do not need to be consecutive. So things like 225757 and 102121. For the 3 pairs: One pair can be any of 10 digits, the second any of 9 (it can't be equal to the first) and the third any of 8. There 6 digits can be permuted in 6!/(2!2!2!) ways so there are 10 * 9 * 8 * 6! / 8 = 64800 possibilities among 10 = 1000000 unique IDs, so 6.48% For one pair one Tris one single digit, with the same reasoning we get 10 * 9 * 8 * 6! / (3!2!1!) = 43200 so 4.32% But things may be different for different number of total digits in the ID
|
[
"how do i simply 8ab - (2ab - 2a2)"
] |
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"math"
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8ab - (2ab - 2a2). Lets remove the brackets (remembering that a minus times a minus is a plus). 8ab - 2ab + 2a2. 6ab + 2a2. Both our elements have 2a in common so let's factorize. 2a(3b + a).
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Homework problems, practice problems, and similar questions should be directed to /r/learnmath , /r/homeworkhelp or /r/cheatatmathhomework . Do not this type of question in /r/math .
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8ab - 2ab. 8 lots of (ab) minus 2 lots of (ab) = 6 lots of (ab).
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ohhh now i think i have a understanding
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got it! thank you for the explanation. this helped me a ton!
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[
"Do infinitely repeating values exist in pi?"
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"math"
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An infinitely repeating sequence would make pi rational, so that's definitely not an option.
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We've all heard that every piece of informational imaginable exists within pi, such as your birthday, social security number, etc. While it is (probably) true that most of us have heard this factoid, it isn't actually true. Most real numbers are disjunctive (meaning that every sequence of digits appears somewhere), but there are very few numbers for which we they are disjunctive. Pi is not among those few. Obviously, the value 3 is in pi. Also 33. And 333. And 3333. Who is to say when this pattern stops? Could I go on infinitely adding 3's? Would this imply that 1/3 is represented somewhere in pi? I've already discussed that pi is not (known to be) disjunctive. But let's assume for the sake of this paragraph that it is. Then, for any natural number n, you can find a patch of n "3"s somewhere in the decimal expansion of pi. This does not mean there is an infinite patch of "3"s anywhere: indeed, if there were such a patch, then the decimal expansion of pi would have to consist of some finite portion - say, k digits - followed by "3" repeating. But then pi would be a rational number (we know it isn't); and also, you would be unable to find any string longer than k that doesn't end in 3 within the decimal expansion, so it wouldn't be disjunctive either!
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You’re actually wrong: it’s not known that every possible string of digits exists within the decimal expansion of pi. Although, if it was true, all it would mean is that YOU get to pre-specify a length, say 7837, and then it would be possible (although perhaps not practically possible) to find the first string of 7837 threes-in-a-row within the decimal expansion of pi. But there’s no guarantee that you can do this for any arbitrary length of string of digits.
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Others have pointed out that your premise is ill-founded: we simply don't know that every finite sequence of numbers is represented somewhere in pi. In fact, we don't even know that every digit occurs infinitely often. We do know that there are no infinitely repeating sequences in pi, or else it would be rational. I want to point out that it doesn't make sense to say that pi contains other values as you put it, but rather that it contains sequences of digits in its decimal expansion. Finally I want to mention that while most mathematicians expect it to be true that pi contains every finite sequence (they in fact suspect that pi is a normal number which is a stronger assertion), this property is in no way unique to pi. In fact "most" numbers have this property.
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The point is that there is a difference between "arbitrary large" and infinite. For example, there are integers with an arbitrary large number of digits. Say, you want an integer number with 3, 7, or 1009 digits you can construct it. But there are no integers with an infinite number if digits. So, even if pi could contain (it has not been proved) an arbitrary large sequence of '3', it cannot contain an infinite sequence of '3', because we couldn't have other digits after those, and this would make pi rational.
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[
"can someone explain"
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This convention ensures that 2 = 2 *2 holds true for all integers.
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Because it’s the natural next step based on the properties of exponents. Notice that 2 / 2 = 2 for positive a and b. You can verify this property for yourself. Now, consider what would happen if a = b. In that case, we see that 2 = 2 = 2 / 2 But since 2 = 2 it follows that 2 / 2 = 1, and as such 2 = 1. This also works for numbers other than 2, however importantly 0 is indeterminate.
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2 to the power of 3 is 8 -> 2*2*2*1 2 to the power of 2 is 4 -> 2*2*1 2 to the power of 1 is 2 -> 2*1 2 to the power of 0 is 1 ->1 2 to the power of -1 is 0.5 ->1/(2) 2 to the power of -2 is 0.25 -> 1/(2*2) Understand?
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Yes I do
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2 /2 =2 =2 But any non-zero number divided by itself=1, so 2 =1
|
[
"Can a phase shift in polar coordinates be translated to cartesian ?"
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"math"
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"zxjnc0"
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cos(theta + pi) is equivalent to -cos(theta) so the formula would just be x + y = -x
|
This feels very homeworky. You can get a general solution by using theta = arctan(y/x), but for that particular function, there's an easier method that you should put some thought into. When you run into a nested function that you don't know what to do with, like cos(theta + 5pi/3), you should try to expand it so the special function is on the "inside" and the elementary operations like addition/multiplication are on the "outside", then you might be able to see other options.
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Thank you! What I’m looking for though is a more generalized way to do this conversation (for example I need to convert an equation with cos(theta+5pi/3). Maybe this isn’t possible.
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The conversion is possible but there isn't a general way to do them. You will have to use trigonometric identities for each one. For example cos(x + y) = cos(x)cos(y) - sin(x)sin(y) With your values: r = rcos(theta)cos(5pi/3) - rsin(theta)sing(5pi/3) x + y = x/2 - (-root(3))y/2 so... x + y = (x + yroot(3))/2 hopefully that makes sense - not sure how to do radicals or fractions
|
It started off in school, but I’m just personally curious about this. I’m doing a project to recreate images with polar and Cartesian functions. Thanks for your help!
|
[
"Career and Education Questions: December 29, 2022"
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This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered. Please consider including a brief introduction about your background and the context of your question. Helpful subreddits include , , , and . If you wish to discuss the math you've been thinking about, you should post in the most recent thread.
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Depends. Some schools stats majors are only stats classes and like one linear algebra class. Other schools the stats major is essentially a math major with a stats concentration, and you have the option to make it as theoretical or applied as you want. Option 1: all stats courses and very light on pure math, option 2: required to take a real analysis sequence, and 3-4 math department electives
|
Probably the best thing to do is to take as many (pure) math courses as possible, specifically the foundational courses like real analysis and linear algebra. If you're a stats major then one good option would be a (measure-theoretic) probability course. Otherwise an ms program would likely list some specific requirements for applicants.
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Is a Math and Statistic major totally different? I've had a few Professors tell me that they specifically majored in Statistics and some said they majored in Math, are those two majors really that different?
|
PhD student here. What are your experiences with completely changing fields after a PhD? I am thinking something like from logic to mathematical models for healthcare. The maths are so different that I am thinking that my PhD is essentially worthless. I would appreciate some insights from people who did drastic changes in their career after PhD.
|
I asked about this a while ago, and the consensus was that it's entirely possible and even happens on occasion. Remember that your PhD teaches you more than just the content of your specialism. Your doctorate isn't worthless; you still learnt how to do mathematical research, and in the end that's the truly valuable thing about having a PhD.
|
[
"Doomed in math?"
] |
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"math"
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"zyces9"
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So recently I been thinking I’m not suited for math. It’s like that moment when Jeff bezos realized he was never going to become a great theoretical physicist because he just couldn’t solve a difficult physics problem. I feel like I just don’t have the raw IQ for it. The reason I say this is because I feel like I work hard to understand the course material, yet I will only be able to solve problems that the professor expects to solve. He will usually put some wazoo really tough question on the final or midterm that I just can’t solve. I just took number theory and got an extra book and did about a third of its problems pertaining to the class and still had no idea how to solve these wazoo problems. But I got the median so I’m assuming everyone else had troubles with these questions too. But here’s the thing: two people got near perfect scores. So how did they do it? I assume they just “saw” the solution during the exam. I’m really beginning to think success in math is just an innate ability thing. What are your thoughts? Have you struggled and become a strong mathematician or known anyone to be a competent mathematician despite struggling or not being a genius level person?
|
Could be bad luck, or maybe number theory isn’t your thing. Number theory is super unintuitive to me; other types of math sit much better in my head. Or who knows, maybe you really are doomed! Just kidding. I wouldn’t worry about it unless you notice a pattern. A single data point is not enough to draw conclusions like that.
|
sounds like you might just be tricking yourself into thinking you understand the material instead of actually understanding it. this is very common.
|
I've always been bad at analysis. I know the various definitions, I can eventually work problems out, but I don't 'see' it or 'feel' the ideas. Indeed, part of my brain just shuts off at integral signs and it takes an active force of will to overcome that. On the other hand, I've always been good at foundational things, so that's the type of thing I studied - in the back of my mind I always felt like I was missing something and that other math folk just had some innate sense I didn't. And while that may be true (or not), I recently had a need for analytic methods - lots of them. As if by magic, results that used to just be formal strings of symbols I mechanically manipulated became much more alive and sensical. Integral signs became intuitive and I could do more with them than work through toy problems. Motivation matters a lot. Not just insight into what motivated the subject (that matters too), but personal motivation. There's a need for some model in which to conceive of the various mathematical objects, something that makes them more than symbols, something that gives them meaning in your mind; if you don't feel a connection to something, building up those models is hard. Not impossible, but hard. For an analogy, most people are capable of lifting weights with increasing intensity over time and strictly controlling their nutrition, there's no law of the universe that's stops them. And, yet, most people aren't a sculpted titan of throbbing muscles with washboard abs nor even an approximation of that. Many people seem to wish they were, many people are capable of getting much closer to that than they are, yet they don't. My point isn't that people aren't willing to put in the hard work and need to just buckle down, but that most people aren't willing to put in the hard work because they have no real meaningful reason to. The result is attractive, some side effects of it are also attractive, imagining the outcome is attractive, but the actual reality isn't and there's no compelling point . So, yes, maybe you aren't good at these subjects for some innate reason, I can't rule that out, no one can really. But I think it's more likely that these subjects don't click with the mental models you have and you don't have an easily accessible way to model the subject from where you're at. That may be changeable, there may be some connection that would make the ideas more attractive and open the door, so to speak, but finding that is a personal endeavour. It's also rather likely that there's some other branch of mathematics - which is a pretty big subject - that will appeal to you more as you are right now. So, perhaps, take a step back from working problems and look at the subject matter conceptually, think about how it connects to ideas you do care about. And, at the same time, look to other parts of mathematics and see how those ideas feel to you.
|
You might just be tricking yourself what
|
I do that too, putting one or two really tough questions in exams and assignments to separate good students from really good students. But I don’t put too much weight on those. May be 5/100. But I feel so proud and satisfied if one or two students gets one or two of them.
|
[
"Is there a field in CS/PhD program in the US that works on the automated theorem proving by using ML?"
] |
[
"math"
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"zxxdqv"
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This is still far reach from what I heard but I'd appreciate any input. I'm going into applied domain but I still love pure math and if there is any tangible field I think I can dedicate my life on it lol. If there is any resource for this, I would very much appreciate it.
|
Talia Ringer at University of Illinois is interested in the intersection of automated theorem proving and machine learning. They might at least be able to point you in the direction of what is going on in the field that might interest you.
|
If you are willing to look outside of the US (my strong suggestion is not to restrict yourself to a single country), take a look at what's going on at Heriot-Watt University in Edinburgh. Katya Komendatskaya , the lead researcher of the Lab for AI Verification , works on formal verification of AI, neural networks, and the like. The group has a lot of expertise with Coq and Agda. Since you're interested in going into the applied domain, you should know there is also a collaboration with people behind Imandra . Katya also did some early work precisely on using machine learning to enhance theorem proving. Take a look at ML4PG and ACL2ML projects. There have been some discussions about picking up that general thread of research. It might also be a good idea to talk to Kathrin Stark and see if her research direction is of interest to you. The current round of PhD applications to HWU is still open (until 23 of January), so if you are curious, reach out. You can also send me a PM here if you want more details.
|
As a non-expert, finding a proof (which is the hard part) can be a non-rigorous process, and then you can verify the proof using formal methods.
|
I don't think Talia's taking students this year, but they would definitely be a good resource to talk to about work in this area. I will also say that as far as I know, there's a lot of overlap between program synthesis and ML-automated theorem proving because a lot of the proof systems we use for automated theorem provers abuse the Curry-Howard isomorphism. As such, if you're interested in the area, program synthesis might also be up your alley.
|
Tim Gowers tweeted today about a PhD position(not in US). He said though that while they do expect to do some ml, it won't be focused on that area. You should look at his Twitter for more info , as I may be forgetting something.
|
[
"Prime Vector Space have a real name?"
] |
[
"math"
] |
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"zybkc4"
] |
[
55
] |
[
""
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[
true
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Trying to find if this coordinate system has a name. Each Prime is an axis/dimension, and each number maps to this p-dimensional space based on its prime factors as coordinates. (x, y, z, ...) as (2 3 5 p 10 → (1, 0, 1) || 6 → (1, 1, 0) || 8 → (3, 0, 0) || 30 → (1, 1, 1) Trying to find if this has a proper name. *Edit, I know it's not actually a vector space, that's why I'm asking what I should be calling this because what I'm calling it is wrong but has proven to be a useful search term to get to similar topics. I am ignorant and asking for guidance, telling me I'm ignorant is redundant 👍
|
Technically what you have here isn't a vector space, since it's not closed under scalar multiplication by elements of R. It is closed under scalar multiplication by integers, so the actual term for what you have is a Z-module EDIT: Actually, it's not that either, since it's not closed under multiplication by negative integers. It's something even more general.
|
I think this is just a commutative monoid. That being said: this a Z-module (or, well, an abelian group) if we allow for all positive
|
a Z-module (or, well, an abelian group) to clarify: those are two names for the same thing.
|
Prime Signature . You can also attach a metric.
|
Don't be so quick to believe that "prime signature" is a real name yet. My internet searching turns up barely any pages that use it in a mathematical sense: Wikipedia, Wolfram MathWorld, PlanetMath (is that site not dead already?), and one math stackexchange post. On MathSciNet, I searched for "prime signature" and there was just one result: exactly the paper in the second link above about a metric. So the term "prime signature" seems to quite uncommon. Of course every technical term in math had to be new at some point, so it began life with only a few references before being more widely used. What you are asking about is used in number theory when you extend the concept to nonzero rationals: associate to a fraction the sequence of exponents (e , e , e , e , ...) in its prime factorization. Most e are 0, some e may be positive, and some e may be negative. This can be extended to nonzero ideals in Dedekind domains (where ideals, or more generally fractional ideals, have unique factorization into prime ideals) and is very similar to the concept of a on an algebraic curve in algebraic geometry. The term in this fancy sense is a very standard in algebraic geometry, having been in use for well over 100 years.
|
[
"Do you have an MO problem related to the number 2023?"
] |
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"math"
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I am holding a weekly problem section in my school's BBS. One problem is posted every Saturday night. Their difficulty is usually near IMO P1/4~P2/5. The next one should be on Dec 31. On this particular day, I feel obligated to post something related to the new year. But I failed to find a problem that I think perfect enough. Have you seen any problems satisfying this topic? ---------- My current selection: Find the smallest integer k: For any 20221231 red points and 20230101 blue points on the plane, there exist k straight lines dividing the plane into parts, and in each part all points have the same color.
|
Well I'm aware of a 2022 I guess. Yitang Zhang's LS zero paper contains the magic number 2022, as an exponent or something. Here: https://mathoverflow.net/questions/433949/consequences-resulting-from-yitang-zhangs-latest-claimed-results-on-landau-sieg Oh you didn't mean MathOverflow but rather math Olympiad lol...
|
I can’t help you with another problem (when I see “MO”, I think of Mathoverflow rather than what you meant), but my only comment is that the numbers 20221231 and 20230101 seem like a very artificial way to put 2022 and 2023 into the problem. Aspire to make actual properties of 2022 and 2023 relevant, if they really aren’t in the problem you posed.
|
The problem is a bit harder if the smaller number is odd.
|
Fair enough. Keep in mind though that the problem is already hard like this and furthermore it nicely contains the number 2023.
|
I actually tried searching for such properties, but I haven't got a good enough one.
|
[
"Is Differential Topology a subject that is ‘easy’ to self study?"
] |
[
"math"
] |
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"zxuy6p"
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[
29
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Obviously no topic is easy to self study, but I’m wondering if anyone here has self studied to a point where you can say you can write proofs and solve problems in Diff. Top., and if so how was the experience?
|
What do you count as Differential Topology? I'm currently self-studying Lee's Introduction to Smooth Manifolds (reading the chapters, doing the exercises and the problems).
|
I am currently in Chapter 10 - Vector Bundles. To be honest I find it quite tedious, but I'm working at a pretty slow pace. I started around last year doing maybe one chapter every few weeks. I don't like rushing through it because I'm kind of obsessive about clarifying everything and doing all exercises / most problems before moving on to the next section.
|
By now I imagine you're sick of manifolds with boundary.
|
Topology
|
I’d second Milnor as a quick first course. But then I really like Hirsch for something more comprehensive.
|
[
"Math Podcasts for “Experts”"
] |
[
"math"
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"zxkq7l"
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Finished my studies a couple years ago. I ended up in a math adjacent job but stay sharp for my next career move. I keep up on the article but spend a lot of time listening to podcasts while working/driving. Does anyone have recommended podcasts aimed at an audience with some mathematical background? I’d like to hear reading series on interesting proofs or novel new results. Background is phd in number theory, andarithmetic/algebraic geometry. Will listen to discussions in any field though.
|
My favourite theorem has really good discussions. Other than that I am starved for quality math content too. Try mathzorro podcast If you want other math recos then there are numberphile, 3b1b etc
|
Steven Strogatz has some nice podcasts: “The joy of X” about mathematics and “The joy of Y” broader sciences. And Quanta Magazine for science and mathematical topics is also nice.
|
What I'm gathering here is that we should crowdsource one from Reddit as a sort of communal podcast.
|
As much as I absolutely adore Strogatz and everything he does, these podcasts just don't scratch my technical math itch
|
My favorite theorem has some really poor production quality. My only gripe with it. Otherwise very good
|
[
"Uses of advanced math in computer science?"
] |
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"math"
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"zy26w2"
] |
[
317
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I don't really know if "advanced" is the correct word for it, but let me explain. I started studying pure math in college a few years ago, then switched to computer science/engineering for a better chance at getting a good job (where I'm from it's particularly difficult to get a job with a pure math degree I think). I haven't finished my computer science degree, and I actually enjoy this field, but I have always loved math. I remember loving learning about topology and being very curious about meaure theory and complex analyisis. I think I already covered all math credits for my degree (linear algebra, single and multi variable calculus and discrete math), and all other courses I see are applications of these subjects to comp sci contexts. I would love to be able to keep learning math but to still apply it to my degree and to any further studies. Is there any use on learning more abstract and advanced math topics besides my own enjoyment? I guess I would like to kill two birds with one stone, study what I like, and also make it usefull for my career. Any help would be appreciated, thanks!
|
Cryptography or automated theorem proving comes to mind.
|
Topology and Differential Geometry are useful in Computer Graphics and Simulations.
|
Type theory? It's the basis for a decent amount of theorem provers (Agda, Coq) and even if you don't count basic dependent type theory as advanced math(Why wouldn't you?), I'd say HoTT is definitely advanced math and is also in part used in automated theory proving (e.g. Cubical Agda).
|
The theoretical aspects of machine learning are math heavy
|
Abstract algebra and category Theory would be useful if you want to play with functional programming
|
[
"What was something you wish you knew before learning so much math?"
] |
[
"math"
] |
[
"zxhg2r"
] |
[
74
] |
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What is a skill, perspective or something you wish you had before learning that helped you learn more efficiently? For me, it was finding interesting things on a topic before diving into the details so i’d be inspired to carry on.
|
That cutting edge mathematics is messier than you'd like. You can't keep doing pretty, elementary proofs alone
|
It’s OK to be a bit slow — and that in fact, being naturally slower and stuck more often can give you all the edge when the research phase starts. Soooo so many people I’ve seen who are way smarter than me, but struggle so much more in research because it’s the first time they really have to struggle.
|
I wish I learned how to study properly instead of just coasting through high school without much effort. But most of all discipline
|
Exposition. I found later in my mathematics education that if I could clearly write out what I was trying to prove, it made errors easier to spot. Writing it out in my native language (English) was in some sense rubber duck debugging myself.
|
General problem-solving skills. Basically, when trying to solve post-highschool math problems, I would just try every single theorem/proposition related, but that requires so much more knowledge, time, and effort than what is generally necessary (even though it is very efficient). By general problem-solving skills, I mean knowing a list of strategies useful to tackle a problem, not just logic/reasoning (induction, deduction, contradiction, absurd, etc.), but also how to argument and move from one step to the next. Some examples of strategies: There are many more, and each strategy requires training. Looking at many puzzles, enigma, problems, brainteasers, and riddles helps a lot. There are great books wholly dedicated to problem-solving skills, I wish I knew about those earlier.
|
[
"What areas of mathematics have a low level of abstraction"
] |
[
"math"
] |
[
"zxhwid"
] |
[
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I'm interested in mathematics without much abstraction. The only area I know that fullfills this is discrete mathematics. Are there some other?
|
Combinatorics. Minimal abstraction, maximal difficulty.
|
Probably category theory
|
isn't abstraction the whole point of mathematics?
|
I'd say euclidean geometry and arithmetic
|
Only beginning number theory! Once you start looking at Gaussian integers, number fields, and automorphic forms, things certainly get abstract.
|
[
"Subtracting any two positive palindrome integers from each other always results in a number divisible by 9 - I can’t figure out how this works, is there any explanation?"
] |
[
"math"
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"urbwn5"
] |
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[
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Are we missing some conditions? 22 minus 11 looks like a counterexample.
|
Are you sure it's palindromic integers? I can think of a few counterexamples to that claim.
|
Do you mean two integers that are palindromes of each other?
|
Maybe what you mean is: Start with any number (any number of digits) and create the number that is its reverse. Subtract them. Then you always get a multiple of 9. Is that what you mean?
|
Try it with bases other than ten.
|
[
"It recently occurred to me that an injection between two finite set of the same size is onto. Would this apply infinite sets as well? I can’t seem to figure it out ."
] |
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"math"
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"ur8kq1"
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"Removed - ask in Quick Questions thread"
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true
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false
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[
1
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No. define a map Z->Z by f(n)=2n. Here Z is the integers. This is injective but not surjective (for instance 3 is not mapped to).
|
Reread the title.
|
My mistake.
|
No worries. It's an easy mistake to make!
|
OP restricted the question to finite sets
|
[
"Why do figures having dimensions less than 1 have volumes much smaller than their actual dimensions?"
] |
[
"math"
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"uqq6z4"
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"Removed - ask in Quick Questions thread"
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In general because numbers below 1 become smaller when multiplied by themselves: 0.5 * 0.5 = 0.25 Note that since the formula is (4/3) Pi r and (4/3) Pi ≈ 4.1888 if the radius is near 1 the volume can be greater than 1, like a sphere with radius 0.9 has volume 3.05. If you have other questions like this you better ask in the Quick Questions thread or in r/learnmath
|
Use other units. Instead of 0.25 inches you can use 250 micro inches. Now the volume is greater then their actusl dimensions
|
Well that's exactly the thing, why does 0.5 * 0.5 gives you 0.25 in practical. If you can share a real-life example where this happens would be great.
|
Think at a Square with a side length of 0.5 meters. How much of these type of squares can fit in a square with length side of 1? 4, so the area of the First Square must be 1/4=0.25 N.B my english skills are not verty good, i hope it could be understandable :)
|
You can understang this better if you use fraction noatation "1/4*1/4 = 1/16" so the denominator actually increases decrasing the value. In real life (1/4 * 1/4) means taking quater part of a quater part of some thing that is if you have a choclate piece take 1/4th part of it and again take a 1/4th part of it so you'll end up with a very small piece.
|
[
"How can I pursue math in a meaningful way this summer since I didn’t get into any REU’s? (Rising Junior)"
] |
[
"math"
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"uqm33w"
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11
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hit up a prof for research opportunity at ur school who does stuff u find interesting
|
REUs aren’t the only way of doing undergraduate research
|
YES! Absolutely yes. You can ask one of your professors for research opportunities that they may have and work on small projects for them. But if you can't find a research project don't despair, there are plenty of other ways to engage. Talk to a professor about doing an informal reading group on an advanced topic: e.g. read an intermediate text in linear algebra like Matrix Analysis by Horn and Johnson; take a research paper in something you're interested in and read it with a professor; grab an inquiry based learning book and present your solutions to your professor e.g. Algebraic Geometry a Problem Solving Approach by Thomas Garritty; to name a few ways. If your plans are to go to grad school, then taking the initiative on a project like this is - in and of itself - meaningful engagement.
|
Try to connect with a professor at your university, either for research or just guided learning. Start looking at different schools for graduate school (if you plan on going). Reach out to professors, try to visit, etc. Do an applied math/data science/coding project. There’s plenty of resources on this sort of thing readily available online. Do all of the exercises in a book related to your interests, whether it’s foundational or advanced Organize and typeset notes from classes you’ve taken so far in your degree. Always good to revisit and reinforce important material Work as a math tutor, teacher, camp counselor, or make YouTube videos.
|
You could try to get matched up with a Mentor in your area of interest... there are various programs out there (per school, or per city, or per country ...) here's an example in Canada: https://stemfellowship.org/stem-skills-development-and-mentorship/stempowerment-mentorship-and-webinars/ )
|
[
"Generalizing the \"remainder\" for integer log and sqrt"
] |
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"math"
] |
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"urahqx"
] |
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17
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0.96
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The standard definition of integer division is as follows: n // d = {q, r}. n = d * q + r So I thought it would be a good idea to generalize it to other functions like the logarithm, like this: ilog_b(n) = {l, r}. n = b^l + r. Examples: This implies r is in the interval [0, n), which means it has a of sizes, and a max size equal to the . This is similar to int division, which has in the range [0, d) (ignoring negative input arguments). We can do the same for Nth roots: iroot(n, a) = {b, r}. a = b^n + r. Examples: I don't know of any practical applications of these remainders, aside from reversing discrete functions and preventing loss of precision during rounding (in the context of computation, like floating-point numbers, fixed & arbitrary-precision alike). I also don't know of any algorithms to compute these efficiently, without computing the "quotients". Any info and opinions would be appreciated. These generalizations could also be done on tetration and other hyper-operations and functions, like the iterated log. But that would require multiplication instead of addition, to avoid astronomically large remainders
|
Euclid algorithm recurses over the integer division to get the gcd of two integers. Similarly, recursing other one of those operations might get interesting results. E.g., applying the binary log to a number, then to the residue, etc., gives the binary representation of the number (it outputs the 1s). Finding the continued fraction for a given real number may also fit into your idea - I have not checked.
|
Really what you are computing is: r = b ) mod n Discrete logarithm is maybe what you are looking for.
|
Nice idea! I'll try experimenting with that in a programming language, to see what results I get with many combinations of values
|
I had the suspicion that those 2 logarithmic functions would be related, but I didn't know they were related in this way. Thank you for the info
|
Could you elaborate? Doesn't this simplify to r = (n + 1) mod n which is always equal to 1?
|
[
"What are the greatest textbooks on Discrete Mathematics?"
] |
[
"math"
] |
[
"ur1q3c"
] |
[
8
] |
[
""
] |
[
true
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1
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I'm aware of Norman Biggs as an author in this subject, his book is quite nice. The text today comes across as somewhat old-timey in terms of language however. Given its roughly 40 years in circulation, that's to be expected. Kenneth Rosen is sometimes mentioned as good, but I can't speak to that. Are there other alternatives this sub might recommend for Discrete Mathematics? What's are your favorites?
|
Graham, Knuth and Patashnik's maybe? Or do I have the wrong level?
|
In what sense is Biggs a "foundational" author? I looked up his book titled , which I'd never heard of before, and the first edition came out in 1985. I assure you that the field of discrete math was well established before the 1980s. A nice book to look at is by Graham, Knuth, and Patashnik. It even has its own Wikipedia page, at which the bottom lists the chapter topics.
|
The text by Susannah Epp is very popular amongst undergrad programs.
|
(In roughly ascending order of difficulty) Knuth's: Concrete Mathematics Bona's: A Walk Through Combinatorics / Cameron's: Combinatorics Aigner's: A Course on Enumeration, Combinatorial Theory (Both Springer's GTM I think) Stanley: Enumerative Combinatorics (hard af)
|
In what sense is Biggs a "foundational" author? Good question. In the sense of providing a wide overview. Not in the sense of pioneering a field a la Newton or Pythagoras. I edited that part out from your feedback. Graham and Knuth have a collaborative book on this subject? That sets a promising expectation.
|
[
"What's the best way to find the roots of an arbitrary polynomial over the complex numbers?"
] |
[
"math"
] |
[
"ur3l39"
] |
[
3
] |
[
""
] |
[
true
] |
[
false
] |
[
0.72
] |
I recently finished a class covering Abel-Ruffini, and I've been interested in factoring polynomials and finding their roots. It looks like is the dominant way to a polynomial over a finite field, but what if I want to find (edit:) a numerical approximation of the zeroes of a polynomial (or even a continuous function) over the complex numbers?
|
I am pretty sure systems find numerical approximations of all the roots of a polynomial by computing all the eigenvalues of the associated companion matrix (using a QR algorithm optimized for the companion structure). per: https://epubs.siam.org/doi/10.1137/140983434 : upon further investigation, many systems also use a cubicly convergent simultaneous iteration technique like Bini's MPSolve https://numpi.dm.unipi.it/software/mpsolve
|
Use Newton's method starting from a close approximation. First make sure your polynomial has no repeated roots. If you're working with polynomials f(x) with rational coefficients and gcd(f(x),f'(x)) is a nonzero constant then f(x) has no repeated roots. In general, if f(x) with rational coefficients has repeated roots then you can pass to a factor with the same roots all of multiplicity 1 by using the ratio f(x)/gcd(f(x),f'(x)).
|
I don’t know if it is standard, but I know winding numbers can be used to find roots of arbitrary complex functions. If you have an initial bounding box, you can divide the remaining area to search in half at every stage of such an algorithm.
|
One popular and efficient method is Aberth method
|
Sorry — numerical approximation
|
[
"Places to buy/sell Math textbooks"
] |
[
"math"
] |
[
"uqqi9i"
] |
[
8
] |
[
""
] |
[
true
] |
[
false
] |
[
0.8
] |
Hope this is the right place to ask this. Apologies if not. Due to my Uncle having a big clearout I've inherited a large number of old Math books, some of which I think are quite rare, or specialised. Can anyone recommend somewhere to sell these, or otherwise provide them with new homes. They are a bit too niche for ebay or the mass book sellers. I'm also told that some of them might be considered too advanced for undergraduate students. Due to their focus I fear any charity shop or used bookstore would just bin them. I've passed a list to the local University Maths department (I'm in the UK) for them to pass around, but was wondering if there was anywhere else I could reach more interested mathematicians. Is there a buy/sell Mathbooks section on Reddit? Thanks in advance. Edit: Full list here - Sorry - didn't want to clog up the screen
|
If you post the full list of books here you might actually find some people that would be interested
|
try university libraries! as an undergraduate, i loved just going and checking out random books on topics way over my head from my university's library just for the sake of learning something new lol
|
Is the classic "casual redditor commenter" attire.
|
Is the classic "casual redditor commenter" attire.
|
Cool. I will investigate and drop you a PM. Thanks.
|
[
"Imposter Syndrome in Mathematics"
] |
[
"math"
] |
[
"ur4ilh"
] |
[
388
] |
[
""
] |
[
true
] |
[
false
] |
[
0.93
] |
Last week I had to put together a qualifier for two of my first PhD students in mathematics. This is actually the first bit where the rest of their committee has to participate, and I found myself getting that old feeling that many of us are familiar with. " " I was worried that the other professor I was working with would see the exam, and declare that it was substandard, and that I haven't been preparing my students properly. I worried that I'd be caught out as a and that the other faculty would realize what I always knew, that I couldn't cut it. I really got me thinking about and how embedded it is in academics. All the way from students to professors. I feel that Mathematics is especially plagued with this, because mathematicians place themselves very high on the academic spectrum. Most of us have probably seen that XKCD comic " " with the mathematician way over on the right. Even bits of mathematics can be quickly discarded as simply something that engineers or physicists do, and not something for real mathematicians. "Applied Math is Bad Math" is a sentiment that has been around for decades (if not more than a century). I recently had a discussion with someone here on Reddit that said an aspect of Theoretical Computer Science wouldn't be considered "applied," since it had proof. Applied math doesn't mean it doesn't have proof, it's just math that isn't done purely for math's sake. There was a post on here recently asking when can someone call themselves a mathematician (wondering where that gate was to enter mathematics), and I know professors that are quick to dismiss nearly everyone else's work as trivial. When you are surrounded by these sentiments like this all day long, you quickly start to worry about fitting in. I took the opportunity to assemble my thoughts , if you'd care to watch. I'll be back on schedule with Real Analysis videos in the next week or so, but I wanted to address it. Video Link: By the way, the other professor simply emailed back " " How do you cope with these feelings of inadequacy or unbelonging?
|
I dropped out of a PhD program because of funding cuts and, after a few twists and turns, got a job in a research division of IBM. I was 1 million percent certain I was in the wrong place -- these were really smart people who were very seasoned software developers -- people who had invented programming languages and operating systems. But I felt the same in grad school, and, to a certain extent, as an undergrad. I got a really good yearly review, a pay raise, and more responsibility and thought, "My boss is just being nice. If he really knew .." Yeah, they promoted me out of ignorance -- that MUST be it. Decades later I realize I will never be done with that sense that I am in over my head, it is always tapping me on the shoulder and reminding me that what we do is hard and someone is better at it than me. But I've done ok and I keep coming back to the question of "why do I do this at all?" Because so many people get their pride caught up in the profession like I do and feel a certain sting when things change -- be it more or less responsibility. I've come to the conclusion that there is a certain craving at the bottom of my impostor syndrome, a craving for validation that both motivates me and criticizes me. That group at IBM was slowly dismantled and I was shown the door like everyone else. It was crushing, I still mourn that loss, and I still feel that sing of doubt. If my craving for validation was amenable to reason, this would have passed long ago -- so it is unreasonable and is no longer helpful so the best I can do is recognize it and talk back to it. I had a psychotherapist (old school Freudian talk therapy, great stuff) who had a quaint office filled with books and a funny little stitch pillow that said, "If all else fails, lower your standards". On our first meeting he pointed to the pillow and asked "What do you make of my little stitch pillow?" I laughed and he smiled and explained there are only ever two reactions to the words -- laughter and fear. The pillow is something of a litmus test. People seeking therapy are all working out how adjust their standards and if they will be swept away by unseen forces if they can't be flexible. But I believe everyone is working that out -- how to be flexible so they don't get swept away by rigid standards -- especially the standards they have adopted to help motivate them in the first place.
|
That is a great story. It must have been incredible working at IBM Research, and it sounds like you did pretty well there. I need a pillow that says "If all else fails, lower your standards." Not a bad way to start introspection.
|
How do you cope with these feelings of inadequacy or unbelonging? Poorly lol. It's not a healthy answer but I kind of just repress those feelings until I hit something that enables me to reason them away but they always come back eventually. When I was in my undergrad for comp sci, it would be from other students knowing how to make GUIs and build things that people outside of the field could recognize as something substantial. This got better when I started looking for jobs and got one almost immediately. In industry, it's that someone else always seems to have an answer to every problem while my skills start to feel useless. For this I just have to remind myself that it's never one person that has all of the answers, but I tend to think of colleagues in this regard as a single unit so it seems like if 6 solutions are accepted then I should feel bad about just 1 instead of 5 despite the team being 6 people. In my first attempt at a master's, I couldn't keep up at all and failed out after one semester. This bothers me much less now as I've realized that CS is not the path for me and that I had undiagnosed and untreated ADHD. Now finishing up my undergrad for applied math, it's back again because I'm simultaneously in upper division classes despite having gotten a D in ODE originally and having forgotten A LOT of calculus. But at the same time, when I really apply myself I can get through it, I notice things and ask questions that don't occur to others, and my professors all seem genuinely happy to have me around. Then there's the ones that are currently looming over me. I just got let go and even though I had been contemplating leaving, it was a huge blow to my confidence. And of course there's the continuous one that is ever present. As a trans woman, I regularly feel this way about my own existence. I'm not sure if that one will ever go away. I wrote more here than I had intended but hey, maybe it'll help someone as a show of solidarity.
|
Pride and ego are the real happiness killers especially in a field like mathematics. IMO you have to take proactive steps to keep them in check because, like you said, the feelings aren't typically amenable to reason. Asking questions you're sure are too embarrassing or dumb to ask is one way. You eventually get used to not caring about being judged. The Fields medalist Tim Gowers has talked about how the whole field would be much more welcoming and, importantly, more productive too if everyone could just ask questions without worrying about what others think. Another way is revisiting topics you're sure you already know about and are too basic. Oftentimes that's pride talking. What's great about this approach is that because every teacher has a different way to explain things, there's usually little bits of insight (or things you once knew but forgot about) regardless how many times you study the same topics. Ultimately these efforts are to stop fighting against imposter syndrome. Embrace that you're a perpetual beginner even if that's not true in some objective sense. Forgetting things is completely natural given the limitations of the human mind and the vastness of the field. There's so much to know about, and so many angles to approach each topic from, that it's unrealistic and foolish to act like you've conquered it all, or expect you one day will.
|
I personally stopped worrying about this when I started thinking that I'm probably not a mathematician. I do math because there are some things I like to know, some contributions that I'd like to make, but that's really it. I started thinking about who really has the authority to deem my work worthy of the "mathematician" label, and there's really only a few people in my life who I'd respect enough to confer that title on me, and then I decided it wasn't exactly worth putting that much work in, and so why worry about it after that point?
|
[
"Random question about shapes"
] |
[
"math"
] |
[
"uqmrug"
] |
[
62
] |
[
""
] |
[
true
] |
[
false
] |
[
0.93
] |
So, I'm in 10th grade Geometry so I'm not too knowledgeable in geometry. Is there any single known shape (3D or 2D) that cannot be cut into two equal halves no matter where you cut it? I know it sounds stupid but it came to my mind and I'd like an answer. Edit: Thank you all for the answers, I honestly have no idea why I asked this. It was like 3am so this was asked while barely awake
|
Do you mean equal in terms of their area? Imagine holding your scissors at the far left edge of any shape. If you cut it there, the right piece will obviously have more area than the left. So you slide the scissors from left to right, looking for a better place to cut. If you slide too far, the scissors will end up towards the far right edge, and then the left piece will obviously have more area than the right piece. If I can slide my scissors along a continuous path from left to right, so that in my starting position there's more area in the right piece and in the ending position there's more area in the left piece, it logically has to be true that at some moment my scissors were over the location where that transition happened - that location is a place where you can cut and have two halves of equal area!
|
Even though the answer is somewhat obviously (once one had analysis) "no", the question itself is very good and should be exactly the thing talked about in high-school math imo.
|
Not only can every shape be cut in half, in n dimensions it's possible to cut n many shapes in half See https://en.wikipedia.org/wiki/Ham_sandwich_theorem
|
gotta love reddit mathematicians pointing a 10 year old to axiom of choice
|
No. https://en.wikipedia.org/wiki/Intermediate_value_theorem Space is continuous, where you can move the knife is continuous, therefore the above link applies. Skim it, don't worry about the gobbledygook, u/alternets has it right; the link above might provide more depth. But, there is a world where the above does not apply. if we're not in continuous space, like we restrict ourselves to a grid, like if we lived in an old video game on the NES. A "triangle" living on this grid shaped so that two blocks are stacked next to one block cannot be cut evenly, since the blocks are the smallest thing that can exist in that universe. Space has no meaning "between" the pixels to someone living in that world. But we, living outside that world, could still view those "points" as the squares that we see, and cut the squares in half. Stepping out one layer further, in our universe, matter is quantized and space probably is too. So we might also be the pixel people who can't really cut every object perfectly in "half" But to reiterate, the real answer to the question you are asking is No. No stupid questions. You can often find a counterexample by suspending assumptions of reality in math. Spherical geometry describes how to calculate motions on a sphere, which is pretty relevant to us living here on earth, using the same rules that you learned in a geometry class, where lines are "straight" when you point in a direction on the earth and keep walking that way, making them arcs in normal geometry instead of lines. Spherical geometry is generated by tweaking one of the axioms of regular geometry, the parallel postiulate. "Paralle lines do not intersect at any point", and change it to "Parallel lines intersect at exactly two points." This is true on a sphere; think about the equator and another circle going between the north and south pole. They must intersect in two places. "Do parallel lines have to never meet" is a stupid question that greatly advanced our understanding of mathematics with plenty of practical uses.
|
[
"What Are You Working On? May 16, 2022"
] |
[
"math"
] |
[
"uqz8hn"
] |
[
14
] |
[
""
] |
[
true
] |
[
false
] |
[
0.94
] |
This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including: All types and levels of mathematics are welcomed! If you are asking for advice on choosing classes or career prospects, please go to the most recent .
|
Figuring out how to organize my research without a dissertation to stick it into, since I finished my PhD this semester :) so far I’ve just got a “misc. results” overleaf file.
|
I'm revising for my numerical methods exam in two weeks' time. It's going to be difficult; I've got hours and hours of lectures to catch up on, plus reading notes for the differential equations bit because there was another strike. Good news is threefold: (1) that's the hump and then it should be relatively easy to make the cheat sheet and practise, (2) I'm going clubbing with a friend this week (which is notable because I'm not really a clubbing kind of girl, and the last time I went I didn't like it, but I also wasn't with people I knew very well), and (3) the Champions League final is the Saturday before the exam and LIVERPOOL ARE GONNA WIN (or at least I fucking hope they do; not only are they my PL team, but if Real win fucking CL it's going to be a right downer for my exam).
|
I am currently "interleaving" between Harvard's online course in abstract algebra (Currently lecture 11) and Abbott's Understanding analysis (Chapter 7, the last of core material). When I finish Abbott's book, I will probably replace it with Axler's LADR, as it is a good companion to the Harvard course. The Harvard course uses Artin, and it is quite LA oriented. I am super excited about what I have learned this year of self-study, and I am looking forward to learning more and more. My reading list has grown so much in the last year that I am not even sure I will finish it!
|
I’m bored of geometry so when I have free time I’m teaching myself higher level things, like solving integrals
|
just reading it now, but one thing popped out at me. where you talk about the ratios approaching 1/3, I think you have a error. You claim 1365/64 = 0.33307...., but I believe that should be 1365/2048. 44 showing in the cycles that is a fascinating result. I wonder if it's ever been noticed before?
|
[
"Is linkedin useful for academic mathematicians?"
] |
[
"math"
] |
[
"uqqo0w"
] |
[
54
] |
[
""
] |
[
true
] |
[
false
] |
[
0.9
] |
Hello, In search for a supportive community, For applied mathematicians employed in the industry, I was recommended to use . As I am keen on pure math and academia. I wonder whether there's a community of pure math academics on . Do you think is useful in anyway for pure mathematicians? For example, in finding jobs, Connecting with other students, or Finding a PhD advisor. Best,
|
No. Topic specific mailing lists is the place to be.
|
Not useful for that imho. It's for "business people" to network. There are special internet groups including math, but they are more for business humans who like/use math. So you will learn about the ramanujan taxi number story or view a humorous gif about the relationship between e pi and i. Or a link to an article in quanta. If, like me, you end up in business you'll want to have an account with your resume. One of the most useful things is to "friend" a new colleague so that the colleague can be sure to know which fancy degree you got from which fancy school, and that you were head of Blah and Bleh Co. You will also see if a former coworker is now in a cool new position, in which case you may want to "reach out". P. S. The other notable thing is that Often the career change notes are of the form "congratulate Joe Schmidlap on being named Managing director and ceo of Schmidco Consulting". This usually means that Joe got fired and just set up an s-corp.
|
As a pure mathematician, one of your roles will be mentoring students. Some of those students will go into applied math or industry jobs, and they will have a use for LinkedIn. On LinkedIn, it looks good for students to be connected to their professors, as it indicates some deeper-than-just-attended connection. So get on LinkedIn, and encourage your students to "Link" to you. Do it for the children.
|
For example, if you are a (broadly interpreted, e.g. including most analysts) applied mathematician, you will usually get a SIAM membership and join the SIAM mailing lists for the fields you're interested in (e.g. PDE, applied dynamical systems, etc.) where you will see conference and seminar announcement, PhD/postdoc positions, etc.
|
For example, if you are a (broadly interpreted, e.g. including most analysts) applied mathematician, you will usually get a SIAM membership and join the SIAM mailing lists for the fields you're interested in (e.g. PDE, applied dynamical systems, etc.) where you will see conference and seminar announcement, PhD/postdoc positions, etc.
|
[
"Can a special case be generalized in more ways than one? Any well known examples?"
] |
[
"math"
] |
[
"uqz9pj"
] |
[
27
] |
[
""
] |
[
true
] |
[
false
] |
[
0.93
] |
Basically, you have a conjecture and someone finds a way to give a generalized conjecture. Could someone later come along and take the original conjecture a different way and give us a different generalized conjecture? Any well known examples of this happening?
|
another way to phrase this, you have two different theorems/objects/etc, but a special case of both is the same thing.
|
Absolutely — any area of research gives more questions than answers :) and those questions lead to new conjectures and definitions! An example in my field is Reed-Solomon codes, which are: an example of a linear code an example of a Maximum Distance Separable code an example of an Algebraic Geometry code an example of an evaluation code an example of a trivial locally recoverable code an example of a self-dual code (at times) etc. RS codes as an object is are examples of many more general types of codes :)
|
I don't know if this is what you're looking for, but if you have finitely many topological spaces then the topology on the product is the topology generated by the set of products of open sets. Many people meeting this for the first time suspect that this might be a way to define the topology on an infinite product as well. You can do it (its called the box topology) but the has better properties. For the product topology you take the topology generated by products of opens of each factor, where all but finitely many elements of the product are the whole of the space they sit inside.
|
I've never seen the box topology used for anything other than counterexamples. The product topology is simply the topology to put on a product of spaces (at least in topology, in algebra we need to be more careful)
|
Not just that: the product topology is the topology with that property. The box topology also satisfies it.
|
[
"What are the classical mathematics textbooks of the 20th Century?"
] |
[
"math"
] |
[
"uqri3v"
] |
[
33
] |
[
""
] |
[
true
] |
[
false
] |
[
0.92
] |
I have some classical books such as Richard Courant Differential and Integral Calculus both volumes I and II (English version). I also have Hardy’s Pure Mathematics. Apostol Introduction to Analytical Number Theory. Also Pure Mathematics I and II by Backhouse and Houldsworth. However are there any other classics in mathematics textbooks from 1900’ to 1970s?
|
Arnold's Mathematical Methods of Classical Mechanics was published in the 70s. Definitely a classic in the area of geometric mechanics.
|
Maybe Steen and Seebach's "Counterexamples in Topology"?
|
Kolmogorov and Fomin - analysis Appel et Goursat - Cours d'Anaylse Lang - Algebra Siegel - complex function theory Dedekind - Was Sind und Was Sollen die Zahlen (1901!) Atiyah and MacDonald - commutative algebra Strang - some linear algebra book Serre - a course in arithmetic Macclane categories for the working mathematicians
|
Hartshorne's AG, unquestionably. Eisenbud's CA in my opinion (not from the 70s but the 20th century nonetheless), but more uncontroversially Zariski and Samuel's. Also Mumford's Red Book. Milnor and Stasheff's book on characteristic classes. Adams's Blue Book on stable and chromatic homotopy, and Ravenel's Green Book on the same (again, the latter was not around in the 70s but is still from the 20th century). Tbh any book that's commonly referred to by its color is probably a classic. Oh, and EGA. Obviously. EDIT: If you count physics (and books from the 80s), Shankar's book on quantum should be on the list. Likewise with Feynman's Lectures in Physics series and Taylor's Classical Mechanics. I would also argue for Purcell's EM text, although people seem to have shifted away from it lately in favor of generic "intro physics" bullshit like Giancoli.
|
Complex Analysis by Lars Ahlfors and Topology from a Differentiable Point of View by John Milnor should both count.
|
[
"What is the best layman interpretation of any mathematical concept you have seen?"
] |
[
"math"
] |
[
"uqrxxl"
] |
[
411
] |
[
""
] |
[
true
] |
[
false
] |
[
0.98
] | null |
I had a student when I was a GTA for a vector calc course describe the average value of a multivariable function as the height of water in a fish tank once the waves have settled. I thought that was a super clear way of putting it.
|
I don't know the 'best' layman interpretation, but the funniest I've seen is SMBC's take on Stokes' Theorem: If you draw a loop around something, you can tell how much swirly is in it.
|
Fundamental theorem of calculus: sum up all the little bits of change to get the total change.
|
"you can't comb a hairy ball flat without creating a cowlick" Condenses a bunch of vector analysis into something very approachable.
|
Cauchy-Schwarz: If you want to push a trolley along a track, it is best to push in the direction of the track.
|
[
"Can a vector space have more than one Basis?"
] |
[
"math"
] |
[
"r90xyy"
] |
[
0
] |
[
"Removed - ask in Quick Questions thread"
] |
[
true
] |
[
false
] |
[
0.33
] | null |
If you're asking for help learning/understanding something mathematical, post in the Quick Questions thread or /r/learnmath .
|
Short answer: yes.
|
How?? Does it have to do with subspaces? Or??
|
Any linearly independent set of n vectors spans an n-dimensional vector space.
|
Absolutely!!!! The plane has a gazillion bases: any two vectors that don't lie on the same line are a basis. The x-axis also has a gazillion bases: any nonzero vector on that line is a basis for the x-axis.
|
[
"Elementary math term question"
] |
[
"math"
] |
[
"r8r3ml"
] |
[
1
] |
[
"Removed - ask in Quick Questions thread"
] |
[
true
] |
[
false
] |
[
0.67
] | null |
I would say “the difference of 7 and 2 is 5”, so to me, the difference is the expression involving the subtraction sign.
|
This is exactly the kind of pedantry that serves only to confuse students and does not teach them anything useful. Teach what subtraction represents, when to use it, and how to calculate it. Whatever you call which term is completely irrelevant to their meaning.
|
Agree, and the word “is” here plays the role of the equals sign. Posed as a question: “What (x) is (=) the difference between 7 and 2 (7-2)?” You get the equation x=7-2.
|
Could you give an example of a problem? It would be helpfull to know what exactly you're tlking about
|
I guess it’s just the fact that the math book says difference=answer. Like ‘I have some grapes. I eat 3. Now I have 5. How many grapes did I start with’ the answer would be 8 but the 8 would have been the minuend. I think the book is trying not to have to teach more terms to the kids but I wonder if the lack of nuance isn’t good…?
|
[
"What does the term Euclidian geometry refer to?"
] |
[
"math"
] |
[
"r8v0ms"
] |
[
1
] |
[
"Removed - ask in Quick Questions thread"
] |
[
true
] |
[
false
] |
[
0.54
] | null |
It's not a variant so much as other geometries are a variant of it. For a long time, "Euclidean geometry" was just called "geometry". It wasn't until people like Gauss discovered hyperbolic geometry and other developments that anybody needed to disambiguate. Occasionally, you will see the term used to mean specifically the geometry developed in Euclid's , rather than just geometry in flat space.
|
Nothing. It's the geometry you probably learned at school. On a plane or in 3D. Certain things do work differently when you walk on a sphere for example
|
It is the non-Euclidean geometry which is "different", Euclidean is the normal thing. Like, there are no rectangles in non-Euclidean geometries, because parallel lines act weird there. You get pentagons with five right angles and stuff.
|
Euclidean geometry == Coca-Cola Classic
|
Oh thanks
|
[
"How many hours would it take to walk 2500km if you are going 5km/h"
] |
[
"math"
] |
[
"r8m66z"
] |
[
0
] |
[
""
] |
[
true
] |
[
false
] |
[
0.12
] | null |
This is one of the millenium problems I can’t help you
|
Zeno is still trying to figure out the first step.
|
Let me get Terence Tao on the phone for this one
|
Sorry I can't help with this one, this is way too far above my pay grade.
|
At least 2
|
[
"Am I stupid or not?"
] |
[
"math"
] |
[
"r8dl0x"
] |
[
0
] |
[
"Removed - try /r/learnmath"
] |
[
true
] |
[
false
] |
[
0.5
] | null |
Others are misunderstanding you due to bad formatting. If you wrote that in reverse, starting with 4=4 and ending up with 2sqrt(2)=sqrt(8), you’d have a convoluted but entirely legitimate proof. Unfortunately you can’t assume what you want to prove and arrive at a true statement and call it done. I can prove any equation that way: Assume 2 = 3 2 * 0 = 3 * 0 0 = 0 But there’s no way to go back from 0 = 0 to 2 = 3. The easier way would be to write: Sqrt(8) = sqrt(2 * 2 * 2) = sqrt(2 * 2 ) = sqrt(2) * sqrt(2 ) = sqrt(2) * 2
|
Your proof is backwards: you start by assuming the conclusion, and you end by deducing something you already knew was true. A proof should start from a premise which you already know is true, and end by deducing the thing you wanted to prove. But you have essentially the right idea, yes. You can simply rearrange the pieces like so: (2√2)(√2) = 2√(2x2) = 2√4 = 2x2 = 4 = √16 = (√8)(√2), so (2√2)(√2)=(√8)(√2). Divide that on both sides by √2 and you're done. Rather than introducing this extra factor of √2, you could also just work directly: √8 = (√2)(√4) = (√2)(2) = 2√2.
|
He’s multiplying by sqrt(2), although the formatting does make that difficult to spot.
|
You assumed the equality you wanted to prove to start. Also, your notation muddies up the algebra. Instead, set x = 2*sqrt(2) and compute x
|
√8 = √2 = √2*2 = 2 √2 Edit: formatting from hell Edit: when dealing with square roots, powers of two are easy Edit: when you multiply both sides by a number, you can’t put that number under the radical.
|
[
"Looking for a formula for a triangle"
] |
[
"math"
] |
[
"r8eyq7"
] |
[
0
] |
[
"Removed - try /r/learnmath"
] |
[
true
] |
[
false
] |
[
0.4
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There is probably a few ways to do this but I would just double the triangle area to find the area of a square and then take the square root of that to find the length of one side.
|
I'm very confused by your question then, you want a formula for the length of the sides, but you don't want that formula to use A? A's all we have.
|
Suppose the area of the triangle is A, and the side length is s. As the triangle is right isoceles by hypothesis, the two legs must both measure s. Construct two sides, completing a square. The area of this square is 2A. Hence s = 2A, or s = \sqrt{2A}. The hypotenuse of the triangle is then s\sqrt{2}.
|
I'm very confused by what you are allowing or disallowing as a solution method. Like... you could do probably something silly with Heron's area formula and the Pythagorean theorem. But that could be begging the question a little when you think about where those are coming from.
|
can you explain this a bit? It might be what im looking for
|
[
"i’m just curious cause my math teacher said it’s not correct"
] |
[
"math"
] |
[
"r84ith"
] |
[
1
] |
[
""
] |
[
true
] |
[
false
] |
[
0.67
] | null |
expand ln(1/3) as much as possible Questions like this are terrible because they are ambiguous: "as much as possible" doesn't really mean anything in this context and a student's interpretation may not be the same as the teacher's.
|
ln(3)-ln(3)-ln(3) = 0 - ln(3) = -ln(3), and ln(1)-ln(3) = 0 - ln(3) = -ln(3), so it's true.
|
Well, the problem is that even though ln(1/3) = ln(3) - ln(3) - ln(3), I really have no idea how you came up with that, and that expansion is overcomplicated (ln(3) - ln(3) is just 0). The way I would do it is ln(1/3) = ln(3 ) = -1*ln(3), which is much more straightforward and uses rules of logarithms which always work. Or you could even do ln(1/3) = ln(1) - ln(3), same thing. So it's not about your answer being wrong, but the method being confusing. I am interested, how did you come up with your original answer?
|
How did you get that? While it has the same value, it is a rather pointless expansion. I mean, you can even write ln(53314) - ln(53314) - ln(3). The first two terms cancel out anyway. Probably the point of the expansion was to use the fact that ln(a/b) = ln(a) - ln(b). If a =1 then this is equal to 0 - ln(b) that is -ln(b), but most of the times it does not help to expand 0 as ln(something)-ln(something).
|
how is it lazy? can you blame them for being uncertain when their teacher is telling them they’re wrong?
|
[
"I finally figured the process of coming up Counterexample's"
] |
[
"math"
] |
[
"r95t1c"
] |
[
13
] |
[
""
] |
[
true
] |
[
false
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[
0.78
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Was working through some problem's and finally got the process of crafting counterexample's. It's very similar to finding bug's/hole's in software KISS goes a long way :).
|
Having clear examples and counterexamples to theorems is arguably more important than knowing the proof of that theorem (depends on the theorem). Everything is much clearer when you have a ton of (counter)examples to test your intuition against :)
|
Now you finally have time to work on your apostrophe’s
|
Pretty much. You have to have a firm grasp on examples and counterexamples before you can make a conjecture and prove it. "Theory-building" I guess is just taking a whole bunch loads of these examples and putting them into a single box that can be used for other objects that have the same properties, without going through the same proof/computation over and over again. Then again, you can have a good theory if it relates two seemingly different subjects (e.g. GAGA). But really, "theory-building" should foremost help with understanding.
|
Everything is much clearer when you have a ton of (counter)examples to test your intuition against :) Indeed :) tho hearing this bring's me to ask what exactly is "theory-building" it seems one starts out with an example and generalizes out ?
|
Keep it simple stupid
|
[
"Philosopher Wins Over €1 Million Grant for Project on Mathematical Knowledge"
] |
[
"math"
] |
[
"r8ab6s"
] |
[
80
] |
[
""
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[
true
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false
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0.94
] | null |
Okay, skimming her actual past research doesn't produce the same "wtf" reaction at all. https://philpeople.org/profiles/silvia-de-toffoli I'm actually looking forward to reading it in more detail.
|
r/badmathematics
|
I’m a little surprised that this evoked any negative reaction at all, did people really think they’re just handing out a million dollars to a Kant scholar and saying “write math stuff now”? Of course she’s going to be phenomenally qualified to win a grant of this size.
|
Given her qualifications (much more than my own), she must know what she's talking about and I would guess that the actual project must discuss some more interesting things than the description lets on. Because some of the stuff in the description just sounds weird. Mathematics can be almost eerily good at describing everything from the universe to the tiniest particles. Therefore, many people believe that mathematics is something that has always existed, that humanity is merely discovering. Is that why people really think mathematics has always existed? It seems like a poor argument. While it is interesting that mathematics often can be used to describe the physical world, this mathematics exists entirely independently from the physical world it's describing. No self respecting scientist will look at a mathematical proof and claim that it actually proves something about reality, until experiments have backed up whatever claims the mathematics points to. According to this view, mathematics is infallible, being the only science to offer us perfect certainty. Perfect certainty of what? Again, a mathematical proof cannot say anything about reality, so it certainly doesn't provide us perfect certainty about anything in reality. Again, I think the general scientific community would readily agree with that. All a mathematical proof says is that given certain assumptions and certain rules of logical deduction one mathematical statement leads to another. Moreover, mathematical proofs are often informal. They are designed to convince mathematicians and cannot be automatically checked by computers. Would being able to check proofs by computers make the arguments any more convincing? How can we be sure the computer is checking correctly?
|
the description by the foundation is horrible, the work might be good but what they are saying is complete nonsense. The things you quote are spot on. The only thing I would argue is that one wants to put mathematics along with the sciences, and I would debate that point, it is infallible in the sense that a proof is immutable in time, it can be generalised/improved but itself it will never be superseded like a physical theory.
|
[
"How do you keep track of important results?"
] |
[
"math"
] |
[
"r936rl"
] |
[
52
] |
[
""
] |
[
true
] |
[
false
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0.97
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Hi everyone, How do you remember important theorems, lemmas, (in-)equalities, etc you found in papers, books, or your own proofs and derivations? And how do you find a helpful result again when you need it? Do you have some index or maybe even some collection? Do you look for useful results from scratch every time? Or do you have a completely different system?
|
I have a binder and plastic cover sheets with dividers. Definitely have needed it throughout my life.
|
Two systems: I have a notebook (yes, paper) where I write down the important and noteworthy stuff. I leaf through it when I'm idle or when I'm stuck, to both distract myself and see what useful stuff I stumbled upon in the past but already forgot about. The paper part is quite important (at least for me), as I found out that I am way less efficient in remembering and re-discovering things when I record it in digital form, even in hand-writing apps. The second is a mindmap app that allows to heavily annotate all entries. I use this mostly to organize topics and things I have not fully understood yet, especially research I am working on. This helps me to recapitulate and see what I already figured out and where the gaps in my understanding are.
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I've tried to do this, but I it slows me down when reading a paper. And I never know how important something is until I see it a few times but by then I can generally remember it or at least remember what paper I have seen it in.
|
Old fashioned paper journal and just general familiarity. I'm long since retired. I still journal and have journals going back decades. These aren't emotional journals, they're my own math journals.
|
It's called vym .
|
[
"Distance between finite set on 2d grid ."
] |
[
"math"
] |
[
"r90j97"
] |
[
5
] |
[
""
] |
[
true
] |
[
false
] |
[
0.86
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Hi everyone, I was trying to find a distance between subsets of point on a 2d grid. You can think of each subset as a collection of points of Z[i] The idea is to create a function D(A,B) -> R (with A,B in Z[i] ) s.t. it satisfies the axioms of distance: 1) D(x,y) = 0 iff x = y 2) D(x,y) = D(y,x) 3) D(x,y) <= D(x,z) + D(z,y) So far I haven't found anything usefull. I looking forward to see what you can came up with :) Thanks.
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Since you didn’t specify it enough, I’m going to cheat and suggest D(x,y)=0 if x=y, and 1 otherwise. It satisfies all of the requirements you have put forward so far. 😈
|
I like the Hausdorff distance . Pick a point in A that's furthest from any point in B, and call this distance d_1. Then pick a point in B that's furthest from any point in A, and call this distance d_2. Then the Hausdorff distance is max(d_1, d_2). Since your points are in Z[i] you can work with the squared distances, and only take the square root at the very end.
|
Or, if you want something more reasonable (not sure what to do with empty sets) For equal sets (X=Y): D(X,Y)=0 For subsets (X is a subset of Y): D(X,Y)=sup{inf{d(x,y): x in X}, y in Y\X} For all other cases: D(X,Y)=sup{d(x,y): x in X\Y, y in Y\X}
|
Hausdorff distance In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set.
|
I think "size of set which contains all points in exactly one of x and y" would satisfy the axioms. It also feels like a good measure of distance since it's exactly a measure of how much change you need to go from one to the other.
|
[
"How many branches of mathematics do you think would be most appropriate to divide?"
] |
[
"math"
] |
[
"r8ohwr"
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0
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When I was studying Olympiad, there were 4 branches of Olympiad - algebra, combinatorics, geometry, and number theory. And these are still 4 branches of IMO. If we use more academic terms, I think combinatorics is a part of discrete mathematics, and other three is just themselves. For example, calculus can be a part of algebra, and topology can be a part of geometry. So I think mathematics could be classified by 4 main branches - Algebra, Discrete mathematics, Geometry, and Number theory. How do you think?
|
calculus can be a part of algebra You have an unusual way of thinking about either calculus or algebra. What you call the 4 "main branches" is a strange-looking list; your viewpoint has been affected by paying too much attention to math contests. Mathematicians often put math into three areas: algebra, analysis, and geometry. See https://en.wikipedia.org/wiki/Areas_of_mathematics and https://en.wikipedia.org/wiki/Mathematics_Subject_Classification for different subdivisions.
|
Try also: Analysis. Statistics. Logic. There are definitely more.
|
Calculus is not geometry. Calculus is not algebra. Descartes, with his creation of coordinates, did in large measure allow elementary geometric ideas to expressed in terms of algebra (algebraic curves), but that is far from saying all worthwhile concepts in geometry can be reduced to algebra. Consider arc length of a curve or almost anything about transcendental curves (those not given by polynomial equations). All of math can be expressed using concepts that come from set theory, but this does not at all mean all of math is (best) thought of part of set theory.
|
I recommend reading the introduction to mac lane's "function and form", where he gives a nice exposition on the big problems with any such division (and also perhaps will help enlargen your view on the scope of modern math)
|
Didn't Gaus believe that geometry could be reduced to algebra.
|
[
"What to do when you can't find a specific paper anywhere, but know it exists?"
] |
[
"math"
] |
[
"r90hei"
] |
[
86
] |
[
""
] |
[
true
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0.96
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The paper in question is Atle Selberg's influential "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series." It's a very important paper in spectral geometry and harmonic analysis and is cited widely. However it appears to be pretty much nonexistant online. I think the journal's page actually went down. I very much need to access this paper for my thesis, so it would help to know if there's an archive where it's available, for free or for fee.
|
Ask your university's library to obtain a copy. It is a standard for university libraries to get copies (from other libraries) of articles someone requests if the journal is not available there. Don't request the journal, just the article. You should be able to get a scanned copy within a week or so. This is well worth the effort to figure out how to do if you have not done it before. Look at volume 1 of Selberg's Collected Papers. It contains all of his publications during 1936-1988, and the paper you are asking about appeared in 1956. I looked there and the paper you want starts on p. 423. If your library does not have Selberg's Collected Papers (volume 1), then you can request this book from your library through their interlibrary loan service. Or figure out how to get a copy of a book online for free, somehow. Many important mathematicians have volumes of collected or selected works, so you can read most or all of their papers through such volumes without having to track down old journals. Learn Russian: http://www.mathnet.ru/links/e0ba03448e5c1adbe5ab6d813121b11a/mat30.pdf . According to footnote 4 on the last page, Selberg's paper has only 1 item in the bibliography, a paper of Maass, and the translator added 5 more papers after that by Gelfand and co-authors. You can read the MathSciNet reviews of those additional papers using their MR number in the MathSciNet search page, which are 0033832, 0033831, 0050595, 0086266, and 0052701.
|
Seeing how Selberg has been dead for a decade, I don't think that would help...
|
Thank you so much! I will check his collected papers! What exactly is the bibliographic information for the volume you're describing?
|
I think you should first try to figure this out yourself, in order to learn by experience how you might do that (teach a man to fish, that kind of thing). After all, you asked us for help and provided only an author and article title without identifying the journal, volume, year, or pages and therefore had to figure out things based only on the author and title. So now you get the pleasure of doing the same thing in reverse.
|
I know your advice was meant to be general, but in this specific case it's kind of funny since the author died 14 years ago. :)
|
[
"Is there a concept dual to that of universal cover?"
] |
[
"math"
] |
[
"r8csp3"
] |
[
14
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""
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Given a based topological space X, we can construct a based universal cover X' whose points are pairs (x, p) where p is a homotopy class of paths from the base point to x. We have f : X' --> X, the projection to the first coordinate. When X is path connected and sufficiently nice, X' is path connected and f is a based covering map. Moreover, f is initial among based covering maps of X by path connected spaces. Is there a concept dual to this one in the sense that homotopy groups (whose elements are homotopy classes of based maps from spheres) are dual to cohomology groups (whose elements are homotopy classes of based maps to eilenberg maclane spaces)? For example, just as we have a good fibration X'-->X that kills first homotopy and preserves higher homotopy, do we also have a good cofibration X-->X'' that kills first cohomology and preserves higher cohomology? By "good" I mean any of several things: defined canonically / characterized by a universal property / related manifestly to the universal cover / useful for computation / visually interesting / etc
|
The dual notion of a CW decomposition of a space X is called a Postnikov tower for X. Like a CW decomposition is an inductive series of cofibers of maps of spheres, a Postnikov tower is an inductive series of maps where the fibers are Eilenberg-Maclane space. The Postnikov tower of a space is homotopically unique, so it makes sense to talk about the kth truncation of the process. This is a space Y so pi_j(Y)=pi_j(X) for i<k+1 and trivial above k. This isomorphism is realized by a map X -> Y, and taking the homotopy fiber of this map gives the kth Whitehead cover of X. By looking at the LES of a fibration, the homotopy groups of this homotopy fiber have exactly the opposite property of Y's. In particular, if we take the first Whitehead cover, all the homotopy groups above the first are the same as X and the fundamental group is trivial. This implies that this is, up to homotopy, the universal cover of X. So the Whitehead covers are the "coskeleta" of the Postnikov tower for X. This means that you would expect the dual notion of the Whitehead cover to be either the skeleton of a CW decomposition for X or the coskeleton of one. Unfortunately, the CW decomposition of a space is very far from homotopically unique and these notions are not homotopy invariant. So the best answer to your question is probably just what we started with: the first truncation of the Postnikov tower which is always K(pi_1(X),1).
|
Universal undercovers. I do not have an actual answer for you but that dumb joke popped into my head so you have to see it as well now
|
Haha thanks. I also like "universal vers" as a name
|
It is called Eckmann-Hilton duality. It is the observation that a lot of results in homotopy theory involving homotopy groups have dual statements involving cohomology groups. For example, the long exact sequence of homotopy groups of a fibration is "dual" to the long exact sequence in cohomology of the dual cofibration sequence where we take maps into K(G,n). It is not really a formal duality, but it is persistent throughout topology. In my point of view there are 3 big tools in topology: homotopy, homology, and cohomology. As you say, the latter two are basically dual in a precise sense. Homotopy and cohomology have Eckmann-Hilton duality, and finally homology can be viewed as a truncation of homotopy .
|
Thanks for this explanation and for your warning about Eckmann Hilton duality being informal. I have a tongue-in-cheek and personal bias against homology (i think having such biases is okay since im not a mathematician) mainly because it isn't representable. But someone told me that such a bias is like disliking tensor of abelian groups, when really all three of [--,G] and [G, --] and (G tensor --) are useful and interesting. So point taken. I humbly accept that nature has given us a formal duality only for homology and cohomology.
|
[
"Minimal amount of abstract algebra required for category theory?"
] |
[
"math"
] |
[
"r8a01t"
] |
[
55
] |
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""
] |
[
true
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0.93
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I dived into category theory recently(reading Basic Category Theory by Leisner), although I don't have a strong background in abstract algebra. I learned some basic group theory on my own and I only know the definitions of the rest of important structures. I understand some of the examples from the book on categories, mostly about groups/monoids, but I think I don't really understand the ones involving more complicated structures well enough. My question is whether my approach is fine considering that I learn CT just out of interest, or I will miss really a lot without further knowledge in algebra? If I need algebra, is there any "crash course", containing just enough information for understanding examples from CT, missing some details which are usually covered in standard algebra courses?
|
There is essentially 0 knowledge of algebra needed to learn category theory. If you want to learn category theory for its own sake, go ahead (most people do not have much success with this). But you should definitely *not* take a crash course in algebra to try to improve your experience with category theory. Algebra is something that needs to be learned properly.
|
I mean… technically you don’t need any abstract algebra to learn category theory. But without any knowledge of abstract algebra it’s going to seem incredibly abstract, pointless, and unrewarding. Category theory in some sense generalizes many of the notions you encounter in abstract algebra and builds them with a new framework. I can’t speak for category theorists, but as a number theorist I think of category theory a language that expresses abstract algebra in a nice way. So you can learn the language on its own. But if you don’t read in it… talk to other people in it… use it… then… it’s going to be hard to understand it. I recommend taking a typical abstract algebra course, then maybe learn some commutative algebra or homological algebra concurrently with category theory. A lot of notions will make sense just with groups and monoids, but having rings, fields, modules at your disposal will expose you to more interesting categorical structures and you’ll gain intuition for many of the categorical notions.
|
I don't know what you would possibly get out of learning category theory without any concrete examples to draw from.
|
I'd say you miss a lot if you don't have a background in either group homology or algebraic geometry, but CT works as a foundation on its own? It just doesn't sound terribly rewarding. That being said, I started out similarly. I had only taken a basic group theory course before I started reading a random CT book I had found on the internet. Didn't understand a whole lot and eventually gave up. This pattern repeated two or three times, until I reached a level of algebra lectures where I suddenly saw loads of things where CT was applicable. I grabbed another book, started all over, and ended up doing my thesis on CT and group cohomology. I have by now forgotten what the point of this story was, but I guess I'm saying there's no real reason not to start like this, but you will probably be incredibly frustrated at some point. I think actually the best way would be to learn "other" algebra and CT concurrently. The AG professors at my uni really dislike CT, so it never featured in the lectures, but I ended up working through the CT book at the same time I took AG and working out the connections myself. Edit: for a crash course in algebra, I (of course) recommend the ever-brilliant Aluffi . I have pitched this book so many times, I should get commission. But it truly is one of the best books on mathematics I have ever read, and written in a very enjoyable style.
|
I highly recommend algebra: chapter 0 By Alufi A category theoretically grounded introduction to abstract algebra. Very readable with tons of exercises.
|
[
"Physics book for a Mathematician?"
] |
[
"math"
] |
[
"r8e5u9"
] |
[
211
] |
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""
] |
[
true
] |
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In particular I'd like a book that uses an axiomatic system to retrieve the theory of Newtonian mechanics onward. I have no physical intuition so I'd prefer something that starts from first principles. After some googling I can't seem to find any such reference and generally it seems like physicists have an aversion to axiomatic systems. Would appreciate any recommendations!
|
I would not say that physicists are averse to axiomatic systems — but you have to recognize that physics is, by its very nature, . In mathematics, you can propose any set of axioms you want, and then explore that mathematical universe. But our goal in physics is to describe what we in our real universe. We do not know the rules yet — we are actively looking for them. If you propose the axioms of a deterministic, classical, Newtonian universe, you will get Newtonian dynamics. But this is only an approximate description, for macroscopic settings (quantum effects negligible), where the effects of spacetime curvature are negligible, etc. This axiomatic system will be incorrect for describing e.g. black holes, or superconductors, or just the humble hydrogen atom. I'm not saying that you can't study an axiomatic approach to physics. I am just saying that you acknowledge the role of empiricism in building your mathematical model of the universe.
|
Arnold's book on the subject is a classic. Brian C Hall has a similar book on quantum theory . Be warned, though. The math involved in these formulations of physics becomes very sophisticated.
|
Thanks, this is the cleanest description of the situation I have seen. The big stumbling block we have in the way at this time is that QM is not classical but the "interpretations of QM" are an attempt to describe QM classically. So, too many people wind up with a weird "understanding" of QM. Getting past that mental block is an interesting challenge. Given that understanding QM is off the table, we need to try to develop a way of taking QM seriously to shed ourselves of the impediment to the further developments in QM. The real nightmare from my point of view is the thinking of an electron as a particle and/or wave introduces such an over abundance of nonsense into the conversation. Consider, for example, "bound state beta decay", just try to wrap your mind around that process and not think silly "thoughts", or "silly mental images".
|
The only way to understand QM is mathematically. The particle/wave stuff is a useful teaching tool, but it's best not to try to visualize it at all. Just treat quantum interactions as purely mathematical objects, because as far as our ability to understand them is concerned that's what they are. https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/lecture-videos/lecture-1/
|
Ichael Spivak has one such book . I think.
|
[
"Is Vertasium’s video on Gödels Theorem incorrect?( are there other formal systems that can raise the twin prime conjecture question?)"
] |
[
"math"
] |
[
"r95hy4"
] |
[
23
] |
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""
] |
[
true
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[
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At the start of the video, it is claimed that Gödels theorem implies that problems may not be provable: namely, the Twin Prime Conjecture. This is incorrect, though, right? I just came out of Computer Science Theory this semester, and individual study has led be down the rabbit hole of Gödels incompleteness and I’m trying to wrap my mind around the concept of the . My understanding is not what is expressed in their video. Claim: Gödels incompleteness implies that the twin prime conjecture may be unsolvable. My claim(?): Gödel’s incompleteness implies that the twin prime conjecture may not be solvable with the formal system (horrifically incorrectly) described a sort of union (z, +) U (z, *) groups. Assuming this arithmetic system with associativity, commutativity, etc; equipped with +, ..., can not prove the twin prime conjecture, this does not mean set theory can’t do it. Now, this is assuming that using set theory or something for proving things about integers is not fundamentally equivalent to the classical arithmetic formal system. Is the twin prime conjecture an artifact of this one formal system?
|
Ugh, here we go again. Sorry, but this kind of question shows up here regularly. Goedel's theorem is in pop-math. It says about specific conjectures in mainstream math being unprovable. In fact, the way it shows there are unprovable statements is by exhibiting one (a self-referential Goedel sentence) with no serious mathematical content whatsoever. Admittedly Goedel's incompleteness theorem was a huge shock when it first appeared, but after the dust settled people realized it's kind of irrelevant: it has had basically no real effect on mainstream math outside of logic. Differential geometry, algebraic topology, algebraic geometry, stochastic analysis, PDEs, etc. all have developed quite substantially (and continue to do so) and none of this has been affected at all by Goedel's incompleteness theorems. To prove that a statement of real mathematical interest is undecidable, such as the continuum hypothesis, you need serious new ideas (like Cohen's method of forcing). It does not come out of Goedel's proof of his incompleteness theoems. And while undecidability of the continuum hypothesis was a major result in mathematical logic, it has had basically no effect on progress in the rest of mainstream math. Anyone suggesting the incompleteness theorem is related to the twin prime conjecture like in the Veritasium video (especially with the massive weasle word "may") is selling pure clickbait. The kinds of statements that have been shown to be independent of ZFC involve some kind of truly infinite set-theoretic phenomena that is totally unlike what you find in the twin prime conjecture. It makes no sense to believe the twin prime conjecture could be undecidable "because" of Goedel's work. You might as well throw up your hands at the prospect of solving any hard problem. Some logicians for a time suggested Fermat's Last Theorem might be undecidable (only because it was unsolved for a long time), and after the work of Wiles they switched to talking about the Riemann Hypothesis or the twin prime conjecture. No doubt someday in the future if RH and the twin prime conjecture are settled, the goal posts will be moved again, maybe to the 3x+1 problem (please don't tell me about Conway's generalization of that, which does imply a resolution of the 3x+1 problem). There is a lot of nonsense in pop-math articles suggesting a solution to the Riemann Hypothesis is going to "break cryptography". RH has nothing to do with finding prime numbers, only with the aggregate count of prime numbers up to a large bound, and even then only to an in that count. There is no sensible way RH is supposed to lead to an improvement on finding primes that can't already be done (and is done regularly in cryptography with probabilistic primality algorithms). Likewise, Goedel incompleteness has no known connection to the twin prime conjecture, and I don't think it's a good idea for pop-math pieces to be suggesting otherwise.
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Firstly, there are systems that prove the Twin Prime Conjecture. Example: (Peano’s) arithmetic plus the statement “The Twin Prime Conjecture is true”. Of course, the catch is that this system might be inconsistent (namely if the conjecture is false), in which case it proves everything, and hence you haven’t accomplished much. It is conceivable that the Twin Prime Conjecture is unprovable in ZFC. However, showing that is way beyond the scope of modern set theory, and would require completely new ideas. We currently have one technique for showing statements to be independent of set theory: the method of forcing, introduced by Cohen in 1963. This is an incredibly flexible tool for constructing models of ZFC where different statements are true, but it is also somewhat rigid in that forcing doesn’t change the truth value of statements about natural numbers. In particular, if you can force the Twin Prime Conjecture to be consistently true, then you’ve actually proven it true.
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This is a great answer, although to be fair to Veritasium most of his video is quite accurate as far as I can tell. And the parts which are misleading are also not technically wrong (in some sense), it's just that it's presented in a way to get the most people interested in the subject rather than to give the deepest understanding. E.g. to give an example of a statement independent of ZFC, he could have explained CH, but that's not so easy to explain in a few minutes to a lay audience, and saying this "may" include the twin prime conjecture is a kinda neat way out of it, I guess.
|
Veritaseum has recently been releasing a lot of smug videos where he claims "X," while despite technically true is just misleading clickbait bs.
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To give a concrete example of this the video about the discovery of imaginary numbers originally had a title with something like "and lead to the discovery of quantum physics". While we do need imaginary numbers for quantum physics, the original title overstated their importantce. Like, you may as well have said that Calculus lead to the discovery of quantum mechanics because we use Calculus in Quantum mechanics too. However, to give them credit they did change the title soon after. I think this shows they do have standards and do care about being accurate.
|
[
"Was looking through my university's library for math books and found this interesting book on lattice theory by Lieber. Ive never seen a textbook formatted like this and with cool illustrations. Thought some people here may be intrigued."
] |
[
"math"
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"r8iucw"
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1474
] |
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""
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true
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0.99
] | null |
If you like this, you will Winning Ways for Your Mathematical Plays by Berlekamp, Conway (yes that Conway) and Guy. It's an absolute delight that doesn't skimp on rigor.
|
This is incredible. I need books like this, somehow I understand words much better in this format.
|
It’s a graphical way of reasoning about things in a partially ordered set. Basically, let’s say I’m trying to decide what to buy at a bakery, and they’ve got a variety of pies. I like apple better than pumpkin, you draw a line from Apple Pie (above) to Pumpkin Pie (below). I also like Apple better than blueberry, so another line down from Apple to blueberry. I don’t really have any preference between pumpkin and blueberry, so no line connecting them, they just sit across from each other on the same level. But I like chocolate cream pie better than Apple, so a line from chocolate (above) down to Apple. Now I can see which pie I should get, because chocolate is on top. I know directly that I prefer it to Apple, and I know indirectly (transitively) that I must also prefer it to pumpkin and blueberry. And as lattices get larger and more complex, there’s an interesting variety of reasoning you can do on them. I believe there’s an alternative voting method for elections (maybe it’s condorcet?) based on paired preference questions and lattice reasoning, but it doesn’t work in practice because, among other things, humans tend to have cyclical preferences.
|
You should look at Fomenko’s stuff—even the three vol GTM is baller as hell
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Short lines does not make reading faster what are they smoking
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[
"Smallest number of areas to see all combinations"
] |
[
"math"
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"r8naby"
] |
[
4
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I have a selection of n colours and want to judge which colours work together. I only have limited paint so I want to paint as few shapes as possible and still want to be able to examine all possible combinations. The practical constraint is that each area should have a similar size & shape as otherwise one colour impression might overweight. My current approach is much alike to scribble where I add a square at a time and try to get the biggest decrease of missing combinations. But is there a better way? Is there a structural approach? Are squares the best option? Where can I find info on work which tackles this problem? In my case, n \in [5,14]
|
Great question! I'll just focus on the topological aspect of it, because that's already mathematically interesting, but that means I'll completely ignore the question of size and shape of the regions (left as an exercise to the artist). Let's abstract out the problem by representing each region by a point inside it, and draw an edge between two points if these represent regions which are adjacent. Of course we can make sure that the edges don't cross each other (so what we get at the end is a planar graph). Now it's a classical result that given 5 points, there's no way to draw an edge between every pair of them such that none of the edges cross (ie. the complete graph on 5 vertices is not planar). So it's impossible to have every pair of colours be adjacent to each other, if we only use one region per colour. So what is the minimum number of regions for each colour that's needed? I could only find one research paper on the topic, by Shirakawa and Ozaki (1967), On the Decomposition of a Complete Graph into Planar Subgraphs . They showed that if you have 4k colours, you only need k regions of each colour. Let me summarise their construction. Say we have 8 colours, call them 1,2,...,8. Place them in a circle in the order 1-2-3-...-7-8-1, so that each pair of colours which are 1 apart are now adjacent. Name this circle C1. Now we can extend all the odd numbered colours into the circle, so they're now adjacent like 1-3-5-7-1. Similarly, extend the even numbered colours out of the circle, so they're adjacent like 2-4-6-8-2. Then each pair of colours which are 2 apart are now adjacent. Next, make a new circle of colours, named C3, in the order 1-4-7-2-5-8-3-6-1. Now each pair of colours which are 3 apart are now adjacent. Finally, in C1, extend 1 and 5 further into the circle, and extend 2 and 6 further out of the circle, so that these two pairs touch. In C3, extend 3 and 7 into the circle, and extend 4 and 8 out of the circle, so that these two pairs touch. Now each pair of colours which are 4 apart are now adjacent. This exhausts all possible pairs of colours, so we're done with the construction. Each colour was used in two regions, one in C1 and one in C3. (Of course, this construction also works for 7, 6, or 5 colours; just ignore the colours we don't actually have.) A similar construction works if we have 12 colours (or 11, 10, or 9). In this case we need C1, C3 and C5, where C1 looks like 1-2-3-...-11-12-1, C5 looks like 1-6-11-4-9-2-7-12-5-10-3-8-1, and C3 actually splits into three circles like 1-4-7-10-1, 2-5-8-11-2, and 3-6-9-12-3. As before, colours which are 2 apart can be dealt with in C1 (odd colours in, even colours out); colours which are 4 apart have to be split between C1 and C5; colours which are 6 apart can be dealt with in C3; and we're done. Each colour is now used in three regions. In general, for 4k colours, we use C1, C3, C5, ... C(2k-1), and similarly split the pairs with even differences between them (carefully). The paper leaves open some interesting problems. For example, they don't show that their construction is optimal, so maybe there's some way to use fewer copies of each colour. Can a K_9 be edge decomposed into just two planar graphs instead of three?
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Squares are optimal considering you can do a regular tiling with them and you'll be probably be using a roll or a flat brush for the painting. You can use squares with horizontal offsets starting with 2 squares that share an edge and a third one underneath them whose upper corners land on the middles of the sides of the other two then keep adding squares so that the new square always shares edges with at least two older.
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Oh that’s a great step! Thank you so much for sharing! I was thinking about making it an edges and nodes problem but was a bit stuck on the thought of shapes. Also to start new regions didn’t come to my mind, I just assumed that adding to the initial cluster would be the way to go. I am looking through the paper and will see if I can apply it!
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The horizontal offset is a good idea, I didn’t think of that! I think if we assume equal shape, the hexagons or pentagons would end up having a more uneven distribution of number of tiles per colour as a single tile has more faces, so the first tile already covers 5 or 6 of its combinations but it becomes harder for the neighbours. Triangles might lead to a more even number of tiles per colour, but potentially more. So squares might indeed be the optimal shape
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Unless I'm misunderstanding, your question is basically "which shape most efficiently tiles the plane?" Clearly any regular polygon can. In that sense we might say a triangle is the best option because it is the simplest polygon. There is a slight drawback to using polygons, though. You might care about how much area to color you get inside your shape given the amount of time taken to draw it. Circles have the greatest area inside for their perimeter/circumference. However, you will lose space in the plane by tiling with circles. Circle-packing is well-understood and you can Google it if interested.
|
[
"How many numbers does a sudoku grid require in order to be completable?"
] |
[
"math"
] |
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"r931e3"
] |
[
385
] |
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""
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I was doing a sudoku earlier and I was curious – what is the absolute minimum number of sudoku squares that would have to be filled in for it to be completable? How do you prove this?
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This is a great example of how the precise wording of your question matters.
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And finally, a solution is only guaranteed to be unique with 78 clues. Two pair of unknown numbers can be ambiguous (e.g. "56" and "65" vs. the other option), so there are sudokus with 77 clues that have more than one solution.
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Fun fact: One of the more common Sudoku tricks is called Unique Rectangle , where you can leverage the knowledge that no sudoku puzzle has more than one solution, meaning no puzzle can have two pair of unknown numbers, so there must be some other number that is in that rectangle. I mean theoretically someone really bad at making sudoku puzzles might have such a configuration, but legitimate sudoku puzzles are required to have unique solutions, so don't contain ambiguous rectangles. If we don't consider garbage quality Sudoku puzzles as true sudoku puzzles, then you don't need 78 clues, since there's no ambiguous rectangles that you have to disambiguate using extra clues.
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If you really enjoy Sudoku and Logic, I highly recommend Cracking the Cryptic on YouTube. Best hour of my day.
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I super highly recomend the android app "Enjoy Sudoku" it has a "hint" feature which teaches you things like this. Also it has a ton of difficulty levels. Ive solved thousands of puzzles in that app but Im still only halfway through the difficilty levels. I thought I was really good at sudoku until I started using that app. Its well worth buying the full version.
|
[
"How do you professional mathematicians work 8 hours in a day? My brain is often spent after 4 hours, if not less."
] |
[
"math"
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"r8bqmu"
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When I'm working on math problems, be it reading or doing exercises, my brain power tends to evaporate after 4 or so hours. So I always wonder, do you professional mathematicians spend 8 hours a day doing focused thinking about extremely difficult problems? And if not, what do you do in the remaining time? I've been doing math for a couple of years now but I've never been able to go at it for the entire day and so this always bugged me. It seems too intense to do for an entire day but maybe I'm missing something?
|
A lot of studies indicate that we have about a four hour limit. For example, with computer coding, after 4 hours you start making more mistakes than corrections. So, if you work for more than about 4 hours as a programmer, you're actually working against yourself.. personally I find this to be true for math, too. I do my best to spend less than 4 hours a day on deep thinking and try to spend the remaining 4 hours on other stuff like copy editing, corrections, and professional communications. Sometimes all you need is good 2 hour lunch, though, and you're good to go for a couple more hours. And often times I find my stamina when I'm working with others is increased compared to when I'm working alone. In an academic setting, we are expected to handle service to the university and teaching and our own research as well as work with graduate students.. there is generally more to do than just math, so finding space in your day for those other things is the way I always handled it in those settings. People don't like to admit it but taking some time to process information in the background is really critical for those 4 hours of good work, though. Having said all that, I had to learn my lesson from years of destroying my stomach with pots of coffee and destroying my sleep schedule by working for 16 hours a day and neglecting all of my laundry and dishes and stuff like that. There really is a balance and a lot of the job is trying to figure out that balance.
|
Most academics don’t work a 9-5 schedule. We take breaks. I’m not a mathematician but a physicist and I worked eight hours straight on something I would literally go insane. Edit: When I say breaks, I mean like hours long stints doing something else like reading, running, watching a movie, meeting with friends/colleagues, etc.
|
Being a PhD student that works mostly with proofs I can say that this is quite acurate for me
|
Let me answer by saying how I ended up doing round 10 hours of maths today. Today I woke up at 10am. I wrote 3 emails immediately about preprint and how to get into journals. Continued trying to work out an example I was stuck on yesterday. Walked to dept in time for 11 where I ran a seminar series till 12. Then I wrote down reports for people who spoke and had lunch till around 1. Then I went back to THIS DAMNED EXAMPLE. I thought I’d made progress but was wrong. I went to a talk 3-4pm. I headed back to office, saw my supervisor/collaborator was in so went and asked him about the example. Took us an hour to sort out. Went back to office round 5. Then walked into common room to restock on water, saw colleagues there and spent an hour discussing future directions of project I’m working on/ generally getting excited about it. Realised to make progress there is a certain paper I should read. Came home ate dinner with flatmate. Looked at said paper after dinner. Decided what I was doing was simpler than their situation (yay). Baked chocolate cake. Ate some of cake. Conclusion: too wet (150ml water? Just why…). While baking I meditated on this DAMNED EXAMPLE. Played computer game for a bit. Discussed maths with flatmate for an hour. He understands langlands and I’m curious about applying some of our ideas to it: managed to get something comprehensible by end of discussion. Sat around with flatmate for last 5 hours. Sometimes we chatted, he largely played a computer game with his gf at one point we had Netflix on. About 70% of time I was meditating on this DAMNED EXAMPLE. Ultimately ive realised the example probably doesn’t break everything in the paper I’m writing. Ive formulated the lemma I need to prove. Tomorrow I’ll go back to my supervisors last paper and try prove said lemma. Then I’ll write it down. (PhD student if that counts as professional mathematician)
|
For many of us, our work gets split up with meetings, teaching, advising, committee work, attending/writing talks… we rarely have 8 hours of uninterrupted time to think about research. And when we do, we take breaks!
|
[
"Suggestions for a good/easy to understand introductory book on Functional/Delay Differential Equations?"
] |
[
"math"
] |
[
"r869qr"
] |
[
14
] |
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""
] |
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true
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Hello! I'm currently an undergrad and have taken intro courses in ODE's and PDE's, and am currently working on a project where I'm going to be using DDE's. However, I've honestly never studied them before in any class. I've been looking for YouTube lectures/tutorials, but I couldn't find much that was useful, and I've seen suggestions for books, but I want to look for something that's easy to initially understand, and explains stuff well. Particularly, I want to look into finding solutions for linear and non-linear DDE's. I've seen mentioned in a lot of places, along with , so I've downloaded them, but I'm not sure which one to start with. Could someone let me know which one of these, or any other books are good intro textbooks that could be understood by an undergrad student? Also, as a side question, is there any other particular area/field of math that I should study well in order to understand the theory behind DDE's better? (Or generally for PDE's as well)? I'd really appreciate if anyone could let me know! Thanks!
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I'm not sure about DDEs, but analysis is fundamental to the study of ordinary and partial differential equations.
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I would recommend dipping your toe into something like Hunter and Nachtergaele's book on functional analysis if you run into trouble in your current research. Analysis extends beyond the real- (or even complex-) valued function stuff you're taught in an introductory course!
|
The theory of linear delay differential equations goes much like the theory of any linear equations, whether they be ODEs or PDEs: view it as a linear equation in the appropriate Banach or Hilbert space and look at the spectrum. A standard dose of analysis, linear algebra, etc. learned by every undergraduate math student should be sufficient background to get going with DDEs. You may want to get into some functional analysis.
|
A solid understanding of ODE theory is quite helpful when thinking about the DDE setting. Most of the existence and uniqueness results are similar but translated to the infinite dimensional state space. Measure theory and some complex variables is also probably useful. Finding solutions to DDEs in general is quite difficult. If you end up looking for numerical methods, Bellen and Zennaro wrote the book on functional continuous Runge Kutta Methods, although that book/subject is dense. Matlab has existing solvers (DDE23 And DDESD) that act very similarly to the familiar ODE suite. I think Hal Smith’s book is a nice introduction to the subject and where I started as a final year undergraduate. If you really want to get into the weeds, Hale and Verduyn Lunel wrote a much more technical book that requires a functional analysis background while Diekmann, …, Walther have a 1995 book on the more general delay equations.
|
Hello, sorry for commenting here again but I had a random, sort of related question. I see that pure math knowledge is clearly really important for DE's. Unfortunately, I don't have the chance to take a lot of courses like these now since I'm a senior, and a lot of courses like further real analysis, functional analysis etc aren't going to be offered at my university again until next year, but I do have the chance to take something like general topology, or another abstract algebra course. And there was something about Lie groups/Lie symmetries that I saw as well. Could you let me know if those are things that I should look into for further study in PDE's? (Or if there's anything else that's important?) I would really reallyyy be thankful if you could let me know!
|
[
"Tao analysis 1 references to exercises unclear"
] |
[
"math"
] |
[
"t5t758"
] |
[
3
] |
[
"Removed - ask in Quick Questions thread"
] |
[
true
] |
[
false
] |
[
1
] | null |
According to Tao's blog , it was a typo in an older edition of the book. In recent editions this has been corrected to Exercise 7.2.6 of Analysis II.
|
Thanks 👍
|
I am also working on Tao's Analysis 1 right now. I agree that it can quite annoying but it seems that whenever a theorem's/lemma's proof cites an exercise Tao believes it to be possible for you to figure the proof for yourself based on the material already covered previously. I agree that it can be annoying especially when you can't seem to figure it out.
|
My problem isn't not finding solutions, it's not finding exercises. When he cites exercise 11.46.6 , where is the exercise, i can't find it in chapter 11 nor in the second book
|
Oh just looked and I see what you mean... I can know assume some it is a typo but I might be missing something.
|
[
"is 1/φ the same as φ-1 ?"
] |
[
"math"
] |
[
"t5rtqn"
] |
[
0
] |
[
"Removed - ask in Quick Questions thread"
] |
[
true
] |
[
false
] |
[
0.5
] | null |
Assuming phi is the golden ratio, then yes, 1/phi = phi - 1.
|
You are correct, but you are being downvoted because φ refers to the golden ratio (although OP didn't make that clear in their post), for which this equation does hold.
|
You are correct, but you are being downvoted because φ refers to the golden ratio (although OP didn't make that clear in their post), for which this equation does hold.
|
If by "φ-1" you mean "phi minus one" then no yes it is
|
i do mean phi minus one
|
[
"Elon tweets: \"Tau > Pi\""
] |
[
"math"
] |
[
"t5oe4w"
] |
[
0
] |
[
""
] |
[
true
] |
[
false
] |
[
0.26
] | null |
Maybe tau is better, but not better enough to warrant a worldwide overhaul of everything. Pi is fine. So is the decimal system. No offense but I feel people care way more about this topic than is necessarily warranted. There's just not much to say about it. "Tau is better." "Okay." Well what now?
|
Oh christ, not this crap again.
|
Tau is more natural than pi, conventional current should flow the other way, maybe we should be using base 12. So what? At the end of the day, no matter how much “better” an alternative convention is in a vacuum, it’s not going to be adopted. In practice the convention actually in use is better because it’s the convention actually in use.
|
Imagine every wasted second by using the wrong constant, multiplied by centuries and centuries into the future. That adds up to a lot of waste. It’s likely still dwarfed by the amount of time wasted trying to convince people on the internet that dropping a 2 from some equations while adding a /2 to others is a huge win for math and society.
|
How often do you use either pi or tau in any sort of calculation?
|
[
"How can you write sin(X) to the negative 1 vs inverse sin(X)?"
] |
[
"math"
] |
[
"t5nbvr"
] |
[
0
] |
[
"Removed - ask in Quick Questions thread"
] |
[
true
] |
[
false
] |
[
0.5
] | null |
(sin x) = 1/sin x = cosec x sin x = arc sin x sin x = sin (1/x)
|
sin } means the multiplicative inverse of sin. No it doesn't
|
sin } means the multiplicative inverse of sin. No it doesn't
|
Yeah I guess so.
|
Oh but if it was sin (X) could you rewrite it as (Sinx) ?
|
[
"Why do these effects happen?"
] |
[
"math"
] |
[
"t6byrs"
] |
[
21
] |
[
"Removed - ask in Quick Questions thread"
] |
[
true
] |
[
false
] |
[
0.86
] | null |
You've got aliasing, once the frequency of your lines is significantly smaller than your pixels you don;t have enough sample points to reconstruct the true signal, and see artifacts.
|
look up "Moirè patterns"
|
Recently I've become obsessed with C++ and I thought I'd try Raycasting. Eventually I got to this point. Originally I just had 36 rays, but I bumped it up to 360 and got two interesting effects. One was that the corners seemed dimmer compared to everything else, but I understand why that is. What I don't understand is the effect around the middle areas of each side. Why does it have that pattern? Looking closer I can see the corners also have the curve pattern.
|
This. You can imagine them as being an analogue of beats. In case of sound, when the frequencies are slightly different, you get alternating loud and quiet sounds. It's the same here. Any repeating pattern in space has a "spatial frequency" and the beats correspond to brightness. In your case, they're your rays and, if I'm not wrong, the pixel array of screens.
|
Aliasing
|
[
"How I See Numbers"
] |
[
"math"
] |
[
"t69531"
] |
[
6
] |
[
"Removed - not mathematics"
] |
[
true
] |
[
false
] |
[
0.68
] | null |
You certainly see things a little differently :) lol
|
It sounds a bit like synesthesia.
|
I have synesthesia, and I see 7 + 8 in the exact same way haha, 6 + 8, 5 + 8, and 3 + 8 all work the same way too.
|
I have synesthesia, and I see 7 + 8 in the exact same way haha, 6 + 8, 5 + 8, and 3 + 8 all work the same way too.
|
Does the guy has synesthesia?
|
[
"Prime fun :)"
] |
[
"math"
] |
[
"t673yd"
] |
[
21
] |
[
"Removed - low effort image/video post"
] |
[
true
] |
[
false
] |
[
0.78
] | null |
Mind a bit of explanation of what's going on please?
|
The prime lines are the second slide The reason it’s symmetrical is because I mirrored the image onto itself, The blanks represent non divisible and the colored represent the divisible
|
A bit of an explanation, I’m not a mathematician, so apologies for my lack of vocabulary, but if you cut this image in half vertically and the turn it 90 degrees you will get an x,y graph that’s upside down Like: 0, 1, 2, 3, 4, 5 1 2 3 4 5 Each cell that is colored represents if the x value is divisible by the y
|
Here's an image 2048 pixels long . I marked in red the areas which are finished, this takes a while to do and the further in you are the slower progress is. You can see some patterns right off the bat; feel free to continue it if you'd like (Edit: Here's a complete 2048x2048 image. )
|
Holy shit you came through, thank you so much man, I’m not to computer savvy so I’m not too sure how I would continue the pattern but this is greatly appreciated once again
|
[
"Timeline of Mathematics – Mathigon"
] |
[
"math"
] |
[
"t5xcgl"
] |
[
86
] |
[
""
] |
[
true
] |
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[
0.97
] | null |
Nice!
|
Seems to indicate geography. Green is America, burgundy is Europe, and there seem to be colors for the near east, far east, Japan. I guess teal is Australia.
|
All of them.
|
Nah.
|
Wow so cool!
|
[
"What is the intersection of economics and mathematics?"
] |
[
"math"
] |
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"t63yx5"
] |
[
28
] |
[
""
] |
[
true
] |
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false
] |
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0.78
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Statistics is a branch of mathematics
|
Is this some sort of riddle? Tons of economics makes heavy use of math. The intersection of math and economics is….all economics since WWII?
|
I like the game theory approach. Although there are many nice topics in economics that involve advanced (ambiguous) mathematics
|
Econometrics is what you're looking for
|
Analysis and Statistics. It’s huge. Macro, particularly is very mathematical. Fixed point theorems and recursive method. Iterative solutions. Time series. Etc.
|
[
"Is a theory decidable in any theory with consistency strength higher than or equal to it?"
] |
[
"math"
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"t5x5aq"
] |
[
7
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true
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false
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For example, could Gentzen's theory of primitive recursive arithmetic with quantifier-free transfinite induction up to ordinal ε0 which has a higher consistency strength than PA, prove that PA proves anything which PA proves? For example, if you have a turing machine that enumerates all proofs of PA and halts when it finds a proof for an arbitrary theorem we've specified, can all of those turing machines that PA proves halts, be proven to halt in Gentzen's theory? Similarly, ZF and ZFC have the same consistency strength, so could zf likewise prove "zfc proves A" for any statement A that zfc proves? On a different note, I've read that you can build a turing machine that halts if and only if a certain axiom system is inconsistent, this has even been done for zf. Can such a "is this axiom system A inconsistent" turing machine be expressed in any axiom system that can express a turing machine, and for any axiom system A? For instance, could the turing machine that halts if zf is inconsistent, be expressed in the language of peano arithmetic (I'm aware it wouldn't be able to prove whether it halts or does not halt, but I'm asking if it can still be expressed, in the same way you could express the Godel sentence of PA or Goodstein's theorem in PA while they remain independent in it).
|
On a table featured in the wikipedia page on ordinal analysis, I𝛥0 is given an ordinal strength higher than Robinson arithmetic. Yeah, this is an example of where some of the definitions of proof theoretic ordinals don't completely capture consistency strength. In general though, the fact that proof theoretic ordinals capture consistency strength is just an empirical observation, not a theorem. I originally assumed interpretability meant at least the same as far as consistency strength but that implies they're unrelated notions. A computable interpretation does imply something about consistency strength, and the interpretation here is computable.
|
For example, could Gentzen's theory of primitive recursive arithmetic with quantifier-free transfinite induction up to ordinal ε0 which has a higher consistency strength than PA, prove that PA proves anything which PA proves? You have to be a bit careful with how you formalize this kind of question. There's some subtlety with placement of quantifiers. For example, if you have a turing machine that enumerates all proofs of PA and halts when it finds a proof for an arbitrary theorem we've specified, can all of those turing machines that PA proves halts, be proven to halt in Gentzen's theory? Yes, but this actually has nothing to do with consistency strength. There are very weak fragments T of PA with the property that T proves that a given Turing machine halts if and only if it actually halts. Similarly, ZF and ZFC have the same consistency strength, so could zf likewise prove "zfc proves A" for any statement A that zfc proves? They do prove the same things about the natural numbers though, but this isn't because they have the same consistency strength. Sorry, I misread your question. Assuming that ZF(C) is "sound," ZF proves "ZFC proves A" if and only if ZFC proves A. Again though, this doesn't have to do with consistency strength. The same is true if you replace ZF with fairly weak fragments of PA. On a different note, I've read that you can build a turing machine that halts if and only if a certain axiom system is inconsistent, this has even been done for zf. Can such a "is this axiom system A inconsistent" turing machine be expressed in any axiom system that can express a turing machine, and for any axiom system A? For instance, could the turing machine that halts if zf is inconsistent, be expressed in the language of peano arithmetic (I'm aware it wouldn't be able to prove whether it halts or does not halt, but I'm asking if it can still be expressed, in the same way you could express the Godel sentence of PA or Goodstein's theorem in PA while they remain independent in it). Yes and yes. PA can talk about the consistency of ZFC. As an example, PA is more than enough to prove that ZFC and ZF are equiconsistent.
|
No I meant like can ZF prove that ZFC proves the axiom of choice for instance, kinda in the same manner as what I said in the first paragraph, suppose that you enumerate all proofs in ZFC, and you have a Turing machine go through and halt if it finds a proof of the axiom of choice for instance. Can that Turing machine be proven to halt in ZF. Based on your answer to the other thing for Gentzen’s theory and PA I would assume the answer is yes here as well, since I assume ZF can do everything that weak fragment of PA can do. Yeah, you need very little to show that ZFC proves the axiom of choice; it's one of the axioms of ZFC. Does this mean that a Turing machine that halts if and only if that fragment T or any stronger theory (in consistency strength) is consistent, does not exist in any theory/is literally impossible to formulate (well unless the theory is already inconsistent but I’m excluding those). Since if it were possible, then T could prove it halts in all cases that it halts according to what you’re saying, which would allow T to prove it’s own consistency. There's some subtlety with how to formalize this kind of statement but yes. There isn't generally going to be a Turing machine that halts if and only if a certain theory is consistent in the same sense that you can encode inconsistency as the halting of a Turing machine.
|
Then again, perhaps my misunderstanding here is that I need to separate (T proving that (T + con(T) proves con(T))) from (T proving con(T)), they’re different right? They are different, yes.
|
How weak is that fragment T exactly, or like the bare minimum weakest theory that contains that property you mentioned about being able to prove a Turing machine halts iff it halts? Would it be the same as the weakest theory that can actually define a Turing machine? Can we even pinpoint it exactly (I’m assuming it has to be stronger than Robinson arithmetic though)? If we can and we know how weak, in what sense would that be classified, with consistency strength or something else? It starts to be somewhat ambiguous what formalizing Turing machines means as you go to weaker and weaker theories. I𝛥 , which is interpretable in Robinson arithmetic, is usually taken to be the bare bones necessary to talk about arithmetical issues like Turing machines, but there are theories that are weaker or incomparable to Robinson arithmetic that are considered in the context of these kinds of questions, like the self-verifying theories of Willard . All of these theories have the mentioned property. I would even say that a theory that fails to have this property can't actually talk about Turing machines. And lastly, I’ve read that there’s a certain point where ordinal analysis has to classify ordinals to theories that are so weak like Robinson arithmetic, using a different methodology than the rest, is this related to that/how much the theory can reason about provability and/or Turing machines, and where is that gap defined in the ordinal hierarchy? I've heard about this, but I don't really know that much about ordinal analysis. It's certainly related to the fact that as you go to lower and lower consistency strengths, you start losing machinery needed to talk about these kinds of things.
|
[
"Being a counselor at math programs like PROMYS and ROSS"
] |
[
"math"
] |
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"t5vmt3"
] |
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""
] |
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true
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[
0.92
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I was a student at PROMYS for two years, and then was a counselor for a whole bunch of years, and later worked for the program in other capacities. It is absolutely wonderful program (as is Ross). You will have some freedom for your own schedule, but will also have a lot of work to do. You do get to go off campus especially in evenings if you want to.
|
I was a counselor at Ross for two years. The mandatory responsibilities I had were meeting pretty much daily with a group of 4-5 students (your "family"), grading and then going over their sets, and hosting Number Theory seminar for a group of ~20 students for around 2 weeks. Outside of that, you mostly have the freedom to do whatever you want provided you're available if students need you. You have the opportunity to do things like attend advanced lectures or host your own dorm talks on interesting topics, but neither of these were heavily enforced by the program when I was there.
|
Oh, the interview is for a fellowship, since I did apply for financial aid. I think I would qualify for the breakout fellowship since I am neither from India nor Europe. Also thank you for your advice, hopefully, I'll get through with the interview.
|
Hi, I applied to PROMYS this year and got finalised for the interview, I wanted to know how rigorous it is and if I should prepare some math topics or not. I would really appreciate whatever you can tell me about it.
|
First of all, congratulations, and good luck. (Although I didn't know they were doing interviews as part of it. Is that PROMYS India or PROMYS Europe specific?) The program is very rigorous. A big part of it is working out essentially on your own how to develop things from an axiomatic perspective. I wouldn't normally say there's anything you should prepare for for the program, but since this is the first I've heard of doing interviews, I'm not sure my advice is particularly helpful or informed as it should be.
|
[
"Is there some notion of how difficult a statement is to prove."
] |
[
"math"
] |
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"t61njb"
] |
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22
] |
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given some formal system, any statement provable from axioms and rules is considered a priori knowledge, but it doesnt seem fair to regard all provable statements as beforehand knowledge. i.e an axiom is known beforehand, but a deduction based on some axioms is less obvious. e.g 1+1=2 is easier to prove than sqrt 2 is irrational which is easier than fermats last theorem. this would be good for just knowing how hard things are, but could also be applied to game theory for example where instead of assuming agents are perfectly ration you could assum some level of difficulty. my first guess would be something like length of the proof, or number of rules used, but i dont really know how you could argue it corresponds exactly to what i want, and it doesnt take into account that some deductions are more surprising than others which doesnt have much to do with length. ive also read about arithmetic hierarchy which is to to with alternating 'forall' and 'there exist' and orders of logic, im not sure of exactly what justification there is to consider these as fundamentally harder but i imagine there is one. but im not sure that 2 statements in the same class are just as easy to prove as each other. there's like a thing in economics called bounded rationality, but the Wikipedia page seems really waffley and not really rigorous or fundamental, and its specific to agents/utility functions and not statements of logic. is there any good metric for this and what justification is there for such a metric
|
Under the conventional axioms, reverse mathematics attempts to do some version of this, where you quantify the strength of a theorem by how much "axioms" are needed.
|
An alternative approach would be to think in terms of computational complexity. The "shortest possible" proof could be defined as the proof with the smallest Kolmogorov complexity. Of course, this depends a bit on which formal theory you prefer for writing proofs. Using standard computer proof assistant such as LEAN or Coq, you can formalize this and there actually have been (non-official) competitions to find the shortest formal proofs of this or that theorem. However, sometimes the shortest proof isn't necessarily the "easiest" proof. Often really short proofs require a lot of really tricky and counterintuitive steps. "Easy" is a very human-specific thing, and I don't know if there is any way of formalizing it. It's like trying to find the "prettiest" proof of a theorem.
|
One thing that might help with measuring is getting into formal mathematics. For instance in metamath it's possible to work out for each statement exactly how many steps were used in the proof and so there can be some measure of like "how far you are from the axioms" or something. However yeah working out if you have the simplest possible proof, that's not easy. And also there are some areas where you can spend a lot of time building up a theory and then get a bunch of results cheaply, but if you just wanted one of those results you could get it more quickly by proving it directly.
|
1+1=2 is easier to prove than sqrt 2 is irrational That rather depends where you start. Peano's axioms make 1 + 1 = 2 somewhat trivial, but in , Russell and Whitehead start from even more fundamental axioms and take dozens of pages to prove it.
|
I think you just want to think about the shortest proof, rather than the one with smallest Kolmogorov complexity. Otherwise you can just use the program that brute forces finding a proof of the statement. Thus the Kolmogorov complexity of the proof is at most the Kolmogorov complexity of the theorem plus a constant. If someone asks you for a proof of Fermat's Last Theorem, answering 'The lexicography ealiest proof of Fermat's Last Theorem' is not very useful to them.
|
[
"Career and Education Questions: March 03, 2022"
] |
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"math"
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"t5w4qn"
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This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered. Please consider including a brief introduction about your background and the context of your question. Helpful subreddits include , , , and . If you wish to discuss the math you've been thinking about, you should post in the most recent thread.
|
I studied physics and mathematics in uni, but haven't used any of it in my work in the last 3 years and I'm realising I've lost a lot of fundamental knowledge. Is there any learning material aimed as a refresher? Or any tips from people who were in the same boat?
|
As a general rule of thumb, no for America (as in no MS degree needed/expected), yes for many other countries.
|
The top students were either people who knew how to study properly and actually did it (quite rare in high school) or people who were able to memorize all of the ways to do a problem without effort (that didnt mean they cared about math though). That being said, none of them did degrees in math. I was a pretty good student but my math grade was at least 15% below theirs, but now I am doing a masters in math.
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For pure math graduate courses are very good to have, and for applied math (or going into industry down the road) the CS major would be very useful. I don't think either choice would lock you into a future if you end up changing your mind though.
|
I doubled in math and computer science. I am now a postdoc in pure math. I’ll say a bit, but I have never served on admissions. I think getting a good letter writer (from research, grad course, etc.) and having strong GPA will be most important. My feeling is taking more advanced grad courses instead of double majoring would have helped admissions for me. On the other hand I did fine with grad courses in grad school and have found computer science back ground help in my research area (combinatorics).
|
[
"Famous Story of an Italian King Throwing Bread Behind his Shoulder to Estimate Pi"
] |
[
"math"
] |
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"t5swok"
] |
[
24
] |
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I remember hearing about a story in which an Italian King (hundreds of years ago) drew a circle and randomly threw bread behind his shoulder, and calculated the percent of bread that landed inside the circle vs outside the circle. Supposedly this was an early attempt to estimation pi. I tried look for this story online, and the closest thing I could find was "Buffon's Needle", which pi can be estimated by randomly dropping needles on to an a rectangular surface marked with rectangular tiles: Does anyone have a reference for the "Bread throwing story"? Thanks!
|
I'm Italian and I never heard this. You are probably remember this story about throwing bread sticks: https://plus.maths.org/content/number-crunching-ants Edit Btw: Buffon was not an Italian king, but many Italian kings were buffoons....
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It's a version probably invented by the person you heard it from
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Outstanding article; thank you.
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-- What does a bread-throwing Frenchman have in common with a tiny species of ant with a brain smaller than a single grain of salt? instant favorite
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Buffon is the undisputed king of all Goal Keepers
|
[
"Differential Geometry"
] |
[
"math"
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[
"t5oo9p"
] |
[
22
] |
[
""
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[
true
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0.96
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Hi everybody! I am looking forward to getting involved in some machine learning courses, so I wanted to get myself some basic notions of differential geometry and topology to get a deeper understanding of the transformations occurring in the models, so I wanted to know if you get to know any good resource to begin with. Also, what are some elemental things I should know/review before going in? I haven't ever had any course about dg or topology, but I am already reviewing some of my Calculus books to get into it and also some linear algebra, but if you think you have anything where I can review this faster or you just like it, please tell me. Tank you everybody in advance!
|
As someone else said, I don't think you have to learn differential geometry to do Machine Learning, unless you are going to specialize in that. Having said that, if you are interested in learning that (such as myself) yesterday I found this online course which seems very promising, and I am looking forward to do myself: https://brickisland.net/DDGSpring2022/
|
Can't think of a good book myself but there is one warning. My experience with Differential Geometry is that everyone have their own set of notations and defining things their own way. As a result you probably want to check out multiple books for different perspectives and develop your own approach to the subject.
|
If you are too brave
|
I'm currently working through Visual Differential Geometry and Forms by Tristan Needham and it's excellent! That said there is no need to study this to be good at ML. I'd view it as a fun side quest that has very little to do with ML.
|
I don’t know anything about ML but a good book about differential geometry/topology for beginners that don’t know any topology is the book Introduction to Manifolds by Tu. You should be absolutely solid with your calculus and linear algebra though.
|
[
"Trying to look for a math documentary video on youtube"
] |
[
"math"
] |
[
"t68jjo"
] |
[
16
] |
[
""
] |
[
true
] |
[
false
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0.95
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Kind of a longshot but, I've been trying to find a math related youtube video I watched quite some time back, I think it was a documentary not a tedtalk, there was a short part of a guy saying there are two types of mathematicians and he proceeded to elaborate a bit.
|
I'm not sure the best YouTube resource for your request, but it sounds like you might be describing what Cambridge mathematician Timothy Gowers has called "The Two Cultures of Mathematics" . Specifically, Gowers makes a distinction between mathematicians whose primary focus is versus those whose main goal is . Does that sound familiar? If so, searching YouTube for a string like "two cultures mathematics" might yield results relevant to what you're seeking. I'd note, though, that there's a C.P. Snow lecture called that's more broadly about differences between science and humanities, so any "two cultures" search results might be more likely to be about Snow's thesis than anything within the mathematical community. Hope this helps!
|
Not a full on documentary, but look for "turn a sphere inside out" on YouTube. It is and old video, but it's such a cool video and it touches some awesome pieces of math.
|
Timothy Gowers Sir William Timothy Gowers, (; born 20 November 1963) is a British mathematician. He is Professeur titulaire of the Combinatorics chair at the Collège de France, and Director of Research at the University of Cambridge and Fellow of Trinity College, Cambridge. In 1998, he received the Fields Medal for research connecting the fields of functional analysis and combinatorics. The Two Cultures "The Two Cultures" is the first part of an influential 1959 Rede Lecture by British scientist and novelist C. P. Snow which were published in book form as The Two Cultures and the Scientific Revolution the same year. Its thesis was that science and the humanities which represented "the intellectual life of the whole of western society" had become split into "two cultures" and that this division was a major handicap to both in solving the world's problems.
|
Related: How to turn a sphere outside in (Warning: lots of swearing)
|
Yea I found those from google, but no it was a small clip of a guy talking in a documentary if I recall correctly, I thought it was from the Erdos documentary, but I scrolled through it and could not find it. I just thought it was an interesting insight.
|
[
"Is there a name for Pythagorean Triples with the same C value?"
] |
[
"math"
] |
[
"efuhj2"
] |
[
4
] |
[
"Removed - post in the Simple Questions thread"
] |
[
true
] |
[
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] |
[
0.84
] | null |
There are just 2 types of triples. Primitive and derivated.
|
It absolutely is! You did so yourself: multiplying two small singles creates a large double if a + b = c and x + y = z then (za) + (zb) = (cx) + (cy) = (cz) And we already know there are an infinite number of singles ;) EDIT: And I just noticed that there are so many more that wouldn’t be covered by this (16, 63, 33, 56, & 65 are coprime) How would one go about parameterizing those?
|
It absolutely is! You did so yourself: multiplying two small singles creates a large double if a + b = c and x + y = z then (za) + (zb) = (cx) + (cy) = (cz) And we already know there are an infinite number of singles ;) EDIT: And I just noticed that there are so many more that wouldn’t be covered by this (16, 63, 33, 56, & 65 are coprime) How would one go about parameterizing those?
|
Here is a list for c less than a thousand: 65 = ... You missed 50 😧 [EDIT: 50 is not a square. For OP’s triples we should use only the squares from the following list.] See http://oeis.org/A007692 “Numbers that are the sum of 2 nonzero squares in 2 or more ways.”
|
Here is a list for c less than a thousand: 65 = ... You missed 50 😧 [EDIT: 50 is not a square. For OP’s triples we should use only the squares from the following list.] See http://oeis.org/A007692 “Numbers that are the sum of 2 nonzero squares in 2 or more ways.”
|
[
"I fricking love statistics"
] |
[
"math"
] |
[
"t5vyoe"
] |
[
408
] |
[
""
] |
[
true
] |
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Im a 10th grade student getting into statistics and I love it. I am not good at any other math persay but for some reson I excel in statistics, I remember in 6th grade we did some of statistics in class and found an imediate likeing. Prior to this, before 6th grade I barely passed any math class I took but for some reson I got straight A's all year from statistics. I bring this up because this has confused and made me think why I would be bad at normal math like geometry, algabra, etc, but have such an understanding of this topic, I was wondering if anyone on this sub reddit could compair and contrast from this.
|
I had a masters in pure math and some years later got a PhD in statistics. They are vastly different disciplines. Some of the students in stats were relatively poor at math, but had a nose for data. In data analysis sometime you get a sense for what's happening, and apply some techniques to formalize it. On the other hand math is just balls hard in a theoretical sense.
|
I'm extremely happy for you! But you should still try to get through calculus. A lot of properties of statistical distributions (like their normalization constants,) can only be found using integrals, and fitting distributions to data requires maximum likelihood optimization which involves calculating first derivatives. A lot of stats can be done black box, but truly understanding stats and probability involves a lot of calculus and linear algebra. I believe in your capacity to learn it though!
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Math degree doing data science. Totally agree. I haven’t encountered anything as balls hard as undergrad stuff (once I see a simplex algorithm at work I’ll go back to school. Not looking likely.)
|
Actuary here. I like math and money.
|
Future Actuary perhaps? Data scientist?
|
[
"How to have infinite area for only an 100 centimetre area"
] |
[
"math"
] |
[
"efti77"
] |
[
0
] |
[
"Removed - incorrect information"
] |
[
true
] |
[
false
] |
[
0.29
] | null |
This reads like an /r/SubredditSimulator post
|
What?
|
The actual solution is a circle: https://en.wikipedia.org/wiki/Isoperimetric_inequality
|
Explain what you mean by "go in".
|
An engineer, a physicist, and a mathematician were all given 100 cm of fencing and instructed to enclose the largest area possible. The engineer built a circle out of the fencing, thus enclosing the maximum amount of area. The physicist found a tall rock to build the fence around, so that the even larger surface area of the rock was enclosed by it. The mathematician built just a small loop of fencing around herself. The others looked at her strangely until she said, "I declare myself to be on the outside!"
|
[
"Would it be a good idea for mathematicians to write down the prerequisites for understanding their papers?"
] |
[
"math"
] |
[
"t5iwt5"
] |
[
639
] |
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For example, something like, "To understand this paper, the reader should know X,Y, and Z, and the reader can learn about these in books A, B, and C." The graduate core material can be taken for granted.
|
I do agree with the answers that the introduction does most of this work. Papers often come with a section that recalls the preliminaries, and gives a nice overview of what you need to know. This will often have references too. I have seen a lot of Algebra + combinatorics papers do this. Unsure about other fields. But also, the "only experts read papers" idea seems borderline elitist as graduate students also read them.
|
This is unrealistic. Research papers are written mainly for other researchers in the same field. This is not specific to math either.
|
I think it's hard to specify what is or isn't needed. There are often several equivalent sets of ideas that achieve roughly the same purposes, and the author might not be aware of all such sets for whatever purposes they're after. Providing a narrow enough description of the prerequisites would also be tough. Sometimes only a small number of tools from a certain area of study are needed. A maximally minimalist description of prerequisites to a paper might just be the paper itself in a lot of cases.
|
Could be done in introduction But also most papers are for professional mathematician who already study in the field so they know all of that, and other people would have difficulty to understand the paper even if they study the other papers first. I'm not sure that it would be very usefull in the end
|
I regularly teach a mixed 4th-year/grad course in matrix analysis. Instead of a final, I give a project that students have to work on during the whole term. The project can be of different types. One of them is to "unpack" a paper: I give the student a paper and ask them to produce an expanded version of the work that bridges the gap between what they learn in my course (or earlier linear algebra courses) and the paper content. I think this is valuable and is part of what one has to do to become proficient in mathematics. And the process itself is important, so I don't agree with OP that it should be made easy. Not because I think papers should be read only by experts, but because part of becoming an expert is this particular journey that takes you from textbook knowledge to being able to understand an advanced concept. Also, it is not always true that there is a book that would cover the material needed to understand research papers. As pointed out, the references in the paper are the place to start and they are often one or two steps away from a book. But also, the "only experts read papers" idea seems borderline elitist as graduate students also read them. I consider that a student who has graduated a 4-year honours degree in mathematics is an expert. They are not a mathematician, the road to that is much longer, but they are an expert. Even more so a grad student. I definitely hope they are experts: not all our students go on to join the "wonderful" life of academia, many of them get jobs and I like to think that they get hired because of their expertise. Because otherwise, this means we are doing something wrong.
|
[
"Thoughts on sin(a*x) as a approaches infinity"
] |
[
"math"
] |
[
"efsnl0"
] |
[
3
] |
[
"Removed - incorrect information"
] |
[
true
] |
[
false
] |
[
0.61
] | null |
sin(ax) is exactly identical to sin(X), where X=ax, that is, a scaling of the x axis. In other words, if you zoom in the x axis, you'll see the same identical function sin(x) for every a. Such function does not seem to have any more mystical properties than sin(x) itself.
|
Sure, first among which that it is a function of two variables and thus defines a surface in space instead of a line on the plane. That didn't seem to me what OP had in mind.
|
Here's a way you could formalize this. Let (S_n)_n be a sequence of closed subsets of a space. We can define what one might call the as the set of all points p in the underlying space such that any neighborhood of p intersects every S_n for sufficiently large n. The is the set of all p having no neighborhood contained within the complement of every S_n for sufficiently large n. (Note that these two notions aren't dual to each other, so we actually get two additional notions, and I'm not sure how to modify the terminology to accommodate all four.) If these two sets are equal, we could call it simply the closed limit of S_n. Then the closed limit of the graphs of sin(nx) is, I believe, the set {|y| <= 1}.
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You are right that f(x) = sin(x) and g(x) = sin(ax) are just rescalings. But h(a,x) = sin(ax) is totally different and potentially could have “more mystical properties” than f and g.
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Is the set not open on the y-axis (except at the origin)? {(x,y) : |y| <= 1} \ {(x,y) : x = 0} U {(0,0)} Edit: you said closed limit. I'm daft.
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[
"What’s the closest polynomial approximation of x^p if you can’t use x^p?"
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The simple answer to your question is that there is no closest approximation. For example, the polynomials of the form (1-epsilon) x get arbitrary close to x But before we can really make sense of the question we need to define what "close approximation" means here. There are plenty of different metrics we could choose here (uniform, L1, etc.) and they will all lead to a different notion of closeness. The fact that there is no closest approximation will be tied to the fact that the set of polynomials you're looking at isn't compact.
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I they might be asking what is the best approximation of x with terms of all lower degree than p. (Even though what they said technically does imply they are talking about the infinite dimensional approximation.) In which case they can obviously use the intuitively defined inner product and get an exact best approximation.
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Given a fixed interval [a,b] and a fixed n, you can compute the orthogonal projection of x onto the subspace spanned by {x ,...,x }\{x } with respect to the inner product ⟨f,g⟩ = ∫ fg. For n = 10 and [a,b] = [0,1] for example, we have Proj(x ) = 60 x - 1170 x + 10920 x - 57330 x + 183456 x - 371280 x + 477360 x - 377910 x + 167960 x - (352716 x )/11 Proj(x ) = 1/80 + (351 x )/16 - (1092 x )/5 + (9555 x )/8 - (19656 x )/5 + ( 32487 x )/4 - 10608 x + (340119 x )/40 - (41990 x )/11 + ( 29393 x )/40 Proj(x ) = -(1/2106)+(5 x)/117+(280 x )/27-(175 x )/3+196 x x )/81+544 x x )/11+(16150 x )/81-(4522 x )/117 and so on. https://www.desmos.com/calculator/wqjz26myjp
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What does "closest polynomial approximation" mean? You have to be clear about what you are trying to measure. There are many different ways to interpret this question.
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Unfortunately, your submission has been removed for the following reason(s): If you have any questions, please feel free to message the mods . Thank you!
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[
"Turning a Line Into a Rectangle"
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"math"
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Since it's unbounded horizontally. how could it represent an actual rectangle which is bounded?
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You may also want to look into space-filling curves. There are sequences of lines, or curves, that in the limit entirely fill a square. If you work through the construction carefully you'll see what exactly they mean by taking the limit of a sequences of curves. https://en.wikipedia.org/wiki/Space-filling_curve
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As I mention you could bound it to be as wide as you want by constraining the values of x that you look at, I'm just wondering if the infinitely large a does indeed fill an entire space with an infinitely thin line.
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In this case, its going to be a tough point to argue. Your argument is essentially that for some point (k, h), as your a increases, sin(ak) spends more and more time equalling h, until when a is 'infinitely' large, sin(ak) always equals h. It almost sounds acceptable as a heuristic argument, but I think it falls down under scrutiny. For example, we can easily see that the number of values of a for which sin(ak) = h is countable, because we can easily order them. Values of a for which sin(ak) =/= h are therefore uncountable, and in this sense it is fair to say that your limit rectangle is more likely to be blank (under a reasonable definition of a limit) than it is to be filled in. With that said, I suggest you look up space filling curves, which are similar in principle to what you describe. Very broadly speaking, it is possible to draw a continuous line that goes through every point of a square, but only if it goes through an awful lot of points more than once. The most famous example is Hilbert's space filling curve.
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Glad you think so, I find it very intriguing.
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"Ideas for mathematics curriculum"
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Have you ever primary school?! 6th-graders are definitely not equipped to handle the entirety of precalculus algebra. And you're trying to teach calculus to 7th graders?! Forget it; plenty of them still don't understand what the "=" sign actually means, or how to manipulate equalities. Even the lower-tier 8th-graders struggle with it. Further, none of your proposed curriculum says about mathematics. How then are you making the claim that it "emphasizes understanding"? The way I see it, it's literally just a sped-up version of the current mathematics curriculum from secondary school through to university.
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I have an even better curriculum https://img.fireden.net/sci/image/1551/03/1551030445079.png
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I agree with the latter part of your comment - there is nothing particularly pedagogical about this curriculum. But I must say that it is also harmful to underestimate the capacity of younger students. I still remember my middle school math classes as a painfully sluggish, at times redundant (to the point of repeating topics we were taught in the first grade!) slog. There is much room for acceleration - anecdotally, my former high school calculus teacher was successfully teaching his primary school daughter to learn single-variable calculus. The basics of calculus are easy to grasp at late primary school ages, and it doesn’t have to be taught all at once!
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Have you ever been to high school or middle school?
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But I must say that it is also harmful to underestimate the capacity of younger students. I am basing this off of actual experience in schools, as a trainee teacher. I do not think it is contentious to say that most 6th and 7th-graders are not equipped to handle the entirety of precalculus algebra, nor do I think it is inaccurate to say that many lower-tier 8th-graders struggle with manipulating equalities. I still remember my middle school math classes as a painfully sluggish, at times redundant (to the point of repeating topics we were taught in the first grade!) slog. There is much room for acceleration I partially agree; there is some room for acceleration, and there is definitely repetition in the secondary-school curriculum (especially at 7th-grade, when secondary schools are trying to ensure parity between the knowledges of all students). But a considerable amount of the problem lies in primary-school education, which primes students with a number of misconceptions that are carried over and have to be unlearned at secondary school. Accelerating the secondary-school curriculum is not going to help with this. anecdotally, my former high school calculus teacher was successfully teaching his primary school daughter to learn single-variable calculus. Then congratulations to said daughter and said teacher. However, not all students are quite so enthusiastic, and not all teachers are quite so knowledgeable. The basics of calculus are easy to grasp at late primary school ages, and it doesn’t have to be taught all at once! Conceptually, perhaps. But working with it more formally, with algebra and mathematical notation, even if you handwave away the epsilon-delta definition of limits and just work with the A-Level curriculum, is definitely not something you can expect the vast majority of 7th-, 8th-, or even 9th-graders to do. That said, I'm not outright opposed to a hyper-accelerated curriculum like this for the really-top performers who really get it. I'm mostly questioning how ridiculously quick this is as a curriculum which I assume is being proposed for high school students.
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[
"If I have a painting, can I find a Polynomial with 2 variables, such that the solution set looks like that painting?"
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"math"
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"effrtz"
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You need a high degree, but 2 variables are enough. Choose a high enough degree and you can approximate any continuous function to any over a finite domain. Hence you can also approximate any bounded level curve to any desired accuracy. Although: a polynomial approximation, especially using the standard base x might not be the most numerically efficient or stable. Even modern computers can‘t handle the numeric evaluation of polynomials of degree 100 or so well, for mathematical reasons. Using a Lagrange basis, or trigonometric functions (Fourier analysis), will almost certainly much better suited for your needs.
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No but look up "Lagrange Interpolation" to see why you need more points for certain paintings
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Well, a lot depends on what do you mean with "looks like". Let's say you turn the painting in a gray tone picture, then using dithering you convert it in a set of black points on a white background (like a b/w inkjet printer would do). At that point you can, in principle, determine a huge polynomial with huge coefficients which has value 0 in the coordinates of the black points. The difficult part is avoiding spurious solutions. From that point of view it is simpler to a use a single variable polynomial and consider the solution modulo some number to virtually plot them on a grid.
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I am sorry but why does that thread belong the simple questions thread. I think a question about the zero values of multivariable polynomials is not simple. The fact that the answers I got was about polynomial functions aproximating this picture which is a different thing confirms this.
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Unfortunately, your submission has been removed for the following reason(s): /r/learnmath books free online resources Here If you have any questions, please feel free to message the mods . Thank you!
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