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[
"Product of two sets"
] | [
"math"
] | [
"10nesrh"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
0.67
] | null | AB would probably be the way to go | AB, as you wrote it, is completely standard. And A+B for the sum. | This is the standard notation. The Cartesian product is with a little x. | I don’t really think this makes me look stupid or arrogant, I missed a word when reading, which happens to everyone lol just an honest mistake | This looks like a subset of the free group generated by A and B. Importantly it is not a subgroup. Edit: totally missed u we’re looking at sets of numbers never mind. |
[
"Logic riddle"
] | [
"math"
] | [
"10n8yy0"
] | [
5
] | [
"Removed - try /r/learnmath"
] | [
true
] | [
false
] | [
0.78
] | null | This is an exercise in Boolean algebra. So essentially, you have 2 variables A and B. Variable A is whether the person you're asking tell the truth. Variable B is whether behind them is the doom door. Your question is a yes/no question asking whether something is true. In other word, your question is a Boolean function f(A,B)->{T,F} Now, the answer you get is A XOR f(A,B). You want to make sure that A XOR f(A,B) determine B itself, which WLOG we assume that A XOR f(A,B)=B. Treating Boolean as polynomial over F2, we are now reduced to the question of finding a polynomial f such that A+f(A,B)+B=0 for all A and B. It would be simpler to just ensure that A+B+f(A,B) equals 0 as a polynomial, in that case f(A,B)=A+B. So the question you need to ask is: "is exactly one of the following is true: you answer honestly to this question, or the door behind you is the doom door?" | So the only threat is picking the trickster's door. Each of the guards guards a specific door. We can ask one single guard a question. What should be noted is this: If we ask a guard a question while also NOT choosing door, the Trickster no longer matters. If we ask the Trickster our question, we won't pick his door anyway! So, we only need to frame our question in a way that matters to the other two. You also mention that the youngest triplet is the liar and the older triplet is the truth teller. So I will go to one guard for my question and ask "Which of the other two guards is younger?" | I see. So what I was missing is that the trickster is required to give an answer leading to no contradictions. | I also interpreted the question like you in that the trickster can answer yes or no as they choose regardless of the question asked. In this case, we can modify the question so that we never go through the door behind the person we are asking. We can ask person A: is exactly one of the following true, you are telling the truth or the door behind person B is the doom door. If the answer is yes we go through door B and if the answer is no we go through door C. This way both the truth teller and the liar will point us to the correct door, and both doors will be safe in the case of the trickster. | I'm not understanding this. If you ask that question to the truthful one, it's the case that they will answer honestly, and not the case that they have the doom door. So they will truthfully answer "Yes". If you ask it to the untruthful one, it's not the case that they will answer honestly (they lie), and it's not the case they have the doom door. Thus the answer to the question would be "No" (neither thing is true). But since they lie, they would therefore answer "Yes". If you ask it to the trickster, there are two cases. If they answer "No", we can rule out that they are the truthful one or the liar, by the above analysis. But if they answer "Yes", we can't distinguish them from the other two, except they have the doom door, so it's no help. What am I missing? |
[
"Hello everyone! I really love math and was wondering if the academic world considers a BA and BSC in math as equal?"
] | [
"math"
] | [
"10n2mxq"
] | [
4
] | [
""
] | [
true
] | [
false
] | [
0.64
] | null | In the US, I honestly don't think anyone cares about the difference between a math BA and a math BS. At some universities, the only difference is one letter on your diploma. At other universities, you won't have a choice -- for example, Stanford offers only a BS, while Princeton offers only a BA. As a result, most people don't pay any attention. If you are at a university where there is a difference, you want to at least do the coursework for the more rigorous degree, even if you don't end up actually having that on your diploma.If you're seriously pursuing an academic career in math, the substance of your education will matter a lot more than what it says on the paperwork. | I’m gonna be real with you dawg acting like a Reddit mod while you aren’t is cringe | I can't speak to the academic side of things, and politics that may occur based on where you went to school or if you took the 'more difficult' course. I would say though, that whatever you end up doing, make yourself known to the professors. Go to office hours. See if you can assist research projects and ask a lot of (meaningful) questions. Be available, show that you are a determined & productive student, and you will be able to get some good recommendation letters from your professors. At least in the professional world, a recommendation will go much farther than a transcript. So I would bet there is a lot of weight put to that as well for the academic side. | Thank you! | This is correct. Nobody notices. |
[
"Is there bigger infinities than other infinities?"
] | [
"math"
] | [
"10msuqk"
] | [
2
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.63
] | null | Look up "Cardinality". The Wikipedia article gives a great overview. If you accept cardinality as the definition of "size of a set", then yes. For example |ℝ|>|ℕ| but |ℕ|=|ℤ|=|ℚ|. In general |P(M)|>|M| for all sets M. | For OP, this means Real numbers (R) (2.34, 5.00, etc) are "bigger" than natural numbers (N) (1,2,3..), but the natural numbers are the same "size" as the integers (Z) and fractions (Q) | The only comment actually answering OP's question | The only comment actually answering OP's question | Well it depends. There are two main categories for infinity. To answer your question, if both sets of lines are countable infinities then the total set would be the same size (that being a countable infinity). If either one is uncountable then the whole set would be uncountable. Hope this helps! |
[
"At the beginning of the COVID-19 pandemic, Joachim Kock began to experiment with epidemiological models. He did not improve upon the predictions, but unexpectedly he made a mathematical discovery that led to the solution of an old problem in theoretical computer science, on Petri nets"
] | [
"math"
] | [
"10n839x"
] | [
260
] | [
""
] | [
true
] | [
false
] | [
0.97
] | null | What was missing in the conventional Petri nets was precisely access to this information of symmetries of Petri nets, provided in Kock's new notion. “With a little bit of homotopy theory and category theory, I could prove that the new version of Petri nets allows a reconciliation of the two semantics, the algebraic and the geometric.” The new definition proposed by Kock has already been used by other researchers (Evan Patterson and his group in Berkeley, USA) to develop Petri-net based software for epidemiological modelling, and in particular for COVID research. “In this way the circle has been closed. | Or the free copy: https://doi.org/10.48550/arXiv.2005.05108 This is the final 'author version', nearly identical to the version published in JACM. | Study | Oh my, this guy continues to exceed all my expectations considering the diversity of his work. | tqfts now this lmao |
[
"Is it true math skill cannot be improved in older age. It must be done between the ages of 1-13."
] | [
"math"
] | [
"10n4z5q"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.36
] | null | No | no, that's crap. The myth probably comes from the fact that if you have a Ramanajun type prodigy, it's apparent by the time they are 13. But 13 certainly is not a limit. | It's just easier to learn when you are young. Anyone can learn more when you're older, it just takes more effort. If that wasn't true, colleges wouldn't exist. | My middle school teacher told me this when I wouldn’t do the work. I’m regretting it now but hope it is not too late. | Neuroplasticity is higher at a young age, but it's certainly not gone at 14. |
[
"I put all the simplex numbers into a sequence, graphed it and found something that approximates a function the more entries you add. What function is this and what properties does it have?"
] | [
"math"
] | [
"10mztu3"
] | [
316
] | [
""
] | [
true
] | [
false
] | [
0.94
] | null | Heuristically, numbers which appear in different simplex sequences should be vanishingly rare, ie there should not be many numbers which are both triangular and tetrahedral. These are solutions to Diophantine equations of the form p(x) = q(y). There is a result of Bilu and Tichy which classifies rational solutions to equations of this type, and essentially except for some degenerate cases(like when the two polynomials are related by composing a linear function) the solutions are very sparse. I haven't checked their conditions for the simplex polynomials but it seems likely. If this holds, this simplifies the problem as follows: let f_k(n) denote the nth k-simplex number. Then f_k (m) is roughly the number of k-simplex numbers at up to m. Let F(m) be the sum over all k of f_k (m). Then F(m) is essentially (because of this rare intersection property) the number of terms in your desired sequence up to m, so the function you have graphed is F . This should enable some further computation, but I'll have to think about it more. EDIT: Some back of the envelope calcs: f_k(n) is approximately n /k!, so f_k (m) is about k! m ~ k/e m The sum of these guys over all k is clearly divergent, but we should also be truncating our sum to account for approximation errors. | To quickly explain a simplex number, [...] this comment doesn't contain any explanation of what the black simplex sequence is. you've described all the other sequences, which a lot of people here probably already know about, but not the other one that I don't recognise. | PARTIALLY SOLVED, SEE COMMENTS (thanks for helping) To quickly explain a simplex number, it is like everybody's favorite triangular numbers but generalized to any dimension. For example, after the triangular numbers is the tetrahedral numbers. You get them by adding up each triangular number, 1+3+6+10, and so on. Repeat for higher dimensions. This is a fascinating beast. It combines a bunch of other sequences in a way I'm wholly unfamiliar with. Unfortunately for me my knowledge of terminology is only marginally better than a layman's so researching what this could be has proved difficult. As of recently I've been on a mission to discover everything I could about "shape numbers" (simplex, square, cube, polygonal, etc.) with as little help as possible and using nothing but basic operations and intuition. But this has stumped me. I would go to higher numbers but I've had to manually compute the individual sequences before copy-pasting them into the final series and doing 100k was torturous enough. My logic for the subtitle is that since the combined series increases slower than the triangular numbers by necessity the leading term can't have an exponent over 2 (which is used in the former sequence) or else it would gradually overtake the tri-nums. That being said, it still looks vaguely quadratic which leaves the question: What is the exponent, assuming it's quadratic? Naturally I have the sequence itself and can share all the terms once I figure out a way to do it without creating a massive wall of text in the comments (I'd appreciate any suggestions on how to do that). | Have you tried fitting a function to it? An approximate value could give a good starting spot. Have you tried the OEIS? As an amateur I found a strange sequence in simplex numbers that I later learned were https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind | We have an imposter. |
[
"Awkward problem, and wrong solution?"
] | [
"math"
] | [
"10mp2mq"
] | [
4
] | [
"Removed - try /r/learnmath"
] | [
true
] | [
false
] | [
0.7
] | null | This is the correct solution. Try some examples, or try all 11 possible examples. | Unfortunately this solution is wrong, because even that would probably take me longer than a minute. In fact, I'd probably knock a coin or two off the table. | I'm dumb... I misread the question. Thanks for the reply. (I thought it wanted equal heads and tails at the end) | Yes but if e.g. you randomly get 9 heads and 1 tail, that means there’s one head in the other group. Flipping the first group gives you 9 tails and 1 head, so both have one head. Maybe what you’re missing is that whatever random selection you get in your group of 10 also determines what’s in the other group. You get exactly as many tails in the 10 as there are heads in the rest. | I have been casually doing some problems in Terry Stickels book "challenging math problems". However I came to this one and worry I'm missing something? If you randomly select coins, you can have lots of different combinations after flipping group 1. |
[
"what do mathematicians do in computational biology?"
] | [
"math"
] | [
"10nl6t3"
] | [
11
] | [
""
] | [
true
] | [
false
] | [
0.7
] | Recently I have been seeing a lot of math majors i know are working or applying in computational biology. I have been wondering why they wanted mathematicians and what they are doing. I thought maybe statistics, but there must be more. | Many mathematicians these days have been saying that, "Biology is the next physics". What they mean is that in the future, new tools and theories in mathematics will be guided by motivations from biology, just like it has been with physics for hundreds of years. There is an endless supply of current problems in biology that people are realizing are amenable to mathematical methods. As another comment here says, the kinds mathematics that are very popular these days in mathematical/computational biology include: | To twist your brain even more... A long time ago in a past life I worked as a bioinformatics software engineer. At the lab there was a post doc. So in biology, there are specific genetic sequences that encode specific binding sites. Binding sites is a loose term for interactions. If you have also learned about combinatorics, you can learn and study a lot about sequences of letters, and specifically give rough approximations of sequences in a finite string of letters. It turns out, combinatorics gives really good mathematically tools for analyzing rough approximations of sequences in a finite string of letters. He was basically studying gene sequences to try and analyze drug/protien binding sites using only combinatorics and information about the DNA sequence. It was actually a really good way to improve discovery because you could basically write off a lot of trials that wouldn't even work just on the sequence level which is basically the symbolic computation level, instead of trying to produce the gene and looking at interactions in the wetlab. If you want to read more about combinatorics, Sedgewick and Flajolet's Introduction to Algorithms and Analytic Combinatorics is really good. There is also Generatingfunctionology that's fantastic. | I am a math PhD student working in comp bio. I'm a bit surprised that all the math majors you know want to do comp bio. It seems like this field is saturated by CS, biology, and stats people much more than math people. Most math PhD students that I know in this field work on the more statistics/data sciencey part of the field, especially involving single cell RNA-seq data. See Lior Pachter's lab at caltech: https://pachterlab.github.io/ . I personally work more on the combinatorial/data structures side of things. It's fun, not necessarily super mathy; some basic probability and combinatorics of strings is the extent of it. The field of sequence alignment and genome assembly was founded by mathematicians (see https://en.wikipedia.org/wiki/Michael_Waterman or https://en.wikipedia.org/wiki/Stephen_Altschul ) and is what I work in. Here's my perspective of my field, which may be biased: while a lot of posters have posted some cool math that can be applied to biology, I hold the opinion that sexy math isn't always the best thing to do in this field. I see myself more as a mathematician working in computational biology rather than working on mathematical problems inspired by biology, meaning the most important aspect of our work is providing useful biological insights. This means that simple models, underlying respect for biology, and scientific thinking take precedence over cool math, which is unfortunate sometimes, but satisfying in a different way. | For dynamics, don't forget models for evolution! Ever since biology has sought to be rigorous about evolutionary arguments following papers like "The Spandrels of Spain", dynamical systems models for evolutionary systems have taken a central role. | Biological systems are often modelized using very complex dynamical system, and the study of such dynamical systems often rely on mathematical analysis. |
[
"Can we write any number as a unique product / quotient of fibonnaci numbers"
] | [
"math"
] | [
"10nduo1"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
0.82
] | For example: 4 = 8/2, 6=3 x 2, 35 = 21 x 55 /(3 x 11). I read some stuff about prime divisors of fibonacci and I get that any number can written like this if we allow raising fibonacci numbers to a power, like 18=2 x 3 But is there also a representation possible without using powers? | My guess is no. Possible attack for proof. One has that gcd(F(a),F(b)) = F(gcd(a,b)). In particular, it will be difficult to separate two primes whose first appear at F(p) where p is prime and F(p) is composite. The smallest such example is F(19) = (37)(113). So 37 and 113 are both good candidates for numbers which cannot appear this way. It seems like that one can combine this with the fact that new prime divisors must show up in F(n) for all but a few small n to prove this. | F_19 = 4181 = 37 x 113 is the first (positive-index) Fibonacci number divisible by either of these primes, so any Fibonacci number divisible by one of them is automatically divisible by the other. As such, we can't get any positive integer which is divisible by exactly one of these primes. | Slightly more reasoning is I think required, although your idea is the same essentially as my suggestion. The subtlety is that one could have Fibonnacis with repeated powers of 37 and 113, which could allow cancelation. So one also needs to think about how bigger primes are getting introduced whenever one does that. | Right yeah, so then the fix is that we still can't get 37 or 113 on its own since it would have to show up alongside another prime (whichever new prime shows up in the largest Fibonacci in a proposed product / quotient representation). Edit: The result about getting new primes (apparently attributed to Carmichael) is doing the heavy lifting here, really. Given any product / quotient, suppose F_m is the largest Fibonacci number that shows up (necessarily in the numerator if we're representing an integer). If m < 19, the number represented is not divisible by either 37 or 113. If m = 19, the number is divisible by both. If m > 19, the number is divisible by more than just 37 or 113. | But that also shows that if such a decomposition does exist, it is also unique? |
[
"Is there a drag and drop mathematics toolbox?"
] | [
"math"
] | [
"10n5dn9"
] | [
26
] | [
""
] | [
true
] | [
false
] | [
0.88
] | Hello, I am a brain cancer patient that has issues writing in general. I can type and use a mouse but holding a pen, or rotating a doorknob or tools, cause me very bad pain in my arm. Every time I extend my arms my nerves dislocate. They can be seen popping in and out on an Ultra Sound. My question is, if there is a program or app that allows me to do mathematic and physics equations? I don't need a calculator, just someway to type my work out. When I was in high school, there was a great app called ModMath on the iPad, but unfortunately, ModMath doesn't support physics or college level math equations. That app was amazing, but I can no longer use it. I've tried to use my iPad to pictures of problems and try to type my work over the picture, but it takes so much time and confusing. Any help would be greatly appreciated. Thank You | https://latex.codecogs.com/eqneditor/editor.php This might help. I pray you recover. | Learn to write in LaTeX. | Use Obsidian if it’s just for notes, people are recommending latex, but that’s taxing to learn for text other than math. Obsidian let’s you use markdown which is much easier to learn. And then for the math you can still learn latex syntax for the math itself and use it in Obsidian. You can also get plugins that allow for shortcuts when typing math symbols. I use it for all my notes and I can normally keep up with my lectures, latex I can’t. | LaTeX is not as difficult as some people make it out to be. It just takes practice. Once you get through the learning curve, it opens a world of possibilities. Particularly for someone like the OP, who has limited use of alternatives. | To add some context, OP: $$x \in R$$ |
[
"Are there any civil engineers that have had significant contributions to mathematics?"
] | [
"math"
] | [
"10nkwi5"
] | [
33
] | [
""
] | [
true
] | [
false
] | [
0.85
] | Just curious.Also tried various versions of it in google with no satisfying result. Edit: What I meant to ask was...is there a branch of mathematics that had significant contribution because a person was trying to solve some probelm related to civil engineering(any subfield...transportation,hydraulics,structure,environment,geotechnics) | Well you could consider many mathematician from the old time as civil engineers because most of the scientists were strong in almost every field(because there were less knowledge at their time). I think if i had to give one it would be Monge the father of Optimal Transport who created it for very concrete problem | Constantin Caratheodory was a military engineer who worked on the Assiut Dam by day and studied math by night. Any math grad student has probably heard of Caratheodory's criterion in measure theory What I meant to ask was...is there a branch of mathematics that had significant contribution because a person was trying to solve some probelm related to civil engineering Finite element methods were anticipated by engineers. For a more recent example, peridynamics in civil engineering stimulated much mathematical work in nonlocal modeling. | Solomon Lefschetz was an engineer but perhaps he doesn't count since he switched to maths after losing his hands in an industrial accident. | I'm surprised nobody mentioned this before, but the Finite Element Method originated from mechanical engineers. This lead to a very rich family of numerical schemes, arguably the most studied and used today. The first use case was for studying stresses and such on solid structures. In fact, I've seen modern mechanical engineering courses introduce FEA as a kind of "analytical approximation", where they carry out computations by hand for simple cases (a beam cut in two triangles, for instance). The same ideas are being used today for numerical simulation in many different fields. In a similar topic, CAD systems were originally developed by Pierre Bézier and Paul de Casteljau, both engineers , though not necessarily civil engineers as they worked for the auto industry. | It was very easy to count his hands though. |
[
"Multiplicity of x^2"
] | [
"math"
] | [
"t4lb3p"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.45
] | null | So it was on my assignment, and it was asking for the zeros of f(x) = x^4-x^2 and I got the answer wrong. When I checked the example problems, they factored out the equation to x^2(x^2+1), then to x^2(x+1),(x-1), and they said that the zeros of the function is x = 1,-1,0. And then their multiplicity are 1 for 1, 1 for -1, and 2 for 0. And their multiplicity is what decides if the graph crosses the x axis or bounces off from the x axis | If you think of x as (x-0)(x-0) or (x-0) you see that the polynomial has the factor (x-0) repeated twice. If it had the factor (x-1) it would be easy to see that the root 1 has multiplicity 2. So it's because the factor appears twice in the factorisation. Also, if a polynomial has a double root (i.e. of multiplicity 2) at x, its derivative will also have a root (not necessarily double) at x. See if you can prove this by factorising the polynomial. | Thanks for the detail I understand now, but the deverites are outta my league kinda, I'm a first year student and taking college algebra. I also checked out ur profile and saw u were doing some marching cubes, and I just wanna say I've done a bit marching cubes with compute shaders, so lmk if I need help with anything :) | Oh, that might be helpful! I gave up on them a while back because I couldn't get the compute shader to do what it was supposed to, but if I try it again I will definitely contact you. Thanks :) | x = x•x If you find the zeroes then you get x•0 and 0•x. Since you get two of the same zeroes it has a multiplicity of two. |
[
"So, I’m in a little predicament. (Can’t remember basic math)"
] | [
"math"
] | [
"t4nxmf"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.5
] | null | No offense but I feel like what I said is being taken to the extremes which may be my fault. To reword it, I’m not as comfortable as I would like with the topics. Sure, taking some time to go over it would in a sense fix the problem. However, I feel like I shouldn’t of forgotten things like exponent rules or the distance formula. | Agreed. Since it all scaffolds and builds upon each other I struggle with believing you are successful in Calc AP and don’t know basic algebraic processes. | Unless you're disabled, you shouldn't have any trouble "remembering" basic math. If you find yourself having such trouble, then you don't need to "relearn" the material, you need to learn it in the first place as you never actually learned it the first time around. Nobody ever thinks of themselves as having forgotten English, or having to relearn how to speak English. That's because for the most part we learn English properly in the first place. When you learn basic math, you need to learn it as fluently as you know English. It needs to be mastered to the point where you literally can't forget it. | https://tutorial.math.lamar.edu/ | https://youtube.com/c/misterwootube |
[
"what are some common mistakes people make while learning math?"
] | [
"math"
] | [
"10n95u6"
] | [
46
] | [
""
] | [
true
] | [
false
] | [
0.87
] | I'm a first year Electronics student, and I'm learning some pure math in my free time (currently algebra by Artin and understanding analysis by Abbot) what are some common mistakes made by people (whether in uni or self study) in trying to learn math I'm not talking about mistakes like getting an answer wrong, I mean mistakes in the way you study or how you study that lead to you either not learning properly or learning inefficiently. | believing that math is about memorizing symbols and procedures, believing that there is nothing to math other than doing calculations, believing that every problem should be solvable in 2 minutes or less or it's impossible, not actually trying to understand anything (or worse, not even realising that there is anything to be understood at all) | Not Doing The Exercises | Reading too fast over apparently minor details without really understanding it, it will come back at you 100% | Blame math teaching for that one. Every single mathematical problem I did in my whole school education was a calculation, intended to be solved in a single way. Formulae were learned by rote. I don’t think anyone ever explained anyone would want to shuffle algebraic symbols or calculate angles. (Though now that I think of it, in primary school they once had us colour in the multiplication tables on 12x12 grids and see the patterns. That was cool.) | Looking at solutions too quickly. Forgetting to play around with concrete, specific examples of the abstract objects you’re learning about. Getting so focused on rigor that you forget intuition and heuristics are important aspects of truly understanding a topic. |
[
"This Week I Learned: January 27, 2023"
] | [
"math"
] | [
"10mpuk6"
] | [
12
] | [
""
] | [
true
] | [
false
] | [
0.88
] | This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn! | Right/left adjoints preserve limits/colimits. Now I need to understand limits/colimits to do my homotopy theory problem set | Oh jeez where do I start... You can prove countability by coming up with a clever way to index or group things and then prove countability rigorously through induction The Cauchy-Schwartz inequality is a generalization of the triangle inequality and has fundamental implications It's in your best interest to understand the properties of prime numbers because taking advantage of their properties is like using black magic The field axioms seem obvious but become really interesting when you consider things like finite fields | This week, I learnt how to win k-pile Nim! By Bouton's theorem, you can force a win if and only if the Nim sum of the number of tokens in each pile is not zero; otherwise, the second player can force a win. (You find the Nim sum of a set of numbers by converting them to their binary representations and going bit by bit; the n bit of the Nim sum is the sum of the n bits of all the addends modulo 2.) Fortuitously, the proof is constructive: there's an explicit algorithm to derive a winning move at each stage if you're in a position to force a win. It was cool. | This week I understood Kan complexes, they actually make so much sense. Infinity-categories feel less scary now. | I learned a host of obvious-seeming statements that are in fact equivalent to the parallel postulate. For example: To be precise, the "angle" AVB is the union of ray-VA with ray-VB where A,B,V are not collinear. The "interior" of the angle is the intersection of the half-plane of line-VA containing B with the halfplane of line-VB containing A. |
[
"Actual application of Combinations?... without anything else..."
] | [
"math"
] | [
"t4kstp"
] | [
1
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.54
] | null | The binomial theorem: (x+y) = sum nCk x y https://en.wikipedia.org/wiki/Binomial_theorem?wprov=sfla1 Since multiplication is commutative (xy=yx) it means that you disregard the order, but still take combinations of all elements whose powers add up to n. | For an even more specific example, this leads to binomial distributions which are everywhere in statistics https://en.wikipedia.org/wiki/Binomial_distribution?wprov=sfla1 | Sort of meta but I do not like the constant influx of "applications of x" threads, where x is some VERY fundamental concept to mathematics, say sequences or topology, or now combinations. Especially when a lot of the replies are some very niche topics. "In how many ways can you group n things" is one of the most natural questions one can ask about mathematics. Thus as expected, without combinations there's really no probability, computer science, number theory and so on. The list is huge. | I'm not sure what you mean by an "actual question," but I think the opposite is true. Combinations are far more important both in pure math and in applications, to the point that there is a special symbol used for them. They just come up all the time. A simple example from probability: You have 20 identical machines on your factory floor. You are about to take on a contract to finish a lot of work on a tight deadline, and you need at least 18 of the machines to be running tomorrow in order to meet your contractual obligations. Each machine has a 2% chance to break down on any given day. What is the probability you will be unable to fulfill the contract? The answer is: 1 - ((20 choose 2) (0.02) (0.98) + (20 choose 1) (0.02) (0.98) + (20 choose 0) (0.98) ) which is about 7 in 1000. Why are combinations used here? Because what matters is how many machines break down, not the order in which they break down. It turns out that in many cases you really don't care about the order in which things happen. | Finally, as the most specific example. Consider a random walk, meaning that you flip a coin, and take a step forwards if heads, backwards if tails. What is the probability that after 20 flips, you end up where you started? This is described by a binomial distribution. Notice that if you have HHHHHHHHHHTTTTTTTTTT or HTHTHTHTHTHTHTHTHTHT or HHHHTTTTHHTTTTTTHHHH etc. You've ended up where you started. The order of H to T didn't matter, just that there is the same TOTAL amount of Hs and Ts (10 each). So this is equivalent to saying "How many COMBINATIONS of Hs and Ts are there out of 20 total, with 10 Hs and 10 Ts?" (You of course divide this number by 2 to get the actual probability.) |
[
"Summle - a little daily math game I made"
] | [
"math"
] | [
"t47imb"
] | [
9
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""
] | [
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false
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0.81
] | null | Looks pretty much like the Numbers Round from Countdown (UK TV Show) Just saying... | Neat! :) | Thanks. | Thanks. Yes, the minimum steps indicated is correct! | After you complete the sum. |
[
"Kinematics and Quaternions — a wonderful monograph by Wilhelm Blaschke"
] | [
"math"
] | [
"t4mqr0"
] | [
19
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"PDF"
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false
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0.87
] | null | Check out geometric algebra too! It unifies exterior algebra and quaternion functionalities. | Quaternions are such an underrated and useful concept, especially outside of computer graphics and quantum mechanics, and I find that terribly disappointing since I think that they are in general the superior way to represent 3D space. | Yeah rotors are much more intuitive and you don't have to drag in a 4th dimension | I would argue that having to involve a supposed four dimensional space to model 3D rotations is unintuitive and even unnecessary. Like another comment mentions, the geometric algebra formulation of rotors supersedes them: https://marctenbosch.com/quaternions/ | That’s their main use today. Hamilton intended to create an entire system for representing 4 dimensions. |
[
"Bifurcation in logistic models"
] | [
"math"
] | [
"t48wte"
] | [
8
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""
] | [
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false
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0.91
] | I am going to teach a course on epidemiological modeling and I am putting together a lecture on forced compartment models. These models bifurcate at certain forcing amplitude thresholds, so I need to explain bifurcation and bifurcation diagrams. I want to use the classical example of the logistic map, however, everything else we do in the course is done in continuous time, so I tried to make the bifurcation diagram based on the logistic differential equation rather than the difference equation. But that does not work, apparently. Can anyone explain to me why bifurcation occurs in the logistic difference equation, but not in the logistic differential equation? | I don't know specifically which equations you are referring to, but one example of the logistic DE with a bifurcation is population dynamics with harvesting. If P is population, k is growth rate, N is carrying capacity of the environment, and h is harvesting rate (animals per time unit), then the DE is P' = kP(1-(P/N)) - h. As h varies, you get a saddle-node bifurcation. I don't know if this is what you are looking for, but I use it to introduce bifurcations in my ODE class. | Yeah. I use Blanchard Devaney Hall as the intro ODE book--it has a good section on bifurcations . | You would have to restrict that statement to first-order differential equations for it to be correct, I believe. https://en.wikipedia.org/wiki/Duffing_equation | Thanks, this is the exact equation I am dealing with. P for proportion of infected individuals, k for pathogen transmission rate and h for recoveries. Maybe I just didn’t explore a wide enough range in k to get the bifurcation? I’ll need to check again. Edit: but the example I refered to in my question was x_n+1 = rx_n(1-x_n), which gives the famous bifurcation tree, versus the differential version x’=rx(1-x), which apparently does not bifurcate. | Yeah, the iterated system has period-doubling bifurcations when the parameter r is varied, but the ODE x' = rx(1-x) doesn't bifurcate when r is varied. There are other parameters that can be introduced to the ODE that create bifurcations, like the harvesting rate in the example above, but there isn't a one-to-one relationship between ODEs and iterated systems with respect to bifurcations. |
[
"Learning individually before learning in school to understand better/maximize gains?"
] | [
"math"
] | [
"t4icni"
] | [
10
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""
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true
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false
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0.92
] | So recently I've been doing poorly in math (adv alg 2). This is really new for me. I seem to understand the ideas in class, but when it's time for a test, I do really bad. I am a good math student and everything so idrk where the problem is... I was thinking about learning topics we are going to learn in class on my own first so that everything after that is just strengthening what I already know. I can also work on my own pace then. What do you think about this? | Mathematician here. I can learn about things in class/lecture/videos. But the only things I really understand, the only things I can *use* creatively, are things that I taught myself by doing problems. Lots of problems. I'm a junky for them. My advice is this: Find a place that has good problems (the problems on Alcumus at "Art of Problem Solving" are fantastic), and do them relentlessly. | That could be helpful! I never did that. My method was take notes in class with all the details I could. Then I would go back to my dorm and basically re-write the notes but with thought and understanding. In class it moved too fast for me to do anything but transcribe. Upon redoing lecture notes, but now with understanding, I would find questions and go to office hours with those. | Yeah my teacher moves really fast. I'd like to understand as she's going through it though She does this thing where she says something once and it isn't on the slides and comes up on the test | I feel exactly the same. I' m having a topology course next semester which probably would be quite challenging for me so I plan to start reading something about it. Initially I followed the abovementioned method. I rewrote everything while fully using my brain when going through proofs but this was very inefficient. A lecture takes about three hours to be understood at very details with this method and sometimes it's a bit hard to do active-thinking all the time (trying to prove the statement) rather than just trying understand it (passive thinking). So now after I take notes during the lecture, go again through proofs, - I try to do some important easy proofs without looking my notes and -I split the long important proofs into smaller parts to have a good idea what's really going on there And I try lots of exercises too. | There is no way you can solve all of them. Atleast where I study. We get so many there is just not time for it.. |
[
"Does the ordinal number given in ordinal analysis have a relationship to the amount of statements that can be proven?"
] | [
"math"
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"t4kw1x"
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2
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true
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false
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0.63
] | So I have no background in this subject, but the wikipedia page briefly states that two theories having the same ordinal number usually is taken to mean that they're equiconsistent. First, am I getting it right that equiconsistency means two theories' can prove each others' consistencies? But secondly is my question in the title, since ZF and ZFC I've been told are equiconsistent which as I mentioned, wikipedia suggests would entail them having the same ordinal number, but of course one of those can prove one more thing than the other (axiom of choice), so would they really? | First, am I getting it right that equiconsistency means two theories' can prove each others' consistencies? Not quite. Note that if you have two theories A and B which can prove each other's consistency, then by applying both results, A could prove its own consistency. That's no good by Godel. Equiconsistency is a little more subtle. It means that under some simpler base system (often Peano Arithmetic or a fragment of PA), the consistency of system A implies the consistency of system B, and the consistency of system B implies the consistency of system A. | You might learn it in a model theory class or an advanced logic class. Most of this sort of thing I've picked up by immersion. Someone else may have a good intro model theory text book to recommend. | model theory Eh, what people call model theory isn't really about consistency results. Basic model theory is certainly relevant to consistency results, but none of the standard model theory textbooks even do Gödel's incompleteness theorem. | I assume this works. Another way would be to say that A proves B has a model and in this model Con(A) holds. But if there is any contradiction in A, then the model of B would also see that contratiction (which is encoded by a natural number), so then the model of B does not satisfy Con(A). Thus A proves Con(A). The reason I was asking it that I usually don't think of consistency as having a model. For example, in PA you can't really make this argument. But I think you can also run it purely syntactically without using models: If A is inconsistent then also B proves A inconsistent. Then if Con(B) and B proves Con(A), then not Con(B). | where do you learn stuff like this? Logic courses? which ones? |
[
"What's a good hobby or entry point to falling in love with math?"
] | [
"math"
] | [
"t4mmea"
] | [
50
] | [
""
] | [
true
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false
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0.86
] | Hi math redditors! What is a hobby or activity—ideally something pretty structured, like a particular type of problem or a particular website or a book series—that can help someone pursue a love of math as a hobby, and develop it as an interest that goes beyond just doing school math and sudoku? Here's why I ask: I have a 15-year-old who is just re-discovering his passion for math. He's homeschooled and is tearing through high school math after a few years of doing almost no math work. But he has already surpassed my own math abilities, so I don't really know what to "feed" him to take his math explorations to the next level. He loved "The Number Devil", and he does a lot of logic grids, and he sometimes enjoys building random number generators or different kinds of calculators in python or Coda...but what else could we introduce him to, that would encourage him to keep exploring? In case it's relevant, I will mention that he has a nearly off-the-charts IQ so he can probably grapple with stuff that is pretty challenging. Past obsessions have included the Fibonacci sequence and the nature of zero. (Actually, he's still pretty into talking about both zero and infinity.) Thanks in advance for your suggestions! | A good source for him might be the Art of Problem Solving website. They have lots of fun and interesting math problems and resources that I think he would enjoy. They also have a series of textbooks that he could learn math with and that could help him prepare for math competitions. I would look up things like the AMC math contest and see if any of the problems on there spark his interest. The AMC is the first level in a series of math competitions for highschool aged students, and if he is interested in it, he could have a lot of fun preparing for and participating in math competitions. | i would caution a bit against this at the current stage. since the kid is at the... let's call it "numberphile" stage of love for mathematics, in my opinion it is very important to introduce him to "mathematical thinking" but without the rigor as a turnoff. this is why i recommended puzzle books. i think the step after would be to pursue your advice if he enjoyed the puzzles | I was hooked on Project Euler in highschool. It's a website with programming/math puzzles. | If he wants to learn what maths is like, then you should give him a book on proofs. Richard Hammack's or Daniel Velleman's are both standard recommendations here. Once he finishes that, or if he finds the example mathematics in those books too dry (which they kind of are) but likes proving things, then you should give him Michael Spivak's . Once he has a good facility with proofs, you may want to give him Paul Halmos's if he likes thinking about infinity. | at this stage, you may find yourself experiencing (but not limited to) the following symptoms: i'm really just recalling my past experience |
[
"how did you get through math you didn't find \"interesting\"?"
] | [
"math"
] | [
"t44hej"
] | [
36
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""
] | [
true
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false
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0.87
] | There's the discipline to do it at the end of the day, but did you find any other approaches helpful? As a bonus, did you ever find any exciting connections later on that renewed your interest? Asking because lately I am not motivated to study. Personally, I've found music to help a bit recently and will continue exploring this avenue. Thanks! | For me, all the math I studied was interesting; HOWEVER, I had some very poor teachers who made the class unbearable. For those, I cut most of the classes and learned from the book. I don’t recommend that, but that’s what younger me did. | Not math related: Do the thing you want to do the least, as early as you can in the morning. That's when you're least burnt out and looking for a way to avoid work, and usually people aren't heavily socializing around the first two classes of the day. Math related: try asking your lecturer what their favorite / most interesting part of the course is. They might put a spin on it you haven't seen yet. Looking at external resources might also shed some new light on things (I probably ended up using different textbooks and lecture notes than assigned ones for >50% of my courses) | Personally, a new topic or field becomes imteresting the moment I stop understanding wtf is going on. Like, in the beginning of a, for example, topology textbook its pretty mundane, even if unfamiliar: something something you can take a union of a family of open sets, its obviously an open set, whatever. But then the field will start throwing curveballs at you and your task is to take the curveball to the face instead of ducking. Why the frick is Niemytsky plane not normal, its so obviously normal? Why are all modules over a division ring free, thats dumb and stupid and I dont get it? Why are there normal and non-normal subgroups, can you tell them apart just by looking? Its very easy to get complacent and adopt a new intuition instead of challenging your existing one; regurgitating the proofs when asked why something works the way it works instead of pursuing the philosophical connection between the rest of the field or just math in general. My personal philosophy is that my task is to feel stupid, and thats the distinct flavor of learning math. If I dont feel stupid enough, I am doing something wrong. | I admit it is likely a symptom of my own short sightedness. Mainly, sometimes the exercises seem excessive but I understand this is necessary for memory encoding. I guess the next thing is sometimes the applications of the information are not given outside of the math itself. Personally, I like finding relations between things so if I am doing something in a vacuum state, I feel a weird negative pressure like I'm about to collapse from the emptiness (joking but you probably get my point). | I tried to find other students to study with. I tend to be more motivated in group settings. With classes that were unbearably boring or just a topic I hate, I just struggled through to the end. A lot of times my motivation was passing because otherwise, you’ll have to do it all again! Sorry this isn’t as practical, but even as a PhD student at the end of my coursework, I still haven’t found a great way to push through! |
[
"How would you break down the sections of math into a \"family tree\"?"
] | [
"math"
] | [
"t48vju"
] | [
0
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true
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false
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0.42
] | This is probably a confusing question but I was wondering if anyone had a good answer of source. Basically looking at math with math as a singular node on top, and then branching down into its children, and children's children, how do you think it should be distributed? I've seen some interesting charts where it starts with pure and applied mathematics, I've seen some where it says algebra, geometry, arithmetic, basically there doesn't seem to be a quality breakdown at least from what I've been able to see. So I was just wondering if there is some decent agreed upon answer, or is there just so much overlap it gets to blurry to say? This is about the most clear cut and comprehensive example I've been able to find: | I have to say i'm always a bit annoyed/confused when people label probability theory and stochastic processes as "non-pure-math". Sure, it's highly applicable math, but it is mainly pure math in my opinion, based on the fact that it's (a) build on measure theory, a subject that's obviously pure-pure-math (b) it is mainly intrinsically motivated, meaning researchers don't necessarily look at real world problems to come up with new research questions and (c) it's connected to various areas of pure mathematics, in both directions, meaning other areas of pure math use probability theory and probability theory uses other areas of pure math. | TBH that tree you linked to seems kinda "off" to me. For instance, how come cryptography is implicitly counted as a branch of pure mathematics rather than applied? Why does statistics branch off from probability--shouldn't it be the other way around, with probability as the more general branch and statistics as a subfield or application of it? Why is calculus separated from the other subfields of analysis; isn't calculus essentially just another name for analysis? (Sure, if you go into a calculus class and real analysis class they'll look pretty different, but the underlying subject matter is the same: "calculus" vs. "analysis" seems to me like a purely pedagogical distinction, referring only to the level at which said subject is taught.) Realistically, yeah, there's too much overlap and too many connections between fields for something like this. At the very least, representing fields of math with a tree rather than a non-tree graph with connections all over the place doesn't make much sense. | Personally I think any tree which attempts to separate math into "pure" and "applied" topics is going about it the wrong way. See also any tree which attempts to organize the topics based on what a subjective observer thinks the topic is "about". The problem is that math is huge, very, very few people have any idea what most of it is about, and virtually every field has links to every other field. We should do to a math tree what the Tree of Life did with the oldschool version of taxonomy. Quit grouping things by their phenotypical characteristics, and instead look at the to find when each field started branching out of its parent topic. However, I am no historian, so I wouldn't even know where to begin with this. Just my opinion based on all the usual criticisms that arise anytime someone makes a "map of math" | The MSC is the closest thing there is, and there’s a reason it’s fairly flat, with a large number of high level subjects and no attempts to categorize further. Basically everything blends together at the ends - go deep enough into some area of analysis and you’ll inevitably run into algebra or geometry or foundations. | Math is funny cause you'll write this and then you find that there is a link between Ring theory and topological space in algebra/category theory/functional analysis. |
[
"The Nyquist Criterion hurts my brain as a Complex Analyst"
] | [
"math"
] | [
"t4f6f0"
] | [
335
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""
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0.96
] | Over the past couple of weeks, I have been working on a video presenting the . And it truly is a clever application of some geometric properties about complex curves that’s related to the poles and zeros of an analytic function. If you are curious to jump into the video and see the resulting presentation, you can find it here: If you don’t know, the Nyquist criterion leverages the argument principle from complex analysis. Essentially, if you integrate f’(z)/f(z) along a simple contour that surrounds the poles and zeros of f(z), then the result is a difference between the number of zeros and poles repeated up to multiplicity, if the contour is oriented counter-clockwise, which is standard in complex analysis. and also gives a two page header breaking down the precise motivation behind the development. Always fun to dig into original works. The Nyquist criterion is all about finding the poles of the transfer function of a linear system, which is the Laplace transform of the impulse response of the system. You take a known plant, G(s), and say a controller H(s), and it is presumed that you know the poles of each and thus the poles of the open loop system G(s)H(s). Then you want to figure out if you have any poles in the right half plane for the closed loop system, whose transfer function is G(s)/(1+G(s)H(s)). He realized that what we really want is to know the zeros of 1+G(s)H(s), since they’d be the poles of the quotient. Since we already know the poles of G(s)H(s), we can use the argument principle with just the right contour to solve for the number of zeros of 1+G(s)H(s). What’s this contour? Well, we’d like to surround the whole of the right half of the complex plane. If we find a zero there, then we know we have an unstable system. However, the argument principle only applies for bounded contours, so we just pick a really big semicircular contour whose flat size travels along the imaginary axis. And then we orient it CLOCKWISE. Which really messes with my intuition. More than that, we don’t actually need to integrate. After a change of variables for the argument principle, you quickly see that the integrand can be written as 1/w, and since the only contributions of an integral along a closed contour comes from encirclements of the poles, we can just LOOK at the image of the contour under f(z) = 1+H(z)G(z) to see what the integral along that contour would give you. So this is what we want: we want zeros of 1+H(z)G(z), and we know its poles a priori. Using the clockwise oriented curve (for a big enough semicircle), we get # of counterclockwise encirclements = # of poles - # of zeros of 1+H(z)G(z). That is, if the number of counterclockwise encirclements matches the number of poles that we know H(s)G(s) has, then the system is stable otherwise it is unstable. . I feel like that happens to me all the time looking at linear controls and complex analysis. | I tell you what, having learned the Nyquist criterion as an electrical engineer before ever touching complex analysis, it was suuuuch a headache the first time around because I wasn't privvy to the motivation behind the test. Looking back I see what's going on, but at the time I just had to chalk it up to "this is a stability test, just learn it". | You are thinking of the https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem but OP is talking about https://en.wikipedia.org/wiki/Nyquist_stability_criterion Control theory was one of the classes I struggled with most as an engineering student, but I never studied this from the complex analysis perspective mathematicians get to. The way I understood the reason for why poles on the RHS of the plane mean the system is unstable is that they give rise to terms that increase without bound with time if you convert the transfer function back to its differential equation form with an inverse Laplace transform (while poles on the LHS will decay). I failed this class the first time, the mathematics got a lot deeper dealing with discrete time systems, many different types of controllers, etc. | I did units on control eng… I remember our lecture said “it just fucking works… don’t need to know how or why! just use it!” | All those Bell Labs guys, Nyquist, Shannon, Hartley, Hamming, have so many things named after them it makes my head spin sometimes | You’re talking about the nyquist frequency, he’s talking about the nyquist criteria. Not the same thing. |
[
"If an analytic function has a continuous extension, is that extension analytic?"
] | [
"math"
] | [
"t405fy"
] | [
4
] | [
""
] | [
true
] | [
false
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0.7
] | Specifically I am asked to prove that if a function can be approximated uniformly by a sequence of polynomials on the unit circle in the complex plane, then it must have an extension that is continuous on the closed unit disk and analytic on the open disk. | Hint 1: The title and body of your post are two different questions. Hint 2: The answer to your title is "no" (think: bump functions) Hint 3: To prove the body of your post, extend the polynomials to the unit disk and use Morera's theorem along with the maximum modulus principle. | f(x)=exp(1/(x -1)) if |x|<1 else 0 is a bump function that's analytic on |x|<1 and smooth on R but not analytic at |x|=1. | f(x)=exp(1/(x -1)) if |x|<1 else 0 is a bump function that's analytic on |x|<1 and smooth on R but not analytic at |x|=1. | Bump functions are not analytic anywhere ? Bump functions have compact support, so will be equal to 0 most places. And 0 is certainly analytic... And if you're unhappy with 0, just add your favorite analytic function | I think that’s what they were saying. The reason the answer to the title is “no” is because bump functions are not analytic. I don’t actually know what you people are talking about but that’s how I read it. |
[
"[Others] Motivation Issue for Those Who Self-Study"
] | [
"math"
] | [
"t44bn4"
] | [
16
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""
] | [
true
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false
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0.81
] | Hi fellows, I am self-studying undergrad graph theory (this bit of info might be irrelevant). The notes provided to me by a super dedicated professor have claims, proofs, and exercises. They are totally self-contained. Sounds perfect, doesn't it? But the problem is that I cannot harness enough self-discipline, focus, and motivation to self-study. I initially planned to complete a paper in the field with this professor by August and I feel that goal is drifting farther and farther away... Note: I already graduated with a math BS from a US college, in case that might be helpful... | Hi, I made a similar post recently. I think a big part is doing it and by that I mean simply starting. Tell yourself you'll do it for 2 minutes. Get started. Maybe start a routine where you pair associate the habit of one thing to another. Personally, recently I found music to help. Not trying to promote but I started a trial for endel and I've been liking it a lot. My usual go to is blume, which is atmospheric ambient type on youtube. Good luck! | Relying on discipline is hard because that takes a ton of decision making. Relying on the power of habit takes virtually no energy. Think about it as a set program; 1)stimulus, 2)routine, 3)reward. 1) Music works as a queue to get your mind into the right place. But so can a timer, or many other triggers (such as setting or time, timers, coffee etc.). 2) Then execute your set routine say 75 min of studying. 3) Then reward yourself for doing it, a small piece of chocolate, coffee, a victory song and dance, lunch etc. then do this routine for a few days and you’ll have a habit. Then just don’t break the habit. Don’t think about the habit of studying, just setup and follow the routine. Then use your mental energy on the math. | For me the biggest thing is having people around you who are working on the same stuff. If you are studying with someone and make a plan, you are much more likely to stick with it because there is some accountability | Thank you so much for the tips! I have already started for 2-3 months now but the progress has been grossly decelerating. | I think that studying from a variety of sources and watching videos online (not necessarily lectures) related to the topic can make things seem more interesting and engaging again. Channels like 3Blue1Brown may help recultivate interest ive found. |
[
"A topology puzzle"
] | [
"math"
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"t43wjt"
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36
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] | [
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false
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0.93
] | View the 2-sphere S as a topological manifold. Suppose we have N embeddings E_i: S -> S , by which I mean maps that are homeomorphisms onto their image. Consider the equivalence relationship on S that identifies, for each i, all points in the image of E_i with each other. That is, the relation E_i (x) ~ E_i (y) for all points x, y in S . Note that a priori, points in the images of different embeddings are not identified with each other. Classify, up to homeomorphism, all possible quotient spaces obtained by this procedure. | The bouquets of spheres will possibly be connected with one another. Take for instance two disjoint circles. The quotient is a "chain" of three spheres, each touching the next at a point. The interesting question is which configurations are possible. For example, I think you can rule out cycles because of the Jordan separation theorem or something similar. | The bouquets of spheres will possibly be connected with one another. Take for instance two disjoint circles. The quotient is a "chain" of three spheres, each touching the next at a point. The interesting question is which configurations are possible. For example, I think you can rule out cycles because of the Jordan separation theorem or something similar. | Not a mathematician, just an enthusiast, so I'm not 100% on all those terms. The mental image I got was putting zipties on a balloon and then contracting them to a point, getting something that looks somewhat like many buns next to each other/touching each other. Is that an accurate mental model? Is it a useful one? So if I put one ziptie, I get two spheres that touch. If I put two disjoint zipties, I get three in a row. If I put two intersecting ones, I get four spheres arranged like sectors of a circle. | First note that the quotient must be simply-connected: every loop on the quotient lifts to the sphere (essentially because circles are path-connected), and the homotopy to a point descends from the sphere to the quotient. Next, assume that the complement of the embedded circles has finitely many connected components. I think something like the above argument shows that the closure of each of these components descends to a sphere in the quotient; hence the quotient is a wedge sum of spheres. Now it is also clear that every simply-connected finite wedge sum of spheres can be obtained from this procedure, by inductively constructing such a "tree" outwards. (Here I say "tree" because the quotient actually contains extra information about which points are identified, eg. a wedge sum of n spheres at one point vs (n-1) spheres stuck onto one sphere at different points. This completes the classification for the "finite case", and perhaps the "infinite case" is more interesting (and it does happen, since embedded circles on the sphere can have infinitely many intersection points). I think in this case the quotient is a countable wedge sum of spheres (countable because the original sphere has a countable dense set S, and the connected components of the complement of the embedded circles give a partition of S). Now it's not immediately clear to me that any simply-connected such space can be obtained; I don't really have a good idea whether this is true or false. | That's the right picture; I visualise it as the "tying my ham/sausage too tight" problem XD |
[
"What’s the best geometry to fill the most space? Or what is the most suboptimal packing solution and why is down so good?"
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"math"
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"t465t3"
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0.67
] | I wonder if anyone here knows anything about packing geometries of unusual shapes. What I’m wondering is if there is a efficient solution to the packing problem. Is there a shape that when randomly added to a bucket will fill it up highest while when added in an ordered way takes up the least amount of space? Is there a term for the ratio of ordered pack size to random pack size? Adding springs into the mix, is there a theoretical spring geometry that does better than down in terms of loft per compressed volume? | You can take a severely non-cubic cuboid, then divide it into two equal non-convex shapes with spikes. That figure will tessellate 3-dimensional space perfectly like the original cuboid, but probably if thrown randomly in the bucket the two pieces composing a cuboid will not fit, resulting in a lot of wasted space. | Not really the same from a mathematical perspective, but a related concept: https://en.m.wikipedia.org/wiki/Tetrapod_(structure) I suspect there is no "minimally effective shape" here, just as there is no solution to the minimum ratio of area to perimeter, for a continuous, closed plane curve (think about very pointy shapes). For example, if you're allowing hollow solids, a spherical shell of extremely low thickness will get you an arbitrarily low packing ratio. If you don't want hollow solids in the topological sense... Just poke one hole into the shell. Also, with the mention of down and springs, you're getting into material properties, is that what you're interested in? Maybe more of a physics question. | Yes! So I’ve seen those structures and am really interested in them from a mathematical perspective but you’re right I’m really interested in the physics of springs. | Using a box as an example if you used a square plane, fitting it diagonally into the box you could only fit one plane, but laying it flat it would fill the whole box. | Yes but randomly packed boxes would end up naturally stacked to be space efficient. |
[
"Einstein said, “Never memorise something you can look up in a book!” Who agrees with this, and why? Need to justify procrastinating my study."
] | [
"math"
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"t3vjt3"
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] | null | “Never memorize what you can look up in books” is a quote often attributed to Einstein, though what he actually said was somewhat different. He was asked, but did not know the speed of sound as included in the Edison Test. When this was pointed out, he said, “[I do not] carry such information in my mind since it is readily available in books. He also said, “…The value of a college education is not the learning of many facts but the training of the mind to think.” Source Sure, it's not a big difference, but it's annoying to have the man's thoughts reduced to "Here's One Weird Trick to Get the Most Out of College!" | Partially agree, partially disagree. As an example for why one might disagree, I can look up any word or phrase I don't know in French and get a translation, but that does not mean I can speak French. On the other hand, the most efficient way of learning French isn't sitting down with a French dictionary and committing every word to memory, but by going out and speaking the language. Memorization naturally comes with use. | I doubt he would support what college Bachelor's degrees have become (extremely expensive certifications for a job) but certainly he supports a strong higher education in general and deep studies | So the next part of the sentence actually supports a deeper education at college? | Einstein was a socialist so he would definitely despise what modern college education has become Can we please resist the temptation to take people out of their time, deify them, and then reflect on how they would react to modern culture? Einstein would not "despise" modern education were he alive today. He would probably have valid criticisms of how education is handled today, just like most of the rest of us. The navel gazing coming from the academic set is remarkable. |
[
"Math student pandemic burnout?"
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"math"
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] | Hi! So this is the only community I know of where I can reach out to other people who are/have studied math and love it too. I'd like to know what other people's opinions are on studying math through the pandemic and how it's affected you? I'm an undergrad, currently in analysis 2. I got my associates in math and graduated mid pandemic, then transferred for my bachelors. I LOVE math. Calculus was amazing and fascinating and beautiful and I wanted to do it and keep going for the rest of my life. Then the pandemic hit, and on cue every class that taught me anything about proofs was totally online. My pure math education was completely online up till fall 2021, where I took abstract algebra 1 and analysis 1 among others. I struggled so much, it took me and my confidence down about 5 notches - but I really enjoyed analysis! Now come this semester, I went part time and the only math class I'm taking is analysis 2 - but I feel so burnt out. I don't know if it has to do with the pandemic, and my foundational knowledge (which because everything was online I think is a little weaker than it should be), but I feel like I can't intake information effectively anymore. It's exhausting because I still have my passion for it, but I just don't know what to do because I look into the future and have a hard time not seeing every summer and fall and spring being filled with classes that make me mentally and emotionally exhausted only to meet the bare minimum or average at best. For any other math students going through the pandemic, or any teachers, what are your experiences? How did you work through what feels like such an unforgiving field and not come out with foundational knowledge issues? Because I look around at my classmates and the large majority of them are thriving and I just don't understand how. | I can not say anything that can possibly make things better (since I'm in the same situation, and I also don't know how to get out), but I started my 1st year in a top math PhD program. Don't ask me how I got in, because if you're in the position to be considering applying to top math PhD programs, your resume/qualifications will almost certainly exceed mine in every possible metric, so I won't have any tips for you. And yes, I have read the posts on Academia.StackExchange about imposter syndrome, but honestly, if you looked over my resume, you too would be scratching your head in incredulity. But by golly, has it been hard. A significant portion of my 2nd quarter (this quarter) was spent online, and my already dirt poor social skills in this remote environment led to me not having anyone to work with on the homework, and graduate level mathematics when done utterly alone over the course of weeks is incredibly soul-sucking. At least for me, mathematics is only sustainable as a social activity. This I think is an ; essentially the only reason I got through undergrad was because I did work with a group of people that met almost every day to work on problems together (as a corollary, my 3rd year of undergrad, the "Zoom year", was extremely hard for me, and I would have failed if not for the friends I had made in person still having some amount of contact with me online). I feel completely burned out, and I feel on the edge of imploding basically every day. I don't hate what I'm doing, but I feel so tired all the time, and I constantly yearn for a "break". But even when breaks do come, like winter break, or last summer break, my mental health does not improve, and I do not feel "refreshed". I've been in "free fall" for a year and a half now, and judging by the classes I'm failing right now, I'm moments from striking the ground. That was all rather dour, and not at all encouraging to you, but I guess I'll justify it by saying it proves that you are definitely not alone in this, and definitely not everyone is "thriving". | You're definitely not alone I'm in a pretty similar situation. I cant offer a lot on how to deal with these problems, I'm honestly looking for that advice as well. I started college autumn of 2020 convinced I was going to do math. I took 5 math classes online last year, including algebra, analysis, and the freshmen proofs sequence which covers a somewhat eclectic collection of topics, and while I did OK I guess it has been a major struggle. I've burned out pretty much every quarter, I've lost a lot of confidence that I'm fit for math as I can barely seem to hit the average. The burn out is particularly horrible since it means that despite struggling I find it harder and harder to maintain or increase my effort. I don't think I really can increase it anymore, I've put everything I can into math this past year and pretty much all I can show for it is a subpar record. I do also feel that I'm probably missing a lot of foundational knowledge. I'm taking Galois theory at the moment and struggling for dear life, I can't help but wonder if I had taken algebra in a normal year where I had people to work with instead of trying to learn it myself in desolate isolation that maybe I would be doing better right now. Last quarter was even worse, I tried to take algebraic topology and it was soul crushing how many things I felt were flying over my head completely. I know that I learn best with other people and the pandemic definitely made that incredibly difficult even if I did manage to make a few online friends who helped me through all my classes last year. One unfortunate side effect of this I think is that I've actually found it pretty difficult to work with people even now that we are back in person. A lot of people seem to have grown used to the pandemic and just power through by themselves as their preferred method? I'm not totally sure what they do but I don't ever see people working together, if I suggest it it never happens, not even review sessions before midterms. People just don't seem to be interested. I don't know how they can do it all by themselves and do so well. I am considering switching my major and making math a hobby. I love math, I think there is something there intoxicatingly beautiful in it that I want to pursue to its full extent but the current set up is hurting my enjoyment of math and life in general. | Thank you, guys. This is unbelievably encouraging to read and know I'm not the only one/there isn't just something wrong with me or my ability. I decided to take the summer off and extend my graduation by a semester, which'll be the longest break I've had in a long time. I'm much more hopeful for the next semester knowing I'll be going into it having actually mentally rested. | You are not the only one having to deal with such feelings. If it is possible, consider do less courses for a while to take some load off. For me, having people to discuss math with really helped as well to get back my appetite for it. Math really flourishes when its allowed to be a social endeavour | Man, I read this and it feels like a carbon copy of the last year and a half for me. All I can say is what people in my life have been saying to me: take a break. You need it more than you think you do. Don't give up on something you love because you can't power through it right now - take some classes over if you feel like they were weak. There's no shame in that, even if you got a good grade in them, and you'll have a stronger foundation for it. And the most important thing is: do not doubt your ability to do this just because you didn't feel confident in material you learned online during a global pandemic. I honestly wish you the best and I hope things get better for you. |
[
"Can't get this one"
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"math"
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"o6um4d"
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3
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"Removed - try /r/learnmath"
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true
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] | null | From the sidebar: 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 . | From the sidebar: 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 . | From the sidebar: 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 . | From the sidebar: 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 . | already got the answer. thanks anyway. |
[
"How to Learn Proofs."
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"math"
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"o6rwan"
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"Removed - post in the Simple Questions thread"
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] | null | Here you go. https://www.people.vcu.edu/~rhammack/BookOfProof/ | This particular proof is by induction and the part after "however then" is the inductive step. The idea is to use the formula for the n = k case to prove the n = k + 1 case. Compare the indices in the sums on both sides of the equals sign right after "however then" and ask yourself what is missing from the summation on the right. | best part of it? it's free!!! | First step: read a lot of proofs, fully understanding them, taking notes of what the important ideas of the proof are, not all the details. Don't memorize all the steps. Second step: try reproducing the full proof using the notes you took in the first step. You'll have to fill in the gaps, for which you'll have to do some thinking. Third step: try remembering the main ideas of the proof (without using your notes) and reproduce the full proof, explain it to someone else. By doing this you'll gain a lot: 1. You'll learn the style in which proofs are written, 2. You'll get a deeper understanding of the subject, 3. When you put the focus on the main ideas, those are going to stuck more in your memory and you can incorporate them to the toolbox for proofs that you'll build. Try to ask yourself: okay, from this proof that I just read, what's the minimum hint that I need to be given in order to reproduce the whole proof, what are the non trivial steps of the proof? Once you incorporate those key ideas to your toolbox, coming up with proofs for things you've never seen will be much easier. | best part of it? it's free!!! No. That's not the best part. The best part is that Hammack's book is straightforward and readable. There are plenty of things that are "free" that will cost you in other ways, some that may not be apparent until much later. |
[
"Infinite mutually exclusive binaries"
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"math"
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"o6otna"
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2
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"Removed - incorrect information/too vague"
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] | null | 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! | I guess the core of my question is how does one ascribe probability in infinite sets? | I guess the core of my question is how does one ascribe probability in infinite sets? | Measure theory can help you here https://en.wikipedia.org/wiki/Measure\_(mathematics) | What method do you use to pick randomly? |
[
"Can someone explain the difference between the infinities?"
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"math"
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"o6j9r4"
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"Removed - incorrect information/too vague"
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] | null | what's indescribable infinity? What's unpredictable infinity? Never heard of those terms, and I've studied a bit of set theory. Perhaps you can provide more context as to where you heard them. There are in fact ineffable cardinals, but those are pretty specialized. https://en.wikipedia.org/wiki/Ineffable_cardinal | Indescribable cardinals are a thing. Never heard of unpredictable cardinals, though. | Studying large cardinals give us an idea of how likely it is that ZFC is inconsistent. For example Reinhardt cardinals are an example of a large cardinal axiom that has been proven inconsistent, while slightly weaker stuff like Vopenska's principle has not yet been shown to be inconsistent. This is useful info if you want to make bets on whether ZFC is inconsistent. Some large cardinals are useful in category theory. For example if you want to annoy your fellow Haskell programmers with the old meme that "a monad is just a monoid in the category of endofunctors", then you better make sure that the category of endofunctors exists. However the category of endofunctors of Set only exists if you have a strongly inaccessible cardinal (and say that Set is the category of sets hereditarily smaller than that inaccessible cardinal). We thus see that without strongly inaccessible cardinals it would not be possible to make funny memes about Haskell. Apparently large cardinals allow you to get models of ZF in which all subsets of the real numbers are Lebesgue measurable. I never understood why anyone ever would want that, but some analytically minded people apparently do. Anyone here who would care to explain to me why we want only Lebesgue measurable sets? | It all sorta boils down to the ordinals, which are sets well-ordered by the set membership relation. Any ordinal is transitive, which means any ordinal is the set of all ordinals before itself. For ordinals and b, we always have a < b, a = b or a > b (and exactly one of these.) When we write 'a < b' we just mean 'a is a member of b.' Thus, the first infinite ordinal is the set of all finite ordinals, the next ordinal is the set containing all finite ordinals and the first infinite ordinal, the first uncountable ordinal is the ordinal containing exactly all of the countable ordinals, etc. Now piggy-backing on the ordinals are the cardinals, which tell you 'how many' elements are in a set. To any set x you can associate it with a unique set card(x), which is the least ordinal in bijection with x. There's a lot more out there- and I'm pretty new to large cardinals- but you could take a look online for more references to get an idea of what inaccessbile cardinals, measurable cardinals are all about. It's a very neat subject | Now I wish I understood large cardinals well enough to be able to introduce a new one, because an unpredictable cardinal would be a fun name for one. |
[
"is it too late ?"
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"math"
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"Removed - post in the Simple Questions thread"
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] | null | naaa, never too late. a bit of motivation and here you go. have fun | I started my math degree at 28 after a refresher course of high school algebra and a lifetime of being "bad at math", u can do it | I think you’ll find that as you learn programming it will contextualize and motivate the mathematics you need to learn. This will make learning it much easier then in high school or middle school. Nobody is just “bad” at math. It is a skill and like any skill can be learned. You might be behind on some of the knowledge some might consider prerequisite. I would focus on the programming, which you hopefully enjoy, and use that to guide your mathematical education. You may want a tutor, teacher or friend to help you understand what you need to learn, after that use khan academy or YouTube to learn it. If you plan in enrolling in a CS program at a university or college speak with an adviser to understand what pre-requisites you need if you don’t think you have them. Other then that just have fun and don’t let a lack of understanding make you feel stupid, you have to be bad at something before you can be good at it. | <3 thnx , I want to get into programming and I know math will make things easier in that field , I hate my current Major | Programming is a fantastic way and reason to get into math. |
[
"Relevant chapters in John Lee's \"Topological Manifolds\" for Diff / Alg Topology respectively?"
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"math"
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"o75abn"
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] | [deleted] | The first four chapters in Lee's Topological Manifolds are essential as they are an introduction to topology motivating the object of manifolds. From my knowledge the book doesn't contain any differential topology in it. Your best book on differential topology I can recommend is "Topology From Differentiable Viewpoint" by John Milnor. These two books should prepare you adequately for Lee's Smooth Manifold book. | Intro to Topological Manifolds covering spaces in much greater depth. My recollection is that the Appendix covers the important results on that topic needed for Lee's Smooth Manifolds book. Of course, knowing more can be helpful. I agree with the other poster that the first four chapters are what is most essential for the Smooth Manifolds book. | He has chapters that cover some of the same material as Hatcher, for instance simplicial complexes, homology, etc. Do you want to just cover the prereqs for those other topics or you could use Lee's book to cover much of the same material as in Hatcher. | Yah I see that most of the later chapters coincide with Hatcher but im curious which chapters are pertinent to Lee’s because I’d like to tackle those first. For example on the appendix to Lee’s smooth manifolds he has a section on covering maps but that’s pretty late into the first book. Should I skip to that or are covering maps not that important in the second book? | All of it is relevant to differential topology. Presumably you're asking what you need to know before starting on differential topology: the bare minimum would be familiarity with basic (point-set) topology, and the essential theorems of differential calculus (inverse/implicit function theorems and Taylor's theorems). If your course is introductory they will certainly define manifolds so you don't need to know what they are beforehand, though obviously prior exposure always helps. The standard introductions to differential topology are Milnor's Topology from the Differentiable Viewpoint, and Guillemin & Pollack's Differential Topology (I like the second one better). There's a new-ish book by CTC Wall (a living legend of the subject) that looks pretty good, and I also like the little-known book of Kosinski, Differential Manifolds (these last two references are graduate level, but self-contained). |
[
"How to Interpret the Ergodic Theorem for Data Science"
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"math"
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6
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] | null | The failure of convergence still only happens with probability zero and in many situations you can actually show a uniform ergodic theorem. | Your argument against applying the ergodic theorem in the real world seems to be that it is only an "almost everywhere" result. But the central limit theorem and nearly every other limit theorem in probability is also an "almost everywhere" result and people don't throw up their hands and say probability theory can't be applied in the real world. | I agree with most of your corrections! I think you're still misusing the idea of L1 convergence. L1 convergence is the convergence of the sequence of integrals of the absolute values of the sequence of functions minus the limit, so it's a convergence of absolute difference averages. Ae convergence is pointwise and has nothing to do with integrals in principle. The case of Von Neumann with p = 1 is NOT birkhoffs theorem. Now, with respect to the theorems where the convergence does happen in norm, there we can agree that the equivalence classes thing shows up, specially since it proceeds by proving an object exists by completeness of the Lp space. I think, but I have no certainty that it might be possible to bypass that issue but it would be in a non trivial way because you'd have to choose a representative of a class that's the limit of a cauchy sequence and that just sounds extremely convoluted. | Clt is a distributional theorem, not an almost everywhere one. | It can happen that the set of feasible starting conditions x_0 is a measure zero set. In which case the “almost everywhere” becomes a problem. For example if you can only start on some embedded submanifold of phase space with dimension less than that of the whole space. Maybe this is what /u/AcademicOverAnalysis meant? Really nice video btw OP! Super cool to see how ergodic theory is applied in real world stuff. |
[
"Mathematics not in english"
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] | I would be interested in learning about the experiences of non english native speakers when it comes with learning math. Did you know english before/independently of studying math? Have you been in a situation where language has been a barrier, even if small, to communicate with people? What about resources like videos, textbooks, notes, etc? Would you have preferred if they were in your mother language? Do you think a large(r) math community in your language would be benefitial to* you? I understand the irony of posting this in english. | I'm from germany. Did you know english before/independently of studying math? Yes, here english is mandatory in school. But I think that education wouldn't be enough for me to understand english without problems. I guess what helped more were a whole lot of english media I started to consume after school. I feel really confident in understanding english nowadays (don't need to speak or write in english often so I'm not that confident in that regard). Have you been in a situation where language has been a barrier, even if small, to communicate with people? Sometimes when talking in english I won't know some words. I then need to find a way to talk my way around it. For example I will usually not remember the word 'cucumber' (or the name of any other vegetable). I guess that's because I usually never need to use that word (most english stuff I read is about vector bundles or magic fantasy worlds - both aren't really known to include a wide cast of cucumbers). But it's a bit emberassing if I notice midtalk that I don't know a word which should be trivial. Funny is that I know the meaning when reading it - I just can't remember. What about resources like videos, textbooks, notes, etc? Would you have preferred if they were in your mother language? Yes and no. I notice that it's harder for me to understand english math texts. Only slightly but I need to pay a little bit more attention when reading. So I think I would be faster if the texts were in german. On the other hand I prefer that research is done in one language hopefully most researchers speak. And then it's easier to inderstand research papers if you already know the terminology. For example the mathematical object 'field' ('Feld') is here called the german equivalent of 'body' ('Körper'). If I only knew the german word and would then read research papers I would need to figure out what the mean by field. There are more examples to this. Really annoying: A sheaf is called a 'Garbe' in german while a gerbe is still a 'Gerbe'. Do you think a large(r) math community in your language would be benefitial to* you? I guess not since I'm confident in (at least understanding) english. For others maybe, especially for beginners with university math it's hard(er) to find good resources in german discussing those topics. So people to discuss these topics with before you will eventually learn math english out of necessity would be a great help for them I think. | I had limited English at the start of my career, so I always tried to use books in Spanish or Portuguese. Using math books in English actually helped me quite a lot improving my English, but when the time of going into a PhD came, there was no way around, so I had to practice a lot in order to fulfill the minimum requirements to apply abroad. That said, when I arrived to my destination country, my English was still very limited, so I forced myself to engage in conversations, give talks, and so on. It took me like solid two years living in an English speaking country to feel comfortable with my spoken English. At the beginning it was very tough tho, ngl. In Brazil for instance there is a big thing for making cheap, quality and accessible education, so there are a loooot of resources in Portuguese. Many Spanish speakers benefit a lot from that in the early stages of their careers. I think while people should work on their English if they want to pursue an academic career, language shouldn't be a barrier to get into math. We need more resources in different languages. | I'm Italian and I'm a researcher in CS. I'm 57 so my experience is not recent. I arrived at the university without really knowing English. I did study it 3 years in middle school, but then I forgot it when I studied 5 years German in high school. Only because choosing German as foreign language was the way to get the best math professor in the school. I hated German language and I forgot it completely as soon I finished high school. When I studied CS (and some math, like algebra, linear algebra, analysis, probability, numerical analysis) at the uni I was lucky that most of the reference books were only available in English, so I was forced to learn it by reading it. In that period I also played a lot of textual adventures on my Commodore Vic20 and 64 and also that helped. Like The Hobbit and Zork, if somebody is old enough to remember them. I was also a fan of P.G. Wodehouse and Raymond Chandler so I started reading their books in original, which was very challenging and slow at the beginning. Anyway, reading technical English was not really a problem. Nowadays I read all my fiction in English at the same speed and ease with which I read in Italian. I've probably read a thousands novels in English, mainly mystery. Unfortunately I have a lot doubts about grammar when I have to write in English since I lack formal studies. I'm also a very poor speaker because my pronounce it usually tentative and often approximate, since for various reasons I had not the possibility to spend one year abroad in USA or UK. This, together with my shyness and social ineptitude has been quite a handicap in my work. Anyway, I don't care about scientific communities speaking in Italian. I always prefer technical books in English because a lot of terminology is in English anyway and translations are sometimes poor. Italian clearly is good for science pop books, since the target is mainly made of young people or persons who did not pursue studies beyond high school. | I had basic ("mid-level") English proficiency when starting the university. These days I'm practically living on the (English-speaking part of the) internet and English is a second nature. Up to university levels, there is material (textbooks, lecture notes) in the local language. From starting your PhD at the latest, you are implicitly assumed to understand English. It probably helps that math textbook English is usually simpler than general English. A few decades back learning Russian was mandatory for everybody here, and there was a large amount of Russian textbooks (either original or English books translated to Russian) available cheaply - at some point I contemplated learning Russian just because of this. Communicating: these days everybody understands English and mixing English words into the local language is perfectly ok. Of course there must be a few students struggling, but it is more-or-less expected that you learn English if you want to study math at higher levels. Most notions have a corresponding word in the local language (in case of latin/greek origin, often sounding very similar), though as you get closer to current research, these often start to be ridiculous. But generally people understand both versions, so if you don't remember one you can just use the other. Videos: I like if there are English subtitles, especially when for example it's more a freeform discussion when sometimes it can be hard to get a word or sentence here and there. But in general I prefer text over videos. The math community is small enough that it makes sense to standardize to a common language, which these days happen to be English. Maybe for larger languages like Spanish or Portuguese it makes more sense to have more native material, I cannot really relate to that. I have a question in return: Do algebraic geometers really read SGA, EGA and all those huge, inpenetrable and volumes? Do all of them learn French just because of this? Or only French algebraic geometers read those. Or maybe not even them? :) | Fun fact: here in Brazil, Russian textbooks have the fame of containing ridiculously hard exercises. But I've never studied any of them, so I don't how true this is. |
[
"Gap month to learn skills?"
] | [
"math"
] | [
"o71jgy"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
1
] | null | I just graduated with my BA in math and have been thinking of studying and taking some actuarial exams. We’re you able to get a job as an actuary? Did you need any internships? | Since I was in a similar boat to you I’ll just throw out my experience. I have a BS in math as of 2017, since then I did some tutoring, some volunteer work, a cubicle physics job that mostly used python, and got my EMT license and worked as a covid vaccinator. I also took some community college classes here and there to explore more options. I finally now have a big kid job that’s full time and with benefits as a lab tech in the biotech field with a promising future in that. I was unable to find a job doing as much math as I would have liked, most of the math jobs I saw wanted a higher degree, were really just computer science jobs, or were teaching, none of which particularly interested me. So if you don’t want to do that for your math fix I would suggest branching out, I still tutor on the side and watch numberphile on youtube to have more math in my life. | Learning about computer programming is probably the most bang for buck, because it's useful at basically whatever you end up to do - almost all areas of work (and life) can benefit from some automation, and programming is all about automating boring stuff. Based on the sentence "I know a bit of R", I'm assuming that you don't actually know much about programming - it's a very very deep field. Now 1 month is not a very long time for this, but it can be a start of a journey. Different people have different ways of thinking and different preferences, so without knowing more about you, it's hard to recommend something concrete, but I will try and give a few options below. But don't worry, especially at your level, much of what you learn is transferable between different languages (it's more important to learn more about the basic principles first). So, some possible choices: SICP And some languages I you from: | Oh, I kind of misunderstood your situation from your description a bit. However, I believe while a month it's not enough to dive deep, it's certainly a good opportunity to start! Programming in general is not a thing you "start and finish". And especially if you teach, programming will be even more important for the next generation (assuming we survive at all lol), so I definitely recommend to use this time to learn more about programming. Maybe if you tell a bit more about what kind of things are you interested in, reddit can give more specific advice on interesting stuff to learn about. | Oh, I kind of misunderstood your situation from your description a bit. However, I believe while a month it's not enough to dive deep, it's certainly a good opportunity to start! Programming in general is not a thing you "start and finish". And especially if you teach, programming will be even more important for the next generation (assuming we survive at all lol), so I definitely recommend to use this time to learn more about programming. Maybe if you tell a bit more about what kind of things are you interested in, reddit can give more specific advice on interesting stuff to learn about. |
[
"Question about actions on closed surfaces"
] | [
"math"
] | [
"o70zfq"
] | [
5
] | [
""
] | [
true
] | [
false
] | [
1
] | What finite groups can act freely on the surface of genus g? | So here are some relevant theorems: All actions are assumed faithful. First, the Nielsen realization problem shows that every finite subgroup of the mapping class group of S_g (for g at least 2) can be realized as an isometry group of a hyperbolic metric on S_g. In other words, even if you're just asking about (orientation-preserving) actions by homeomorphisms, you can actually get actions by diffeomorphisms, isometries, conformal maps on Riemann surfaces, etc. So the actions are as nice as can be. Second, by Hurwitz's theorem , any such finite group acting on S_g has order at most 84(g-1). Third, for finite group G, there is a genus g such that G acts on S_g. This presents a problem for you. What could a "list of finite groups" look like if it could in principle contain any finite group? How would you want that data presented and how would you "recognize" a group? Having said all that, here's a MO page that gives some answers. It's possible to describe a computable algorithm that will calculate all automorphism groups for genus g, and for small genus, has been carried out. The data can be stored in GAP SmallGroup names, or in general, presentations for the groups can also be given. | Thanks! And for the usual torus? I know about those that if I quotient by I get another torus (those preserve orientation) and those that if quotient by I get the Klein B. Are there more? | for every finite group G, there is a genus g such that G acts on S_g Do you mean acts *faithfully* on S_g by homeomorphisms? Or ...? | Yes, all actions here are assumed to be faithful, or else this is a very boring question. | Right, for sure. But it shouldn't be a random restriction that the reader has to guess. It would be very interesting if you can make any finite group act faithfully irreducibly on a surface! |
[
"Career and Education Questions: June 24, 2021"
] | [
"math"
] | [
"o73tyv"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
1
] | 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’ve thought of reading the textbook thoroughly but that would be a major time Reading the textbook (or your lecture notes) thoroughly is about the bare minimum you can do to study in higher level math classes. If you can't make time to do it, take less classes at once. Math is really hard; don't rush it. | It's definitely a good idea to reach out to someone before applying. Your chance of getting noticed and having your application stand on its own two feet are much higher if there is a person in the department looking out for you. When I was applying for PhDs I wrote a short report (1-2 pages) about what I'd learned/worked on that I attached to an email I sent to each person I thought I could work with. I didn't ask them about their work. To be honest I don't think asking professors for personalised responses explaining their work just for the sake of a PhD application will serve you much better than just expressing interest and demonstrating that you have the background and commitment to work with them. If you've got good grades and have demonstrated a specific interest to specific people in the department, they'll make sure your application gets properly considered and you will be in with a good shot. In many cases even if I didn't get in/decided not to go with an offer, I had good communication with the people I contacted at different departments, so it was clear they were tracking/paying attention to my application. I was applying in Europe though so YMMV. | If you haven't taken introduction to analysis or algebra, you've missed probably the introduction to what math majors would beyond variations on computational calculus--have you written proofs before? If not, I'd be very careful before enrolling in a masters program. | When you get asked "what can you do with a math degree other than academia?" by family or friends, what is your favorite way to respond? I generally have bland responses that tend to go something like "well, if I pick up some programming, I can go into tech", so sometimes I wish I knew of some cool "flashy" examples that would satisfy such people. It's a little difficult for me to explain the value in studying mathematics that don't have an immediate application to those that haven't had much exposure to math in a long long time. | Hi, I'm a 2nd year (going into 3rd year) maths BSc student at a russell group uni in the UK. I'm considering applying for maths masters, preferably in Europe but outside the UK and am looking for some advice. My favourite area of maths so far is probability theory, and this is the key area I'd like to further pursue. So in my final year I'll be taking units like complex networks and martingale theory. My first question is are there any areas of pure maths I should make sure I have? Specific examples would be Metric spaces which I avoided last year but now think might be worth taking. Other units I'm considering are dynamical systems / functional analysis in order to build up the analysis side of pure maths which I believe is important to probability reseach. My other questions is about universities themselves. I'm keen to explore Europe outside the UK but am struggling to determine the relative strengths of maths departments, in particular their strengths in probabiltiy research and so I was hoping someone could shed some light on this. One particular course I've noticed is the mathematics masters at the University of Oslo - does anyone have any experience with this course/university? If anyone could help with any of these questions I would be most appreciative. |
[
"Nontrival digital root of balanced ternary numbers"
] | [
"math"
] | [
"o6md76"
] | [
8
] | [
""
] | [
true
] | [
false
] | [
0.79
] | Hey guys, I decided to apply to numbers to see if anything interesting happens. Here are the first 10 values: The digital root is 0 for all even numbers. The interesting one here is when we get -1. Starting from 0, here are the first few list of numbers that get a digital root of -1: 5, 15, 17, 23, 41, 43, 45, 47, 51, 53, 59, 69, 71, 77, 95, 97, 121 Taking the difference of each of these, I get: 10, 2, 6, 18, 2, 2, 2, 4, 2, 6, 10, 2, 6, 18, 2, 24 I couldn't find an obvious pattern to these, and thought it was interesting, decided to share with you guys. | That is a digital sum. Digital root involves adding the numbers until you reach 1 digit. In balanced ternary, the available digits are T (-1), 0, and 1. So when we get 2 (1T), we continue to add those together, 1+T=0. | A134452 : Balanced ternary digital root of n. 0,1,0,1,0,-1,0,1,0,1,0,1,0,1,0,-1,0,-1,0,1,0,1,0,-1,0,1,0,1,0,1,0,1,... I am OEISbot. I was programmed by /u/mscroggs . How I work . You can test me and suggest new features at /r/TestingOEISbot/ . | Thank you! Ediit: here are the numbers for -1: https://oeis.org/A134453 | This is OEIS sequence A134452 for reference. | Ohh this is like the second time I've seen balanced ternary numbers. I didn't even process that the digital sum is in balanced ternary my bad |
[
"Research Paper Help?"
] | [
"math"
] | [
"o6mva2"
] | [
11
] | [
""
] | [
true
] | [
false
] | [
0.84
] | Hi all, I have been working on a research paper regarding multivariable calculus applied to ellipses and rational functions. I'm at a point now where I could use some guidance from someone who has more experience in the field than I do, but am unsure where to go. I have tried my university, but many of the professors I have tried aren't interested or don't have the time. Does anyone know of organizations or resources I could turn to in order to get some help? Thanks! | Is this a real, original research paper with intent to publish, or more of an undergraduate project/thesis? It is strange to hear about research happening without faculty guidance, usually the process is reversed: a faculty member agrees to take a student, then gives them a research problem to work on. If you just want to bounce ideas off of someone, you can post things here, on math.SE/mathoverflow, or maybe even like AoPS, but if you want to publish anything you should really try harder to get faculty from your university to mentor you. | If you're an undergraduate and doing this on your own, it's very likely that the reason that professors won't help you is because what you're writing about has already been done. It's great that you're taking an independent interest in math and I'd be willing to take a look at your paper to give you some outside feedback, but keep in mind that the research that an undergraduate in math has the background to do is very unlikely to be new. | It’s more something that I’ve been working on as a side project. It’s by no means an official research paper. I was just hoping to get some feedback on it to see where it goes. | That’s mostly what I figured. I’m okay with it not being new, it’s been a fun problem that I have been solving and I’ve been pursuing it because of the joy of solving. I can PM you the draft! | Why don’t you post it here? |
[
"Math programs not supported on MacOS?"
] | [
"math"
] | [
"o6weum"
] | [
74
] | [
""
] | [
true
] | [
false
] | [
0.89
] | I hope this is the right place to ask. I am looking forward to buying a Mac, which will mainly be used for university. Are there any programs that a math student needs that are not supported on MacOS? Mathematica, Spyder, MatLab.. all seem to be supported. TIA | As others have said, (almost) every program should be supported, but one thing you might want to check first is whether the version available via your university is compatible. For example, my university offers Mathematica 8 to students and faculty, but only has a 32-bit version for MacOS. Now MacOS doesn't allow you to run 32-bit applications anymore, so I can't use Mathematica unless I buy it myself. | AIMMS officially only supports Windows, but I suppose you could run it in Wine (I didn't try though, I'm on a dual boot machine with Linux and Windows, so I switch to Windows whenever I need AIMMS). | As far as I know everything should work fine. At my math department the IT for some reason chose to only do support for Mac. So most people have an Apple Laptop or Mac, i.e. it can't be a big problem. Edit: I chose a Dell Powerstation for myself as the raw computation power was much cheaper compared to bying an apple product. Keep this in mind if you work on computationally heavy problems. | Mathematica is closed-source, so I don't think you can compile it yourself unless you work for Wolfram. Unless they're started open-sourcing old versions? | I've never heard of AIMMS being used in a university program. |
[
"I discovered a pattern, e.g. 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2. There are more. Are there infinitely many? Why does it arise? How does it relate to triangles?"
] | [
"math"
] | [
"o72g41"
] | [
792
] | [
""
] | [
true
] | [
false
] | [
0.97
] | Preface with I am not a mathematician. I set out to find if there are bigger objects like the 3,4,5 triangle just out of curiosity on a coffee break. I found: 10 + 11 + 12 = 13 + 14 21 + 22 + 23 + 24 = 25 + 26 + 27 I set out to find more... then found: 36 + 37 + 38 + 39 + 40 = 41 + 42 + 43 + 44 55 to 60 = 61 to 65 Does anyone have any insight into what I found? I don't see any good reason why there should exist consecutive squares whose sum is the next set (less one object) of consecutive squares. Bonus: I'm almost certain that to someone (or maybe most of you) on here this will be trivial and obvious, but for me randomly stumbling on this it feels personally like I've discovered the atom or something. | You are giving examples of n and n+1 consecutive squares with equal sums, where all the terms on both sides are consecutive squares The smallest term being squared in the formulas for n = 1, 2, 3, 4, 5 is 3, 10, 21, 36, 55. A polynomial pattern for that smallest term can be found using discrete differences: the first discrete difference is 7, 11, 15, 19 and the second discrete difference is 4, 4, 4, which the original list is a quadratic polynomial in n and it's not hard then to work out the coefficients for the pattern to be an + bn + c: it is 2n + n. That is not a proof of anything, since it's based on finitely many data points, but once we find such a rule in examples it's not hard to check it works in general: for every positive integer n, show x + (x+1) + ... + (x + n) = (x + n + 1) + ... + (x + 2n) when x = 2n + n. Without knowing a solution, by careful algebra the equation x + (x+1) + ... + (x + n) = (x + n + 1) + ... + (x + 2n) can be simplified to x - 2n x - (2n + n ) = 0, which is quadratic in x with roots x = 2n + n and x = -n. So for each n there are exactly two solutions x, but one of them turns the original equation into the silly equation n + (n-1) + ... + 0 = 1 + ... + n . Doing a web search, papers on this topic can be found going back to the 1960s: see https://www.tandfonline.com/doi/abs/10.1080/0025570X.1962.11975326 (Mathematics Magazine 35, 1962) and https://www.tandfonline.com/doi/abs/10.1080/00029890.1964.11992315 (American Mathematical Monthly 71, 1964). These quadratic equations are closely related to Pell's equation. When you make the exponents greater than 2, then the situation changes: Pell-type equations have lots of integral solutions if they have one at all, but when the exponent is above 2 then such equations typically have no integral solutions or just finitely many integral solutions. For example, x + (x+1) = (x+2) has no solution in integers since expanding both sides and simplifying converts the equation into x - 3x - 9x - 7 = 0, which has no solution in integers. | That gives me lots of jumping off points to look further into it. Thanks! | That math moment when you think you discovered something, but find papers from the 60s | Let me rearrange your pattern a bit, e.g. [; (41^2 + 42^2 + 43^2 + 44^2) - (39^2 + 38^2 + 37^2 + 36^2) = 40^2 ;] . We have, in general, the identity [; (n+k)^2 - (n-k)^2 = 4nk ;] . Summing over [; 1\leq k \leq l ;] gives us [; \sum_{k=1}^l (n+k)^2 - \sum_{k=1}^l (n-k)^2 = 2nl(l+1);] . You're asking about the situation where this equals [; n^2 ;] , in other words [; n = 2l(l+1) ;] So we can find such an identity exactly when n is 4 times a triangular number. In particular, there are infinitely many solutions. | Those damn busy people from the 60s who didn't have reddit to distract them! |
[
"Is the terminology \"Hawai'ian earring\" problematic?"
] | [
"math"
] | [
"o6g0bu"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.4
] | To those unfamiliar, there is a certain shape referred to as the Hawai'ian earring, which is made of infinitely many circles with lengths that decrease to zero which have a common point of tangency. This shape is an important counter-example in topology. There was a recently a discussion on twitter, sparked by . In it, Jeremy makes the claim that the the space in question was named "Hawai'ian earring" because the space itself is exotic, and people also view Hawai'ians as exotic. If that claim is true, then the terminology Hawai'ian earring would be culturally appropriative, as it was named only because white people though Hawai'ian culture sounded fun. I find the claim dubious; I think that the shape is called a "Hawai'ian earring" because it resembles a multi-hoop earring, and someone thought such earrings were associated with Hawai'ian culture. However, I can find no evidence that multi-hoop earrings are Hawai'ian, so I ultimately agree that the name Hawai'ian earring is nonsensical at best, and offensive at worst. My questions to everyone else are, | Ask some Hawaiian people and see what they think. | If that claim is true, then the terminology Hawai'ian earring would be culturally appropriative, as it was named only because white people though Hawai'ian culture sounded fun. That's not how you define cultural appropriation. "Exotic" is not facet of a culture, it's an external aesthetic judgement. This conversation feels pretty silly. Just invoking the name of another group or culture doesn't automatically make a reference faux pas. What is someone defines a space as the "American Garage Band T-Shirt" space? Is that going to set off alarm bells? | And not just one Hawaiian person, because no one Hawaiian person speaks for everyone. With that said, I don't see anything wrong with "bouquet of circles." I think that's what it was called when I took topology. | A bouquet of circles is a wedge sum of circles, which is not the same thing. If you take a countably infinite wedge sum, it's still a different space from the "Hawaiian earring". (For example one is compact and the other is not.) | Not sure if this is true since Polish spaces are a thing. |
[
"Opinions on Axler? Should determinants be deferred when learning/teaching linear algebra?"
] | [
"math"
] | [
"o6gtxb"
] | [
390
] | [
""
] | [
true
] | [
false
] | [
0.98
] | Sheldon Axler's book, , banishes determinants from most of the exposition. Axler also makes the case for this approach in his article . Ultimately, I think I need to read the book (or at least the article) to judge the approach, but before I invest the time, I am curious what thinks. The introduction to Down with Determinants! isn't super convincing to me. Here are some of Axler's main arguments and my initial reactions: Thoughts? If you've read Axler, what are some good things about it? | I think Axler’s fear is that students will gravitate towards some easy mindless determinant-based criteria and unintentionally avoid the hard work of developing good mental models for notions that are deeper than finite dimensional linear algebra. Basically I think he’s afraid of students treating a proof based class like computation based calculus. For example, take invertibility. I’ve run into students who thought invertibility means that the determinant is nonzero. While these are equivalent statements, it’s a bit misleading: in infinite dimensions, invertibility is still an important thing but the determinant is no longer viable. And given that many of the students in math, engineering, or physics WILL run into Hilbert spaces a ton, problems can arise from this myopic determinant-only view. I don’t think I agree with Axler though. Determinants, especially if a teacher takes the time to unpack their interpretation for volume scaling, are quite useful, intuitive, and fascinating. Any object with rich geometric structure like that is bound to be worth the time of properly learning it. Personally, I like it when linear algebra classes provide multiple proofs of the same theorem using various flavors of proof techniques—really clarifies the core ideas. | Axler is interested in functional analysis (infinite-dimensional linear algebra, with heavy doses of topology), and in that setting determinants do not play a significant role. You need to find new methods to prove theorems in functional analysis that are handled with determinants in ordinary linear algebra, and completely new phenomena can appear too (continuous spectrum of a linear map). I think Axler tries to avoid determinants in part because they are not used in functional analysis. I think trying to minimize determinants is not a good idea since determinants show up in many places that use linear algebra and it is better to confront these topics than to shy away from them. For example, the matrix groups GL_n(R) and SL_n(R) for rings R (fields, Z, Z/mZ) are nice examples in algebra, at least for n = 2. I think it would unfortunate to leave a linear algebra class without having seen characteristic polynomials. | I put them off until we can think about where they come from. Otherwise students use them for everything, and it obfuscates the actual definitions I want them to grasp. | I agree with the volume scaling definition being very useful for intuitively justifying the determinant, but I don't know how useful it would be in a first year linear algebra class. I knew the volume scaling fact, but I didn't really internalize why it was true or useful till second year multivariable calc when we covered the change of variables formula and stuff. It's definitely useful intuition, but I don't know if a first year lin alg course has the tools to prove why it's actually true. | I don’t think determinants are bad necessarily, but I think sometimes people use determinants as a crutch in proofs when they really aren’t necessary. For example, proving “for square matrices A and B, AB is invertible iff A and B are invertible” using determinants is bad imo. The proof with determinants doesn’t generalize at all to infinite dimensions or to e.g. continuous linear maps between Banach spaces; it doesn’t generalize to non-square matrices (of course in both these cases, the original statement had to be modified to make it work); but most of all, determinants feel unnecessary or too “brute force”-y, too complicated for such a simple theorem. |
[
"Is it possible to learn Statistics without having applications constantly being crammed down your throat?"
] | [
"math"
] | [
"o701pi"
] | [
122
] | [
""
] | [
true
] | [
false
] | [
0.82
] | I'm the type who prefers studying math just for the fun of it, and I really don't like that every 2 slides in my teacher's presentation has some real-world application mentioned. I'm aware that statistics is very useful, but I feel like that usefulness has prevented me from ever really appreciating the subject for what it really is. | You might find studying more interesting. That is closely related to statistics, but it can be developed and explored as a branch of pure math more readily than statistics after you learn real analysis and measure theory so that you know the rigorous foundational material for the subject. Probability theory is concerned with the mathematical consequences of probabilistic models (limit theorems, etc) while statistics is concerned with determining what the right probabilistic model is for a given situation, and the latter point of view is closely tied to some intended application. Within pure math there are probabilistic models developed to describe some kind of random phenomena (like a model to describe the statistical behavior of prime numbers), but I don't think that is what mainstream statisticians are interested in. It sounds strange to want to study statistics without applications, like wanting to studying experimental physics without doing experiments. | I don’t understand some of the comments here. No, statistics is not just applications. You can spend your whole life doing theoretical statistics without ever working with actual datasets. So, the OP’s question makes total sense. Some universities even offer separate statistics courses for statistics majors and maths majors. The one for maths majors require a solid background in measure-theoretic probability, do not spend any or at least not much time on applications, and are often called “mathematical statistics”. Try looking for those courses taught by a maths prof and aimed at maths majors. For self study, I would also recommend the book by Schervish. It is a great textbook on mathematical statistics without much focus on applications. | Yes, look for classes that cover "measure theory" and "martingales". Typically these classes are advertised as grad-level probability classes. | Look up "Mathematical statistics." | Alright so allot of people are commenting on this post saying that you should either study probability theory or just accept the fact that statistics is focused on real world data. I have the complete opposite opinion on this. Statistics is a mathematically rich discipline that can be formulated and studied without any mention of real world data.Some might say that what I am talking about is mathematical statistics and not statistics. Alright, fine, but wouldn't it be better to mention mathematical statistics rather than just reject OP's question and say they need to study a different field of math?Anyways, perfectly abstract points of view apply to both Bayesian and frequentist approaches to statistics.Furthermore, I would summarize that statistics, in frequentist tradition, is concerned with rates of convergence for estimators that well approximate the theoretical nth moments given in probability theory. This seems an abstract enough of a definition to avoid the necessity of real world problems. Some might say that statistics doesn't exist without real world problems because there is no place to get the data. I would respond by saying that the data comes from probability distributions, which seems obvious to me. I think every course I have ever had in statistics usually follows with data from a PDF then see if the model works for real world problems. To answer the questions. Yes it is entirely possible and it is how I learned stats. Just look into mathematical statistics for a less applied look at stats. |
[
"Recommendation on where to read for math history?"
] | [
"math"
] | [
"xsc1mh"
] | [
10
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.82
] | null | I enjoyed Morris Kline’s trilogy: Mathematical Thought from Ancient to Modern Times, Vol. 1 https://a.co/d/5DmROd5 | 'Number' by Tobias Dantzig "This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of mathematical thought from the earliest times to the latest constructions is presented here with admirable consistency and originality and in a wonderfully lively style" -- Albert Einstein | Stillwell has a book on the history of maths which is well regarded. I don't know how comprehensive it is though. | by Burton is probably a good start | I will check it out, i would like something to not tell me that xxx person discovered this but i want also to know why they discovered it, what were they trying to do what problems were they trying to solve cause aside from interest i think math in general will stick in my head better this way if when i look at a theory or a formula and i just know its history |
[
"Quick Questions: June 23, 2021"
] | [
"math"
] | [
"o6frox"
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""
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1
] | This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread: Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example consider which subject your question is related to, or the things you already know or have tried. | Every cube is equal to 0, 1, or 8 mod 9. You can't get 5 mod 9 by adding three of these, so 5 is not the sum of three cubes. | You can do this in a number of ways. One simple way is to simply take the tensor product of sections of a vector bundle with differential forms, to get vector-valued differential forms . These come with the same grading as differential forms, but if you have an E-valued p-form and feed in p vector fields you get a section of E instead of a function. This is a very commonly used notion once you get beyond just the tangent bundle and start considering other vector bundles. Another thing you can do is just directly define the exterior powers of a vector bundle E. The vector bundles \Wedge E, \otimes E, Sym E, \Wedge E, etc. all make good sense, and for example sections of \Wedge E will be wedge products of sections of E. If you were to take the vector space of all sections of all exterior powers of a vector bundle E, you would get a grading based on the degree of those p-forms (really p-vectors in this case, p-forms would be sections of the exterior powers of the dual bundle). If you do this when E=TM or T*M you get back the standard construction of differential forms. You can do a similar thing for symmetric tensors and get a grading, although as you note there isn't quite a grading on general tensors (although there is obviously a bigrading into (p,q)-tensors). EDIT: In terms of turning these things into algebras, if you for example take the End(E)-valued forms then you can define a wedge product where you compose endomorphisms. If instead you have a Lie algebra bundle then you could take Lie brackets along with wedge product. You could also consider taking wedge product of sections and get a bi-graded algebra between differential forms and the exterior algebra of sections of a vector bundle for example. | You can define a complex structure by specifying which continuous functions are holomorphic. This is equivalent to using an atlas, so in some sense you are right. I don't personally view sheaves as way to put structure on a topological space, because I think there are many structures where sheaves are not the natural way of describing them (even if you can describe them with sheaves). For example whilst sheaves do encode smooth or complex structures, a Riemannian structure is much more elegantly encoded by a principal O(n) bundle, which is a similar but still different construction to a sheaf. I would say sheaves serve the following purposes: I like to think of sheaves as one of the many tools algebraic geometers invented to deal with the fact that they aren't differential geometers: they let you glue things together when you don't have partitions of unity, just like flat or smooth morphisms let you use the inverse function theorem even though the inverse function theorem doesn't exist in algebraic geometry. | is it just an alternative to "a)" and "(b"? Yes, that's exactly what it is. | Yes, I believe this notation is common in France. So [0, 1] means with endpoints and ]0, 1[ means without endpoints. Edit: apearantly it's due to Bourbaki https://hsm.stackexchange.com/q/142 |
[
"I really want to involve myself more in math"
] | [
"math"
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"xs8x6v"
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] | null | You said you’re in high school, so I’d suggest doing some summer math programs. My personal favorite is mathcamp , but there’s also mathily , SUMaC , PROMYS , HCSSIM and ross . Keep in mind these programs are very selective. My understanding is that mathcamp and SUMaC are generally the most selective of them all, but there is no “easy” one. | I’m in hs haha | I’m in hs haha | r/learnmath has stickied posts with good information. If you plan on studying math in college, proofs will become important. Here are a couple of links that introduce you to concepts you will use. https://math.berkeley.edu/~hutching/teach/proofs.pdf https://www.cs.sfu.ca/\~ggbaker/zju/math/proof.html | Unfortunately, your submission has been removed for the following reason(s): Career and Education Questions /r/mathematics /r/matheducation If you have any questions, please feel free to message the mods . Thank you! |
[
"Why divide by (n-1) for sample standard deviation?"
] | [
"math"
] | [
"xrvwde"
] | [
1
] | [
"Removed - ask in Quick Questions thread"
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false
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0.57
] | null | Why dividing by n gives you an underestimate: Standard deviation is a measure of objects’ distance from the mean. But you don’t know the actual mean, you just have an estimate. Your sample will be closer to the estimated mean than the actual mean, because the estimated mean is calculated from that same sample. Or at least that’s what one of my physics professors told me. Why the solution is specifically to divide by n-1 instead of n: because the algebra works out that way | By dividing by (n-1) we get unbiased estimate for variance. This is not true in general, it only holds if you assume the data follows a normal distribution. | By dividing by (n-1) we get unbiased estimate for variance. This is not true in general, it only holds if you assume the data follows a normal distribution. | Basically it is what it is. Lol | Nice! |
[
"I have never been very good at math but I enjoy patterns and I found a pattern involving multiples of 3."
] | [
"math"
] | [
"xrty7p"
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56
] | [
"Removed - ask in Quick Questions thread"
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true
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false
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0.82
] | null | Yes, this is a well-known divisibility rule. Good job discovering it on your own! | Yes, there are practical applications. We study these in modular arithmetic. Another one, albeit more commonly known, is for 9. The repeated sum of digits, also known as the digital root, of any multiple of 9 is 9. In fact, the digital root of any number is nearly congruent modulo 9. The term modulo X in basic terms means the remainder after division by X. Nearly, because sometimes the remainder is 0 which implies that the digital sum is 9 not 0. | "I'm a fan of numerology..." Oh. Oh dear. | "No matter what number you multiply by 3, you can add the sum of thenumbers together until you have 1 digit and you will always be left with3,6, or 9." But how do you know for sure it will work? Can you prove it? | Fun fact: you can construct a similar result like this for any number. For illustrative purposes, consider divisibility by 7. Calculate successive powers of 10, starting from 10^0, and compute their remainder mod 7, until you enter a repeating cycle: The cycle repeats here, giving us the infinite sequence (1,3,2,6,4,5,1,3,2,6,...). We now use this as a sequence of multipliers. To test for divisibility, take the digits of the number in reverse, and multiply by successive terms in the sequence. E.g., is 8694224 a multiple of 7? Reverse the digits and use the sequence: 4 2 2 4 9 6 8 x x x x x x x 1,3,2,6,4,5,1,3,2,6,... ---- 1x4 + 3x2 + 2x2 + 6x4 + 4x9 + 5x6 + 1x8 = 112 Repeat on 112, for 1x2 + 3x1 + 2x1 = 7 We've arrived at 7, so this means 8694224 is a multiple of 7. If you do the same trick with 7 instead of 3, the sequence is an infinite string of 1s. So there's no 'multiplier' for each digit. That's why adding works for multiples of 3. It also yields all the usual divisibility tricks: You can also replace 10 with any base, to turn this into a divisibility test for any number in any base. And if you're truly daring, for any given n and b, you can use this to construct a finite state machine that takes base-b representations of a number and accepts only if that number is divisible by n, which then proves by Kleene's theorem that you can represent divisibility by n in base b using a regular expression. |
[
"Term for the \"orderedness\" of a set/list?"
] | [
"math"
] | [
"xsewp8"
] | [
9
] | [
""
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true
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false
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0.85
] | I'm trying to find a mathematical term to described how well-ordered a list is (i.e. if a list is completely arranged in ascending order, maybe this value would = 1.0, and for a random list it would = 0.5 on average?). I couldn't find what I was looking for with Google searches, so I was wondering if anyone here would have any ideas? | Another possibility is counting the number of inversions ) in the list as a measure of how out-of-order it is. | Maybe the Pearson product-moment correlation coefficient (see wikipedia) would give you the sort of measure you want. It compares two sets of data. One of these would be the list you have and the other would be the standard list 1, 2, 3, ... n. | There's a field of machine learning that scores error of ranked lists. There's no perfect error metric but you can see a bunch here https://stats.stackexchange.com/questions/159657/metrics-for-evaluating-ranking-algorithms You can score a set with the equivalent ordered set with these metrics. | What is more ordered? If you have a list of 10 integers arranged in order 0-9 is this as ordered as those same numbers reversed, 9-0? Is 1,0,2-9 as ordered as 2,1,0,3-9? In this case, first 0 and 1 are swapped, in the second, numbers with a greater separation are swapped but it is still one swap. What about 0-4,9-5 you only have to reverse one range of numbers to get them in order? There's a lot to consider and it seems the metric chosen needs to be understood and right for the application. | For a relatively general (but likely computationally impractical) method, you can view the list as a permutation of the ordered list, i.e. an element 𝜋 of the symmetric group on however many elements your list has. Now, pick some set of generators 𝜋_i, i€I of the symmetric group that you want to consider as basic/not-too-bad errors and assign to each 𝜋_i some weight w_i (some non-negative real number, or, more generally, element of a fixed, linearly ordered monoid) and define the orderedness of the list (viewed as the permutation 𝜋) as the minimum of the w_(i0)+...+w_(ik) taken over all factorisations 𝜋=𝜋_(i0)*...*𝜋_(ik) (this can be computed using Dijkstra's algorithm or, if you have some nice set of generators 𝜋_i for which you can define a heuristic, something like A*). If you choose your 𝜋_i's to simply be the transpositions of the form (i(i+1)) swapping two adjacent elements, and assign each of them weight 1, then this should give something similar to /u/A_UPRIGHT_BASS 's answer, but there is, of course, nothing to stop you from choosing other generators or weights, depending on what your problem requires. |
[
"any interesting math related field trip destinations in Europe?"
] | [
"math"
] | [
"xs8b2k"
] | [
6
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] | My grade 13 class would like to go on a trip to a country where we can visit 'math-related' places. It has to be in europe and accessible via bus. Since it's my math class and also normal class we are encouraged to visit a place that has museums or other activities where we can learn more about math or math related topics. We'll be traveling from Germany so we'd prefer to go to a country besides Germany but it can't be too far. | International Bureau of Weights and Measures in France and Graph Theory with the rail system | Dublin, Ireland: the quaternions | There is some stuff dedicated to Euler in Basel, and some stuff dedicated to Einstein in Bern. | The seven bridges of Konigsberg are the first thing to spring to mind, but then I remembered Konigsberg is now Kaliningrad, so that's probably not a great idea. | Incidentally, some of the bridges have been destroyed in WW2, while others have been demolished. They've repaired some, but with the changes, an Eulerian path is now possible. |
[
"Bachelor thesis in numerical methods/mathematical modelling"
] | [
"math"
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"xrzhde"
] | [
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] | Hello, I am a year 4 student of mathematics. Next semester I am writing the final project to graduate. I have started thinking about the upcoming task. I wrote a letter to the professor of numerical methods. She agreed to take me in under her wing for the paper saying it could be about something from mathematical modelling problems/numerical methods/applications of math. We are meeting to talk upcoming week and to consider possible topics. We have a month to come up with a topic. I don't want to come in empty handed so I'm trying to come up with proper directions for the project. Though I am having quite some difficulty arriving at anything tangible. What are some resources to hack and slash through to find ideas for a BS paper in numerical methods and/or mathematical modelling? Or should I tune down my initiative and be guided by the professor more? To provide some background about the professor - her expertise is in numerical analysis, mathematical modeling and finite difference methods. Her publications are on finite difference schemes, Sturm-Liouville problem, elliptic equations and mathematical modelling of medical problems as far as I can see. She has been involved with the faculty of medicine. To provide some background about me - I enjoyed the numerical methods course, enjoying the mathematical modelling course currently. Ready to put in a lot of effort into this work. Recently I have kindled a newfound interest in data science. I would love for the project to be in or at least related to data science very much. I realize that perhaps I should go towards the statistics professors for this, but unfortunately in my courses, the ones I've encountered - they are not the ones I would like to go to. So is there a possibility of connecting numerical methods/mathematical modelling and data science somehow? What could be some examples? Or would it be like trying to juggle on a bicycle while on a rocking boat, as in too big of a bite for a simple bachelor student? Appreciate any response and input. Have a great one. | First, it's okay to approach the meeting in part as a brainstorming session. It's good to go into the meeting with ideas, but you don't need to go into the meeting with the answer. Your professor has a lot more experience and expertise, so take advantage of that. You professor will likely have ideas of a project thats appropriate for a Bachelor thesis. That being said, you have a say in the project. And you should work with the professor to find topic that interests both of you. If you want ideas for topics, my recommendation is to read/skim journal articles. You don't have to read them in depth, but start by reading the titles and abstracts. When something peaks your interest, give the paper a quick read to see if it's interests you. I'd also read some of the papers authored by the professor. | I did my bachelor's in biomathematics. Got the idea for my thesis (really just re-deriving and somewhat expanding a model I found in the Bulletin of Mathematical Biology. Skimming abstracts to find interesting stuff is a good habit! | This is a little tangential to your question: Because computers can only do finite applications of addition, subtraction, multiplication and division any numerical method eventually reduces to a problem in linear algebra expressed in code. Iterated application of a linear problem then frequently solves the problem. Conceptually then numerical analysis is the study of linear problems expressed with perturb values due to the deficiencies of computation finite precision number systems. From this point of view the core problem is understanding how the result of a perturb linear problem relates to the result of the unperturbed linear problem. A numerical method is called "stable" if the two results have some appropriate "closeness" relation (expressed in a variant of different ways). One of the most basic algorithms for solving a linear system is Gaussian elimination. We still don't know why solutions of perturb systems solved using Gaussian elimination are close to the solutions of the unperturbed system. Be warned this is an extremely hard open problem in numerical analysis... but a fascinating one never-the-less. See https://people.maths.ox.ac.uk/trefethen/nov12.pdf for a very readable and short discussion. I'm not suggesting that you should suggest this problem. I'm only trying to pique your interest. But... perhaps you are more interested in problems to which numerical methods can be applied? | It is understandable to me that many of the numerical method ideas boil down to the linearity you have mentioned. I have some fear I am not capable enough of studying and working on such theoretical problems, so applications seems to be a choice of comfort. But perhaps I could try to challenge myself and try to work through something like that? It's been very curious to me to see the theoretical background of certain numerical methods and why they work. Beautifully written. Thank you for your response and the link. I will update on if I would consider going a bit deeper from the side you have mentioned. | If you insist on Data Science, which if you squint your eyes enough is some ML model applied to a problem, then pick something like the least squares problem. Not the simple OLS that have exact solutions but ones without a closed formula, so you need some approximation tools. That's where the numerical method comes in. It's still a completely solved issue as far as the proven, regular solvers are concerned but for a BS thesis it could just be a fun project to explore. There are some slightly more complicated stuffs like SVT for Nuclear Norm Minimization or Iteratively ReWeighted Singular Value Minimization etc, for which you can just do a literature analysis type of thing for their numerical stabilities for example. Besides, I would not advise you to come to your first meeting as if you had already chosen your own project. She will know better and 99% chance you will change your mind after meeting her. Bachelor's thesis is 100% exploratory and it is for your own education. |
[
"Research topic suggestions for an advanced high schooler?"
] | [
"math"
] | [
"xrskx7"
] | [
3
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true
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0.59
] | Hey guys, I’m a 15 year old doing a science fair for my school but am struggling to come up with an “experimental question” to base my research on. I have a book about writing research math that has given me some basis on formatting and where to look for source material. Some ideas i’ve looked into include: -geometric interpretations of the Fibonacci Matrix -methods of solving non-linear diophantine equations (but what new findings can I expect to get?) -when is the derivative of a figure’s area equal to its perimeter? -goldbachs conjecture My background only includes high school mathematics, multivariable calculus, linear algebra, and Olympiad level competition math. I am also interested in kinematics and optical physics, but I’m not sure how I can apply my math background to create or model experiments in those subjects. Any help would be appreciated, thanks. | Well, I am no expert on science fairs, but I do know a huge number of projects aren't particularly novel. Maybe don't worry about doing something "new", and maybe focus on explaining something underappreciated. The Fibonacci matrix is intimately connected to the Stern-Brocot tree, which is a highly underappreciated topic. The Stern-Brocot tree is also very useful for Diophantine equations and Diophantine approximation. | You aren't going to uncover any exciting new math theorems, but you can find some fun stuff. Maybe look at continued fractions of the square roots of numbers, and then see if you can find a connection between that and the pattern of folds (in vs out) when you repeatedly fold a piece of paper. Or use continued fractions to find some startlingly good approximations of numbers (e.g. what's a good approximation to pi, that's better than 22/7 ?). | Always remember your audience, and pitch your expectations appropriately. In your case, you are preparing for a high school science fair. Your audience should be the other students who are participating in the fair. Not just the other students who have studied all of the math that you have studied. This also includes that student who is fascinated by ecology, but has taken the minimum level of mathematics that they could to get to this point. This is not the place to look for novel research -- there most likely will be very few people involved in this fair who would be able to understand or appreciate work at that level. Instead, find a concept that you find particularly enjoyable or interesting, and work on a way to present it so that others -- including all of the other students who participate in the fair -- can understand why you found it interesting. | What is your goal for this project? If you want to do original research, as in prove things that have never been found before, I am not sure any of your suggested topics would be good; the unfortunate truth is that all of the math you know is very very old, so it is very very hard to find new results there. If your goal is smaller, then any of those could make fun projects. | Thanks so much for the suggestion! I’ll take a look at the Stern Brocot tree, I never realized it has connections with the Fibonacci sequence. |
[
"What is the most interesting or strange formula or identity you know of?"
] | [
"math"
] | [
"xs3t9k"
] | [
147
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] | Though I've recently graduated with a Bachelor's degree in pure math, I remember that one of the primary things that got me interested in pure math as a high schooler was being exposed to some of the strange and interesting formulas that populate many areas of pure math (It was the that really got me.) Now, I'm still interested in learning about obscure identities, and would like to see more of them (so I can add them to my collection). Also, when someone asks me why I enjoy mathematics, or why it interests me, I would like to have something accessible and cool to show them. Therefore, I'm looking for interesting and strange identities that relate two values, functions, or perhaps other objects that you would not on the surface expect to be related. The more obscure the formula, the better (The fact that ζ(2)=π /6 and e =-1 is cool and all, but everyone knows about these.). Though I'm more so interested in identities that are easy to understand without a huge amount of explanation or background knowledge, if you have something that requires high level knowledge in a specific field, please do post it. Here are some examples of what I'm looking for: This that can be used to calculate the greatest common factor of two numbers. This which evaluates the Euler totient function as the sum of weighted cosines (it’s the third formula). This for the number of ways to tile an m×n rectangle with 2×1 dominos. This that relates π to the expected value of a random infinite nested radical. Anyways, what are the most interesting or strange formulas or identities you know of? Edit: The experiment is over. Thanks for participating. | If A, B and C are the angles of a triangle then tan(A) + tan(B) + tan(C) = tan(A)tan(B)tan(C) | My favorite odd formula is one David Mumford chose for an exhibit on the favorite formula of some distinguished mathematicians. https://home.dartmouth.edu/sites/home/files/styles/max_width_720px/public/news/mumford-590.jpg?itok=4YmrNHeG Mumford’s explanation is here: “My contribution was a startling identity that arose studying moduli space, most peculiar in having the number 13 appear in it. As I said in the accompanying blurb, the only numbers bigger than 2 that are likely to appear in a math article are usually page numbers. This one has also the merit that it has been used by string theorists.” | Draw a secant line through a parabola. Next, draw the tangent line that is parallel to the secant line. Next, make a triangle with the 2 intersections of the secant line and the tangent point. The area of the parabola enclosed by the secant line is 4/3 the area of the triangle! | Though this isn't too surprising, I do really like the symmetry. Nice example. | No, sadly. It's tan(A) + tan(B) + tan(C) + tan(D) = tan(A)tan(B)tan(C) + tan(B)tan(C)tan(D) + tan(C)tan(D)tan(A) + tan(D)tan(A)tan(B) |
[
"Venting?"
] | [
"math"
] | [
"kbyt8v"
] | [
18
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.91
] | null | The score on the Putnam is an irrelevant measure of mathematical skill. It is only a measure of how well you can answer timed math problems, which is not what real math is about. Therefore if you want to vent, your score on the Putnam is a bad example to illustrate anything about career goals in math or physics. | According to Wikipedia, the median score in the Putnam test is often 0. It doesn't feel fair to say you bombed a test that hard. | I have an honest question. When you say that nothing has ever “clicked” even back to calculus what exactly do you mean? I’m confused how you could have made it all the way through an undergraduate math major if you literally understood nothing even back to calculus. Your post gives me the impression that you are selling yourself a little short and perhaps expecting too much out of studying math. I mean, I am confused when studying math and I’ve been studying it for a decade now. | There's something about that word understand that's actually really quite nebulous. What does it really mean to "understand" something? I think it's actually quite common to reach a point where you think you "understand" something only to find later that your perceived "understanding" was flawed somehow. In other words, I don't think your experience is particularly atypical. I think the productive way to address this is to identify areas where you think your understanding is deficient and then seek to address those deficiencies/fill-in those gaps. Try to turn nebulous feelings like "I don't understand maths" into more specific questions like "I don't understand why, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides?" To some extent, you're not helped by the school system of teaching maths. A lot of the reasoning as to why things are the way they are has been stripped out of the curriculum, or never really put in in the first place. There's a series of books by Professor Hung-Hsi Wu: where he takes frequent aim at the mathematical quality of the (American but his criticisms are not really limited to the US) school curriculum from a mathematical perspective and seeks to try and remedy the deficiencies from a (research) mathematician's perspective. You might find those books helpful and some "chicken soup for your tortured soul." Warning - the books are written for a particular audience and they specifically aren't meant to be textbooks per-se so you may find them a somewhat different experience to what you are used to. For a long while, I wondered how the situation Professor Wu describes came to pass and why the improvements he talks of, which make a lot of sense to me, hadn't already been made. It wasn't until I was looking at the last book, that a paper he references: A revolution in mathematics? What really happened a century ago and why it matters today, Frank Quinn. really helped me to understand what was going on. I also found some of the stuff referenced from that paper helpful to my understanding of what was going on (this is "pre-print"): Contributions to a Science of Contemporary Mathematics , Frank Quinn. There's a good article on the "Foundational Crisis in Mathematics" in the Princeton Companion to Mathematics: The Crisis in the Foundations of Mathematics , by Jose Ferreiros, which provides some more background to what happened and why contemporary mathematical practice diverged from school practice. | I took it, I didn't do well (but did make an honest effort to think about the problems while taking it), and I really didn't care about the bad score since I already knew that (i) the Putnam is ultimately pointless and (ii) the average grade is 0 anyway. Have you considered pursuing a career in the technology field instead of law, if you want to do something more mathematical? I know two people who got undergraduate degrees in math and now design programming for flight software for planes or rockets. One of them went to law school and practiced law for several years before going to graduate school in computer science. |
[
"Surface Area of a Planet: A New, Non-Arbitrary Definition"
] | [
"math"
] | [
"xs0rp1"
] | [
421
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""
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true
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0.96
] | Since the surface of a planet is fractal-like, it doesn't make sense to measure its surface area directly. Instead, scientists approximate the shape of a planet using an ellipsoid, also known as a . By measuring surface area on a datum, we can represent planetary surface area (for instance, the area of the United States) in a meaningful way. Despite their usefulness, datums are still arbitrary to an extent, as planets are not perfect ellipsoids. As such, many datums with slightly varying parameters have been proposed over the years, resulting in minor disagreements in areal measurements depending on the datum used. However, a much greater problem arises on asteroids, many of which are not even remotely ellipsoidal in shape. In such cases, datum approximations are inherently crude. What if, instead, we create a definition of surface area that is based not on a datum, but rather, on the actual physical properties of a planet? Introducing , a new definition of planetary surface area that doesn't rely on a datum, and is instead based purely on the actual planetary surface and gravity. Datumless surface area provides a universal standard of measurement that can be applied to any terrestrial planet, moon, or asteroid. Conveniently, on fairly round planets such as Earth and Mars, datumless surface area yields very similar values as surface area on an ellipsoid. So how does it work? Before defining surface area, let's first define the . Let the planetary surface consist of all points on a planet can be directly hit by falling raindrops, colloquially speaking. (In more technical terms, these raindrops are infinitesimally small and fall along the , or curves that are tangent to gravity along their lengths.) As shown in , the planetary surface assumes the form of many disconnected surfaces. Calculating the area of the planetary surface directly doesn't make sense due to its fractal-like nature. However, consider another kind of surface: the , or surfaces of constant gravitational potential surrounding a planet, to which gravity and the plumblines are perpendicular to. Unlike the planetary surface, the equipotential surfaces have a defined surface area, as they are smooth. One may think of choosing a particular equipotential surface to represent the surface area of a planet. However, this approach is still arbitrary, as the choice of equipotential surface remains open-ended. The non-arbitrary solution has a slight twist: instead of choosing a particular equipotential surface to represent surface area, let's add up bits and pieces of different equipotential surfaces that intersect the planetary surface! More formally, the is the sum of the surface areas of infinitesimally small portions of equipotential surfaces that intersect the planetary surface. Here's a This diagram approximates datumless surface area in a similar manner as a Reimann sum in calculus. Since the equipotential surfaces are actually infinitesimally close to each other, datumless surface area is equal to the limit of the quantity depicted in the diagram as the distance between equipotential surfaces approaches 0. Alternatively, datumless surface area is simply equal to the of unit vectors pointing in the direction of gravity through the planetary surface. Altogether, datumless surface area provides a physically meaningful, non-arbitrary definition of surface area that can be directly applied to any planet or asteroid, even the most . To learn more about datumless surface area, along with other datumless topographic measures that may be of interest, I invite you to check out my . It also introduces datumless formulations of path length, as well as mean value / other statistics of a numerical surface attribute (such as surface temperature, elevation, albedo, etc.) within a region. Please put all your questions in the comments! I'm happy to answer them. | One downside of this, which you may be aware of, is that even for very simple surfaces the datumless surface area will differ from the true surface area. For example, the datumless surface area will not give the true surface area for a cube. Perhaps an easy thing to see would be to consider a perfect sphere and then take a small circle on the surface and replace it with a hemispherical cap. That replacement should double the true surface area contributed from that small circular region on the surface, but the contribution to the datumless surface area is unchanged, up to first order. If you’re aware of this already that’s fine, datumless surface area will just always be an underestimate to true surface area, with larger discrepancy wherever the landscape is tilted. If you weren’t aware of this and wonder why this is the case, look up the pi = 4 paradox to see an example of this in two dimensions. The point is that even if one curve or surface limits to another, their lengths or surface areas don’t limit to that of the other unless their first derivatives limit nicely as well. | This isn’t an issue of fractals having an area, it’s an issue of fractals (usually) not having perimeter, but ramped up 1 dimension higher. This is just the coastline paradox. | Ah, that's a good point to clarify! When we talk about the surface area of land on a planet, we usually mean surface area in a normal to gravity context, also known as , as that tells us how much horizontal land we have to build on. If you wanted to build a house, a 1000 m^2 plot of horizontal land would be far more useful than a 1000 m^2 cliff face. In the case of the cube, the points of the cube would be considered mountains. Mountainous land on Earth doesn't add to a greater horizontal surface area (i.e., we wouldn't say Switzerland is bigger because it has more mountains). The π = 4 paradox is real interesting, thank you for introducing me to it!! | The fractals that have areas generally don't have a well defined perimeter. In that sense, I'm assuming that while the surface of planets are fractcal-like, they might have a well-defined volume, but not a well-defined surface area. | Interesting concept. I don’t see where equipotential surfaces are important. As I understand, you only count the outermost surface along each plumbline. That can be done without defining equipotential surfaces. The main issues I can see are: This definition is susceptible to the fractal surface problem. As your data resolution increases, so does the surface area. As a thought experiment, imagine the surface is a sine wave and then gradually increase the frequency of that sine wave. The calculation of plumb lines requires a simulation of gravity. This could be particularly problematic for asteroids, which could have a dense metal core on one end but a low density ice spike on the other. Thus, changing the center of mass for an object changes the plumb lines and therefore the surface area. |
[
"Formula for: The lower the input, the higher the output."
] | [
"math"
] | [
"kbxlbl"
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1
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""
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true
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false
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0.56
] | null | S = 6/X To add, if you want it less extreme, do something like S = 16/(X+10) | S = 6 / X | Okay, so I don't develop games, nor am I an expert in maths by any stretch of the word but have been looking at this post for a while now and have decided that I may aswell say something. Firstly, I understand what you are aiming for with the formula but would it not be easier to have a set speed for each of the characters that you as the Dev decide on? I might not be understanding something though. Secondly would it not be easier if you are looking for a formula to do it backwards where 1 person is set at 100% speed and it lowers as more people join? | Have you considered using a Gaussian distribution? (Or some other distribution) By varying the standard deviation it would be much easier to fine tune gameplay too | Why do you need a formula? Just test values for every number of players. |
[
"Looking for suggestions"
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"math"
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"kbwm5z"
] | [
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 | You could try sanity checks at intermediate steps so you catch things earlier. Mathematica or something like it might help with that. But the big thing is not to take the mistakes too seriously I think. Google Grothendieck's prime for some fun. | Sanity checks are important. For example, I recently had an example with my precalculus students where I messed up the algebra on a step and I had something like x(tan 1.2 - tan 1.5 ) = 20 and I knew that x was supposed to be a length. I stopped and pointed out to them that I had to have made a mistake because tan is an increasing function from 0 to 90 degrees (or if in radians from 0 to pi/2) so the quantity in the parentheses had to be negative. But then x wouldn't be positive. I don't know what level of math you are doing, but it can help to repeatedly look for warning signs that one has gone astray. If one doesn't see them, then proceed. | Stop skipping steps Write all your steps neatly and line by line and use as much space as you need (trying to save paper on your math homework is dumb) | A calculator with a good CAS can go a long way. A TI Nspire CAS CX or an HP Prime G2 | I don't know how much this will help. I'm just thinking out loud. Once, I was watching an NFL game with my roommate at the time. The quarterback threw the ball to the receiver. The receiver was all alone and it was a good throw. It should have been an easy catch. But the receiver mishandled it and lost control of the ball. My roommate looked at me and said "Do you know what he was doing wrong there? He was thinking too far ahead. He got excited and got ahead of himself. He was thinking 'Oh boy, I'm going to catch this ball and probably get to run it into the end zone' but he was trying to skip steps. He was thinking too much about the *next* thing to do, instead of doing the one thing he should be doing right now." So, it might help to remember not to outsmart yourself or get ahead of yourself. Live in the moment more. Do the one thing you're supposed to be doing right now, instead of thinking ahead to the next thing. |
[
"Someone explain."
] | [
"math"
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"kbtgoh"
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0
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""
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true
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false
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0.38
] | null | if you were to write pi out but without the decimal, you'd end up with a number just as long, and exactly the same as pi, only it wood be a whole number. Not true. There are infinite digits of pi so it’s not possible. | I think you are making the classic mistake of confusing "unbounded" and "infinite". A whole number can have an unbounded number of digits, but it can't have an infinite number of digits. | Let's take your objection to the statement that the decimal expansion of pi without the decimal point is a whole number. In other words you claim that 3141592653589... is a whole number. If you say this is a whole number, then there must be a number preceding it. What is it? Do you see the fault in your reasoning? To your second objection. Cantor's proof doesn't work for whole numbers. Here's how the (false) proof would go: Assume the set of whole numbers is countable. Then it is possible to write down all whole numbers in an infinitely long list. Now construct a new whole number whose n-th digit is different from the n-th digit of the n-th whole number in our list. The reason this doesn't produce a new whole number is because you are going create an infinite sequence of digits by following the construction above, which is not a whole number. EDIT: In fact you write the whole numbers in a list. Here's how it begins 1 2 3 4 5 6 7 8 9 ... | Integer numbers can have an arbitrary lenght, but is always a lenght, so you cannot turn pi or any other irrational number into an integer by removing the decimal point. | In Cantor's diagonal argument the point of adding digits to the right of the number is that there is an easy way of checking that the number we are constructing is indeed different from every number in the list. But even if you could add new digits left and right you would still get an infinite sequence of digits. In fact this time the sequence of numbers would extend infinitely far possibly in both directions. Also, I dont see the flaw in my reasoning for making pi a whole number. Your reasoning states that there is a problem with the normal pi as there is no number proceeding it. I said that if 3141592653589... is a , then some number should precede it. 3,141592653589... is not a whole number so we do not expect there to be a number preceding it. But the issue we should be focusing on is what a whole number is. This might sound like a silly question but it turns out that it is not at all that easy to give a completely rigorous answer. If you're interested in that, then check out the Peano axioms . If we have a notion of counting, then intuitively we can say that a whole number is one that can be reached by counting from 1. Notice that this definition doesn't say anything about digits. In fact we need digits only in order to represent whole numbers (or real numbers) but not in order to define them. However it follows from the definition (for example by induction) that every whole number must have finitely many digits in its decimal representation. |
[
"Book suggestions for higher order derivatives"
] | [
"math"
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"kbqkkl"
] | [
3
] | [
"Removed - post in the Simple Questions thread"
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true
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false
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0.62
] | null | These things won't be in a book about higher derivatives. These are more application/context based than anything. So for your example you'd probably find information about that in a statistics text | You just take the first derivative of the first derivative, and so on. The second derivative measures the convexity of a function, whether it curves up or curves down. | And the third derivative? I have read in my statistics book that it is used to test the smoothness of function but i don't know how that works. | All smooth means is that you can keep differentiating without getting to a function where the derivative isn't defined. | As far as I know, beyond the third derivative there's no natural interpretation of higher order derivatives. The third derivative is already quite rare. I did a quick search and found that it is sometimes interpreted as "instantaneous jerk", whatever that means. But I've never heard anything about notable uses for the fourth derivative and beyond, except for Taylor series. Higher order derivatives (when they exist) can be used to give bounds on the error in the approximation given by Taylor polynomials, a generalization of the tangent line at a point. If the function is infinitely differentiable you can usually represent it as an infinite series involving the derivatives of the function. But other than that I'm not aware of any widespread uses of higher order derivatives (except of course the second derivative), so I doubt you'll find any books that specifically deal with that. But If you just want to learn to work with derivatives in general, any calculus book will do. |
[
"Does Everyone Have To Be A Genius?"
] | [
"math"
] | [
"kbeqfr"
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0
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""
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true
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false
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0.38
] | null | This sub isn't into mocking others (* except for math crackpots). | DoD and DoD contractors hire software engineers and data scientists with lower GPAs pretty regularly. Will you be able to go from a 3.2 undergrad GPA to cryptography research at the NSA? Almost definitely not, but if you're just looking to go into industry directly, you should be more focused on getting internships at this point | Learning math really should be teaching you how to learn. The education can apply in almost any (semi technical) field. But you've gotta do something you enjoy if you want to excel at it. | I’ve had some work at the U.S Attorney’s office. But that was not technical. I’m trying to break into the government to pave myself a path | Don't worry. DoD would be glad to have you. |
[
"What's a great christmas present for someone with a PhD in Mathematics?"
] | [
"math"
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"kbg4ja"
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17
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""
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true
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false
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0.91
] | null | My advice is never give a gift about something the receiver is more knowledgeable/passionated about than you. The person is probably very opinionated and you could very probably pick the wrong gift. (It's less likely for something like math, but it can still happen.) | Alcohol | A job. | Was going to say coffee but this is probably better | I’m a mathematician, and I think most of the suggestions are great. |
[
"Approach to approximation of statistics of euclidean distance between two positions with a degree of measurement error."
] | [
"math"
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"kbkc8h"
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1
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""
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true
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false
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0.6
] | Lets say, hypothetically we have two points, position of which was measured with error. What would be the best approach to calculating the probability of their actual positions being within a specific range? The measurement errors aren't non-linear, can be position dependent, they are also not normally distributed. The measurement errors of each of the position states are independent of one another. I looked into statistics, but I could not actually find anything like this in the textbooks. I am open to a machine learning approach however. | I would think you need at least some assumption on what kind of measurement error can be generated... otherwise there will be nothing to calculate | you only have one measurement for each point? | It's a position measurement, so I guess 3 measurements, X, Y and Z directions. | ok, but only one of each per point? | for both points. |
[
"Why is it that almost 300 years after Euler found the solution to the Basel Problem, that we still don't have a closed form solution for Apéry's Constant?"
] | [
"math"
] | [
"kby44w"
] | [
14
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""
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true
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false
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0.78
] | Euler solved the Basel problem in 1734. The 300th anniversary will be in 2034. It seems that every method to arrive at the closed form of 𝛴 1/n fails for . Why is that? What makes the finding the exact sum of the reciprocal odd powers so difficult. | I want to put forward the idea (without necessarily claiming that it's correct) that you're asking the wrong question, and that the right question is "what makes finding the exact sum of the reciprocal of even numbers so easy?". That's a question that I presume we have an answer to, and you can extract it (and maybe even a nice short summary of it) from whatever argument we have that computes those values. Judging by this wikipedia article there's a relationship between the even case and the Bernoulli numbers. A sensible follow-up question would be "why doesn't this generalize to finding the exact sum of the reciprocal of odd numbers?", and I suspect there's a nice answer to that that someone can hopefully provide in this thread. But in general, I think questions like "why is this so hard" are coming from a place where most tasks of that form are presumed to be easy, and "hard" is the exception. In my experience, this is a bit naive, and in fact the world looks more like the opposite of this. "Most" tasks of this form are hard (often so hard that there is no elementary expression at all), and there are islands we've come upon (and some islands we haven't) where the task is easy for some reason. So "huh, why is this easy?" is often a more useful and informative question to ask, and leads you to more "meat", than asking "huh, why is this hard?". | Along these lines, construction of p-adic L-functions kinda tells us that it is the values of the Riemann Zeta Function which are important. The only reason that we know anything, then, about the positive values is because of the reflection formula. The equation is of the form The three functions on the right interact in different ways based on the parity of s when it is an integer. For each integer s>1, Gamma(1-s) has a pole but sin(3pi*s/2) is zero. The general formula for Z(2n) is then, basically, the value of Z(1-2n) - which we know - and then the interaction between the zero of sine and the pole of the gamma function (a L'Hopital's relationship, actually). That is the information content for the formulas for Z(2n). This is "why" the even values are easy. For Z(2n+1), sine is no longer zero, but the gamma function is still infinity and Z(-2n) is zero. The information content of a formula for Z(2n+1) would then not relate to the value of Z(-2n) but the value of the of Z(s) at s=-2n. This is very non-trivial information. | Have you at any methods to derive values with an even exponent to try to understand what goes wrong if you try to extract values for odd exponents in a similar way? If not, you really should. It would be more educational than having someone else explain anything about this to you first. For some methods to derive the formula for even exponents you eventually wind up with a formula like ax = b where x is the sum you want to compute and a and b are known positive numbers, so you can write x = b/a to express x in terms of the other numbers. But the same method, if used for odd exponents, results in such a formula where a = 0 and b = 0. The equation 0x = 0 is valid but doesn't let you solve for x. That is the practical problem. Conceptually, the value of the sum for an odd exponent bigger than 1 is not expected to be related in any reasonable way to known numbers: they are pi are collectively expected to be algebraically independent. And of course your question has been asked many times before. Did you try looking it up yet? For example, read this: https://math.stackexchange.com/questions/12815/riemann-zeta-function-at-odd-positive-integers . You could ask a similar question about why after 300 or so years we still can't prove that Euler's constant 𝛾 = .577.... is irrational. The truth is that, as Henry Cohn once wrote on an MO page ( https://mathoverflow.net/questions/129364/why-is-it-hard-to-prove-that-the-euler-mascheroni-constant-is-irrational ), all of these kinds of questions are hard by default and nice answers or solutions should not generally be expected. | Intuitive/high-level explanation: The natural numbers are not a group. The integers are. This means that sums over integers are much more approachable via algebraic methods, often leading to closed exact forms. This way one can show that the sum of 1/n over all non-zero integers has a nice closed form for integer k (for which it converges), odd or even. It just so happens that this implies a nice closed form for the sum over natural numbers when k is even, because (-n) = n . But for odd powers the sum cancels, so we can't extract information about the sum over natural numbers anymore. | And if you look closely at the factors that are 0 by using limits and residues, then you can get a formula for the zeta function at an odd integer n greater than 1, but the formula is in terms of the derivative of the zeta function at -(n-1), which is nothing nice at all, e.g., 𝜁(3) = -4𝜋𝜁'(-2). That is hardly a closed form expression, since it expresses one mysterious thing in terms of another mysterious thing. |
[
"What is Lattice Maths?"
] | [
"math"
] | [
"kblbr4"
] | [
2
] | [
""
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true
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false
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1
] | What is Lattice Maths I have been doing some maths with my sister and she has been doing work called lattice maths for longer forms of multiplication, I hadn’t seen it before so I thought I’d ask you guys. Is there anything wrong with it or some way it doesn’t work, just wondering why I hadn’t been taught it, thanks. | Are you perhaps thinking of lattice multiplication ? I'm going to assume "yes" for the rest of this answer. Is there anything wrong with it or some way it doesn’t work, just wondering why I hadn’t been taught it, thanks. No, not really. Curricula differ from school to school, so it's entirely possible that your own school taught you a different way to multiply numbers. Both are fine. Wikipedia gives a host of other ways to multiply. | I think they generally don’t teach wrong math that doesn’t work... if you didn’t get the chance to learn about it your classes and you’re curious, you could ask your sister nicely to show you how it works. | Yeah I have, it’s a really interesting thing, I’m curious how someone even came up with it | Thanks | You should check out the history section of that Wikipedia article... it’s pretty cool! Sounds like it dates back to medieval times and was developed in by Chinese and/or Arabic mathematicians. I can imagine having a reliable way to multiply would be important if you’re running a global trade center before calculators. |
[
"Unit circles and Infinity."
] | [
"math"
] | [
"kbfrbe"
] | [
17
] | [
""
] | [
true
] | [
false
] | [
0.92
] | I was doing this for fun to see what I would find. I'm a self-taught amateur "mathematician" and I just do things like this for fun in my spare time. I wanted to share this. If my thoughts here are flawed, please correct me. If anyone here has references that are related to what you see here, please let me know down in the comments below. Let have fun! What I wanted to do is describe infinity as a location in a coordinate system. I wanted to treat infinity as a polar location like you could with zero. I wanted to see where the locations of zero and infinity would be depended on what equation you use as a reference. In this snapshot, I use x=0. In the picture, imagine you are an observer located at ( -1,0 ). You are facing the positive integers of x perpendicular to the y-axis. Now, imagine that there is a transparent line that is simply x = 0 (The black line). Light passes through the line and hits your eyes. As the light hits your eyes, you locate where that light passes through the unit circle. For example, you look exactly perpendicular towards the y-axis and you will see that 0 is located at (1,0) on the unit circle. With 1, the location in the unit circle is (0,1). You do this with all integers on x=0. Here is 0 through 7 (some are approximate values): 0: (1,0) 1: (0,1) 2: (-0.6,0.8) 3: (-0.8,0.6) 4: (-0.882352941176..., 0.470588235295...) 5: (-0.923076923077..., 0.384615384616...) 6: (-0.945945945... , 0.324324324... ) 7: (-0.96,0.28) The reason I'm not typing more values is that I couldn't notice any more patterns and I pretty sure there are more. I just don't have the competence to figure this out in detail at the moment. I'm sure you have noticed the symmetry with numbers 0 through 3. On 4, you will see a pattern of ...8823529... in both the x and y coordinates (maybe just coincidence). With number 5, I can't see anything in particular. Number 6 is very interesting to me. Both x and y have a repetend number. Number 7 is a precise decimal number. As for the negative integers, It's going to be the same, but with the y sign flipped. The goal here is to locate the coordinates of infinity as you approach infinity on the unit circle for x = 0. It's obviously going to be (-1,0) where the observer is located: e^(i*pi). But, now I have more questions. There is obviously a pattern here and want to know how it is the way it is I guess? Is there a way to explain this? I guess what I am doing is projecting all the integers on the y-axis on the surface of a circle. In this perspective, there are no positive or negative infinities just like there are no positive or negative zeros. Negative infinity is just infinity, but you're approaching infinity in the negative direction. If you want all integers on the x-axis, you can make the unit circle into a 3d unit sphere. Has there been anyone that's done something related to this? Anyone that is from the past? I'm pretty sure someone has thought about projecting an entire coordinate system on the surface of a sphere. Because of my lack of college education, I don't even know if this is useful. Anyway, this post is a bit too long now. I don't know how to end this other than thank you for reading this post. | One point compactification, stereographic projection and Rieman Sphere are all things that come to mind. | It sounds like you’re describing a Riemann sphere in 2D https://en.m.wikipedia.org/wiki/Riemann_sphere | You are describing the Real Projective Line . On it +infty=-infty, as you say, and it is very related to projections onto the circle. You can create the Real Projective Plane , which extends the plane in a way that makes infinity make sense, and it has a "line at infinity", every point of which corresponds to the slope of some line and, in this plane, parallel lines intersect at the line at infinity at the point corresponding to their common slope. | Congratulations, you invented 2D stereographic projection. As you can see, most people are more familiar with 3D stereographic projection which comes up with complex numbers. If you want to learn more: https://en.wikipedia.org/wiki/Stereographic_projection | For 4 your coordinates are ( -15/17 , 8/17 ) As a consequence we also get 8 + 15 = 17 . In particular, rational points on the y axis map to rational points on the circle. |
[
"Coordinate free definitions of derivatives and integrals?"
] | [
"math"
] | [
"kbj7m7"
] | [
51
] | [
""
] | [
true
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false
] | [
0.9
] | The function y = x does not have a derivative at x = 0 because the tangent is vertical, but if you rotate the curve then the tangent has a defined slope, and if you do other transformations like inversion y=x then the derivative is very easy. This seems wrong somehow -- it is as if our imposition of a particular coordinate system alters the "rate of change" associated with the curve. Is there a better definition that talks of the intrinsic smoothness of the curve? Could there be a similar intrinsic notion of integral too, although there would be no "under" the curve in the absence of a coordinate system? | I think the fallacy might be here: our imposition of a particular coordinate system alters the "rate of change" associated with the curve. Curves don't have rates of change. have rates of change with respect to other --- that's what "rate" means! The notation dy/dx is intrinsically written in coordinates x and y. Even tangent vectors to a curve depend on choice of orientation... Although I think the span of the tangent vector might be intrinsic to the curve. So there's some truth to what you're saying. This wikipedia article on differential geometry might be a good start to reading about intrinsic geometry. | Basically what you have noticed is that there are two separate fields of study: The study of functions, especially the analysis of their rates of change The study of curves, surfaces, and their geometry However, often you can study functions by geometrically studying the curves and surfaces that are formed by forming their . This helps us analyze a function, but ** remember that the graph of a function is not a function itself . These tools are so powerful that it can feel like it's all just one field (analytic geometry), but really they are two fields (analysis of functions and differential geometry) that overlap and interact. But the interaction is not always seamless, as they are indeed different field of study. You do indeed run into edge cases where something perfectly well-defined in one perspective becomes ill-defined in another. You are not crazy. The curve in the plane parametrized by ( t, t ) is totally not weird-looking at t=0. But if you ask what that point tells you about the cube root function when you view this curve as a graph, then yeah suddenly that seemingly normal-looking part tells you that the function behaves weird, even though the curve itself doesn't behave weird. Why? Because the tangent line is perpendicular to the independent-variable axis, which makes the rate of change ill-defined. You are right, this is not a coordinate-indepent feature. But should it be? I think that there is an coordinate-dependent nature in this process of moving between functional and geometric perspectives. It makes sense that this whole situation is not coordinate-independent! You weren't analysing the curve without context, you were analyzing it the graph of the function, and so the independent-variable axis and the dependent-variable axis are made special by this context. | Yes, here is a start: https://en.wikipedia.org/wiki/Curvature | What's off? It sort of sounds like you don't want arc length to be defined. Maybe the mighty Theorema Egregium is what you're after. The coordinate-free definition of an integral is probably measure ). | Again that is just a consequence of decreeing that the x and y axes are special, which makes sense in this context because this is the graph of the function. But if we just cared about the curve without this context, specifying a parematrization based on distances from these two axes would be a weird and arbitrary-seeiming choice and you do indeed get artifacts from this choice. The second coordinate of that parametrization is a function that measure how far from the independent-variable axis you are when at time t. After taking the derivative, all that the singularity at t=0 is saying is that this function changes arbitrarily fast around this t value. Here's the way a differential geometer would think about this. At any point of an N-D topological manifold (i.e. a continuous shape that locally looks like N-D space), we want to construct a "chart" around that point. This is a homeomorphism (abijective continuous mapping) from a neighborhood of the point to an open set R -- essentially, a construction of coordinates. Why do we want to do this? So we can specify points near our pre-specified point by using N-tuples of numbers. Here N=1, and you know well this special point we want to make charts around. So we want to make some charts near this point, some continuous association between points and real numbers. One person might construct the chart u, and another person might construct the chart v. Perhaps the two people get together to compare their charts and realize that their charts are related in the following way: if u associates a certain point to a number t, then v associates that same point the the cube root of t. Of course, we can also have worded this discovery as "if v associates a certain point to a number t, then u associates that same point to the cube of t". This is essentially what we have here -- we have two different number-association-systems... I mean, charts...around our point. Both of them are perfectly fine charts, but what we have determined is that if you try to measure how v changes in terms of how u changes, you get a function with a singularity. That does mean That either of the two charts are bad charts That this point is in some way special or singular Concluding 1. is wrong because what we have determined is only the the two charts are bad charts . A differential geometry would say that the charts are"incompatible". Concluding 2. is wrong because this can actually happen around point! Around any point, we can construct two perfectly-fine charts that together are incompatible. It just happened that at our point, these charts ended up being the first couple we thought of. Why was that? Only because for previous reasons we were already considering this shape pre-embedded in the Euclidean plane in a certain why and already especially cared about a certain pair of lines in the plane (our axes). The reason being is that we were thinking of this curve as the graph of the function. If we weren't doing that, we would not have already had this Euclidean plane embedding in our mind, nor would we have cared about that specific pair of lines. Without those, that particular pair of charts would not have seemed more "natural" than any others. |
[
"What Journal(s) To Submit to"
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"math"
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"kb8c3w"
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16
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""
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true
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false
] | [
0.85
] | Hello, I'm a recent PhD graduate and I have two papers I'm finishing up, one in graph theory and one in number theory. I have never published before, so I'm not sure how to determine which Journals I should be applying for. Given the nature of the problems solved, I'm definitely not looking at an Annals paper. Do I submit the graph theory paper to the Journal of Graph Theory and work my way down a list of graph theory journals? I'm new to this whole process especially since the papers are not in topics I did my thesis on. Perhaps one could answer the more general question of figuring out what journals to submit to for other first time submitters like myself. | It's pretty common in pure math for the first publishable piece of original research to be the PhD thesis itself. In that case the student may graduate and then modify the thesis into one or two papers to be submitted soon afterwards. | A key part of a journal article is describing how your work fits into the existing scholarly research. Therefore I would think its almost mandatory for your paper to cite at least 1-2 journal articles (although I dont write papers about pure math) | Yeah it's fairly different in pure math. It takes quite a while for most people to get up to speed with the current state of their field and to find positive (or useful negative!) results. Similarly, pure math PhD theses can sometimes be as few as 60 pages. They're likely to be very dense, but still. Mine was about 100. | have you looked at what journals have published similar papers on the topic? or in which journals the papers you are citing were published in? | Yes. Even if you can prove your theorems by citing only textbooks, you need to mention related work. |
[
"What would be the ideal outcomes of a PhD in terms of personal development?"
] | [
"math"
] | [
"kbjoci"
] | [
404
] | [
""
] | [
true
] | [
false
] | [
0.97
] | I would soon be starting a PhD and I wanted to gather a list of things that I want to (or rather should) achieve at the end of the PhD. I am mostly looking for points on personal development. Currently, the only goal I have in mind is that I want to develop a level of independence of thought as I reach the final few months of my PhD. The point of a PhD for me is to be able to learn to be an independent researcher. I also aim to know and understand more than my guide in the very narrow topic that I would be working on, although I am not sure whether that would be an achievable goal. I must note that although I am a physics student, the area of my study is at the boundary of modern maths and physics, and in a lot of aspects I consider myself to be more of a mathematician than a physicist. | beyond the math stuff, you should have learned about independence, how to find research directions more or less on your own, some sort of “mathematical self-reliance”. But also expect to learn about grit and late nights lol. And oh, you’ll probably get an intro to department politics! | Kind of generic Become an effective communicator in general and within your field of study. Especially if your work crosses departments. Establish meaningful relationships. Develop good life-long habits wrt finances, nutrition, exercise and sleep. Have more solid career goals and a plan. No one to, but this includes thinking about an "exit plan". | Self confidence should be high on your list. Most of the people throughout your program will look to tear you down, to fit you in the ranks as one of their own. The ability to look back upon your work and say "this was my own" is, honestly, super rare these days. Don't let anyone on reddit or anywhere else tell you differently. | Oh I think I could have well done without the politics. It’s a nightmare. | > Develop good life-long habits wrt finances, nutrition, exercise and sleep. Hah, I find it difficult to believe you'd completed a PhD with that on the menu |
[
"Gift for Math-Minded"
] | [
"math"
] | [
"kb7bnx"
] | [
9
] | [
""
] | [
true
] | [
false
] | [
0.92
] | I am looking to get a Christmas gift for my brother-in-law who self-identifies as a “math nerd.” I’m thinking maybe a t-shirt with a math pun/joke. What gift ideas come to mind that someone who loves math would be proud to own (that maybe the rest of the world would not quite understand)? Thank you in advance for all of your help! | I would go with a book. Dover has a ton of mathematics books for reasonable prices. There's also tons of mathematical books that aren't textbooks. A favorite of mine is "Magical Mathematics" by Persi Diaconis and Ron Graham - it's a book on magic tricks that are all based in (decently advanced) mathematics | Check out Martin Gardner. Not all of his stuff is math-related, but he wrote a of math and puzzle books and he's amazing. | Nice. That sounds quite interesting. Thank you! | I think you should check amazon. Otherwise you should just get a custom made one if you can. | Maths gear ( https://mathsgear.co.uk ) has some pretty neat stuff - including my personal favourite, the utilities problem mug. Or one of Cliff Stoll’s Klein bottles. |
[
"This Week I Learned"
] | [
"math"
] | [
"kb6lac"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
1
] | This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn! | This week I learned that calculus is where math gets fascinating, not scary. I am 31 and contemplating going back to university to get a degree. I never found math difficult but it was really boring. People always said how tough higher math was so out of lazyness so I just avoided all together. Now I decided to watch some videos on YouTube about what calculus is and it turn out, it's where you use all the boring math for the interesting stuff! Looks like I'm going back to university! Edit: I know this might not be the most relevant place to share it, I'm just excited and wanted to share it somewhere. | My master's thesis is about multivariate recurrence relations and fractal structures in their values when viewed modulo a prime. The most well-known example of this is how the odd entries of Pascal's triangle form a discrete version of the Sierpinski triangle . Consider an affine transformation of Pascal's triangle, defined as a function f : ℕ → ℕ with values given by the recurrence relation f(x, y) = f(x-1, y) + f(x, y-1) with the boundary condition f(0,0) = 1 and terms dropped from the sum that would include negative arguments. (Combinatorically, this function counts the number of lattice paths from (0, 0) to (x, y) consisting of the steps (+1, +0) and (+0, +1).) Color-coding the values of this function modulo 2 produces a nicely symmetric right-angled Sierpinski triangle, and similar unnamed fractals appear in the nonzero residues modulo any prime. This is fascinating and compelling because it relates a "local" phenomenon, in which values of a recurrence relation only "know" about their immediate neighbors in an array of numbers, to a "global" phenomenon possessing structures of arbitrarily large scale. Amazing! Now, the closed-form expression for values of f is given by f(x, y) = (x+y)!/x!y!. So here's the thing I learned this week . Imagine that instead of the usual factorials in the equation above we use a kind of pseudo-factorial, replacing the values 1, 2, ..., n in n! with values of a one-dimensional recurrence relation (with two parameters). If we define a new function that way, its values ! (Given some simple restraints on the parameters.) That means this path-counting, Pascal's-triangle discrete fractal phenomenon connects to a huge subterrain of even more widespread number arrays that exhibit similar patterns. My thesis is currently organized around the path-counting approach, but this makes me wonder if I'll have to reorganize the whole thing before I'm done. (You can show by exhaustion that no variation of a path-counting recurrence relation in three dimensions produces the Menger sponge modulo 3 — but maybe these pseudo-factorials will allow me to finally spear that white whale.) | I just found out that Brian and Keith Conrad are brothers, I'm shook. | I learnt the weak law of large numbers in probability. It's not just an important theorem that I'm glad to know the precise statement of, but it also has an awesome name. I also learnt about the inverse of a matrix is its adjugate divided by its determinant. I didn't notice until the lecturer explicitly pointed it out, but then I was like "ohhh...!" Makes a nice change from school, where even my brilliant teachers just spoonfed us the formula. | that comes from the cofactor expansion, I think it is a very nice formula. |
[
"Am I just burned out?"
] | [
"math"
] | [
"kbwvuq"
] | [
517
] | [
""
] | [
true
] | [
false
] | [
0.97
] | Hi everyone, I’m sorry in advance if this is not the appropriate place to post/ask this. I’ve been doing quite a bit of math throughout the last few months, up to 5-6 hours a day all week. It now comes to a point that whenever I sit down and attempt to do more math, I just cannot focus anymore. Is this normal to everyone? I know that the obvious answer is taking a break but I just feel extremely guilty for doing so. Whenever I attempted that, my mind kept wandering back to the last problem that I was struggling with as well as the last theorem that I read and then I started to feel bad for not studying. Does anybody have suggestions or advices on what should I do in this case? | I hear you, i did 3 semesters of math just now (physics and chem too) and im so burnt out from math. Last week of classes and I'm having a hard time grasping the topic. Too anxious to end these stressful online classes. All i do is study | Yo this is totally normal. It is always good to take breaks, even if you feel like you really need to focus or grind out some math work. For one, your work is probably not worth as much if you are totally burned out. Your thought processes will be different, and you could be tricking yourself into thinking you are doing something right (or good enough) when you really need a fresh set of eyes. On top of that, your brain actually works on problems subconsciously, so taking frequent breaks and trying to get out of your mindset might not set you back as much as you might think. | Fresh air and exercise will have you good as new in no time. Get some good, healthy food into yourself as well. | Yes, you are burned out. Perform the following experiment: Schedule yourself breaks of at least an hour, and make daily notes of whether you took a long break, and whether you had any "ah ha" moments, any sudden breakthroughs in your understanding, or learned something "big." (Yes, this is subjective, but you get the idea). If you're like me, you'll find that all of the best positives come after a significant break (at least an hour, but I really mean several hours or even a whole day). And I don't mean "time working on something else", I mean scheduled time away from all math and not feeling guilty about it. You'll find that you actually get more done if you get into the swing of it with regular scheduled breaks, not grinding away after the initial impetus has run out. | And lots of sleep. |
[
"Suggestions for Extended Essay Topics"
] | [
"math"
] | [
"kbhbel"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.75
] | I am in my first year of doing the IB diploma program and I have to write an essay. The problem is the commitment to said essay. I really need something I want, so I would really appreciate suggestions. I really want something that applies mathematics to literature or poetry. Currently the only guiding question I have is “How applying fractal mathematics to poetry derives complexity in writing. But I’m not even sure if I want to keep that I would appreciate any suggestions even if they don’t align with the applied math to literature. If you have any good recommendations to read as well that would be incredible. Thank you to anyone that took time to read this. | The extended essay is supposed to be a research paper right? You might struggle to find resources for your topic. The schools I’ve been to refuse to let students do extended essays in math. I applaud you. | Hi there. I double-majored in English and mathematics for my undergrad, so I’ve thought about the connections before. Personally, I think the best connections you can make between mathematics and poetry are motivational rather than material in nature. Both mathematics and poetry This is more true of poetry before, say, the middle of the 20th century. Contemporary poetry can harbor a lot of suspicion of the above ideas, so adjust your literature search accordingly. Keats’s famous line about beauty and truth has a lot in common with the way people describe aesthetically pleasing mathematics, even providing the title for a book by Ian Stewart. How applying fractal mathematics to poetry derives complexity in writing. Can you say more about what you mean here? In my opinion, the most robust connection between fractals and literature is Wallace Stevens’s poem "Connoissuer of Chaos" which I believe contributed in inspiration to the subdiscipline sometimes labeled "chaos theory." (This might be akin to the James Joyce connection with particle physics. I don’t have my copy of James Gleick’s excellent book at hand, so I leave the verification of this connection as an exercise to you, the reader.) Going a little more abstract, the trope of a “story within a story” could be considered the first iteration of a fractal structure, and maybe there’s some fractal-like infinite regress in the work of someone like Borges … but ultimately “fractal mathematics” is a geometrical concept. At the risk of sounding like a spoilsport, I would caution you against being too freewheeling with specific mathematical ideas. Jacques Lacan, for example, is regarded as a prominent thinker of the 20th century, but when he compares the imaginary unit to a penis it’s just weird. (I read this excerpt in undergrad many years ago, again my apologies for not having a direct citation ready.). Beware of stretching your comparisons too far, lest they fall into “howler” territory. Good luck with your essay. I hope you can cross-pollinate your interests fruitfully and make the world a more delicious place. | Thanks. I did change the guiding question to this: > How does applying sequence mathematics to poetry derives complexity in writing? And then detailed things I wanted to touch base on. *Mandelbrot Set *Types of Meter *Hausdorff Dimension I will take all of ur recommendations into account. Thanks. I’m familiar with chaos theory, but all of this is self research not learned in class, so it is all very loose. | I don’t think any can be considered broadly the “best” to use as interpretations are different, but the meter suggestion did help ground me a bit, so thank you for contributing. This is all very helpful. | Some members of Oulipo were mathematicians. Perhaps their work would interest you |
[
"Prime test"
] | [
"math"
] | [
"to48pf"
] | [
3
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.64
] | null | It's pretty cool but not new. It's equivalent to saying that either p%6=1 or p%6=5 for p>3, which is a known fact. (n+(n±2))%12≠0 (2n±2)%12≠0 (n±1)%6≠0 n%6≠∓1 | So your code there basically says that 2n+2 and 2n-2 can only be prime if they're divisible by 12. You can divide that all by 2 and you get that n±1 can only be prime if it's divisible by 6. Basically you've figured out that all primes are either one more or one less than a multiple of 6. It's touched on in this numberphile video, as well as a related concept with the squares of primes. Starts talking about it at around 2:00 in. https://youtu.be/ZMkIiFs35HQ | Besides what Martino106 said, to check divisibility by 3 of large numbers one just compute n%3. Obtaining the digits in base 10 of a number stored in base 2 (as usual) is by itself costly in terms of operations. | Yes, you are right to say that barely_sentient's comment require a lemma saying that the sum of the digits of n is congruent to n modulo 3 (and this is even true modulo 9), this is a separate lemma that you can check pretty easily. | Exactly. I meant that if you have for example the number n=545666522 stored in a computer it is much more computationally costly to extract the digits 5 4 5 6 6 etc and adding them than computing directly n%3. |
[
"How do you round?"
] | [
"math"
] | [
"tocbey"
] | [
1
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.67
] | null | Your teacher is wrong. | Yikes on your teacher. You don’t round on intermediate steps. | Your teacher is wrong. That figure rounds to 12.3. Rounding does not cascade; you're supposed to do it once. In school sometimes you have to deal with authority demanding the wrong thing. That's practice for real life, too. Psst. Tan(51) = (2 - sqrt(3) + sqrt(5 - 2sqrt(5)))/(1 - (2 - sqrt(3))sqrt(5 - 2 sqrt(5))) | Speak to a department head | Yea I thought so |
[
"Rock, Paper, Scissors, is non-associative!"
] | [
"math"
] | [
"kb6j4m"
] | [
50
] | [
""
] | [
true
] | [
false
] | [
0.92
] | Hey, I made this short video explaining associativity and how the "verse" operation of Rock, Paper, Scissors fails to be associative. I love explaining this fact to my non-mathy friends | tl;dw (RP)S=PS=S R(PS)=RS=R S=\=R | You might find this paper interesting: https://arxiv.org/pdf/1903.07252.pdf | This paper would have been useful to me a couple of years ago when I was trying to help someone create game mechanics for an extended version of rock-paper-scissors. (Not so much any more, we've already done the research on our own, and the project fell through for unrelated reasons.) Thanks for sharing. | This is why I only play rock-rock-rock. | Name checks out. |
[
"I'm trying to do the indefinite integral of sinx*lnx"
] | [
"math"
] | [
"tnzz78"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | null | It's been awhile since I've solved an integral but dv=x and v=-cosx seems strange. How are those both possible at once? | I dont see what you're talking about I mean unless your asking saying dv cant equal x if v=-cosx, well since they were different problems i just assumed we could use the same variables so just pretend they are different anyways i don't think it effects the answer (sorry for bad English) | oh yeah you're ima edit that so people don't get confused | oh yeah you're ima edit that so people don't get confused | oh that makes sense thank you I didn't know this |
[
"Where can I find a proof of this result?"
] | [
"math"
] | [
"tnxxu2"
] | [
7
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.82
] | null | Basically 1) show that the group of units mod p is cyclic (of order p-1) and 2) show that a cyclic group of order k has phi(k) generators | Gauß is the correct way of writing it (Gauß was German after all). Writing Gauss is common in English as the ß is typically not available. Weierstraß is another example of this. It's the same as writing Chebyshev instead of Чебышёв transcripting Cyrillic to English. | ß = ss in German so Gauss | I'm pretty sure this goes back at least to the 1800's. Check Ireland and Rosen or perhaps Serre's "A course in arithmetic" (or some larger volume on algebra or number theory like Dummit and Foote or Lang). | German has few more letters (namely ä,ö,ü,ß) than English. They are used to express sounds unusual to native English speakers but very common in German. For example, ß is very akin to to ss but pronounced slighty differently (despite being German I cannot really describe it). |
[
"I have an interesting dilema"
] | [
"math"
] | [
"tnjfqs"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.2
] | null | I don't understand : pi minutes is less than 4 min, so how can it be longer than 4min? | The fact that π runs on forever doesn't make it infinite. 3<π<4, so a video of length π would last longer than 3 minutes but less than 4. Your confusion is the same as that caused by Zeno's paradox. In fact, the scenario you're describing Zeno's paradox. Adding up infinitely many numbers doesn't always result in infinity. | No, it would run a sum of infinite lengths where each part of this sum gets a lot smaller. | You should check out Zeno's paradox! | The choice of a unit of time is in principle arbitrary, so pi times a unit of time doesn't seem to be a particularly special quantity... |
[
"Why is it that Radians are a more \"natural\" unit than degrees?"
] | [
"math"
] | [
"tnj1sb"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.5
] | null | radians are based on constants (π) and degrees are based on how we count them (why the circle is broken into 360 degrees, instead of, for example, 1000 degrees?) | π=l/d where l is a circumference of a circle and d is a diameter. Both of them are constants for a unit circle, no matter how we measure, l/d would remain π, or, l/2r=π where r is a radius of a circle, therefore l/r=2π. Some people prefer using tao=2π but it doesn't change the fact that both π or tao are constants for all circles. | 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! | But how did we discover there were 2 pi radians in a circle. Isn’t it the same thing as arbitrarily deciding a circle contains 360 degrees. | Think about the formula for the circumference of a circle, C = 2πr. 2π already naturally features in the equation, so let's say 360˚ = 2π radians. Well then, half a circumference would be πr, a quarter would be πr/2... in general an arc has length r times the angle in radians. Why does this mean calculus requires radians? Imagine a right-angled triangle with a very small angle (good old Wikipedia!). You can show that sin(x) and tan(x) are approximately x in the limit, and cos(x) is approximately 1 - x /2. This leads to the way we are taught to differentiate and integrate these trig functions. |
[
"Why don't programming IDE support typing math expressions the same way math does?"
] | [
"math"
] | [
"tnqg14"
] | [
8
] | [
"Removed - not mathematics"
] | [
true
] | [
false
] | [
0.67
] | null | As a CS researcher (and as amateur programmer since 1978), I have never experienced the problem you want to address, so I have difficulties in assessing the usefulness of your proposed solution. Expressions in code may be ugly (?) but usually you need to read/write/check them only once, once they are checked you can just read the nearby comment that surely clarifies the expression. In general I would not be happy to be tied to a specific IDE. Anyway, do you know Knuth's literate programming ? There may be aspects in common. | I was doubtful if you were serious until you proposed that: A different naming convention. For some reasons, programmers use variable name to comment to what the variable do, instead of separating them like mathematicians. An IDE could have easily make it so that it is easy and natural to use a one-letter name for variables and check the comment on it to see what it is supposed to do. Now I am 90% positive this is a shitpost, but on 10% chance you are actually serious let me give you a single objection: not every programmer uses the same IDE and as such, it is critical to understand what variable does when it is used (be it using it's value or assigning it value), not when it's defined. A way to write math expression the same way math does it. If I want to write a sum over a set, I already use multiple lines for that in practically any programming language, just like I would writing \sum_{} {...} in LaTeX. | Then standardize certain features https://xkcd.com/927/ Webpage designing don't have this problem, people are not expected to type HTML by hand. Webpage designing != Webpage creation. Even if these days you are not expected to type HTML by hand, you are always expected to write the JS that will inevitably be part of said webpage mostly by hand beyond using external libraries. In LaTeX you can see the expression in math form. Yes, and half of things I write when I do programming can't be written in math form no matter hard you try. You can try: write a math form for some simple algorithm like quicksort. The most likely out come is you will get something vaguely resembling pseudo code for the same quicksort, and now your thread transforms into a more reasonable question of why nobody ever made a compiler for pseudocode. People don't just read an expression written in LaTeX as is if they want to check the logic of it or check for error, that's way too cumbersome. I never got into advanced TeX, but fairly positive you are not going to automatically check some of the more advanced macros there without diving into the code either. | For starters, I think you recognise real pain points with scientific computing, and we could do better. Maths equations can be quite complicated and there is less chance to introduce bugs when the language syntax matches the maths notation. This is why Julia allows Unicode operators and symbols, although it is still flat. I've been working on a language with inline equations, Forscape , which is still in an early stage but has some nice promise IMO. As to why languages haven't taken that approach: 1) you need a tree doc of some form, 2) this requires special IDE support for rendering and editing, 3) this requires special code model support to do syntax highlighting etcetera on the tree doc, 4) which is all a lot of effort for a niche feature, and 5) carries some burden to typical flat text use cases, at least in terms of design complexity even if IDE performance impact is negligible. Juypter Notebooks kind of targets this, but naturally in a way that is more concerned with inline typeset documentation than actual executable equations, because it is after all using Python. You'll see lots of CAS tools with typesetting, but usually the scope is smaller than a full programming language. I think it's a question of effort vs impact. | Furthermore, a lot of programmers don't use IDEs at all (myself included). I once had to rewrite code that someone else had written that had variables names such as ss and ch. It was a nightmare to understand. |
[
"4 colour theorem"
] | [
"math"
] | [
"tnkydr"
] | [
128
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.92
] | null | Blue on the outside, green on the big inside piece, alternate purple and orange for the small ones | This probably belongs in the simple questions thread. But in any case, you can do this with three colors. The top row of regions can be colored using two colors (just alternate colors: R B R B). Then use a third color for the region below them. Edit: This is assuming that you aren't considering the "outside" of the figure you drew to be a region that needs to be colored. As other people have noted in this thread, if you do want to color the outside, you'll need a fourth color for that. | If there are only 3 regions, you can 3-color it but not 4-color it | It’s r/math fam | This is so pedantic and I'm all for it. |
[
"All men are created equal. All people who are created equally are women"
] | [
"math"
] | [
"tngc5x"
] | [
0
] | [
"Removed - low effort image/video post"
] | [
true
] | [
false
] | [
0.27
] | null | I'm not mother tongue, but the second sentence does not seem to make sense. | How is it a valid argument? | Being created equal doesn't imply being created equally, so no, it doesn't. Unless your definitions of being created equal / equally differ from the commonly used ones | 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! | I am not making it political at all, what are you talking about? |
[
"An integration bee. Like a spelling bee but with integrals."
] | [
"math"
] | [
"tofn7p"
] | [
41
] | [
"Removed - low effort image/video post"
] | [
true
] | [
false
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0.93
] | null | Love these. Came in 4th place the year I participated, many years ago :) | Integration by parts, i-n-t….. | Since x lives in the interval e to e , x is positive, so you can just ignore the absolute value sign. | Do you stand on a stage, do all your work in your head, and present the answer orally? | Not even, you can multiply by 3/4 and change that x to x and get the answer easily from there |
[
"When does your field get to \"the good stuff\"?"
] | [
"math"
] | [
"tnn1kh"
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263
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""
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true
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false
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0.98
] | null | Differential geometry. The entire first course is probably like this. Messy formula, really long and mostly obvious definitions that you have to check, and homework problems feel like more checking definition and algebraic manipulations. Part of the problem is because nobody really study differential manifold by themselves, but many of them do stuff on differential manifold with additional structures, so you end up with a catch-all class full of general concepts but no applications and few non-trivial results. Once you get to Riemannian geometry, symplectic geometry, complex algebraic geometry, or general relativity then things started getting interesting. | First courses in manifolds unfortunately have this issue, so much dry technical machinery building before anything much happens. But if you get to do some classical differential geometry of curves and surfaces (Frenet frames, first fundamental form, theorema egregium, Gauss-Bonnet theorem) it's absolutely bangin' from the beginning. | Proof theory has a similar problem - the early stuff is a lot of different formal systems (natural deduction! sequent calculi! one-sided sequent calculi!) and translations between them (ugh), and I know people who gave up on the subject entirely before getting past that. But I think it's pretty clear that the good stuff starts with cut elimination (at least for propositional logic) and its applications. | I’m in (algebraic) number theory. I feel like number theory has kept me engaged thoroughly through my education. I rarely felt like I was building up boring background. When you start out you have to learn modular arithmetic and such, but there are always neat applications to things like solving Diophantine equations that I found interesting. A first course usually culminates around quadratic reciprocity which is an interesting result itself. Algebraic number theory techniques are interesting, which leads into class field theory that is brilliant. I was blown away by the group law of an elliptic curve. Even in the (graduate) analytic classes I took, we only spent a couple weeks building up tools like Abel summation before diving into complex analysis to prove the prime number theorem. The only time I was frustrated building background was having to learn so much commutative algebra and scheme theory getting to research level. | Yeah - it would be interesting to see how a course that tries to manifolds would be like. Instead of starting with definitions and moving towards examples, you go the other way - starting with examples and moving towards definitions. You lead the students on a journey of how we started from the mathematical objects we all know and love ( , S etc.) and how we get to those technical definitions, why they're useful, and why manifolds really are the right object of study to get the most useful and powerful theorums out. Of course, running such a course would be difficult and would probably run into lots of practical issues - but at the same time, it feels far more principled than the way we usually do things teaching mathematics. |
[
"Mathematicians, in which stage of your academic path were you able to devote the most time to focused research?"
] | [
"math"
] | [
"togjnv"
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50
] | [
""
] | [
true
] | [
false
] | [
0.92
] | e.g. during PhD program, research/non-research postdoc, assistant prof, post-tenure, etc. (since there may be confounding variables due to other commitments in life, such as family responsibilities, suppose we're interested in a reasonable maximum number of hours per week one could possibly allocate to research and publication if those weren't a barrier) | I am not a mathematician (my field is machine learning), but during my PhD years I did nothing else but research. I learnt a lot of stuff during those years. Now I spend most of my time teaching, searching for money, and supervising PhD students. I also spend an inordinate amount of time with administrative duties (committee meetings, organizing things, etc). I specifically try to put aside a few hours each week for reading papers, to keep up with recent advances in my micro-tiny-sub-sub-field. | The last fall semester of grad school when I had no teaching load, no kids, and I didn't have to start writing my dissertation yet. Two publications actually came out of the work I got done in that short window. Now, between grading, teaching, service, and admin work, I only manage about 3 dedicated hours per week, so I can't have too many outstanding projects (and the one I do have is coming up on its second birthday; I may have overestimated it a bit). | professor emeritus | I had the most time when I was a postdoc. Life was busy, and it didn’t seem like it then, but that’s where I had the most time for research. Funny enough, now that I’m an Assistant Professor, with a family of four, I have much less time, but I’m way more productive. Just more efficient. | This is why I'm heading to industry :>). |
[
"Is there a name for this pattern?"
] | [
"math"
] | [
"tofvr2"
] | [
35
] | [
""
] | [
true
] | [
false
] | [
0.91
] | For fun, I was trying to find a pattern in pi! In that pursuit, I was trying different things, what I tried tonight was I looked at each value of pi, and subtracted the next value in the sequence from it, and replaced that value with the absolute value of difference. Then I just repeat the process so for example 3.1415 I would take 1-4 = 34-1 = 31-5 = 4etc... I applied this formula to the first 12 digits of pi, What I found is there is a max number of iterations, before the cycle falls into a repeating pattern. This would happen for any numerical value, not just pi. I was wondering if this has a name and if there are any applications for it? | In general, if S is a finite set and f:S->S is a function, and s\in S, then the sequence s, f(s), f(f(s)), f(f(f(s))),...... will be eventually repeating. The reason is that by finiteness of S the terms of the sequence can't be distinct. Then once f^i(s) = f^j(s) with i>j, all f^{a+b(i-j)}(s) equal f^a(s) for all b=1,2,3,... | So in a nutshell, it's the pigeonhole principle. | Just to be clear, this isn't about the value of pi, but the practice of taking a value, replacing each digit with a new digit which is the difference between it and the neighbor to the left, then just repeating the process when you get to the end. | What rule generates the 12th value? Edit: Because to generate the new kth digit requires the kth and the (k+1)th digit from the previous generation, but you have no 13th digit visible. The fact it oscillates between 5 and 2 is making me a little suspicious. This could occur if you always append a 7 as the 13th digit. | just to be clear, the first line you see is only the first 12 digits of pi that I am applying this formula to. The values below, are what the new modified values represent |
[
"How to analyze independent single data points that were modified by some function by comparing them to a non-modified standard?"
] | [
"math"
] | [
"tofepk"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
1
] | Hey all, I know the title is confusing and I apologize in advance for that, because I'm unable to explain the problem more concisely. But here it is: For example, I have 5 different animals (one animal of each of 5 different species for a total of 5 distinct samples) and they're all being fed a new nutritional supplement that's supposed to help them gain weight. They all gain substantial amounts of weight such that they're more than 2 standard deviations from the normal for the age/sex. Because they're different types of animals, I can't simply pool them together and run a t-test or z-test to show that they're significantly different in weight. What's an alternative way for me to analyze this data to come to a conclusion? Thanks for any input or insight you can provide! | Predictor variable for amount of nutritional supplement fed + interaction term for each animal type | Thanks for your comment! Do you mind elaborating on this or providing a source with more info? Sorry, my familiarity with this topic is limited! | Here are some examples: https://are.berkeley.edu/courses/EEP118/fall2010/section/10/Section%2010%20Handout%20Solved.pdf You'll probably find a more receptive audience on r/statistics or r/datascience | I'm probably not understanding that concept in complete detail, but it seems to me that I'd need more data points than are available to me currently in order to come to some conclusion? I only have single data points from 5 distinct entities and want to compare to a control. Is there a way for me to come to some conclusion from this "collection of single data points"? I wish I could explain the issue more elegantly but my math/stats knowledge is relatively rudimentary! I really appreciate your input!! | If you only have a single data point for each entity type, then it's not possible to do any meaningful inference on the entity-specific interaction with nutritional supplement. You can estimate a value, but we have no idea what the distribution looks like for the errors. |
[
"What comes between Algebraic numbers and Computable numbers?"
] | [
"math"
] | [
"toaehe"
] | [
57
] | [
""
] | [
true
] | [
false
] | [
0.95
] | Algebraic numbers (𝔸) are defined as roots of polynomial equations with integer coefficients. Computable numbers can be defined as limits of ratios of computable sequences. 𝔸 has decidable equality, computables do not. Are there any well-studied number systems situated "between" these? (excluding dumb examples like 𝔸[π] etc.) | Periods ) come to mind. | fixed link It's always a fun exercise when coming across a new number to try and see if it's a period :) | The rationals and algebraics have computable equality yet in general it's very hard to show numbers are irrational and transcendental. It is conjectured periods have computable equality but it seems to be open. | It's kinda too bad that they only form a ring. | You can also take the algebraic closure of the field generated by the periods, and can then iterate this, allowing your coefficients of your "algebraic" functions to be elements from that. If I recall, it is open whether the tower one gets from doing this keeps eventually terminates in the sense that you aren't getting anything new by iteration. |
[
"How to practice proofs?"
] | [
"math"
] | [
"tolcex"
] | [
25
] | [
""
] | [
true
] | [
false
] | [
0.81
] | I'm reading a discrete math book for CS students, but the book has no problems or assignments. I know reading the material alone won't be enough to learn a topic. Could you suggest a resource or problems that I can use to practice? Thanks a lot! | Hammack's Book of Proof is available online. | How to Prove it is awesome for a beginner to proofs and it goes all the way from basic logic to countable and uncountable sets, and it has tons of exercises. | Great, thanks! | Discrete Mathematics and its Applications by Kenneth Rosen is also a good book for beginners. It has lots of proofs and exercises. A free pdf is also available online. Do go through it to check if it satisfies your needs. | Concrete Mathematics by Donald Knuth is a good book for wrapping your head around the process of doing proofs. Other great resources include Art and Craft of Problem Solving and Putnam and Beyond. |
[
"This Week I Learned: March 25, 2022"
] | [
"math"
] | [
"tnp2ke"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
1
] | This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn! | This week I learned that I got accepted as an undergrad at Michigan State! Not sure whether I'll end up going (that depends on how my other admissions decisions go), but they have a really solid math program (and a particle accelerator!) so it's certainly a great option. I also learned about the multiple reduction copy machine , a cool (and surprisingly simple) recursive algorithm for generating fractals. I'm planning on writing my own implementation of it (based mainly on a more detailed description of the algorithm in Gary William Flake's ); anyone here have recommendations for graphics libraries in Python, especially ones with good tools for handling linear transformations? I've been thinking about Pygame but I'd like to know what other options there are. | This week I learned the proof of Dirichlet’s theorem from Davenport’s . It states that for any coprime integers a,b, there are infinitely many primes in the form an+b. The proof hinges on certain zeta-like functions called L functions, and proving that a wide range of these are nonzero when evaluated at 1. While I believe I fully understand each part of the proof and why it works, I am left completely lost as to how someone would come up with this. I can sort of understand why you prove it by showing that the sum of the reciprocals of those primes diverges, but I don’t understand the motivation for any of the rest of it. I can’t understand how Dirichlet thought of using L functions for this, nor can I think of how he came up with the proof that they’re nonzero at s=1. | I sent off all my applications for summer research! 🥳 And I even played a video game by myself for the first time in real actual years. The Stalingrad level of CoD: World at War is genuinely one of the all-time classics, and I had such fun playing it. | This week I learned of the Stirling approximation. The fact that x! ≈ x e √x √2π . Numberphile video came out and I thought it was quite interesting | I learned (not this week but recently) that the number of primitive roots of a prime p is equal to phi(n-1), which I find really interesting. I came across this fact working on a problem in the Stanford high school math tournament from 2020. I would love to read a proof of this fact, although I’m not sure where to look for it. |
[
"Is there something deeper in Galois correspondence of covering spaces?"
] | [
"math"
] | [
"tnnenb"
] | [
14
] | [
""
] | [
true
] | [
false
] | [
0.86
] | I was thinking about this recently. Suppose we have a functor from Top to Grp, which we kind of do via the fundamental group. We know that in Galois theory, we solve polynomial equations over fields and try to find roots up to permutation ie get a Galois group. Additionally, the fundamental group has an action on the fibers of the base space and if I recall correctly, it can be identified with the symmetric group (deck group and Automorphism of L that fix K seem like basically the same thing (L,K are fields)). If we can find some sort of connection between fields and topological spaces, which is not immediate to me, would we somehow be able to solve polynomial equations using covering space theory? | There’s a book by Tamás Szamuely, Galois Groups and Fundamental Groups, about the relation between them. I highly recommend you to check it out! | There is the Etale Fundamental Group which, in a way, generalizes both topological and algebraic "Galois" groups. At this level, we don't just work with field extensions, but schemes. Schemes make algebra geometric. If you have a field K, then the scheme associated with K is Spec(K) and it is just a point but implicitly has "rational functions" defined on this point and this field of rational functions is exactly K. This is how Spec(K) and Spec(L) are different (when L is a different field), even if they are the same as topological spaces, both just being a single point. In a way, classical Galois Theory and field theory in general is the geometry of spaces that are a single point. Anyways, if L/K is a field extension, then this means that there is an inclusion homomorphism K->L which, in the world of schemes, turns into a topological map Spec(L)->Spec(K). If L/K is separable, then we can think about this topological map as being a covering map and, in this situation, we call it an "etale map". Now, in Galois theory we usually pick and fix some algebraic closure of K and imagine that all our spaces live inside that algebraic closure. We want to still do this, and this is done by what is known as a "geometric point". If F is an algebraically closed field containing K, which means that there is an inclusion K->F that we fix, then this means that if x=Spec(F) then there is a fixed morphism x->Spec(K). We then call "x" a choice of a "Geometric Point" of Spec(K). It should be noted that a geometric point will always be a fixed point and so if X is a general scheme (with more than one point), then a map x->X is a selection of a "good" point on X. But, in general, such a choice makes the pair (X,x) a pointed space. So, with our setup so far, we have a pointed space (Spec(K),x). In topology, the fundamental group arises from a pointed space (X,x) by first finding a universal cover U->X and then lifting x to fibers in U. The deck transformations of the cover U->X then correspond to the fundamental group of the pointed space. We want to do the same but in this more general setting universal covers and paths don't necessarily exist. If Spec(L)->Spec(K) is an etale map (read: finite cover), then we can already talk about "Deck Transformations" as being automorphisms of Spec(L) that fix this cover. Ie, Galois transformations. But there are some subtle points here. In a way, the "point" in a pointed space for the fundamental group doesn't matter, but it also kinda does. The analogous situation here is that the choice of algebraic closure doesn't "really" matter, but it actually kinda does. Especially since we still don't have a "fundamental group", we just have "relative" fundamental groups. So if I have my etale map, then I can take the point x->Spec(K) and "lift" it to Spec(L). Algebraically, this is just done by noting that both L and the algebraically closed field F are K-algebras and so I can make the tensor product space L⨂F over K. L⨂F acts like "F for L" since we both have a map F->L⨂F (since L/K is finite, this will be an isomorphism so it is algebraically closed) and L->L⨂F as an inclusion. This will give a new space, y=Spec(L⨂F) where we have y->Spec(L), so it is a geometric point, but also y->x and so it acts like a fiber normally would. Things like Spec(L⨂F) are often called fibers for this reason ("the fiber of x over Spec(L)->Spec(K)"). Note, there are more complicated etale maps whose fibers split x into multiple copies, but these do not come from field extensions alone. As with the topological case, we want to lift the cover "all the way". In topology, we have the universal cover to do this, but in this situation we don't. We want to think about x->Spec(K) a universal cover, but etale maps must be finite (for the work we do with them and the categories the constitute to make sense) so this does restrict us a bit. But what we do is find a bunch of finite extensions which is finite at each step, but the union of all these covers is the algebraic closure of K. Now, in our general setting, we can't really talk about unioning abstract fields, so we use the slightly more flexible notion of a "projective limit". We have a bunch of covers Intuitively, since each of these maps has a fiber for x, we can think about this infinite sequence as being a replacement for the map x->X (that this actually is the case has to do with representability of functors and stuff). So if we want to make Aut(x->X), which is the fundamental group in the topological case, then we can instead look at the successive limits of the groups Aut(Spec(L_i)->Spec(K)). This "Inverse Limit" gives us a new group, which we call the fundamental group of (Spec(K),x). It is isomorphic to the absolute Galois group of K in the algebraic closure F. Though, this should be thought of a geometrizing field theory, rather than algebratizing topology. While most everything boils down to polynomials, the more complicated setting and more complicated objects allow us to do work more easily than if we were stuck with polynomials. | Look at algebraic curves. For every curve, we get the field of rational functions on that curve, and finite extensions of this field give you branched coverings. The automorphisms of this extension permute the sheets of this cover. | These notion are actually the same thing if you look at it through the lense of Grothendieck topology. A good source for this is Milne's Etale Cohomology, but scheme theory is a prerequisite. Long story short: Grothendieck topologies are object defined using category theory that generalise the notion of a topological space. In this setting, one can see a field as a space with one element, and algebraic field extensions are connected coverings of the base field. Then, the algebraic Galois group is the same as the Galois group of the covering. Furthermore, the algebraic closure (unless I'm mistaken and it is the sebarable closure?) is the universal covering and the Galois group is the fundamental group of the "space" corresponding to the field. | Thank you, scrolled through the table of contents and this is exactly what I was looking for! |
[
"Mathematics books that follow breaking and building style of narration?"
] | [
"math"
] | [
"tnn23f"
] | [
55
] | [
""
] | [
true
] | [
false
] | [
0.97
] | I was reading through the Amazon reviews of the book by T.W. Korner when I came around this particular review: This book derives calculus in the style of a Physicist, but with the understanding and clarity of a Mathematician whose job it is to understand exactly how the mechanism works. In the closing stages of the book we begin to learn how the Physicist's approach eventually fails. It would be great if the author would now write an introductory book in between this one and his "companion to analysis", which explains which problems are being fixed by rigorous analysis as this new analysis is introduced for the first time. Most books on Quantum Mechanics start by describing how the "old theories" break down - perhaps analysis should be taught in this way too - as a story of enlightenment. It struck me that I don't know of any such book when it comes to analysis. Yet it seems to be a very good teaching style. Developing some theory then showing why and where it breaks down and how later theories have tackled the problem by developing more mathematically tight theories. In fact, I would like to know about books that teach subject ( ) like this. Not only analysis. I think one example of such a book is by Knuth et al. If you know about such books please let us know! | Bressoud has done a few books in this style, "A radical approach to Real Analysis" and "A radical approach to Lebesgue's Theory of Integration". Radical here means historical. He starts with the same point of view as the original developers of the theory and explains the problems they were trying to solve. | An Introduction to Measure Theory by Terence Tao takes an approach that I think is similar to what you are describing. | As a physicist, even I can say that it's much different studying eg. De Rham's cohomology theory from a physics book (where motivation and important results are clearly presented) than from an intended math-grad audience book, where it's more like theorem->proof->lemma, rinse and repeat. | Why do people still assume physicists are mathematically illiterate? | I found it kind of hard to follow the real analysis book, but I also may not have been ready for it. Ambitious high school student at the time. |
[
"what problems seem most mysterious in Math to you?"
] | [
"math"
] | [
"tnfqvi"
] | [
25
] | [
""
] | [
true
] | [
false
] | [
0.89
] | Sure...any problem that you can't solve can be considered mysterious, but some seem different than others. Considering how wide the field is, I'm curious what problems appeal to you the most? Some spend their lives solving these problems, which is crazy to me, so I'm always interested in hearing how these things captivated you and potentially any work you've done. A common one would be prime numbers. Is there a pattern or any way to figure out, efficiently, on whether a number is prime? Or in general, number theory, but it can be any branch. Just trying to spark the imagination here... ✨️ Or have you personally dealt with a problem only to figure out later it was relatively simple? I always find such things interesting. Thanks for sharing if you do! | An elliptic curve is a type of algebraic curve (roughly meaning a curve defined as the set of solutions of a polynomial equation) on which the set of points admits a group law. An isogeny between two elliptic curves is a map defined by polynomial equations and that is also a group homomorphism. An isogeny from a curve to itself is an endomorphism, and the set of endormophism of a given elliptic curve is a ring, for pointwise addition and composition. Now, for every curve, its endomorphism ring is known to either be the ring of integers, the ring of algebraic integers in an imaginary quadratic extension of Q or a maximal order in a quaternion algebra over Q. The last case only happens in some settings, and in particular in can happen for a curve defined over a finite field. Such a curve is said to be supersingular. It is fairly easy to identify a curve as supersingular. However, it is very difficult to compute the actual structure of the endomorphism ring of a given supergsingular elliptic curve. In fact, this is used as a hard problem in the field of isogeny-based cryptography. I find it terribly insulting that we don't have a clear algebraic method to determine the ring of endomorphism of a supersingular elliptic curve. | This is genuinely fascinating; I feel like I'm being tempted to the dark side lmao. | The volume of a torus has a pi squared. Stefan’s constant has a pi to the fifth. The normal distribution has a square root of pi. Same for Stirling’s formula. | The volume of a torus has a pi squared. Stefan’s constant has a pi to the fifth. The normal distribution has a square root of pi. Same for Stirling’s formula. | I think the Riemann zeta function is particularly fascinating. When you think of the critical strip and how all the nontrivial zeroes lie within the strip. I think of the impact of solving the Riemann hypothesis and it’s exciting. |
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