Questions tagged [big-list]

Questions asking for a "big list" of examples, illustrations, etc. Ask only when the topic is compelling, and please do not use this as the only tag for a question.

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74 views

Short, yet powerful proofs

One of my favorite proofs is the following: Claim: There exists irrational numbers $\alpha$ and $\beta$ such that $\alpha^{\beta}$ is rational. Proof: Let $\alpha = \sqrt{2}^{\sqrt{2}}$ and $\beta = \...
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What are some sources of recommendations of Mathematical books - both Recreational and Popular Mathematics as well as Technical Mathematics?

I'm a lover of books about Mathematics. However, I have noticed it's quite hard to stay in touch with new releases or book reviews in the world of Mathematics. I use GoodReads for recommendations in ...
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The connection between Kolmogorov complexity and mathematical logic

We know that Kolmogorov complexity has connections to mathematical logic since it can be used to prove the Gödel incompleteness results (Chaitin's Theorem and Kritchman-Raz). Are there any other ...
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272 views

Interesting/Useful tricks in linear algebra. [closed]

This may be an opinion based-based question, since everyone understands things differently, but what are some interesting/useful tricks in linear algebra? For example, if we know that the determinant ...
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1answer
92 views

What are some questions that seem easy but aren't?

I've only just started to study a degree in Mathematics and I find it extremely satisfying to have a seemingly easy question that is incredibly difficulty or tricky to work out. I've looked into heaps ...
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77 views

Mathematical proofs that are neither fully understood nor verifiable?

In 1948, Von Neumann mentioned that some processes in nature might be irreducibly complex. “It is not at all certain that in this domain a real object might not constitute the simplest description of ...
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76 views

Major unsolved problems in mathematics which were first solved by abstract algebra

I've been wanting to ask this for a while now and hope that it will enlighten me on why precisely abstract algebra is powerful. What were some major unsolved problems in mathematics which were first ...
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1answer
216 views

Rigorous and comprehensive textbooks on precalculus [closed]

I am looking for comprehensive and rigorous textbooks on precalculus that provide proof for all the formulas and theorems they mention. You can suggest multiple books on different topics like ...
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25 views

Noncommutative probability puzzles?

Probability is fun mainly because of the many puzzles. Are there any puzzles in noncommutative probability? I would guess that quantum mechanics could offer puzzles like those, but I can't find any.
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146 views

Math problems that are easier to solve using an exponential-exponential coordinate system

Are there any math problems that are easier to solve on a exponential-exponential coordinate grid? (as opposed to a normal x-y cartesian coordinate grid). Some problems are easier to solve in ...
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Examples of simple but highly unintuitive results? [closed]

QUESTION: What are some simple math problems whose answers are highly unintuitive, and what makes them so? There are plenty of unintuitive and frankly baffling results in math, like the Banach-Tarski ...
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Where to do PhD-level work on topos theory and “categorical mathematics” in 2020/2021? [closed]

I'm doing my master's in mathematics, and I will be graduating next year. I'm looking to start a PhD soon, perhaps next year, most likely in two years. I'm really interested in category theory and ...
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20 views

List of Laplace transform differential equations

I’ve been starting to learn about solving differential equations using the Laplace transform, and I wanted good, non obvious examples. I think this can help beginners(like me) get where it is ...
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1answer
54 views

What are the most interesting math riddles you know? [closed]

I am talking about riddles that do not involve play on words. The riddles should be logically and mathematically correct too.
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1answer
56 views

Examples in number theory where a heuristic argument fails

Many conjectures in number theory are motivated by heuristic arguments, and many results that are known to be true can be predicted by heuristic arguments. To give an example, consider the Euler ...
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84 views

What is something incredibly interesting/insightful about mathematics that requires only a basic understanding of mathematics? [closed]

As a fun project, I want to create an interesting, insightful, yet educational video about some fascinating topic/concept/idea in mathematics that only requires a basic understanding of mathematics so ...
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1answer
55 views

Accessible examples of using ideas in mathematical logic to solve problems in “main-stream” mathematics

It is perhaps well-known that ideas from mathematical logic (esp. model theory) can help solve problems in "main-stream" mathematics, e.g. using ideas from model theory to solve problems in algebraic ...
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60 views

What is the best book for the following mentioned courses?Internet resources too please.

1.Motivation:Learning physics and computer science and of course the enchantment of maths itself. 2.Background:A little bit of algebra,Trigonometry utmost basics and basic geometry 3.Why it is ...
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138 views

Examples of problems/proofs which can be (surprisingly) represented in terms of graphs

I am looking for examples of problems or proofs in mathematics which have a equivalent representation in terms of graphs, which makes solving the problem easier. For example, the problem of finding ...
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3answers
164 views

Examples of additive categories

There are a lot of interesting and creative examples of categories, such as for example, the category whose objects are the positive integers and the set of morphisms from $n$ to $m$ is the set of $m \...
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1answer
13 views

Measure of error in smoothness of approximation of sphere

I'm meshing a sphere and am solving a physics problem on this. What I want to show is that the error in the model scales like$$ \varepsilon = \epsilon^p, $$ where $\epsilon$ is the "error" in the ...
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42 views

Examples of false conjectures suggested by probabilistic evidence

Background Recently I read about some attempts of finding the exact order of Mertens function. It is conjectured that $$0<\limsup_{x\to\infty}\frac{M(x)}{\sqrt x(\log\log\log x)^{5/4}}<\infty\ (...
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1answer
51 views

First class mathematics that is useful

In Oksendal's Stochastic Differential Equations he writes in the first few pages: The Kalman-Bucy filter is an example of a recent mathematical discovery that has proved to be useful - it is not just ...
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4answers
373 views

Simple understanding of advanced math

Title might be a bit vague, so I will explain further here. I am compiling a list of examples of how a person may realize some mathematical result is either obvious or unsurprising from the ...
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1answer
840 views

What are some good university level texts with solutions?

What are some good university level texts for which the exercises have solutions? Any textbooks, books, lecture notes etc. whose problems and exercises have solutions would be great. Please indicate ...
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3answers
142 views

(Non-trivial) Maths problems that don't require any formal maths education [closed]

Does anyone know good brain teasers that show the nature of mathematics to people with little or no mathematical knowledge? Or do you know a book with such problems? Criteria are that the problem ...
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4answers
181 views

Mathematical discoveries that have occurred by cleverly linking two initially unrelated topics?

I have heard mathematical proofs often require cleverly linking two areas of maths which initially seem disconnected. Could anyone provide an example of this, as I feel at my level of study, many ...
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0answers
41 views

How do the various classifications of $p$-divisible groups relate?

There are several dozen 'classification' theorems of various notions of $p$-divisible groups in terms of crystals (where 'crystal' can seemingly mean various things), thanks to theorems of Dieudonné, ...
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21 views

Examples of parametric equations for compact $n$-manifolds embedded in $\mathbb{R}^m$, $m>n\ge3$?

I've been messing with some computational topology texts and want to do some manifold/surface reconstruction on various data sets. I'd like to test on some purely geometric examples. Are there any ...
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4answers
86 views

Examples of triangles, which related ellipses are perfectly packed with circles.

Ellipse can be perfectly packed with $n$ circles if \begin{align} b&=a\,\sin\frac{\pi}{2\,n} \quad \text{or equivalently, }\quad e=\cos\frac{\pi}{2\,n} , \end{align} where $a,b$ are the ...
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1answer
86 views

Simple applications of higher derived functors

I was surprised to not have found this question discussed before on this or any other forum, hence I am posting it myself. From an abstract point of view of homological algebra derived functors are ...
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1answer
22 views

How to find the difference of two arrays.

I am working to find if a list has changed in python. The list contained a list inside each of its elements. Like this, [[0,1,2,3],[2,7,3,4]] Now image the list is longer and much bigger than this. ...
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1answer
46 views

Another big list with integrals involving Lambert's function Omega,$\pi$,$e$ and others constants…

Hi I would like to create another big list always with Lambert's function , but this time with the constant Omega and pi (don't mind if there is others constants as $e$) .I have : $$\int_{1}^{e} \frac{...
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Real life applications of the Lefschetz fixed point theorem

I'm looking for real life applications of the Lefschetz fixed point theorem. I encountered this link: https://mathoverflow.net/questions/127045/fixed-point-theorems However, I was hoping for ...
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2answers
120 views

reference request: Plouffe's Lambert-type series for $\zeta(2n+1)$

According to Wikipedia, Plouffe gives the series $$\begin{align} \zeta(5)&=\frac1{294}\pi^5-\frac{72}{35}\sum_{n\ge1}\frac1{n^5(e^{2\pi n}-1)}-\frac2{35}\sum_{n\ge1}\frac1{n^5(e^{2\pi n}+1)}\\ &...
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1answer
78 views

A little game around Lambert's function and simple and beautiful integral

I like very much to play with the Lambert's function so I was wondering if we can find integrals collecting famous constant and involving just logarithm and Lambert's function . I have proposed an ...
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37 views

Simple examples of rings with non-trivial Picard group and infinite Brauer group

I am working on a certain problem about commutative rings which has an obstruction involving the Picard group and the (algebraic) Brauer group of the ring. The obstruction is trivial at least when the ...
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20 views

$s_1-\max{|\lambda_i|}$

Let $M$ be some square matrix with entries uniformly bounded, not necessarily normal. Let $s_1$ denote the largest singular value, and let $\lambda_i$ be eigenvalues of $M$, $|\lambda_i|$ is the norm ...
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2answers
131 views

Integral representations of the Fibonacci numbers

I would like to assemble a thorough list of integral representations of various number sequences, and the Fibonacci numbers are naturally my first choice. So, my question: What are some integral ...
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1answer
70 views

Can you list all the finite series that can be solved in a closed form?

I'm interested to know all the finite series that can be solved in a closed form (e.g. the geometric series)
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1answer
67 views

Set Identities in which Venn Diagram Proofs do not work

I want to give my students some example problems for proving set equality about why you cannot take the general venn diagram proof at face value. I read this post that stated venn diagrams are not ...
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27 views

Summation of series of functions by Fourier methods

Last week I tried to find a proof for the so called partial fraction decomposition of the cotangent, that is $$\pi \, \cot(\pi k) = \sum_{m=-\infty}^{\infty} \frac{k}{k^2-m^2}$$ For the proof I ...
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9answers
434 views

Unexpected use of polynomials in combinatorics

Can someone please post some (relatively easy, say high school level) combinatorial problems which can be solved with polynomials but NOT generating functions. Edit 30. 1. Related to this post in ...
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21answers
5k views

Unexpected appearances of $\pi^2 /~6$.

"The number $\frac 16 \pi^2$ turns up surprisingly often and frequently in unexpected places." - Julian Havil, Gamma: Exploring Euler's Constant. It is well-known, especially in 'pop math,' that $$\...
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3answers
76 views

Big list: collecting results of diagram chasing

I plan giving a homework problem on linear algebra for my students and my idea was to collect many results of diagram chasing. Could you help me and give me some results of diagram chasing you know? ...
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2answers
88 views

Rational approximations for $\pi$ using Fibonacci numbers?

It is (well?) known that $$\frac{\pi} 4 = \sum_{k=1}^\infty \arctan \left ( \frac1{F_{2k+1}} \right )$$ Where $F_k$ denotes the $k$-th Fibonacci number. However, any truncation of this sum is ...
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2answers
220 views

Recommendations for differential geometry textbooks that develop geometric intuition.

I'm currently self-studying complex analysis (CA), and reading "Visual Complex Analysis" by Tristan Needham. I'm absolutely fascinated by how much geometric intuition he provides for the key findings ...
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0answers
51 views

Mathematical Consequences of $P=NP$ or $P\neq NP$ [closed]

It is common in mathematics to assume some unknown hypothesis or conjecture to be true or false, and then prove results dependent on said assumption. This can lead to many interesting developments ...
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1answer
32 views

Application of linear transformation and matrices

I have been studying linear algebra for some time now, and I have seen some really interesting applications of linear transformation and matrices such as finding the integral $\int x^2e^x dx$. Does ...
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79 views

Identities about matrices that are useful in contests / olympiads [closed]

I would love for students ( such as myself ) that want to participate in mathematics competitions or olympiads to have a place from where they could gather theorems, identities or maybe instructive ...

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