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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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Ways of geometrical multiplication

There are at least five ways to multiply two natural numbers $a$ and $b$ given as integer points $A$ and $B$ on the number line by geometrical means. Two of them include counting, the others are ...
0
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0answers
10 views

Errata in Convex Analysis and Minimization Algorithms by Hiriart-Urruty and Lemaréchal.

Hiriart-Urruty and Lemaréchal have written Convex Analysis and Minimization Algorithms I and II. Is there an errata list for these books?
5
votes
7answers
85 views

Examples of the Pigeonhole Principle

As most of you might know, the Pigeonhole Principle basically states that If $n$ items are put into $m$ containers, with $n>m$, then at least one container must contain more than one item It ...
1
vote
0answers
34 views

List of Integration Formulas [duplicate]

While going through some of my notes, I came across a formula for evaluating an integral of the form: $$\int_0^\infty {\frac{dx}{x^n+1}} = \frac{\pi}{n\sin(\frac{\pi}{n})}$$ I was wondering if any ...
3
votes
0answers
51 views

Algebra & Artificial Intelligence (AI)

Artificial intelligence, especially deep learning & neural networks for image processing and classfication, are related to statistics and physics e.g. as decribed in below papers. Statistics and ...
2
votes
1answer
106 views

Category Theory & Artificial Intelligence (AI)

Category theory turns out to be useful in more and more areas. (see e.g. MSE - Category Theory & Biology) Question. Does anyeone know of some connection of category theory to (convolutional) ...
0
votes
1answer
85 views

Make a list of algebraic, topological and dynamical properties of the circle.

For example, there is only one cyclic group of order $m$ in the circle, the circle is compact and connected, all its proper compact and connected subspaces are arcs, is a minimal dynamical system ...
0
votes
2answers
63 views

Multiple proofs of $\sum_{d|n}{\phi(d)}=n$ [duplicate]

I am looking for multiple proofs of that statement: here $\phi(n)$ denotes the Euler’s totient $$\sum_{d|n}{\phi(d)}=n$$ Here’s one: By unique factorisation theorem: $n=\prod_{k=1}^{m}{p_k^{\...
4
votes
0answers
81 views

Collection of less well-known, non-trivial, elegant story proofs (ie, “double counting proofs”) of combinatorial identities

By story proof I mean proving a combinatorial identity by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. The ...
2
votes
1answer
53 views

Pathological, continuous functions

Today in my introduction to measure theory course, the professor said that often when we think of continuity, what we're actually thinking about is smooth functions. We've studied the Cantor set and ...
12
votes
2answers
577 views

Important Olympiad-inequalities [duplicate]

As an olympiad-participant, I've had to solve numerous inequalities; some easy ones and some very difficult ones. Inequalities might appear in every Olympiad discipline (Number theory, Algebra, ...
7
votes
2answers
132 views

The “I wish I had read that first” textbook list

Here, I am trying to collect as many "I wish I had read that textbook first" textbook names as possible, for as many different topics in mathematics as possible. The "I wish I had read that textbook ...
5
votes
2answers
177 views

“Milk” the integral $\int_0^\infty\left(\frac{x^2}{x^4+2ax^2+1}\right)^r\frac{x^2+1}{x^2(x^s+1)}\mathrm dx$

I found the following integral in chapter $13$ of Irresistible Integrals, and I would like to see which conclusions you can reach from it. My goal in asking this question is to see which methods I can ...
8
votes
2answers
87 views

What natural ways of partitioning a group $G$ are there?

What natural, or at least useful, ways are there to partition a finite group $G$? The two examples that come to mind are: Partitioning $G$ into all the left (or right) cosets of a subgroup $H$ of $G$....
1
vote
1answer
71 views

Html5 Math Applets. Interactive free online

I was an aficionado at collecting links from websites with math java applets that allowed interaction to learn mathematical concepts visually and interactively. My favorite was http://www.ies-math.com/...
17
votes
1answer
292 views

Killing a butterfly with a bazooka [duplicate]

Let $n\ge3$. Prove that $\sqrt[n]2\notin\Bbb Q$. Let us suppose that $\sqrt[n]2=p/q$, that is $2q^n=p^n$, so $q^n+q^n=p^n$, against FLT. Do you know similar examples, in which simple problems are ...
10
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1answer
89 views

Collection: Results on stopping times for Brownian motion (with drift)

The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties: distributions of stopping times (Laplace ...
1
vote
0answers
23 views

Sequence $\rightarrow$ Generating function simple practice problems

I am trying to teach myself more about generating functions and how they "generate" sequences. Specifically I am trying to learn more about how to go from sequence to function and I would like some ...
1
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0answers
47 views

Forward-backward induction

I've seen the famous proof presented by Cauchy for the AM-GM inequality but what other neat proofs use forward-backward induction? Is it fundamentally inextricable from ordinary induction (are there ...
2
votes
0answers
395 views

Theorems in the form of “if and only if” such that the proofs of BOTH directions are nontrivial

In Theorems in the form of "if and only if" such that the proof of one direction is extremely EASY to prove and the other one is extremely HARD, bof suggested the question in the title is ...
12
votes
9answers
2k views

Theorems in the form of “if and only if” such that the proof of one direction is extremely EASY to prove and the other one is extremely HARD [closed]

I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. ...
3
votes
3answers
76 views

What are some applications of mathematics whose objectives are not computations?

In mathematics education, sometimes a teacher may stress that mathematics is not all about computations (and this is probably the main reason why so many people think that plane geometry shall not be ...
2
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1answer
74 views

Books like Terence Tao's An Introduction to Measure Theory

This is an attempt to get a list going of books in mathematics (algebra, topology, geometry etc.) that in the spirit of Terence Tao's An Introduction to Measure Theory: According to my viewpoint, the ...
0
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0answers
17 views

List books with end of chapter summaries

I am looking to compile a list of maths textbooks with end of chapter summary notes on each chapter. I have found that using such summaries makes revision much easier, especially if I have not taken ...
1
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1answer
97 views
+500

Fractional/Integer Based integrals

I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?
2
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1answer
70 views

Importance of Axiom of Choice (or its weak form) in Real Analysis

By Wikipedia article on Axiom of dependent choice, it is necessary to have it for development of real analysis. The ${\mathsf {DC}}$ says: Axiom (${\mathsf{DC}}$). For any nonempty set $X$ and ...
8
votes
6answers
376 views

A list of Rudin-style textbooks

I'm fond of baby Rudin: elegant presentation, "clever" proofs, certain terseness, difficult exercises, etc. I'm working up my enthusiasm to seek similar textbooks in other areas of math, e.g. linear ...
3
votes
1answer
91 views

I am looking for a modern and thorough exposition for presentations of groups

Conider the following abstract description for the Quaternion group: $$\langle x,y\mid x^{4}=1,x^{2}=y^{2},y^{-1}xy=x^{-1}\rangle$$ This description is called a presentation of the Quaternion group ...
6
votes
3answers
304 views

$\int_{0}^{\infty} \frac{1}{1 + x^r}\:dx = \frac{1}{r}\Gamma\left( \frac{r - 1}{r}\right)\Gamma\left( \frac{1}{r}\right)$ [duplicate]

As part of a recent question I posted, I decided to try and generalise for a power of $2$ to any $r \in \mathbb{R}$. As part of the method I took, I had to solve the following integral: \begin{...
17
votes
2answers
317 views

$\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{ax^2+bx+c}=\pi$ similar identities

I recently found that $$\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{ax^2+bx+c}=\pi$$ iff $$b^2-4ac=-4$$ I found it by integrating $$I=\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{ax^2+bx+c}$$ If the ...
2
votes
2answers
51 views

What properties of a linear map can be determined from its matrix? [closed]

I am currently taking a proofs based linear algebra course for math undergraduates. It's been almost two years since I took a more computational linear algebra course (solving matrix equations, ...
10
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0answers
117 views

Peculiar pictures in advanced maths books

I have recently started reading Introduction to Symplectic Topology by McDuff and Salamon and I came across this picture: I find it very funny and really interesting. I read on Wikipedia that Ian ...
5
votes
6answers
335 views

Request for crazy integrals

I'm a sucker for exotic integrals like the one evaluated in this post. I don't really know why, but I just can't get enough of the amazing closed forms that some are able to come up with. So, what ...
1
vote
1answer
55 views

Exercises for commutative algebra

I'm currently studying for my final exam of commutative algebra. In the class we covered Artinian rings, Dedekind domains, Integral closures, Grobner basis, (discrete )Valuation rings... I'm ...
12
votes
1answer
326 views

Has the Riemann Rearrangement Theorem ever helped in computation rather than just being a warning?

A decent course in elementary analysis will eventually discuss series, absolute convergence, conditional convergence, and the Riemann Rearrangement Theorem. However, in any presentation I've seen, in ...
1
vote
2answers
459 views

Could every Hausdorff space be induced by a total order relation [closed]

Let $(H,\mathcal T)$ is a Hausdorff space then is there any total order relation on $H$ such that the topological space induced by the order relation be the same $(H,\mathcal T)$? Thanks a lots ...
2
votes
0answers
37 views

Is there any research problems in number theory an undergrad can give a try?

I am an undergrad student and have a deep interest in number theory and now I am learning analytic number theory own for 3-4 months. I have also covered group theory. I want to work on research ...
0
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0answers
34 views

Books like “Jeux de l'esprit et divertissements mathématiques” [duplicate]

I have enjoyed the second edition of the title's book (Nouveaux jeux de l'esprit et divertissements mathématiques) and now I have bought the first edition. To set some context, the book has some "...
4
votes
2answers
106 views

Short mathematical proofs for teaching

I am looking for short proofs in order to illustrate undergraduate notions. Most of the time students struggle with technical exercises without having the time, before going on with the following ...
1
vote
0answers
97 views

How does the locally compact group play a role in many branches of math? [closed]

It seems that Fourier analysis/harmonic analysis plays an important role in math, at least in number theory and statistics. It also seems to me that talking about it is talking about locally compact ...
5
votes
1answer
122 views

What other uses are there for Prime numbers? [duplicate]

Simple question out of curiosity... Beside the use of cryptographic safety and prime factorization, what other uses are there for prime numbers? Thank you. Edit: To clarify and not confusing with ...
41
votes
16answers
4k views

Possible definitions of exponential function

I was wondering how many definitions of exponential functions can we think of. The basic ones could be: $$e^x:=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ also $$e^x:=\lim_{n\to\infty}\bigg(1+\frac{x}{n}\...
19
votes
1answer
454 views

Is there an integral for $\frac{1}{\zeta(3)} $?

There are many integral representations for $\zeta(3)$ Some lesser known are for instance : $$\int_0^1\frac{x(1-x)}{\sin\pi x}\text{d}x= 7\frac{\zeta(3)}{\pi^3} $$ $$\int_0^1 \frac{\operatorname{...
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votes
1answer
92 views

What mathematics cannot be reduced to pigeonhole?

Pigeonhole is a fundamental principle without which state of mathematics will be much different. However what examples of good mathematics has not yet been proved and cannot be proved with pigeonhole ...
0
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1answer
50 views

Major Mathematical Results Understandable by An Intelligent Lay Person [closed]

Question: What are some explicit examples of major mathematical results understandable by an intelligent lay person ? Here are few I managed to google: Euclid - infinitude of primes. Cantor - the ...
12
votes
1answer
177 views

Martingale theory: Collection of examples and counterexamples

The aim of this question is to collect interesting examples and counterexamples in martingale theory. There is a huge variety of such (counter)examples available here on StackExchange but I always ...
14
votes
4answers
2k views

Easy to understand examples of category theoretic theorems that are useful

Every now and then I hear a category theorist saying that category theory is a unifying language for mathematics, and that category theory proves general theorems that some people would prove ...
2
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0answers
40 views

General findings about Galois groups

To get a better feeling for Galois groups I'd like to know some general cases that allow to tell a Galois group from a polynomial and vice versa. The most simple example I came about is $P(x)\in \...
3
votes
1answer
78 views

Metrics on families of functions

Let $\mathcal{F}$ be a family of functions $D\subseteq \mathbb{R}^n\rightarrow \mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally ...
1
vote
1answer
40 views

Problem that is very compute-intensive but also has a good approximated solution?

For my work I'm looking for good examples of problems that are very compute-intensive and at the same having good and fast approximated solutions. Could you give me some examples? I'm unsure if this ...