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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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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
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9answers
1k 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
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3answers
60 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
56 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 ...
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0answers
16 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 ...
2
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0answers
51 views

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
59 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 ...
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6answers
285 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
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1answer
88 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 ...
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3answers
276 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{...
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2answers
282 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 ...
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2answers
40 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
101 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 ...
4
votes
6answers
291 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
49 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 ...
6
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3answers
108 views

prove $\sum_{n=0}^{\infty}\frac{\Gamma^2(n+1)}{\Gamma(2n+2)}=\frac{2\pi}{3^{3/2}}$

I am seeking alternate proofs for $$\sum_{n\geq0}\frac{\Gamma^2(n+1)}{\Gamma(2n+2)}=\frac{2\pi}{3^{3/2}}$$ Here's mine: Recall that, for $x\in(0,2)$, $$\frac1x=\sum_{n\geq0}(1-x)^n$$ Hence we have ...
12
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1answer
317 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 ...
2
votes
2answers
432 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
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0answers
35 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 ...
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0answers
33 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 "...
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2answers
102 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 ...
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0answers
95 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
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1answer
111 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 ...
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16answers
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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}\...
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1answer
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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
86 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 ...
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1answer
49 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 ...
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1answer
109 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
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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
38 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
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1answer
77 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 ...
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1answer
36 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 ...
10
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3answers
315 views

Categorical characterizations of ring properties

Looking at the interesting list of ring properties that are inherited from a ring $\mathcal{R}$ by its polynomial ring $\mathcal{R}$[X] and remembering a question I once asked I want to repeat the ...
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2answers
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Which properties does the polynomial ring $\mathcal{R}[X]$ inherit from the ring $\mathcal{R}$?

I've learned that $\mathcal{R}$ is commutative → $\mathcal{R}[X]$ is commutative $\mathcal{R}$ has no zero divisors → $\mathcal{R}[X]$ has no zero divisors $\mathcal{R}$ is ...
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0answers
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Applications of the compactness theorem. [closed]

It's well know that the compactness theorem has many aplication in model theory, its main shows existence of nonstandars models of aritmetical and the real numbers, and not elementary of some theories ...
0
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2answers
31 views

Total Cost of List of Items

Suppose you have a list of items. The end items cost 1 dollar each. The items next to the ends cost 2 dollars each and the items next to those (towards the middle) cost 3 dollars each and so on. Hence ...
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0answers
49 views

Group theoretic Characterizations of Large Cardinals

Vopenka's principle is characterized (in the category theoretical definition) by no large full subcategory of a locally presentable category being discrete. In other words, proper classes of types of "...
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11answers
4k views

BIG LIST: Statements that look obviously false but cannot be disproved

I'm looking for statements that look obviously false but have no disproof (yet). For example The base-10 digits of $\pi$ eventually only include 0s and 1s. To make this question a little objective, ...
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2answers
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What are some properties proved to have largest elements?

I know of various properties of numbers that are known to have no largest element (like natural numbers, primes, etc.) and I know of unproven conjectures that certain properties have a largest element ...
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4answers
267 views

Category Theory & Biology

Category theory is becoming more and more used in the following fields (besides others) (1) Quantum physics (e.g. C. Isham and B. Coecke et al) (2) General relativity (e.g. A.K. Guts et al) (3) ...
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0answers
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What are some examples of fascinatingly complex diagrams in mathematics?

Usually, in mathematics, diagrams appear to be pretty simple, even if they do look cool. Occasionally though, there are the instances where the diagrams are counterintuitive to what one should think ...
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7answers
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Short but Substantial Math Papers [closed]

I am looking for short papers that made a significant impact on the mathematics community. I have already seen: interesting-but-short-math-papers and, What is the Shortest Ph.D. Thesis? on math ...
0
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0answers
129 views

Modern Mathematical Crankery: A List.

I'm sorry if this question is off topic. I'd also like to apologise if this is deemed too broad or otherwise a poor question. The Question: I'm interested in examples of modern mathematical ...
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votes
1answer
52 views

Significance of $\int_a^bf(x)dx=(b-a)\int_0^1f\{(b-a)x+a\}dx$

I learnt this property of definite integrals- $$\int_a^bf(x)dx=(b-a)\int_0^1f\{(b-a)x+a\}dx$$ Now, is there any nice geometrical interpretation of this property (maybe using areas) that would give ...
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1answer
99 views

Combinatorial problems that were solved using the representation theory of finite groups?

Question: What are some examples of problems in combinatorics that were solved using the representation theory of finite groups ? I am aware the representation theory of finite groups plays a role in ...
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23answers
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Non-associative operations

There are lots of operations that are not commutative. I'm looking for striking counter-examples of operations that are not associative. Or may associativity be genuinely built-in the concept of an ...
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8answers
971 views

Interesting questions (with answers) about concepts in topology for an amateur audience

I have been asked to hold an introductory math quiz for the Freshmen batch in my college. It entails interesting questions about different areas of mathematics presented in such a way so that it seems ...
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0answers
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What are the properties of abundancy numbers?

Define abundancy numbers as the rational numbers that are equal to the abundancy index of some integer (not to be confused with «abundant numbers», which are natural numbers with abundancyindex ...
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0answers
122 views

What interesting properties does $A^3$ have?

Suppose $G$ is a finitely generated group. Suppose $A^1$ is a finite subset of $G$, such that $\langle A^1 \rangle = G$. Let’s define $A^n \subset G$ for $n \in \mathbb{N}$ using the recurrent ...
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7answers
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What problems have been frequently computationally verified for large values?

Although any theorem (or true conjecture) can be computationally checked, many long-standing open problems have been computational verified for very large values. For example, the Collatz Conjecture ...