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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6
votes
3answers
65 views

Tricks for quickly reading off the eigenvalues of a matrix

I noticed that some mathematicians have an uncanny ability to identify the eigenvalues of matrices without doing much in the way of computation. For instance, one might notice that all the rows have ...
2
votes
1answer
60 views

Classical results of Algebraic Geometry using cohomology.

I am looking for classical results of Algebraic Geometry that can be proved using cohomology. For example, Riemann-Roch Theorem and Bezout Theorem admits short proofs (providing that you know enough ...
1
vote
1answer
27 views

Comprehensive handbook of formulas

I'm looking for comprehensive, free and online handbook of mathematical formulas, in the form of pdf file. I mainly aim at undergraduate mathematics. Do you know any link? Thanks.
5
votes
1answer
85 views

List of matrix properties which are preserved after a change of basis

Lately I encountered such a problem. Which of the properties of matrices are preserved after a change of basis ? (orthogonal basis and square matrix are preferred in the first place) Maybe it is a ...
5
votes
2answers
78 views

Book about intuition behind Lebesgue measure

I recently completed a course in Real analysis covering Lebesgue and Borel measure, Fourier series, $L^p$ spaces and such. I can solve problems in these topics but am afraid that I do not truly ...
6
votes
4answers
238 views

'Almost rational' integrals with no known closed form?

I recently stumbled upon an 'almost rational' integral, namely: $$\int_0^{\pi/2} x \frac{\sqrt{\sin x}-\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} dx=0.231231222\dots \approx 0.231231231\dots= ...
1
vote
1answer
77 views

Advanced examples of categories

I'm looking for some example of categories which requires some effort to prove that it is a category (For example it is straightforward to prove that $\mathbf{Set}$ is a category, I don't want that ...
6
votes
0answers
140 views

Separating Heavier from the Lighter Balls

This was posted Here and received a good answer, solving the general questions in either $n$ or $n+1$ moves, which is by just half a move on average "less good" than my manual solutions here. ...
0
votes
1answer
16 views

Calculating the amount of times a binary search could run (worse case) without a calculator/calculating base 2 logs without a calculator.

Ok so I had a question on a test that I had to do without a calculator. And I can not figure out how in the world I am supposed to do it without a calculator. The question asked to find how many ...
1
vote
1answer
41 views

What are some pairs of mathematically-important functions that differ only at a few points?

Examples would include things like $$f(x, y) = \begin{cases} x^y & \text{ if } (x, y) \neq 0 \\ 0 & \text{ else} \end{cases}$$ versus $$g(x, y) = \begin{cases} x^y & \text{ if } (x, y) ...
2
votes
3answers
295 views

How many ways are there to prove Cayley-Hamilton Theorem?

I see many proofs for Cayley-Hamilton Theorem in textbooks and net, so I want to know how many proofs are there for this important and applicable theorem.
1
vote
1answer
32 views

Results on “subtraction” of measures and outer measures?

Most results I have seen involves addition of measures For example, let $m^*$ and $m$ be Lebesgue outer measure and Lebesgue measure respectively, then given $A = \bigcup\limits_{n = 1}^\infty E_n, ...
3
votes
1answer
44 views

Foliations vs Laminations

What's the big difference/similarity between foliations and laminations? What kind of theorems hold for both of them? Is there something that makes them essentially the same/different?
1
vote
1answer
45 views

Desirable properties of statistical estimators?

What are some of the properties that people will consider when designing a statistical estimators? For example, unbiasedness and sufficiency are some of the factors considered. Please give some ...
13
votes
2answers
131 views

$a^x+b^x=c^x$ in geometry

The Pythagorean theorem. Let $A$, $C$, $B$ be three points on a line in this order, and let $D$ be another point, such that $\angle ADC =\angle CDB = 60^\circ$. Let $a=AD$, $b=BD$, $c=CD$. Then, ...
1
vote
4answers
45 views

What are the principal (different) mechanisms of infinite descent proof?

I’m interested in building a list (including, where possible, links to proofs/papers/examples) which presents all known mechanisms of infinite descent (ID). I think this list would best be presented ...
2
votes
0answers
27 views

On the alternative stamentes of the famous Sperner's Lemma.

The Sperner's lemma can be stated as follows. Lemma of Sperner. Let $\Omega$ an fintie set with $n$ elements. If a family $\{ A_i \}_{1\leq i \leq N}\subset \Omega$ of subsets satisfies the ...
0
votes
0answers
30 views

How to write the mathematic formulation of a pairwise sum?

In the normal summation, we can do this: And that would sum a list X as such in Python: ...
8
votes
0answers
65 views

Categorical formulations of basic results and ideas from functional analysis?

I'm taking a first (undergrad) course on functional analysis. Though the material is nice, the approach seems very ad hoc and in a sense, near-sighted (?). I was wondering whether the/a big picture ...
5
votes
2answers
183 views

Ugly solutions to easily stated problems [closed]

I recently saw a very hideous closed form for a quartic equation here: Does a closed form solution exist for $x$? For fun, I'm wondering about surprisingly ugly solutions/ complicated machinery ...
0
votes
0answers
43 views

What are some interesting results that could be derived if some conjectures were true/false?

I recently came up on Djikstra's idea of a fictional company called Mathematics.Inc which produced proofs as trade secrets which could then be used by customers.. For example, it produced the proof of ...
0
votes
1answer
30 views

Exceptions in infinite-dimensional spaces

What are the properties that are true in finite-dimensional spaces but fails in the infinite-dimensional space? For example, the closed unit ball is compact only in finite-dimensional normed space.
6
votes
1answer
142 views

When adding zero really counts …

Note: Although adding zero has usually no effect, there is sometimes a situation where it is the essence of a calculation which drives the development into a surprisingly fruitful direction. Here is ...
5
votes
0answers
52 views

Curvature and topology

I am studying Riemannian Geometry and I came across various Theorems which give conditions on the topology of a manifold given conditions on curvature, and vice-versa. Just to mention a few of them: ...
2
votes
1answer
44 views

Books with SAGE portions

I recently finished working through Adventures in Group Theory and really appreciated the use of SageMath it employs. I considered myself moderately proficient with Sage, but I found working through ...
5
votes
4answers
132 views

Questions that started new mathematics [closed]

Most mathematical theories (Exceptions are unknown to me) were created by answering one or more open ended non-trivial questions. For example, The Brachiostome problem lead to the calculus of ...
9
votes
1answer
122 views

Results in mathematics whose only proof is model theoretic

What are results in mathematics, for example in algebra, whose only proof so far used model theoretical arguments?
0
votes
3answers
69 views

How many ways are there to prove that there is no largest prime? [duplicate]

Is there any other proof by which I can show that there is no largest prime? I saw an example where it is proved with contradiction.(Idea is basically that of Euclid's proof) Imagine that the ...
1
vote
2answers
46 views

What is a cool topic regarding differential equations that you would suggest for self-studying?

I took a basic course in differential equations and I loved it. I'd like to study them more in depth. However, I'm taking another, more advanced course in a while. For this reason, instead of ...
8
votes
0answers
127 views

Recreational problems in set theory?

Most areas of maths that I can think of have a number of fun, recreational problems that come under their category. Nothing deep: number theoretic stuff in olympiads, integrals, limits, products, ...
1
vote
1answer
84 views

Indefinite Integral challenge problems

This year I am going to participate in an olympiad of indefinite integrals. The level is very high, I would like to know some (hard, olympiad) Indefinite integrals challenge problems Note: Here is ...
0
votes
2answers
98 views

Interesting real life applications of elementary mathematics

If you teach mathematics to future highschool teachers, you often feel that they are bored because what they learn at university does not have much to do with what they will have to teach in school, ...
3
votes
0answers
45 views

Complex Analysis with differential forms

I'm studying a little of Complex Anlysis and I have seen that I can thing the integrals of complex functions as integrals of differential forms in $\mathbb{R}^n$. For example I know that Cauchy ...
1
vote
1answer
21 views

Example of series of functions that converges uniformly but whose series of uniform norms does not converge

Analysis two is a very heavy exam, and many people try it thousands of times. Thus, I still have friends to help about that, and today I have been asked, "What do I do when investigating the ...
89
votes
24answers
8k views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
27
votes
20answers
972 views

What are some surprising appearances of $e$?

I recently came across the following beautiful and seemingly out-of-the-blue appearance of $e$: $E[\xi]=e$, where $\xi$ is a random variable that is defined as follows. It's the minimum number of ...
3
votes
1answer
122 views

Do any mathematican still reserach about trigonometry?

Do any mathematican still reserach about applied trigonometry? If so, what are the subject area called in the PhD level except fourier analysis? In many area, you could see a lot of trig and ...
-2
votes
2answers
86 views

Is the area of linear programming dead right now? [closed]

By dead i mean not much/completely no research there . Is the area of linear programming dead right now? If it is not dead, what are the active area called for example except computer science?
0
votes
0answers
26 views

What are some easy to prove results on the density of primes?

Bertrand's postulate states that for any integer $n>3$, there's always a prime $p$ between $n$ and $2n-2$. That result sets a reasonable 'lower bound' on how often we can expect primes to show up, ...
1
vote
1answer
47 views

Examples of holomorphic, complex differentiable, always positive functions

I am looking for classes of functions which are: 1) holomorphic 2) |f(z)|>0 for all z 3) complex differentiable (i.e. f(z)=mod(z) is not valid) ...
2
votes
2answers
43 views

Areas where closed form solutions are of particular interest

Assuming the definition of 'Closed Form' given in the table of: Closed Form Wikipedia entry, what areas tend to have problems that are traditionally expressed in closed form? EDIT: Given the comment ...
4
votes
0answers
60 views

Open problems in Lie theory

I been studying lie theory for some time. Beside classification related problems what are some examples of open problems in the lie world? Especifically in the topological/differentiable structure of ...
0
votes
0answers
25 views

Examples of sans serif Greek used in published mathematics

Unicode contains several thousand mathematical symbols, including individual code points for different maths alphabets. For example, U+1D434 is "mathematical italic capital A" (𝐴), U+1D63C is "math ...
1
vote
0answers
30 views

Uniform continuity with respect to parameter.

Let $\mathbb{X},\mathbb{Y}$ and $T$ metric spaces. A family $\{f_t\}_{t\in T}$ of maps $f_t:\mathbb{X}\to\mathbb{Y}$ is uniformly continuous with respect to parameter $t$ if, $$ (\forall ...
-2
votes
4answers
25 views

Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers

(Note: This question has been cross-posted from MO.) Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) ...
3
votes
1answer
44 views

What are the most common handwavy calculus rules used in statistics?

I am two weeks into my first stats course and already I have noticed that, because my class ignores measure theory, the instructors are being sloppy about explaining which kinds of functions are ...
3
votes
2answers
92 views

Simple, stable $n$-body orbits in the plane with some fixed bodies allowed

I'm working on a visual simulator for the $n$-body problem in the plane (here). The goal is to show how complex behavior can arise from the simple inverse-square law of gravity. To that end, I want ...
9
votes
8answers
961 views

Examples of fallacies in arithmetic and/or algebra [closed]

I'm currently preparing for a talk to be delivered to a general audience, consisting primarily of undergraduate students from diverse majors. My proposed topic would be Examples of fallacies in ...
15
votes
1answer
379 views

Most wanted reproducible results in computational algebra

I am interested in suggestions for major computational results obtained with the help of mathematical software but not easily verifiable using computers. "Most wanted" could refer, for example, to ...
33
votes
7answers
2k views

What are some math concepts which were originally inspired by physics?

There are a number of concepts which were first introduced in the physics literature (usually in an ad-hoc manner) to solve or simplify a particular problem, but later proven rigorously and adopted as ...