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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2
votes
5answers
140 views

Good true-false linear algebra questions?

Can you suggest me a collection of true-false linear algebra questions, like the ones found in the MIT exams, if possible with solutions (i.e. explanations)? Sorry if it turns out that my request is ...
-4
votes
1answer
32 views

Can one extend the analogy between mathematics and art beyond a just “pleasing” result? [on hold]

This is a soft question. It's extremely commonplace for mathematician's to refer to work as "elegant," "beautiful," and I've seen many compare the process of doing mathematics to painting, or playing ...
4
votes
2answers
42 views

Continuous functions with orbit of period $3$

I would like to build some continuous functions $f : E \to \Bbb R$ (where $E \subset \Bbb R$ is an interval), such that $$\exists x \in E,\;\; [f(x)≠x≠f(f(x)),\;\; f^3(x):= f(f(f(x)))=x]$$ I tried ...
1
vote
1answer
55 views

In which topologies do open sets maintain open under countable or arbitrary intersection?

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology. I am curious if there is well known classes of ...
2
votes
4answers
60 views

Proving convergence or divergence of series: Tips and Tricks

I currently write an article where I collect some tips for students for proving the convergence or divergence of series. What tips and tricks do you know or use or teach? Remark: I will add some ...
2
votes
1answer
40 views

Seeking advice from the more experienced on which trig identities are crucial to memorize and which can be derived quickly

This is a bit of a two part question. I also have read some of the related questions, but I think mine is different as whether they can be derived quickly, rather than whether they can be derived, is ...
58
votes
10answers
3k views

Big List of Erdős' elementary proofs

Paul Erdős was one of the greatest mathematicians of all time and he was famous for his elegant proofs from The Book. I posted a question about one of his theorem and got a reference, and I have other ...
6
votes
2answers
80 views

Noteworthy examples of finite categories

So far all the finite categories I have encountered fall into one of these c̶a̶t̶e̶g̶o̶r̶i̶e̶s̶ sets: finite monoids finite preorders just formal devices to explain, what a "diagram" in another (...
3
votes
3answers
189 views

What are the most obscure or advanced mathematics with practical application

Throughout my engineering studies there were jokes made by my professors (mostly mathematics professors) that referenced the fact that pure mathematicians strive to create mathematics with no ...
4
votes
2answers
89 views

What properties of the real numbers are almost always true and there are no (or very few) known examples of?

What properties of the positive real numbers are almost always true and there are no (or very few) known examples of? Two that come to mind are numbers that are normal in every base and numbers ...
3
votes
1answer
22 views

Seeking Additional Solutions for the Number of Network Links

The Problem Show that the number of possible links in a computer network of $n$ computers ($n \in Z \land n \geq 1$) is $\frac{n(n-1)}{2}$ in as many ways as you can. My Work Solution 1 Given $n$ ...
2
votes
0answers
61 views

Problem sets on Abstract Algebra

Many times we ask about what books should we read to learn or know more about a math topic (Abstract Algebra, in this case). However, I would like to get a list of the exercises what should we solve ...
2
votes
2answers
63 views

Proving/Disproving $M$ has the structure of an $R$-module

Given an abelian group $M$ and a ring $R$, how can one prove or disprove that $M$ has the structure of an $R$-module? When proving $M$ is an $R$-module, if it is not obvious how to define an action $R\...
11
votes
1answer
171 views

Lonely theorems [closed]

What are some instances of theorems which are especially unique in mathematics, i.e. for which there are not many other theorems of a similar character? An example I have in mind is Pick's theorem, ...
0
votes
0answers
33 views

Conflicting conjectures [duplicate]

I feel like when two conjectures are inconsistent with one another, it's a clear sign of our misunderstanding of deeper mathematics. I was wondering if anyone knew of a comprehensive list of ...
1
vote
0answers
47 views

Suggestion of books on Integral Calculus of Several Variables

I'd like recommendations of books on integral calculus of several variables (double integral until Gauss's theorem) that contains challenging(hard) problems. And I'd like books in languages other than ...
17
votes
4answers
655 views

Tough integrals that can be easily beaten by using simple techniques

This question is just idle curiosity. Today I find that an integral problem can be easily evaluated by using simple techniques like my answer to evaluate \begin{equation} \int_0^{\pi/2}\frac{\cos{x}}{...
0
votes
1answer
75 views

Large, small but a useful number. [closed]

Today we were discussing in our class about usefulness of a number no problem how large,small may be it's value. As per my knowledge (till grade 11) Avogadro number $N_A=6.022\times 10^{23}$ is a ...
3
votes
2answers
188 views

Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various 'three-number means' with as many different elements in it as possible. So far I've got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+...
163
votes
20answers
32k views

What are some examples of when Mathematics 'accidentally' discovered something about the world?

I do not remember precisely what the equations or who the relevant mathematicians and physicists were, but I recall being told the following story. I apologise in advance if I have misunderstood ...
5
votes
3answers
78 views

What are all the uses of the determinant?

I've learned how to calculate the determinant but what is the determinant used for? So far, I only know that there is no inverse if the determinant is 0.
12
votes
1answer
165 views

Fake proofs using matrices

Having gone through the 16-page-list of questions using the tag (fake-proofs), and going though Best Fake Proofs? (A M.SE April Fools Day collection) and https://en.wikipedia.org/wiki/...
1
vote
0answers
23 views

Machine Learning: are there other functions similar to the softmax?

Recall in probability and machine learning softmax is defined as: $\sigma(\mathbf{z})_j = \dfrac{e^{z_j}}{\sum_{k=1}^K e^{z_k}}$ for $j = 1, ..., K.$ where $\sigma: \mathbb{R}^k \to (0,1)$ ...
8
votes
3answers
84 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
64 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
37 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
106 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
108 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
252 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= \frac{...
1
vote
1answer
81 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
160 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
18 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
43 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
321 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
35 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
51 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
54 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 ...
14
votes
2answers
134 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, $$a^{-...
1
vote
4answers
49 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
29 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
32 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: ...
9
votes
1answer
82 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
188 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
48 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
31 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
2answers
171 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
61 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
49 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 ...
6
votes
4answers
135 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?