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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25
votes
19answers
485 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
102 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
74 views

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

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
19 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, ...
0
votes
0answers
42 views

Record-holding mathematical proofs [closed]

Which mathematical theorems admit proofs that are extreme in some sense? Here is what I have in mind: The classification theorem for finite simple groups is the longest proof mathematics has seen. ...
1
vote
1answer
40 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
36 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
45 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
19 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
26 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
20 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
39 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 ...
2
votes
2answers
75 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
923 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 ...
14
votes
1answer
351 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 ...
0
votes
0answers
40 views

Projects like Stacks?

I am interessted for other projects like Stacks Project which works on algebraic geometry.My questions are : Are there other projects like that? Are there projects like that which are not on research ...
1
vote
0answers
80 views

Obscure/hard definite integrals evaluating to 2016? [closed]

I wish to send a tricky integral to my math-loving friend which evaluates to 2016 owing to the advent of the upcoming year. What are some interesting integrals which evaluate to 2016? Something ...
1
vote
1answer
15 views

Index notation of tensors and mnemonics

I've been trying to learn to manipulate tensors but I've got probably too comfortable with all the matrices in my Linear Algebra course, that it gets really difficult beyond rank-3 tensors. So, ...
47
votes
17answers
2k views

What are some math books written in dialogue or story form, e.g., a teacher explaining to a student?

Good examples would be The Square Root of 2 by David Flannery or Math Girls by Hiroshi Yuki.
13
votes
8answers
380 views

The proofs of the fundamental Theorem of Algebra [closed]

There are many proofs of the fundamental theorem of algebra. Which are the most beautiful proofs?
98
votes
19answers
4k views

Past open problems with sudden and easy-to-understand solutions

What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved ...
1
vote
4answers
87 views

Reference request for set-theoretic foundations of geometry

My question is, Is it possible to define geometrical concepts (say, of Euclidean Geometry) like 'point', 'striaght line' in purely set theoretic terms? So far, I could think of the following ...
3
votes
2answers
31 views

Condition for inverse of quadratic function - alternative solutions

I was helping my friend teacher to assemble a list of exercises to their precalculus students. So I came up with this problem: Let $f$ be a quadratic function, i.e. $$f(x) = ax^2 + bx + c,$$ ...
23
votes
4answers
1k views

What's so special about characteristic 2?

I've often read about things which do not work in a field with a characteristic $2$, mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of ...
2
votes
0answers
63 views

Algebra 2 Test-Taking Shortcuts

Intro: I will be taking an Algebra 2 multiple choice final very soon. The material covered will be, a review from Algebra 1, composition of functions, variation, polynomial division, logarithms, ...
2
votes
1answer
42 views

Applications of Nagata's Lemma

In the spirit of this MO question I would like to ask for applications of a (somewhat lesser known?) lemma. Lemma. (Nagata) Let $R$ be an atomic domain. TFAE: $R$ is a UFD There exists a ...
7
votes
0answers
71 views

Undergraduate group theory Aha facts [closed]

I am learning group theory. I found out that there are a lot of (more or less) simple facts that are completely trivial for working mathematicians (e.g. my professor), but that are nowhere explicitly ...
0
votes
0answers
15 views

Is there more RBF's kernel?

I am working on a sfde(stochastic fractional differential equation),I use some of Radial basis function to find solution.like this list ...
36
votes
10answers
958 views

What are Different Approaches to Introduce the Elementary Functions?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the ...
1
vote
0answers
41 views

Recommendations on visualizing basic linear algebra

I am teaching linear algebra this semester, and I would really like to recommend my students some cool youtube videos visualizing some simple stuff like the span of a set of vectors, linear ...
0
votes
0answers
50 views

Learning multivariable/vector calculus through guided discovery

I am asking this question as a question similar to what has been asked previously for other topics as well as math in general. But I'd like to ask for text references specifically in the domain of ...
5
votes
1answer
92 views

Unexpectedly uniformly continuous functions

The other day in a exam, I was given the following exercise: Given $f : [0,1] \to \mathbb{R}$ continuous and such that $f(0) = 0, f(1) = 1$, let $g : \mathbb{R} \to \mathbb{R}$ be $g(x) = [x] + ...
4
votes
2answers
167 views

Definitions of “linearity” across branches of mathematics or levels of math education

Linearity is a ubiquitous concept in mathematics; however, each branch of mathematics appears to have its own definition of what a linear map (function, functional, functor, transformation, form, ...
6
votes
2answers
62 views

Uses of vector spaces over $\mathbb Q$

I know of two applications of vector spaces over $\mathbb Q$ to problems posed by people not specifically interested in vector spaces over $\mathbb Q$: Hilbert's third problem; and The Buckingham pi ...
6
votes
0answers
96 views

Using multiple integrals for tough single integrals

I'm just getting started on double integrals, and I recently saw the super cool way to use double integrals to arrive at $$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$ So, I am wondering if ...
1
vote
1answer
108 views

Humorous mathematical essays

Even though there are plenty examples of mathematical jokes, the mathematical literature is (in many cases) pretty dull. Nevertheless, examples exist in which an essay makes you smile with a nice pun ...
1
vote
0answers
38 views

Evaluating definite integrals using complex contour integrals

In most cases one only has to consider the complex function where we put $z$ instead $x$. For example, calculating the integral: $$\int_{0}^{\infty}\frac{dx}{x^{4}+1}$$ Here i just integrate the ...
10
votes
7answers
703 views

What are your favorite relations between e and pi? [closed]

This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for ...
2
votes
2answers
30 views

Books in the spirit of A. Cox “Primes of the Form $x^2+ny^2$”

David A. Cox "Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication." has a very good (at least to me, and many) methodology. He starts from page 1 asking a simple ...
7
votes
4answers
142 views

A new approach to find value of $x^2+\frac{1}{x^2}$

When I was teaching in college class ,I write this question on board . if we now $x+\frac{1}{x}=4$ show the value of $x^2+\frac{1}{x^2}=14$ Some student ask me for multi idea to show or prove that ...
0
votes
0answers
27 views

Sufficient conditions for finite-dimensionality?

Suppose we are given a subset $A$ of a complex vector space $\mathcal V$, and we are asked to look at the subspace $V$ (Hamel-) spanned by $A$. Of course, a spanning set may be very large compared to ...
9
votes
1answer
143 views

Counting number of mathematical objects and structures

Regarding the numbers of certain mathematical objects and structures, especially sets, relations and functions, I've compiled a list of the counts from various sources: Partitions of a set with $k$ ...
3
votes
2answers
45 views

Different proofs for the $L^2$ isometry of the Fourier transform on $\Bbb R$

Over the years I have come across several different proofs for the $L^2$ isometry of the Fourier transform (as it exists as an integral operator on $\Bbb R$). Often the traditional proofs hinge on the ...
2
votes
0answers
20 views

Analogies between finite groups and Lie groups

I believe there are some striking analogous facts between finite groups and Lie groups. One analogue almost too basic to mention is the appropriate notion of subobjects. In elementary group theory ...
5
votes
2answers
109 views

Applications of Hodge theory to topology and analysis

I am going to give a talk for the PhD students' seminar at my university. The audience is composed mainly by algebraic topologists, algebraic geometers and analysts. I have decided that I'm going to ...
6
votes
1answer
172 views

What does Hartshorne do wrong?

I'm currently trying to learn algebraic geometry from Hartshorne's Algebraic Geometry. I've often heard it said, both on MathOverflow and in my department, that Hartshorne's treatment of certain ...
1
vote
0answers
15 views

Examples of generalized geometric series.

I'm preparing a short presentation on the convergence of the geometric series of matrices, and I'd love some examples of their uses. I've encountered them when approximating inverses of matrices ...
1
vote
2answers
59 views

Theorems relating to the limitation of mathematics

At one point, mathematicians believed that they may be capable of expressing all of mathematics in one system of ideas, and that their abilities were unlimited. Unfortunately, things like the Godel's ...
4
votes
1answer
117 views

Examples of bi-implications ($\Leftrightarrow$) where the $\Rightarrow$ direction is used in the proof of the $\Leftarrow$ direction.

[I'm asking for examples of proofs with a certain structure. There is quite a lot of text before arriving at the questions. This is because asking for examples of a phenomenon is best carried out by ...