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
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0answers
30 views

GRE Mathematics Practice Exams

I will be taking the subject test in the near future. Can you recommend me some sources (online or print) from which I can find realistic practice exams? I would like to get my hands on as many ...
1
vote
2answers
60 views

Doing Michael Spivak's Exercises

I am doing Spivak's Calculus, and I find it EXTREMELY difficult. I usually ask questions here because I cannot do the problems on my own. How long should it take to do a Spivak problem? Is it ...
0
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0answers
33 views

What area of maths to study now? [closed]

My very subjective questions is what to learn next? I have been self studying category theory (up to Yoneda, Adoints...), abstract algebra, commutative algebra, topology and algebraic topology ...
3
votes
1answer
33 views

What are the subjects an analytic number theorist must be well versed with after undergraduate studies?

I am a mathematics major and I aspire to be an analytic number theorist. In general, what are the subjects an analytic number theorist must be well versed with after undergraduate studies (i.e. in ...
3
votes
3answers
91 views

What is an interesting math topic that an advanced high schooler could self study? [closed]

So right now I am in AP Calculus AB and the only math background I have is the standard high school curriculum. But I am extremely interested in math and I want to self study a new topic in math that ...
12
votes
4answers
231 views
+50

Books in the spirit of Problems and Theorems in Analysis by George Pólya and Gábor Szegő

In the Preface of the first German Edition of the book Problems and Theorems in Analysis by George Pólya and Gábor Szegő, one can read $_{\text{[emphasis mine]}}$: The chief aim of this book, ...
1
vote
3answers
58 views

Different Types of Waves

I am making a basic 2D rigid body simulator as a hobby. It involves springs. Naturally, I need to render them. Rigid body simulators, such as Algodoo, render them simply like this Another (more ...
0
votes
1answer
56 views

Learning Olympiad Level Combinatorics

Combinatorics has always been my weakest point, I want to improve it. There are such problems like: "Five friends should give each other gifts. They have made a gift each, as they should give away ...
1
vote
0answers
53 views

What are some really cool problems that involve “least squares and Eigenvalue problems”?

I am required to find a research topic in this domain, so I'm really interested in finding out what kind of problems are covered in this domain, and how others are using these techniques to solve ...
0
votes
0answers
19 views

Other Diophantine problems that use a Pell equation

What Diophantine equations employ Pell equations in their solutions? A well-known example is the case of Pythagorean triples where the legs differ by 1, like, $$20^2+21^2 = 29^2$$ These are ...
1
vote
3answers
73 views

Higly axiomatic geometry book recomendation

Recently I have started dipping my toes in mathematical waters besides calculus,and with varying success I have started learning bit of something about "everything". But I have one issue,namely I can ...
0
votes
2answers
63 views

differences between complex and real analysis [closed]

I am hoping to learn some of the key differences between complex and real analysis; but only within the basics, concepts that could be understood with only knowledge from a first course in these ...
0
votes
1answer
24 views

Different ways to prove convexity of quadratic form associated to rank 1 matrix

Let $v \in \Bbb R^n$, and $f:\Bbb R^n \to \Bbb R^n$ with $f(x)=\langle x,(vv^T)x\rangle$. Show that $f$ is convex. I'm looking for different approaches to solve this (rather simple) problem. ...
4
votes
3answers
153 views

Do you know any almost identities?

Recently, I've read an article about almost identities and was fascinated. Especially astonishing to me were for example $\frac{5\varphi e}{7\pi}=1.0000097$ and ...
45
votes
10answers
1k views

Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
20
votes
9answers
1k views

Surprising applications of topology [closed]

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The ...
2
votes
0answers
59 views

What computations would advance math knowledge a lot?

Suppose we where given a super computer that would be capable of computing anything, but only for one day. We could for instance compute many of the Ramsey numbers. What would be some computations ...
6
votes
1answer
82 views

Theorems discovered without observation

Can you name me a few theorems that were discovered without first observing some special cases? In other words, by brute logic: Starting from the known and logically deducing the unknown? EDIT: As an ...
0
votes
0answers
40 views

Different ways to prove Fundamental Theorem of Algebra

This is just a curosity .I know some proofs of the fact that Every non constant polynomial with complex coefficient has a complex root via using Liouville's theorem in Complex Analysis.Proof goes as ...
3
votes
3answers
82 views

Examples of open problems solved through short proof

Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with ...
0
votes
0answers
64 views

New proofs of the Fundamental Theorem of Calculus

Apart from the standard one, are there any other proofs of the Fundamental Theorem of Calculus which have been published recently?
4
votes
2answers
333 views

An example of a great explanation or freely accessible article on a math concept [closed]

Question: Give an example of a great explanation or freely accessible article on a math concept (suitable at the undergraduate or lower level), and explain why you think it is great. Possible ...
99
votes
43answers
12k views

What's your favorite proof accessible to a general audience?

What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in ...
4
votes
2answers
115 views

Toy examples for Kan extensions

Background: If $\mathcal{C}$ is a cocomplete category and $f : I \to J$ is a functor between small categories, then $f^* : \mathrm{Hom}(J,\mathcal{C}) \to \mathrm{Hom}(I,\mathcal{C})$ has a left ...
2
votes
3answers
74 views

Examples of orthogonal/orthonormal functions which are not finite degree polynomials?

I've been reading "Fourier Series & Orthogonal Polynomials" by Dunham Jackson. Great introductory read for anyone interested by the way! My question is, what are other examples of Orthogonal ...
1
vote
0answers
42 views

Methods for evaluating polynomials quickly

I am wondering what methods exist for effectively evaluating polynomials (manually or in the head) in a quick, efficient fashion. For example, one of my favorite methods is the "nested form of a ...
23
votes
8answers
396 views

Big list of “guided discovery” books

K. P. Bogart wrote Combinatorics through Guided Discovery, available freely online. In the preface, he writes (emphasis mine): The point of learning from this book is that you are learning how to ...
64
votes
38answers
8k views

A fan, a horn, and a snowflake - unusual math terms [closed]

From time to time, I come across some unusual mathematical terms. I know something about strange attractors. I also know what Witch of Agnesi is. However, what prompted me to write this question is ...
1
vote
0answers
33 views

Applications of Splitting Lemma and Exactness

I'm looking for nice applications of exact sequences, the splitting lemma, and exact functors in algebra and topology (i.e not using the five lemma to get long sequences in homology etc..). For ...
8
votes
4answers
582 views

What are some elementary results (number theory) using theorems that went long-unproven?

Firstly, I do not mind if there are examples from fields other than number theory! This was just the first field, and where I think the richest examples, may come from. Now to elaborate on the title, ...
7
votes
1answer
82 views

For finding limits of functions, when are graphs deceiving?

What are some examples of limits which exist of functions $f:A \to B$ where $A$, $B \subseteq \mathbb{R}$ that require by-hand, "analytical" methods and the value of the limit is seemingly ...
1
vote
2answers
96 views

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, ...
3
votes
1answer
89 views

Innocent looking open problems in real analysis

Are there any apparently easy problems or conjectures in basic real analysis (that is, calculus) that are still open? By apparently easy, I mean: so much so, that, if it was for the statement alone, ...
3
votes
2answers
129 views

Connectedness arguments in elementary mathematics?

To begin, let me explain a proof strategy (which I'll call the connectedness principle for want of a better, more canonical term): One way to prove that a mathematical object $O_1$ has some property ...
8
votes
0answers
128 views

What is the most cited mathematical paper?

Just out of curiosity: What is the paper with the largest number of citations in all of mathematics? I think it is Shannon's A Mathematical Theory of ...
-13
votes
6answers
294 views

How many mathematical identities that equal $1$ [closed]

I usually see some identities equal to $1$ , for examples $$\sin ^2(x)+\cos^2(x)=1$$ $$\sec ^2(x)-\tan^2(x)=1$$ $$\csc ^2(x)-\cot^2(x)=1$$ $$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta ...
68
votes
9answers
4k views

Besides proving new theorems, how can a person contribute to mathematics?

There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example: Organizing known results into a coherent ...
2
votes
0answers
56 views

Generalizations of de l'Hospital rule

Are there any useful generalizations of de l'Hospital rule? Could you point out some references?
0
votes
1answer
81 views

Common conditions on functions to be morphisms. [closed]

When coming in contact with the concept of morphism one may start to wonder what makes different structured objects of the same kind to be similar in a "morphical" way. At least I did. Below ...
1
vote
0answers
46 views

Open problems for which all cases except one have been solved

Keller's conjecture states that in any tiling of Euclidean $n$-space by identical hypercubes there are two cubes that meet face to face. The conjecture has been shown to be true for $n<7$ and ...
3
votes
1answer
48 views

Group structures on Hausdorff space

Could anyone give me some practical (and possibly intuitive) examples of Group structures on Hausdorff spaces? Let us say you had to get freshmen university students interested into fields of maths ...
35
votes
12answers
3k views

What are some theorems that currently only have computer-assisted proofs?

What are some theorems that currently only have computer-assisted proofs? For example, there's the four colour theorem. I am very curious about this and would like to generate a list.
15
votes
1answer
226 views

Learning roadmap request: compiling a “Mathematics Stack Exchange Undergraduate Bibliography” [closed]

[Book recommendation] questions are quite popular on this website, which is, at least for me, one of the best places to get useful and insightful suggestions ...
3
votes
0answers
76 views

Big list of references [divided by categories] that collect commented open problems and conjectures [closed]

The aim of this question is to collect a big list of books or survey papers or websites which collect an up-to-date, comprehensive, well-organized, and possibly commented list of open problems. I ...
2
votes
1answer
100 views

Integration by nonobvious substitutions

The standard technique for evaluating the integral $$\int \sec x \,dx$$ is making the nonobvious substitution $$u = \sec x + \tan x, \qquad du = (\sec x \tan x + \sec^2 x) dx,$$ which transforms the ...
2
votes
4answers
114 views

Reference request: self-contained rigorous introductions to “cool” topics

I am looking for some self-contained (i.e., providing all necessary background information) rigorous introductions to topics perceived as "cool" to propose to (really) advanced high school students ...
15
votes
4answers
388 views

Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that ...
0
votes
0answers
41 views

Books on contemporary set theory [duplicate]

I have gone through Halmos' Naive Set Theory. Now, could you recommend me a good follow-up book for a rigorous treatment of contemporary set theory? (For example, I've been suggested to look at ...
2
votes
3answers
185 views

“Methods of Theoretical Physics for Mathematicians”

I read in the Princeton Companion to Mathematics that even pure mathematicians should know some theoretical physics. However, I see that there are many reference books of mathematical methods for ...
11
votes
1answer
180 views

Websites that promote co-operation and social networking among mathematicians

Are there some websites that could be defined as social networks for mathematicians and scientists? What I have in mind is something similar to Academia.edu or ResearchGate, but more specific towards ...