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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1
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0answers
90 views

Hard-to-put-together but easy-to-prove results

What are the most important examples of theorems and definitions which are post factum obvious, i.e., hard to put together but easy to understand and use (and prove, in the case of theorems) once you ...
2
votes
2answers
86 views

Alternate proof for $a^2+b^2+c^2\le 9R^2$

As I studying geometric inequalities, one of those famous inequalities is $$a^2+b^2+c^2\le 9R^2$$ I did some research and I found that there is a proof (not exactly the this inequality but an useful ...
6
votes
8answers
245 views

Calculate $\pi$ By Hand?

All over the internet the only hand equation i found was $$\frac\pi4 = 1 - \frac13 + \frac15 - \frac17+\cdots.$$ But this takes something like a thousand iterations to get to four digits, is there a ...
8
votes
0answers
146 views

Known exact values of the $\operatorname{Li}_3$ function

We know some exact values of the trilogarithm $\operatorname{Li}_3$ function. Known real analytic values for $\operatorname{Li}_3$: $\operatorname{Li}_3(-1)=-\frac{3}{4} \zeta(3)$ ...
6
votes
3answers
427 views

Undergraduate mathematical magazines to improve mathematical knowledge

I'm sorry my ignorance, I don't know very much about mathematical magazines. I'm finishing my master degree in pure mathematics and I'm looking for mathematical magazines which could improve my ...
-1
votes
3answers
106 views

A big list of non-trivial examples of functions from outside mathematics

I will be teaching my students about functions, and want to stress that functions are not only the usual mathematical ones (linear, logs, exponential, ...), but that function is fundamentally a ...
5
votes
0answers
44 views

Books with “project”-like questions

I'm looking for a big list of resources for advanced undergraduate - beginning graduate (and even beyond, really) with a particular feature. Namely, I really like solving "project"-like problems that ...
8
votes
3answers
343 views

Resources for Integrals?

I want to learn to solve integrals of some type, probably definite integrals with results involving various constants such as Catalan's, Euler-Mascheroni,Golden-ratio etc. and involving various ...
4
votes
0answers
178 views

What do group automorphisms fix? [closed]

I have often found it useful to sit and contemplate what kinds of elements, subsets, or structures do the automorphisms of an object fix or permute. Sometimes the observations do not have immediate ...
2
votes
3answers
327 views

Abstract Algebra Book Request

I am looking for a good undergraduate level book on Abstract Algebra. By a 'good book' I mean a book which gives equal importance to both, rigor and the historical perspective of the subject. For ...
4
votes
6answers
124 views

Finite sequences of prime numbers

There is a lot of prime sequences: prime numbers in a special form. For example Mersenne primes are primes of the the form $2^n-1$, or Pythagorean prime are primes of the form $4n+1$. Even primes are ...
7
votes
2answers
202 views

Theorems with one-line proofs [closed]

Inspired by this very concise answer, which proves that $$\sin^2(\theta)+\cos^2(\theta) \equiv 1 $$ as follows: $f(\theta)=\cos^2\theta+\sin^2\theta \quad;$ then it's simple to see that ...
0
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0answers
30 views

What are some [mostly trivial] Pell transformations?

Euler looked at some transformations which turned one Pell[-type] equation into another. Example 1: $$u^2-av^2=-1 \quad\iff\quad (2u^2+1)^2-a(2uv)^2=1.$$ Example 2: $$u^2-av^2=-2 \quad\iff\quad ...
10
votes
3answers
473 views

Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
7
votes
1answer
159 views

Handwaving gone wrong

My motivation for this question is twofold: On one hand, I'm studying algebraic topology, where - at least in the book written by Hatcher - there is quite a lot of handwaving (e.g. maps are continous ...
0
votes
0answers
101 views

What advanced methods in contour integration are there?

It is well known how to evaluate a definite integral like $$ \int_{0}^\infty dx\, R(x), $$ where $R$ is a rational function, using contour integration around a semicircle or a keyhole. Most complex ...
37
votes
10answers
3k views

Mathematical literature to lose yourself in

H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics: ...I have tried to say to students of mathematics that they should read the ...
13
votes
5answers
746 views

I need help finding a rigorous Pre-calculus textbook

I dislike modern textbooks; their cookie-cutter approach and appearance, over reliance on breaking things down into little boxes, the general spoon-feeding they engender and most of all the poor ...
2
votes
2answers
139 views

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $ for positive integer $n$ and real $a, b$? You can use any techniques you want. My proof just uses algebra, ...
4
votes
5answers
384 views

Examples of advancement in mathematics due to war

It's not a lie that, in most sciences, some of their advancement comes from war. A couple examples would be the Haber process in chemistry and none other than the Manhattan Project in both physics and ...
3
votes
0answers
115 views

Undergraduate Schools for the Mathematically Inclined

I'm a rising senior and working on generating a list of colleges to apply to, but it seems to me that (with few notable exceptions) my two main criteria are mutually exclusive. Are there any schools ...
1
vote
1answer
156 views

Video/audio lectures on differential topology?

Do there exist decent online video lectures, or even audio lectures, covering differential topology? I'm aware of Milnor's talk, but it is more like exposition and doesn't go very far.
21
votes
5answers
660 views

Who are some blind or otherwise disabled mathematicians who have made important contributions to mathematics?

Two prominent mathematicians who were disabled in ways which would have made it difficult to work were Lev Pontryagin and Solomon Lefschetz. Pontryagin was blind as a result of a stove explosion at ...
5
votes
2answers
290 views

Properties of reflexive Banach spaces

I just want to see the importance of reflexive Banach spaces and what is special about them compared to other Banach spaces. What kind of properties hold in reflexive spaces that do not necessarily ...
3
votes
2answers
118 views

What are some good questions for this trick, if $\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots=\alpha$ then $\alpha=\frac{a+c+e+…}{b+d+f+…}$?

I need some good algebra questions that are applications of this trick, often in a non obvious and elegant way: $$\text{If } \frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots=\alpha \text{ then } ...
3
votes
1answer
53 views

General lists of techniques to prove whether a set is a generator of a matrix group

It seems like a rather common problem in group theory, at least in undergraduate maths, to check whether a set is a generator of a group. The question is usually as follow: Given a group $G$, and a ...
3
votes
2answers
153 views

An introduction for integral tricks.

I wonder if there's a good book or internet page introducing integral tricks? For example integration by parts, and Feynman's trick. I'm not looking for an exercise book such as "Problems in ...
32
votes
5answers
3k views

Examples of “Non-Logical Theorems” Proven by Logic

I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. Question: What are some examples of mathematical logic solving open problem outside of ...
4
votes
1answer
67 views

Are there more examples of functional equations which are also valid for the identity map?

I find the co-incidence of the identity: $$\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B$$ very pleasing. So, I was wondering if there are more of these type of identities. To make my question precise: ...
8
votes
2answers
892 views

Video lectures of algebraic geometry (Hartshorne, Shafarevich, … )

I am a commutative algebra student. I wonder if there is some video lectures of algebraic geometry courses available online for free? I'd like the lectures to cover main topics of the books ...
-1
votes
1answer
221 views

Crazy Set Theory Analogies

I think the following analogies are too interesting to be ignored: Union = Least Common Multiple If $G_1,...,G_n$ denote a number of sets of points (either linear or in any number of dimensions), ...
2
votes
1answer
228 views

Seemingly hard integrals which are made easy via differentiation under the integral sign a.k.a Feynman Integration [closed]

I recently discovered Differentiation under the integral sign a.k.a Feynman Integration and I read an article which says it can be substituted for contour integration. Therefore, I am assuming this ...
4
votes
7answers
343 views

Classic Counting Problems

Does anyone have some good, classic, counting problems? I want things that are interesting, as well as instructive- more than just compute the number of way to get a flush, etc. (Not that those aren't ...
0
votes
0answers
49 views

Text on Witt vectors that are accessible to undergraduate students

I am looking for a thorough text on Witt vectors that is accessible to an undergraduate student that have completed the following courses: Calc 1, 2, Linear Algebra and Abstract Algebra. (In Norway, ...
1
vote
1answer
59 views

Identities involving adjoint action

I'm looking for list of identities involving adjoint action $\mathrm{ad}_A X = [A,X] = AX - XA$. For example, it can be easily shown that: \begin{equation} e^{\mathrm{ad}_A} X = e^A X e^{-A} ...
15
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7answers
596 views
1
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1answer
46 views

Interactions between geometry and graph theory.

I'm looking for some nice theories or just exercises, with both geometrical aspects and graph theoretics aspects. Example may include for instance the 4-color theorem or Euler characteristics, maybe ...
4
votes
1answer
131 views

References for mathematical theory of summability of divergent series

Once in a while, I can't help it to ask very broad questions. I have read (a portion of) Hardy's Divergent Series. Back then, I think besides in mathematics, divergent series and the need to assign ...
0
votes
2answers
124 views

Important applications of the Uniform Boundedness Principle

There's like three applications of the uniform boundedness principle in wikipedia: 1) If a sequence of bounded operators converges pointwise to an operator, then the limit operator is also bounded, ...
7
votes
2answers
367 views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
3
votes
5answers
1k views

Real life examples of order relations.

It's easy to find examples of equivalence relations (for example, A shares room with B), but I can't seem to find a real life example of an order relation (that is, a relation that's reflexive, ...
2
votes
1answer
117 views

Infinite families of prime numbers

What interesting/useful infinite families of prime numbers are there? Right now it would be useful if I could find one with arbitrarily many 1's in its binary representation, but I am doing a larger ...
4
votes
1answer
230 views

Calculus and Real Analysis: open source lecture notes ready to be edited

I would like to collect a big list of good open source lecture notes for a course in calculus; real analysis. Such notes should be in .tex format, that is, ...
2
votes
0answers
100 views

Is there any visual animation to show the basic concept of algebraic geometry? [closed]

Is there any visual animation to show the basic concept of algebraic geometry? There are rarely pictures in textbooks, so are there any animation to show basic but important concepts?
12
votes
3answers
335 views

Textbooks, lecture notes, and articles from arXiv for undergraduate students

I have found some interesting textbooks and articles on arXiv, such as the following one, that are accessible to an undergraduate student: Course of linear algebra and multidimensional geometry, ...
4
votes
3answers
2k views

Daily exercises to speed up my mental calculations?

When I was a kid in school my father prevented me from using a calculator when solving my math homeworks. However at that time I was not convinced as of why not to use such a useful tool! So I kept on ...
4
votes
4answers
333 views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
3
votes
1answer
120 views

What are the problems that you tried to find their solutions and you did not know that it is impossible?

Tell us your story about Mathematics. Have you dream one day to do a big contribution in Mathematics because you are curious and love challenges. What are things that you tried to prove which then ...
1
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0answers
72 views

Things you've believed for a long time were true, but are false in reality [duplicate]

Do you have any things (mathematical statements, statements about mathematics) you've believed for a long time were true, but now with enough mathematical knowledge you realize were wrong? For ...
2
votes
1answer
111 views

Example of a Problem Made Easier with Skew Coordinates

Skew or oblique coordinate systems are coordinate systems where the angle between the axes is not 90 degrees. The second answer to this question has formulas to convert between these systems with an ...