# Tagged Questions

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.

307 views

### Books that use probabilistic/combinatorial/graph theoretical/physical/geometrical methods to solve problems from other branches of mathematics

I am searching for some books that describe useful, interesting, not-so-common, (possibly) intuitive and non-standard methods (see note *) for approaching problems and interpreting theorems and ...
106 views

### Is Legendre’s solution of the general quadratic equation the only one?

Legendre famously solved the general quadratic equation $$ax^2+bxy+cy^2+dx+ey+f=0$$ by noting that \begin{equation*} 4a(b^2-4ac)(ax^2+bxy+cy^2+dx+ey+f) = 0 \tag{$\star$} \end{equation*} along with ...
1k views

### Concepts in mathematics which are referred to as 'generalizations' [closed]

I am curious to know some theorems usually taught in advanced math courses which are considered 'generalizations' of theorems you learn in early university or late high school (or even late ...
6k views

### Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
91 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 ...
87 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 ...
248 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 ...
155 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 $\operatorname{Li}_3$: $\operatorname{Li}_3(-1)=-\frac{3}{4} \zeta(3)$ ...
430 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 ...
111 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 ...
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 ...
346 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 ...
184 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 ...
338 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 ...
126 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 ...
211 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 ...
499 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 ...
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 ...
104 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 ...
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 ...
786 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 ...
148 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, ...
386 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 ...
127 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 ...
168 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.
684 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 ...
311 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 ...
118 views

942 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 ...
228 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), ...
236 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 ...
355 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 ...
50 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, ...
59 views

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: e^{\mathrm{ad}_A} X = e^A X e^{-A} ...
597 views

### Are there any well known mathematicians who published very little?

Say no more than 5 publications.
48 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 ...
135 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 ...
129 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, ...
378 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 ...
2k 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, ...
118 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 ...
239 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, ...
103 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?
350 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, ...
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 ...