# 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.

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### Complete course of self-study [closed]

I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study ...
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### 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 ...
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Recently, I was studying articles that display results on mixed determinants (for exemple in the Milman's Article A characterization of the mixed discriminant). I am familiar with good references of ...
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### equations, formula, or proofs that cause one to feel “sad,” or negatively in general. [on hold]

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 music. I think that for ...
3k views

### Mathematics, Philosophy and writing.

Do you know of any famous mathematicians who were also philosophers? I have heard of Descartes, Plato and Leibniz. Are there other good examples, especially more modern examples? Also welcome are ...
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### 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 ...
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### Collections of undergraduate research projects [closed]

I would like to compile a "big list" of undergraduate research projects in the following areas of mathematics: calculus; analysis; abstract algebra; linear algebra; number theory; geometry; ...
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### 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 are, ...
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### 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 ...
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### 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 ...
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### Are there any interesting semigroups that aren't monoids?

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)? To be a bit more precise, I guess I should ask if there any ...
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### 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 ...
4k views

### Intuitive Understanding of the constant “$e$”

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say "...
18k views

### A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
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### 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 ...
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### Unconventional mathematics books

I've recently purchased Oliver Byrne's reproduction of Euclid's Elements. It's a beautiful tome, that's rather unique in its presentation of the material as it represents many of Euclid's proof as ...
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### 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 ...
49k views

### Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
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### 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?
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### What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
31k views

### Real life applications of Topology

The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here What ...
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### 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 ...
9k views

### Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
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### 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 (...
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### Recommend book Taylor expansions

I've taken up self-study of math and i start using the book called : Mathematical Analysis I Authors: Canuto, Claudio, Tabacco, Anita I would like to start from zero to understand taylor ...
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### Analytical mechanics book

On my PHD I have to learn the following subject: Analysis on manifolds and Analytical mechanics. But my book is really not good to read; it is too hard. So I need some book that explains to me ...
176 views

### Intuitive functional analysis book

I would like a functional analysis book like Terence Tao's Real Analysis and Measure Theory book, full of intuition. I am ...
167 views

### Representation Theory Symmetric Group Book?

I'm looking for a nice book that discusses the representation theory of the symmetric group. My background is an introductory class in representation theory.
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### Good book on Lebesgue Theory [duplicate]

I am a graduate student and I need a suggestion for a good book in Lebesgue Measure Theory with good exercises and if its possibly with hints or solutions. Thank you.
17k views

### Good Book On Combinatorics

What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions ...
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### Graph Theory for Dummies Book [duplicate]

Does anyone have a good book on Graph Theory that will introduce me to some of the basic concepts without being so filled with terminology that it's hard to read? I have taken an introductory course (...
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### Abstract Algebra book

Has anyone read the book "Abstract Algebra: An Introduction" by Thomas Hungerford? I'm in the class Intro to Abstract Algebra, and my professor is, to say the least, not very great at teaching. Great ...
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I am looking for the book about advanced stochastic process . It may cover the following content: Stochastic matrices. Ex: $A(k)$, where $k$ is the time index. Stochastic process in space (...
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### Distribution theory book

I'm looking for a good book on distribution theory (in the Schwartz sense), I have the basic knowledge as given in Grafakos' Classical Fourier Analysis, but I want to know more about it. Is the ...
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### 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 ...
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### Proofs of Sylow theorems

It seems that there are many ways to prove the Sylow theorems. I'd like to see a collection of them. Please write down or share links to any you know.