Questions asking for a "big list" of examples, illustrations, etc. Please do not ask too many of these. Please do not use this as the only tag for a question.

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0
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1answer
250 views

If $X$ is complete and totally bounded, then $X$ is compact [closed]

Let $X$ be a metric space. Whar is your favorite way to show: If $X$ is complete and totally bounded, then $X$ is compact? Thanks for your help.
2
votes
5answers
2k views

How many types of functions are there [closed]

We have the following types of functions : a) Logarithmic function b) Rational Function c) Irrational Function d) Piecwise or modulus function e) Smallest integer function or cieling function f) ...
2
votes
1answer
47 views

Problems where SPD linear system arises

I know some of the places where SPD linar systems arises such as elliptic PDEs and normal equations. Can I have a more comprehensive list of scientific applications which require solving SPD linear ...
6
votes
3answers
142 views

Use of infinity as an “idealistic approximation”

There have been several recent posts about the work of N. J. Wildberger, a finitist who seems to think that mathematics should only focus on things that have some sort of "real world" connection, ...
16
votes
11answers
3k views

How to prove $[a,b]$ is compact?

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
138
votes
21answers
12k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
2
votes
0answers
51 views

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable? Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space. ...
3
votes
1answer
129 views

Worst category with first isomorphism?

I am no expert in category theory, but from VIII of Algebra: Chapter 0 I learnt that In an abelian category every $A\xrightarrow{\phi}B$ can be decomposed into \begin{equation}A\twoheadrightarrow ...
3
votes
1answer
196 views

Convergence Counterexamples

I'm trying to compile a list of counterexamples for convergence implications (or rather, the lack of). I have an incomplete list and I hope to get it all together in one piece. I'm currently working ...
58
votes
20answers
4k views

The Best of Dover Books (a.k.a the best cheap mathematical texts)

Perhaps this is a repeat question -- let me know if it is -- but I am interested in knowing the best of Dover mathematics books. The reason is because Dover books are very cheap and most other books ...
14
votes
8answers
8k views

What is the real life use of hyperbola? [closed]

The point of this question is to compile a list of applications of hyperbola because a lot of people are unknown to it and asks it frequently.
2
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0answers
73 views

Books similar to “Primes of the form $x^2+ny^2$”

Are there any other books which are similarly to the book "Primes of the form $x^2+ny^2$"? Basically, I want a book which starts with a very important classical problem ( in this case which primes can ...
3
votes
2answers
146 views

Definition of “Up to” (homeomorphism,isotopy, etc), and Examples?

I've tried googling this usage and understanding the results but I'm struggling to make intuitive sense of it. So my question is, what is the phrase "up to" understood to mean, and what are some ...
4
votes
0answers
107 views

The mathematical heritage of Lewis Carroll

Which mathematical results has Lewis Carroll, the author of Alice's Adventures in Wonderland, produced? Wikipedia is very vague with regard to this topic and gives us little more than a matrix ...
115
votes
27answers
14k views

Best Fake Proofs? (A M.SE April Fools Day collection) [closed]

In honor of April Fools Day 2013, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen. I've posted one as an answer below. I'm also thinking of a ...
14
votes
1answer
362 views

Examples of falsified (or currently open) longstanding conjectures leading to large bodies of incorrect results.

In general, the way that modern mathematical research is conducted isn't the way that many would assume is the ideal method of research. That is, mathematics is not the linear progression of ...
4
votes
2answers
114 views

Foundation on Diophantine Analysis and Number Theory

I want to read particularly about diophantine Analysis and Elementary Number Theory from a novice level. The books which I found on net: A Guide to Elementary Number Theory by Underwood Dudley ...
-1
votes
1answer
164 views

Best graphing program for Mac or PC?

I just bought the highest end iMac, with a student discount, of course, and was wondering what is the best graphing program out there. A program that can graph any equation that I throw at it AND one ...
4
votes
4answers
4k views

How does linear algebra help with computer science

I'm a Computer Science student. I've just completed a linear algebra course. I got 75 points out of 100 points on the final exam. I know linear algebra well. As a programmer, I'm having a difficult ...
5
votes
1answer
116 views

Differences in worlds with and without $\aleph_0<|S|<2^{\aleph_0}$

Paul Cohen told us that whether or not there is $S$ with \begin{equation} \aleph_0<|S|<2^{\aleph_0} \end{equation} cannot be decided within ZFC, and hence it is reasonable to work in two ...
6
votes
2answers
307 views

Counterexamples in algebra

I got the feeling that whenever a subject gets so sophisticated that Zorn's lemma is needed, a book of counterexamples in that subject would probably benefit researchers/ students a lot. Zorn's ...
5
votes
8answers
436 views

Final year project ideas - complex analysis

For my final year, I have to do a project for a module. I want to investigate something in the complex analysis area. I've only covered the basics of analysis, like Cauchy's IT/IF, residue theorem ...
8
votes
5answers
242 views

High-School Level Introduction to Dynamical Systems

In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory. I'm having trouble coming up with a topic compelling enough ...
1
vote
1answer
257 views

Is there a list of all known Sophie Germain prime numbers?

Is there a list of all known Sophie Germain prime numbers available anywhere for download? I found a small list from OEIS and the top 20 biggest of such primes, but I can't find a list that would ...
28
votes
3answers
536 views

Alternative proofs that $A_5$ is simple

What different ways are there to prove that the group $A_5$ is simple? I've collected these so far: By directly working with the cycles: page 483 of ...
23
votes
10answers
4k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
44
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8answers
3k views

Are there real world applications of finite group theory?

I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as ...
385
votes
131answers
23k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of Mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
6
votes
2answers
86 views

Collecting definitions of continuity.

Let $X$ and $Y$ denote topological spaces and consider a function $f : X \rightarrow Y$. I'm collecting possible definitions/characterizations of the statement "$f$ is continuous." Here's two to get ...
6
votes
3answers
258 views

Why are ordered spaces normal? [collecting proofs]

Greets This is a problem I wanted to solve for a long time, and finally did some days ago. So I want to ask people here at MSE to show as many different answers to this problem as possible. I will ...
10
votes
8answers
236 views

Examples of “transfer via bijection”

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then ...
0
votes
2answers
372 views

List Table(s) of Series Here

I've been interested in series expansions of all types of mathematical functions. I was wondering if anyone has ever created a large list of all types of series. For example, Wolfram's Mathworld's ...
18
votes
7answers
4k views

Any open subset of $\Bbb R$ is a countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
18
votes
19answers
1k views

Elementary books by good mathematicians

I'm interested in elementary books written by good mathematicians. For example: Gelfand (Algebra, Trigonometry, Sequences) Lang (A first course in calculus, Geometry) I'm sure there are many other ...
10
votes
7answers
874 views

Applications of the Isomorphism theorems

In my study of groups, rings, modules etc, I've seen the three isomorphism theorems stated and proved many times. I use the first one ( $G/\ker \phi \cong \operatorname{im} \phi$ ) very often, but I ...
4
votes
1answer
131 views

What other math fields wouldn't require learning a huge amount of material in advance?

From An Introduction to the Theory of Surreal Numbers: [...] Thus the reader has the opportunity which is all too rare nowadays of getting to the surface and tackling interesting original ...
11
votes
6answers
2k views

List of problem books in undergraduate and graduate mathematics

I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, ...
44
votes
6answers
4k views

Studying for the Putnam Exam

This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the ...
0
votes
3answers
491 views

What are your favorite proofs using mathematical induction? [closed]

I would like to get a list going of cool proofs using mathematical induction. Im not really interested in the standard proofs, like $1+3+5+...+(2n-1)=n^2$, that can be found in any discrete math ...
113
votes
6answers
4k views

Best Sets of Lecture Notes and Articles

Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting ...
2
votes
3answers
149 views

Computer Assisted proofs apart from the 4 color theorem

I recently read about the 4 color theorem and that it was proved using help from computers. Does anybody know of some other 'good' computer-assisted proofs apart from the 4 color theorem?
0
votes
1answer
215 views

Functional Analysis - Where to go from here?

The short version of this question is this: I like functional analysis and want to learn more. I've taken a class on it and I've read the books by Brezis and Conway. Where can I go from here? Do ...
12
votes
6answers
368 views

Good examples of Ansätze

Frequently, when talking to mathematicians, I have some trouble when I mention, use, or try to explain what an Ansatz is. (Apparently it is more of a physics term than a maths one, for some reason.) ...
1
vote
1answer
695 views

what is the most traditional abstract algebra textbook? and [Linear algebra & Abstract algebra] [closed]

I have listed 3 textbooks i have in my mind to buy Herstein - Topics in Algebra Artin - Algebra Lang - Undergraduate Algebra Unlike Lang's Algebra is the most traditional abstract algebra text for ...
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votes
4answers
114 views

Noetherian and Artinian rings (reference) [closed]

I started to study localization of rings and Noetherian and Artinian rings. Do you know any good references for these subjects? I'm using the one by Atiyah and Mcdonald. Is there another one? Thank ...
1
vote
2answers
100 views

Types of numbers.

Is there a comprehensive list of real-number-describers that allude to the properties of that number? For example: Amicable numbers, Abundant, Deficient, Perfect, Carmichael, prime, transcendental, ...
8
votes
6answers
9k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
2
votes
2answers
147 views

How many ways to construct a dense subspace?

For any topological space $X$, as the title explains, how many ways to construct a dense subspace of $X$? For example, we can construct a dense subspace which is the union of disjoint open subsets of ...
1
vote
0answers
111 views

Example relations: pairwise versus mutual

There are by now several questions on math.se asking about pairwise versus mutual relations, eg: • When does “pairwise” strengthen and when does it weaken? • Relation: pairwise and mutually • ...
91
votes
33answers
8k views

Can you provide me historical examples of pure mathematics becoming “useful”?

I'm trying to think/know about something but I don't know if my basis premise is plausible, here we go. Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because ...