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.
8
votes
4answers
95 views
Examples of nonlinear ordinary differential equations with elementary solutions.
I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with ...
16
votes
5answers
262 views
Examples of properties that hold almost everywhere, but that explicit examples unknown.
In measure theory one makes rigorous the concept of something holding "almost everywhere" or "almost surely", meaning the set on which the property fails has measure zero.
I think it is very ...
4
votes
0answers
58 views
Good examples of proofs in mathematics exemplary of creative reasoning [closed]
Just what the title says. I'm not looking for any proofs that require specialized knowledge past the very fundamentals of real analysis. I'm looking for proofs for important results (don't have to be ...
17
votes
2answers
126 views
+500
Conjectural closed-form representations of sums, products or integrals
What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
1
vote
1answer
27 views
Special numbers in patterns and the reasons they are special
I know there are several big list questions out there (e.g. Patterns that break down at certain numbers) that touch on classifications of mathematical structures where certain numbers don't fit in, ...
9
votes
1answer
107 views
List videos of interesting courses at the doctoral level.
Many mathematics departments has provided video lessons their courses (usually one semester) that are offered in their doctoral programs in mathematics. Most often these courses total average of 26 ...
12
votes
8answers
305 views
Free online mathematical software
What are the best free user-friendly alternatives to Mathematica and Maple available online?
I used Magma online calculator a few times for computational algebra issues, and was very much satisfied, ...
0
votes
0answers
31 views
Find two linearly independent functions with a zero Wronskian but a nonzero product.
Known to me examples of L.I. functions having Wronskian=0 have also product=0. One such example was manufactured by Peano about 1890. By the way: Analytic functions with a zero Wronskian are linearly ...
0
votes
0answers
36 views
Intuition on matrix multiplication and algorithms
Yesterday, I was watching Strang's lectures on Matrix multiplication. He mentioned five different ways of looking at the multiplication A x B = C.
Classic way (row of A x column of B).
Column (of B) ...
32
votes
1answer
402 views
Unexpected approximations which have led to important mathematical discoveries
One often finds at MSE approximate numerology questions like
Prove $\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$,
Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$,
Comparing ...
7
votes
1answer
334 views
Practical Tips: Mathematical research and discoveries [closed]
How to behave when you have the feeling of working on something innovative? What to do if there is a chance (even the $1\%$) that your
work is leading you to something original?
For example ...
9
votes
7answers
352 views
Why is it important to study combinatorics?
I was having a discussion with my friend Sayan Mukherjee about why we need to study combinatorics which admittedly, is not our favourite subject because we see very less motivation for it(I am not ...
10
votes
5answers
185 views
Applications of Character Theory
Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
7
votes
2answers
104 views
Algebraic geometry in representation theory?
I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
0
votes
2answers
111 views
Big Topics in Mathematics [closed]
My question is as follows: It is now the year 2013 as we know it, and I'm wondering what the "big topics" in mathematics are. What fields are of utmost interest and foundation in the modern era? How ...
3
votes
1answer
70 views
Tychonoff Theorem and the axiom of choice
How to show that
The Tychonoff Theorem and the axiom of choice are equivalent?
Here I want to collect ways to prove it.
Thanks for your help.
1
vote
1answer
63 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
91 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
29 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 ...
5
votes
3answers
76 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, ...
13
votes
10answers
1k 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 $[a,b]$ ...
103
votes
18answers
8k 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 ...
3
votes
0answers
31 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
80 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 ...
4
votes
1answer
96 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 ...
43
votes
17answers
1k 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 ...
2
votes
0answers
53 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 ...
4
votes
2answers
55 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
80 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 ...
85
votes
28answers
10k 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 ...
6
votes
0answers
66 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
61 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
76 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 ...
2
votes
4answers
191 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
102 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 ...
4
votes
2answers
130 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
193 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 ...
5
votes
4answers
114 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 ...
0
votes
1answer
75 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 ...
25
votes
3answers
312 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 ...
13
votes
9answers
527 views
Examples of nonabelian groups.
Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
39
votes
8answers
722 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 ...
265
votes
115answers
13k 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
77 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 ...
4
votes
3answers
107 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
7answers
194 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
1answer
63 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 ...
11
votes
7answers
533 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 ...
16
votes
19answers
752 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
242 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 ...
