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.
32
votes
1answer
415 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 ...
4
votes
1answer
75 views
Problem books in different languages
I simply love problem books in mathematics, though you have to know how to use them properly. I think they are useful to me because most of time I study on my own. I'm thinking here at MSE, since we ...
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, ...
10
votes
0answers
117 views
projective profinite groups
I'm reading the first chapter of Serre's Galois Cohomology.
On p. 58, He gives two examples of projective profinite groups:
(a) the profinite completion of free (discrete) groups;
(b) the cartesian ...
6
votes
0answers
67 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 ...
6
votes
0answers
283 views
Positive definite function zoo
A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$.
For a definition and discussion of ...
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 ...
4
votes
0answers
104 views
Known algebraic geometry's results in Characteristic $p$
What are the most well known results in classical (à la Weil) algebraic geometry in characteristic $p$, which are thought to be true (but not yet proved) in characteristic 0?
Thanks
4
votes
0answers
221 views
Structuralist slogans
I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
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
0answers
432 views
Lists of open problems in set theory
Are there any publicly available lists of open problems in set theory besides the following ones? (And if so, what are they?)
http://www.math.wisc.edu/~miller/res/problem.pdf
...
3
votes
0answers
42 views
Basis-dependent results on matrices
It is common fashion to try to formulate results about matrices in a basis-free way, using linear algebra. What are some good examples of situations where this is impossible? I illustrate what I have ...
3
votes
0answers
520 views
Common tricks to compute series and integrals (common substitutions)
Tomorrow, I will have a test about Calculus 1 and 2 and up to now, I was solving exercises over and over again but I have the impression that I don't learn a lot from doing this. I believe that when ...
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 ...
2
votes
0answers
92 views
Properties about Matrices that can be proved by only using Block Multiplication of Matrices
I recently proved the property that product of two upper triangular matrices is an upper triangular matrices by using the block multiplication of matrices. The basic fact that was required to prove ...
2
votes
0answers
153 views
Motivating questions for some topics in undergraduate calculus
Being a grad student I'm going to teach a whole class for the first time the coming summer and I'm looking for some motivational problems which I could use to introduce different topics. In other ...
2
votes
0answers
139 views
Examples of proofs by induction with respect to relations that are not strict total orders.
I have read this Wikipedia article and found it fascinating. I came across it after I tried to prove a certain statement with a method resembling induction in the set of natural numbers but ordered by ...
2
votes
0answers
113 views
Interesting approximations? ($\pi$, $e$, etc.?)
I think it'd be cool to see a list of neat approximations of mathematical constants, interesting convergent sequences of functions, links to Ramanujan's notebooks...! Post your favorites!
2
votes
0answers
92 views
List of large classes of functors providing morphisms
I recently posted this question on mathoverflow and it was closed as being too localized. I am hoping to more precisely say what I mean here.
I recently learned, through my Topology coursework, that ...
1
vote
0answers
38 views
Easy Independence Statements
Proving a first order sentence is independent of Peano Axioms can be very difficult. An example of such a statement is the Paris-Harrington theorem:
...
1
vote
0answers
69 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
• ...
1
vote
0answers
57 views
A gratifying re-encounter with a piece of math that was out of my mind
A series of real numbers is said to be conditionally convergent if it is convergent but not absolutely convergent.
By rearranging the terms of a conditionally convergent series we can make the ...
1
vote
0answers
115 views
A rigorous book (or preferrably set of notes) on classic multivariable calculus-analysis?
This is different to (Theoretical) Multivariable Calculus Textbooks as I want a classical treatment of line and surface integrals without the notion of a differential form.
Prerequisites: Paths, ...
1
vote
0answers
56 views
Partition of open sets in $\mathbb{R}^d$.
Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets.
In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
1
vote
0answers
134 views
Problems and conjectures that have positive practical consequences for society, once solved
This question made me think a bit about how mathematics can be used in such a way that society benefits from it.
I think there are quite a lot of good answers to the aforementioned question. Still, ...
1
vote
0answers
98 views
Potential computational questions that could be asked about p-adic numbers and Galois Theory
I have an exam on P-Adic integers (and a bit on Galois Theory) that my professor said would be very computational, but he never does any examples of the theorems he proves in class. He said the exam ...
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) ...
0
votes
0answers
408 views
Working through Math 55 problem sets as self-study
I am not a professional mathematician, but have learnt Engineering Mathematics in college and worked through parts of maths textbooks myself. The latter include the first few chapters of include Real ...
