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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7
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1answer
83 views

Create a Huge Problem

I am wondering if any problems have been designed that test a wide range of mathematical skills. For example, I remember doing the integral $$\int \sqrt{\tan x}\;\mathrm{d}x$$ and being impressed at ...
4
votes
1answer
27 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
3
votes
1answer
70 views

Second order linear differential equation

I have to teach the following methods to my juniors at college to solve differential equations: 1) partial fractions 2) reduction of order 3) variation of parameter 4) power series 5) green's ...
2
votes
1answer
87 views

Hard problems book in linear algebra

Could you suggest me a book where I can find hard problems in Linear Algebra for an undergraduate student? Thanks in advance.
1
vote
1answer
114 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
-1
votes
1answer
64 views

A big list of examples that a power of a prime ideal is not primary in an algebra of finite type over a field

Let $k$ be a field. Let $A$ be an integral domain which is a $k$-algebra of finite type. I would like to know examples that a power of prime ideal of $A$ is not primary. The more example, the better. ...
19
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0answers
180 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: the profinite completion of free (discrete) groups; the cartesian product ...
10
votes
0answers
210 views

Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
10
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0answers
264 views

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 ...
8
votes
0answers
169 views

Interesting but short math papers?

Is it ok to start a list of interesting, but short mathematical papers, e.g. papers that are in the neighborhood of 1-3 pages? I like to read them here and there throughout the day to learn a new ...
6
votes
0answers
399 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 ...
5
votes
0answers
82 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
5
votes
0answers
115 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
5
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0answers
299 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 ...
4
votes
0answers
93 views

What are some great graduate textbooks with solutions in the back to the problems?

I can think of Aubin's A Course in Differential Geometry, as well as Knapp's books. Any other great ones you know of? Especially in the GSM series from AMS (blue and yellow covers).
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 ...
4
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0answers
1k 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
86 views

Which mathematical game or puzzle did you invent?

A couple of weeks ago, a friend of mine showed me a extension of a game we are all familiar with that he was working on. The game we know is called Tic-Tac-Toe, and he was working on his own version ...
3
votes
0answers
61 views

Math for kids with Cuisenaire rods

I work with kids and i am searching some cool stuff to do with Cuisenaire rods. Thinking about an application i thought that i can show to my students what will be the sum of first $N\in\mathbb{N}$ ...
3
votes
0answers
49 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 ...
2
votes
0answers
58 views

List of most useful coverings and their applications?

I've heard that many problems may be simplified when looking at covering spaces, but I haven't been able to find a good list. What are the most common covering spaces one should understand by heart? ...
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. ...
2
votes
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 ...
2
votes
0answers
132 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
198 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
190 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
120 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
101 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
33 views

Matrices of the form $A^p=(a_{ij}^p)$

I am wondering if there is a name for these kind of matrices and if they are interesting or not? Do they even exist? Let $A$ be a $n\times n$ matrix with elements $a_{ij}$. $A= (a_{ij})_{i,j\in\{1, ...
1
vote
0answers
67 views

List of Common or Useful Limits of Sequences and Series

There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. ...
1
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0answers
143 views

Lecture Notes in Real Analysis

I understand that this question was partially addressed here but I would like to have a question dedicated to just real analysis. I am looking for both elementary real analysis (advanced calculus type ...
1
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0answers
28 views

limit formalisms

Let $f:\mathbb R \to \mathbb R$ be a function and $a\in \mathbb R$ a point. The Cauchy definition of the limit $\lim _{x\to a}f(x)=L$ is well-known. For pedagogical reasons I'm interesting in a ...
1
vote
0answers
88 views

What is a good source of problem-solving type problems?

I am not looking for contest problems where there is a clever trick or a standard approach, I am looking for more creative and open-ended problems such as this , and I am not looking for questions ...
1
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0answers
368 views

What are real life applications of Diophantine equations?

Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.
1
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0answers
59 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 $\mathbf{AB} = \mathbf{C}.$ Classic way (Row of $\mathbf{A} ...
1
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0answers
110 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
61 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
184 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
66 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
163 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
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0answers
605 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 ...
1
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0answers
104 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
30 views

The term $rank$ in methematics

Reading wikipedia's disambiguation page about the "rank" word I see many concept of rank of many different matematical object. I only know about the rank of a graded poset and the rank of a set that ...
0
votes
0answers
29 views

What are $\Gamma$-semigroups?

I have some problems with $\Gamma$-Semigroups, the definition that I've found is A $\Gamma$-Semigroup is a pair $(M,\Gamma)$ defined as follow If $x,y$ and $z$ are in $M$ and $\alpha$ and ...
0
votes
0answers
24 views

List of crucial results deserving more attention for first course in Real Analysis

Can it help to form a list of crucial results for basic courses that are concealed as exercises or neglected? I don't know of other resources for this, as I wrote here. I am happy for this to be ...
0
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0answers
81 views

What are some great Bachelor's Project subjects in the field of Mathematical Optimization Theory?

Currently, I ought to pick a subject for my third year mathematics bachelor's thesis. I would like to research something in the field of mathematical optimization theory. I have a background in basic ...
0
votes
0answers
12 views

How to suggest the intractability of a problem that is not known to be $\mathcal{NP}$-complete

If a proof of a a decision problem in $\mathcal{NP}$ being $\mathcal{NP}$-complete can be found, it is a strong evidence that the problem is intractable: people have not found efficient algorithms for ...
0
votes
0answers
106 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 ...