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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204 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 ...
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244 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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301 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 ...
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420 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 ...
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69 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
5
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87 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 ...
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104 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).
5
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117 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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307 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 ...
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111 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 ...
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2k 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 ...
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66 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
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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
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42 views

Good topologies on $\mathcal{P}(X)$

Let $X$ be a topological space, and let $\mathcal{P}(X)$ (resp. $\mathcal{P}_0(X)$) be the set of all subsets of $X$ (resp. the set of all non empty subsets of $X$). Finally, let ...
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34 views

Revise high school material

Can you suggest me a comprehensive book to revise high school mathematics (up to besic calculus)? It should be extremely clear and complete and "scientific" (not like most high school books). Thank ...
2
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0answers
30 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 ...
2
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214 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. ...
2
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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? ...
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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. ...
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74 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
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139 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
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210 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
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0answers
199 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 ...
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103 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 ...
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37 views

Video lectures of algebraic geometry (Hartshorne, Shafarevich, … )

I am a commutative algebra student. I wonder if there is some video lectures of algebraic geometry courses available online for free? (I want to download) I'd like the lectures to cover ...
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23 views

What [precisely] is special, algebraically, about fundamental Pell solutions?

I'm asking for a list of algebraic identities which "uniquely identify" (or nearly so) the fundamental solution (or those immediately around it) of a Pell equation. As a concrete [numerical] example, ...
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41 views

Are there such prime giving functions?

Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us ...
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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, ...
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34 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 ...
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90 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 ...
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415 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.
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65 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} ...
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120 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 • ...
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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 ...
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194 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, ...
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0answers
67 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 ...
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170 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, ...
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685 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 ...
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109 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 ...
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29 views

Text on Witt vectors that are accessible to undergraduate students

I am looking for a thorough text on Witt vectors that is accessible to an undergraduate student that have completed the following courses: Calc 1, 2, Linear Algebra and Abstract Algebra. (In Norway, ...
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61 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
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19 views

Reference request for papers proving the existence of a special function satisfying the following conditions

Previously in this site it has been proved that there exists at least one prime between $c_n$ and $n$ where $c_n$ denotes the $n$-th composite (see the question Prove that there exists an $m$ such ...
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46 views

Has this property for algebraic structures got a name?

Given a binary operation $*:M\times M\rightarrow M$. I define the extension of $*$ to the subset of $M$ in the usual way (with $A,B \subseteq M$) $a*A:=\{a\}*A$and $A*a:=A*\{a\}$, ...
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23 views

A compendium of proof-techniques per objective

Please consider this as an on-going list of techniques preferably per objective or subject. Many mathematical books (at least lately) are focusing on "design patterns" if you like of proof-techniques ...
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37 views

List of prime graphs

Prime graph under action $\circ$ (where $\circ$ can be Cartesian, strong, direct, lexicographic, etc. product of graphs) is a graph $G$ for which if $G=H\circ J$ then $H=E\vee J=E$, where $E$ is ...
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22 views

Trigonometric identities for angles that sum up to $2\pi$

Given $A + B + C + D = 2\pi$, are there special trigonometric identities that concern these four angles? If possible, I would like to know whether the above relation has any implications on any of the ...
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14 views

Discrete and Continuous analogues

What are some famous theorems who have both discrete and continuous versions but only one of them is well-known?
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34 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 ...
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53 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 ...
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92 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 ...