All Questions
542 questions
16
votes
2
answers
1k
views
Teaching Steenrod Operations
I am teaching a class on topology and want to introduce Steenrod Operations. I have talked about simplicial sets and classifying spaces of groups but have not talked about Eilenberg–MacLane spaces. ...
- 3,439
6
votes
8
answers
2k
views
Mathematical objects whose name is a single letter
(Not research-level, but perhaps not easily answered elsewhere — you decide if MO can afford the innocent fun. If so, it should likely be “community-wiki” i.e. one object per answer.)
I am seeking ...
- 18.4k
22
votes
13
answers
8k
views
Category theory sans (much) motivation?
So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...
- 1,228
5
votes
1
answer
548
views
Question about denoting/designating of algebraic structures
I saw this image on Wikipedia (Template:Group-like structures, current revision):
Since there are five "properties" that we can have (in this context), namely: totality, associativity, identity, ...
- 11.8k
24
votes
11
answers
8k
views
The role of the mean value theorem (MVT) in first-year calculus
Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...
CommunityBot
- 1
9
votes
5
answers
3k
views
Assessing effectiveness of (epsilon, delta) definitions [closed]
There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...
- 16.6k
16
votes
1
answer
2k
views
A conjecture in which both "if" and "only if" are near misses
[Migrated from Math Stack Exchange]
More than a year ago, I posted the following on the Math Stack Exchange.
Consider $2^n-1$. Based on checking a few small numbers for $n$ (in
fact, the first ...
- 32.5k
1
vote
1
answer
211
views
Notation for the restriction map in Galois cohomology
My coauthors and I are writing a paper based on MO questions and answers:
Friedrich Knop's answer,
my answer 1
and
my answer 2.
For a linear algebraic group $G$ over a perfect field $k$, I consider a ...
- 47.4k
15
votes
3
answers
2k
views
When was the "arrow notation" for functions first introduced?
When was the "arrow notation" $f: X \to Y$ for functions first introduced? Who introduced it and with which motivation?
I ask this question in order to understand whether it was, in part, this ...
- 31.5k
13
votes
2
answers
2k
views
teaching higher algebra
Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)?
I'm asking out of curiosity (and also hoping for more resources).
The kind of ...
CommunityBot
- 1
7
votes
3
answers
3k
views
Problems reducing to a graph-theory algorithm
This is essentially a question in pedagogy -- the answers could be useful to teach (or rather, motivate) graph theory, and especially the algorithmic side of it.
I have been very impressed with this ...
CommunityBot
- 1
11
votes
3
answers
448
views
Easy proof that reflections generate $N(T)/T$ for connected compact group?
I'm teaching a course on Coxeter groups and I'd like to provide an overview of the connection to compact Lie groups. Let $G$ be a compact connected Lie group, $T$ a maximal torus and $N(T)$ the ...
- 878
39
votes
4
answers
2k
views
Important open exposition problems?
Timothy Chow, in his article A beginner's guide to forcing, defines an open exposition problem as a certain concept or topic in mathematics that has yet to be explained "in a way that renders it ...
CommunityBot
- 1
3
votes
1
answer
793
views
Are traditional notations for elliptic integrals/functions in Latin or Greek letters?
I am doing some calculation involving elliptic integrals/functions, and find the notations confusing.
In Wittaker-Watson, the "Jacobi's earlier notation" H(u) is called the Eta-function, so the "H" ...
- 26.3k
14
votes
0
answers
919
views
Grothendieck construction and coends
In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively:
$$
\int F
$$
for a functor $F:C\to\mathbf{Cat}$,
and:
$$
\int^x G(x,x)
$$
...
- 2,129
7
votes
5
answers
6k
views
Advantages of the sequence definition of limits
I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of ...
- 16.6k
2
votes
2
answers
247
views
Technical term for representing object of a presheaf determined by a left-adjoint?
If $\mathcal{D}$ is a locally-small category, then a functor $F\colon\mathcal{C}\rightarrow\mathcal{D}$ has a right-adjoint if and only if for each object $d$ of $D$, the presheaf $$\mathcal{C}^{\...
- 21.3k
2
votes
0
answers
240
views
What does the $\pi_1(\mathsf{C})$ really mean?
Assume that $\mathsf{C}$ is a small category (in my case with finitely many objects but this is probably irrelevant). In a paper I'm studying at the moment there is a notion used constantly, this of $\...
- 441
2
votes
0
answers
99
views
Spectral multiplier and Littlewood-Paley projection
I am trying to understand this paper, and have some basic question, and hope this is OK for the MO.
Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz Space).
We know that $\widehat{\nabla f}(\xi)= 2 \...
- 31
3
votes
0
answers
146
views
Local system corresponding to induced representation
Let $p\colon Y\to X$ be a finite covering map of path-connected "good" spaces (e.g. manifolds), and let $L$ be a local system on $Y$, and let $V$ be the corresponding representation of $\pi_1(Y)$. ...
- 3,079
9
votes
1
answer
420
views
notation for $(a-b)(a-qb)\dots (a-q^{n-1}b)$
I wonder whether there is a notation for such thing, which I denote $[a;b]_q^n$ for a moment:
$$
[a;b]_q^n:=(a-b)(a-qb)\dots (a-q^{n-1}b)=a^n(b/a;q)_n,
$$
this last equation uses $q$-Pochhammer symbol ...
- 5,622
5
votes
1
answer
1k
views
Generalizing Big O notation to arbitrary vector spaces
I'm constructing a Coq library for Big-O notation. Naturally, I'd like it to be as general as possible. The Wikipedia page on Big-O notation says
The generalization to functions taking values in ...
- 47.6k
21
votes
7
answers
3k
views
What should be taught in a 1st course on Riemann Surfaces?
I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...
CommunityBot
- 1
7
votes
0
answers
214
views
Notation: Why Ω for the based loop functor?
This is just a question about notation - probably useless, but it's always baffled me:
Why was $\Omega$ chosen to denote the based loop functor?
I once heard someone speculate: "It's because $\Omega$...
- 113
-1
votes
1
answer
124
views
Typed Values in Formulas
Question:
are there any "standard" ways of indicating the meaning of numerical values in formulas, resp. general mathematical texts (theorems, proofs, etc.)?
I am especially looking for ...
CommunityBot
- 1
6
votes
1
answer
462
views
How to talk about certain "free" categories?
Given two categories $\mathcal{C}$ and $\mathcal{D}$, we can describe the following category $\mathcal{E}$. It is the initial category whose object set contains $\mathrm{Obj}(\mathcal{C}) \times \...
CommunityBot
- 1
13
votes
17
answers
3k
views
Short Course Suggestions For High School Students
I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are ...
CommunityBot
- 1
2
votes
0
answers
323
views
Is there standard notation for restriction partial functions?
Given a partial function $f : A \rightarrow B$, and a subset $S \subseteq A$, we get a new partial function $$f \restriction_S : A \rightarrow B$$ by restriction. However, I prefer to analyse $f \...
- 20.5k
2
votes
1
answer
215
views
Notation for the automorphisms of a $S$-scheme over automorphisms of $S$
Here is a slightly anecdotical notational question.
Let $S$ be a scheme and let $X$ be a scheme over $S$, with structural morphism $s\colon X\to S$. Is there a good suggestive notation for the group $...
- 14.2k
8
votes
2
answers
6k
views
Equal signs with fancy marks
Some people use $\stackrel{\mathrm{def}}{=}$, $:=$ or $\stackrel{\Delta}{=}$ for definitions.
In more informal contexts, I have also seen $\stackrel{?}{=}$, for "I wish to prove this equality, which ...
- 1,697
49
votes
14
answers
21k
views
Applications of the Cayley-Hamilton theorem
The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...
CommunityBot
- 1
3
votes
1
answer
169
views
What is the function space $H^1_{m, \sigma}$?
I am reading Hildebrandt's and Widman's 1975 paper on "Some regularity results of quasilinear elliptic systems of second order".
Theorem 3.1 is the first time in their paper that the function space $...
CommunityBot
- 1
13
votes
7
answers
35k
views
Real analysis has no applications?
I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications ...
39
votes
6
answers
6k
views
Who invented diagrammatic algebra?
There is a strong and growing trend to do mathematics via diagrammatic algebra, which involves constructing and manipulating equations whose elements are diagrams drawn in the plane. The manipulations ...
CommunityBot
- 1
12
votes
12
answers
2k
views
What are fun elementary subjects in probability?
I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just advertisement....
- 39.9k
1
vote
0
answers
112
views
Notations - Hardy and Sobolev Spaces [duplicate]
After some confusion on my part, I wanted to know is there a profound mathematical reason why both Hardy spaces and Sobolev spaces are denoted by $H^p$(1). Is it just coincidence? Does it have any ...
- 11
1
vote
0
answers
224
views
Does the LaTeX $\eqslantless$ symbol, or the comparable Unicode ⋜, have a well defined meaning for binary numerical relationships? [closed]
At first this appeared a simple question; Unicode defines the symbol as "equal to or less-than", which would appear to be the same as "less-than or equal to". But on investigating a bit, I found very ...
- 31
27
votes
10
answers
4k
views
What (fun) results in graph theory should undergraduates learn?
I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph ...
CommunityBot
- 1
6
votes
0
answers
622
views
How necessary is the knowledge of Lebesgue integral for non-analysts? [closed]
Recently I have learned that at some math department the introductory course to Lebesgue integration not obligatory. Thus in another course on introduction to Hilbert spaces the $L^2(0,1)$ space is ...
- 21.8k
5
votes
0
answers
361
views
Notation for calculus with measures?
One of the strengths of ordinary multivariable calculus is that you can use notation where functions are expressed pointwise (e.g. $\int_a^b x^2 \, \mathrm{d}x$ rather than merely $\int_a^b f$), and ...
CommunityBot
- 1
1
vote
1
answer
185
views
Using Ordinal Notations in Computability Theory Is There A Standard Notation For The Notations Below $\alpha$
I find I frequently have to refer to the set of ordinal notations below some given notation. For instance given a notation $\alpha$ I often need to refer to the set $\lbrace \beta \mid \beta <^{\...
- 764
6
votes
2
answers
588
views
Applications of isotropic quadratic forms
I will soon be teaching an introductory course on bilinear algebra and quadratic forms. I will likely spend most of the time and effort on positive definite quadratic forms and euclidean spaces. These ...
- 79.9k
13
votes
3
answers
1k
views
Teaching polarisation formula
When teaching about Hilbert spaces, one begins with a polarisation formula, which allows us to reconstruct the scalar product from the norm:
$$\langle u,v\rangle=\frac14(\|u+v\|^2-\|u-v\|^2+\imath\|u+\...
CommunityBot
- 1
8
votes
2
answers
3k
views
What is the standard notation for reversing the order of vector's components? [closed]
If we have a vector $x=(x_1,x_2,\ldots,x_n)$, is there any standard way to denote the vector $(x_n,x_{n-1},\ldots,x_1)$?.
I think that $x^{-1}$ could be a good option.
- 2,834
2
votes
1
answer
340
views
Name of a lattice-property
Assume that we have a complete lattice $(L,\leq)$.
I would like to know whether the following property has a specific name and whether lattices with this property have been studied somewhere:
For ...
- 4,713
3
votes
1
answer
771
views
Stochastic Process Notation
Note: I'm not an expert on stochastic processes. Please use small words and speak real slow.
I'm reading a paper [1], which uses a notation for stochastic processes that doesn't seem to be standard.
...
CommunityBot
- 1
2
votes
1
answer
293
views
Notation and reference for polynomials with coefficients not commuting with the indeterminates
Let $R$ be a noncommutative ring (with unit). Then a "fully noncommutative" (for a lack of better wording) monomial over $R$ in the single noncommutative indeterminate $X$ of degree $d$ is given by a ...
- 5,482
5
votes
0
answers
2k
views
A course on modern algebraic geometry from "The Stacks Project"
I hope this question is viable for this site. I'm sincerely sorry, if you think it isn't.
For a lot of time, "EGA" by Alexander Grothendieck and Jean Dieudonne was "the" reference on the basics of ...
- 59
55
votes
16
answers
16k
views
Why do we need random variables?
In this MathStackExchange post the question in the title was asked without much outcome, I feel.
Edit: As Douglas Zare kindly observes, there is one more answer in MathStackExchange now.
I am not ...
CommunityBot
- 1
7
votes
2
answers
1k
views
Two different kinds of definitions of $C^k(\overline{\Omega})$ — extension and restriction
This is cross-posted in MSE.
I have seen two different kinds of definitions of the notation $C^k(\overline{\Omega})$ — by "extension" of functions on $\Omega$ or by "restriction" of functions on $\...
CommunityBot
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