All Questions
90 questions with no upvoted or accepted answers
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
11
votes
0
answers
2k
views
Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective
$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...
- 156k
9
votes
0
answers
887
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.9k
8
votes
0
answers
416
views
Pedagogical question on Lie groups vs. matrix Lie groups
There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...
- 28.1k
8
votes
0
answers
554
views
Lower semicontinuity of naive fiber size
I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
- 156k
7
votes
0
answers
366
views
Why are fundamental weights denoted by omega?
In my field (and many others, I believe) the absolutely standard notation for the fundamental weights of a root system is lowercase omega: $\omega$. Recently I was taken aback to receive a copyedited ...
- 3,513
7
votes
0
answers
1k
views
Conventions for Riemann curvature tensor
I am aware of two conventions for the Riemann curvature tensor, namely the expression
$$\langle\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W\rangle$$
is either declared to be $R(X,Y,Z,W)$ or $...
- 18.7k
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
6
votes
0
answers
466
views
What is the "permanence relation" really?
I have come across the words "permanence relation" in a 1969 paper by Keith Hannabuss The Dirac equation in de Sitter space. The only other similar google hit for this phrase appears in ...
- 33.3k
5
votes
0
answers
131
views
Why $f^\lambda$ in the hook-length formula?
This is my first question on this site so I apologize if it’s not adequate for it.
I just learned the hook-length formula for the number $f^\lambda$ of Standard Young Tableaux of shape $\lambda$:
$$f^\...
- 151
5
votes
0
answers
121
views
Adjunction symbol
What are the reasons for the adjunction symbol $F\dashv G$ for a pair of functors $F:C\to D$ and $G:D\to C$? There is no explanation or motivation in the article of Kan where adjunctions are ...
- 16.5k
5
votes
0
answers
186
views
Examples of partial adjoints
Recall that a functor $$R: D \to C$$ is said to have a partial left adjoint $L$ defined at an object $X \in C$ if the functor
$$D \to Sets, Y \mapsto Hom_C(X, R(Y))$$
is corepresentable by some object ...
- 2,040
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 ...
user13113
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
4
votes
0
answers
160
views
Proof of Theorem 9.2 of the book Cubic Forms by Yu. I. Manin (end of page 37)
I warn that I first posted this question in Mathematics Stack Exchange but it got no attention at all. I think that it fits better there by its explanatory nature but maybe the book being reference is ...
- 573
4
votes
0
answers
180
views
Ideals with certain properties
I recently isolated the following definition, which I believe it should have appeared somewhere.
Let $\kappa$ be a cardinal, and let $X$ be a set with $\kappa^+\leq |X|$.
Definition: An ideal
$\...
- 2,381
4
votes
0
answers
197
views
Who introduced the heart ($\mathcal{C}^\heartsuit$) notation in the context of $t$-structures on triangulated categories?
In the context of $t$-structures
([Wikipedia],
[nLab],
[Notes I],
[Notes II],
[HA, Definition 1.2.1.11)],
[BBD, Définition 1.3.1]),
one often writes $\mathcal{C}^\heartsuit$ for the heart of a ...
- 11.8k
4
votes
0
answers
111
views
Is there a name for groups of the form $Sp(1)^n$?
A (compact) torus is a Lie group isomorphic to the product of finitely many circles: $T^n = S^1 \times \cdots \times S^1$. Such groups are extremely important in Lie theory, Differential Geometry, ...
- 4,729
4
votes
0
answers
4k
views
Pronunciation of ¡ (inverted exclamation mark, historically used for subfactorial)
For anyone who uses ¡ (inverted exclamation mark) in a mathematical context, how do you pronounce it?
Background: I have privately been using ¡ in a couple of notations for a while, and am ...
- 21.3k
4
votes
0
answers
176
views
Are injective modules flabby on basic open sets?
In order to give a simple proof of a basic fact about quasi-coherent modules (see below), I'm interested in knowing whether the following statement holds:
Statement: If $A$ is a commutative ring and $...
- 1,064
4
votes
0
answers
795
views
Almost linear ODE: how node becomes a spiral
Most introductory ODE books contain a discussion of almost linear systems, and there are two cases when the behavior of an almost linear system near an equilbrium point can differ from the behaviour ...
- 29.1k
3
votes
0
answers
167
views
Suitability of formal type theory for mathematical thinking (vs. traditional set theory)
Type theory has advantages over set theory for the (computer) formalisation of mathematics, but has anybody who does mathematics with pen and paper found proof assistants or automated theorem provers, ...
3
votes
0
answers
176
views
What is the meaning of big-O of a random variable?
I encountered this problem in a book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. I excerpt it below:
screenshot of the book
In the excerpt, the big-O notation $O(\xi^...
- 31
3
votes
0
answers
873
views
Hard problems solving tricks
This question is motivated by this one that I posted on math.stackexchange.
When I fail to solve a hard math problem (like the ones I presented in the linked post), I read a solution and I noticed ...
- 161
3
votes
0
answers
238
views
How to denote a partial derivative?
This question is related to Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix? and Suggestions for good notation .
When there are two ...
- 6,891
3
votes
0
answers
649
views
Does the Polish character Ł have an established mathematical meaning
I was suggested to use the slashed letter $\L$ (the European character Ł, which looks like the English letter L with a small bar crossing its vertical part) to denote the left half-plane. To avoid ...
- 81
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
3
votes
0
answers
264
views
Nesting big-O with big-Omega $O(g(\Omega(h(n))))$: is it $O$ for all $\Omega$ or for one $\Omega$?
I want to express the following statement about a function $f(n)$: there exists $f_\Omega\in\Omega(h(n))$ such that $f\in O(g(f_\Omega(n))$. What's the correct notation for this? Is it $f\in O(g(\...
- 163
3
votes
0
answers
311
views
Tensor power- Notation question
Hi everyone
I have a notational question, which is written usually in papers, but I can not figure it out what could be. Let $M$ be an $A$-module. I have seen this notation
$$M^{\otimes -n}$$
I ...
- 2,508
3
votes
0
answers
131
views
Isomorphism modulo the residual
Given a group $G$ let $R(G)$ be its residual, that is the intersection of all the normal subgroups of finite index. Is there a name for the relation between $G$ and $H$ defined by $G/R(G) \cong H/R(H)$...
- 5,450
2
votes
0
answers
182
views
What do you call $x$ such that $\textrm{dim} f^{-1}(f(x))>0$?
Let $f:V\to W$ be a morphism between varieties, with $\dim \overline{f(V)} = \dim V$. What do you call the closed proper subvariety $S$ of $V$ consisting of points $x$ such that $\textrm{dim} f^{-1}(f(...
- 20.2k
2
votes
0
answers
311
views
Degree of a morphism between affine varieties
(Context: rewriting a joint paper with a coauthor.)
We are defining the degree of a morphism $f:A^m\to A^{n}$ to be $\max_{1\leq i\leq n} \deg(f_i)$, for $f_1,f_2,\dotsc,f_{n}$ the polynomials ...
- 20.2k
2
votes
0
answers
234
views
Why is $H$ the standard notation for mean curvature?
I am curious about the origin of the notation $H$ to denote the mean curvature of a surface in $\mathbb{R}^{3}$.
I suppose that the symbol $K$, which is commonly used to denote the Gaussian curvature, ...
- 985
2
votes
0
answers
100
views
Name for the theory of words with equal length, prefix, successors
I've worked with this theory for a while, but I've never been quite sure what to call it:
$$(\Sigma^*, =_{el}, \preceq, (S_a)_{a \in \Sigma})$$
Where
$\Sigma^*$ is the set of finite words on finite ...
- 429
2
votes
0
answers
74
views
Terminology and notation for generated subgroups
I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
- 12.5k
2
votes
0
answers
316
views
Higher order Leibniz rule and ordered multiindex notation
Although I think this is probably known, I am making here a short exposition on the multiindex notations I am using to make this question self-contained. I note that there is at least two different ...
- 3,307
2
votes
0
answers
177
views
Can NBG be interpreted in this system that use new notation for class-abstractions?
We introduce a new symbol $\lambda$ to denote class-abstractions, and we add the following rule:
if $\phi$ is a formula that use $``\mu"$, and in which the symbol $\sf y$ doesn't occur; then: $\lambda ...
- 11.3k
2
votes
0
answers
124
views
Good notation for finite partial functions from $\omega$ to 2
I'm working in computability theory and need to use partial functions with finite domain from $\omega$ to 2 as approximations in my current paper. Normally this is simply done using $2^{< \omega}$ ...
- 3,029
2
votes
0
answers
905
views
Confusing notation for sets of unordered vs ordered pairs
Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$.
One may also consider ...
- 1,444
2
votes
0
answers
221
views
What is the p-adic Plancherel measure?
What I know as the Plancherel measure for a group is a measure on the spectrum of $G$, aka the set of irreducible representations - at least for finite groups, this makes perfect sense.
Now, this ...
- 7,746
2
votes
0
answers
314
views
Notation for induced subgraphs
For a graph $G=(V,E)$, is there a standard notation for the induced subgraph on $V \setminus \{v,w\}$ where $v,w$ are the endpoints of some edge $e$? I know $G[V \setminus \{v,w\}]$ is an option, but ...
- 19.7k
2
votes
0
answers
323
views
Marcinkiewicz-Mihlin-Hormander Fourier multiplier theorem
I'm trying to understand the hypothesis of the Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in this paper of Elias Stein.
Theorem A: Assume that $m: (0, \infty)\to \...
- 161
2
votes
0
answers
115
views
Name for generalization of property: $f^n(x) \ne x$ for all $n > 0$
I am curious about how to specify with standard terminology that a certain function is non-periodic, in the following sense:
In the simple case of a unary operation $f: X \to X$, this property would ...
- 193
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
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 \...
- 3,793
2
votes
0
answers
657
views
Mixed tensor index position significance
What is the significance of tensor index position?
For example the fourth order Riemann curvature tensor
\begin{align}
R^m_{ijk}
\end{align}
or
\begin{align}
R^{\phantom{i}m}_{i\phantom{m}jk}.
\end{...
- 219
2
votes
0
answers
812
views
Products between metrics in a product of manifolds
In the "Einstein Manifold" book written by Arthur Besse, chapter 16, there is a notation of a manifold composed by the Cartesian product between two others:
$(M_1\times M_2, f^p(g_1 \times g_2))$
...
- 21
2
votes
0
answers
179
views
Notation for a canonical quotient of an abelian variety in positive characteristic
This is a light question about notation, but I received no answer in Stackexchange.
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian ...
- 797
1
vote
0
answers
106
views
The proposition associated with a set
Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
- 19.7k