All Questions
542 questions
2
votes
1
answer
740
views
What does the subscript 'x' of a matrix mean? [closed]
The 3x6 matrix G is as follows,
$\text{G} = [\text{V}_\times| I_{3\times3}]$
$\text{V}$ is a skew matrix of a vector with 3 elements about a 3D point. The dimension of $\text{V}$ is 3x3.
$I$ is the ...
- 23
1
vote
1
answer
163
views
The meaning of $L_p^l(\Omega)$ in a paper of Bogovskii on Sobolev spaces
On the first page of the old paper Solution of the first boundary value problem for an equation of continuity of an incompressible medium of Bogovskii, the notations $W_p^l(\Omega)$ and $L_p^l(\Omega)$...
- 13
0
votes
0
answers
148
views
About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
8
votes
1
answer
1k
views
Why aren‘t op and co switched?
When reading through Loregian and Riehl - Categorical notions of fibration, on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^...
- 355
4
votes
1
answer
1k
views
Chalkboard eraser [closed]
I just started my first year of university and because I'm visually impared I have trouble seeing what's written on the chalkboard.
I've partially solved this problem by purchasing chalk from hagoromo ...
48
votes
8
answers
5k
views
Ideas for introducing Galois theory to advanced high school students
Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (...
1
vote
0
answers
113
views
Common notation for function over infinitely many variables? [closed]
For a document about reinforcement learning, I want to write the joint probability density over the entire trajectory of states and actions like $p(s_0, a_0, s_1, a_1, s_2, \dotsc)$. However, this ...
- 61
22
votes
1
answer
3k
views
What is so special about Chern's way of teaching?
First of all sorry for this non-research post.
I was watching Jeffrey Blitz Lucky documentary movie and it was interesting to me that a winner of Lottery was a math Ph.D. from Berkeley.
In the movie ...
- 4,195
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
25
votes
6
answers
3k
views
What is the standard 2-generating set of the symmetric group good for?
I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...
- 4,492
0
votes
2
answers
250
views
Is it improper to define matrices as being $n \times m$ rather than $m \times n$? [closed]
For whatever reason, I have always defined matrices as being $n \times m$, and that is how I have been defining matrices throughout my dissertation. Recently however, I have noticed that nearly every ...
5
votes
2
answers
445
views
About the maximum number of leaves adjacent to a vertex in a tree
Let $T$ be a finite tree graph with the set of vertices $V(T)$. For an arbitrary vertex $ v \in V(T)$, I define $l(v)$ to be the number of leaves connected to $v$.
In my study, I need to define the ...
5
votes
0
answers
640
views
What does $\omega^*$ mean? [closed]
I've recently found in some short article (source below) the symbol $\omega^*$ (generally, starred ordinal number), but without explanation what that symbol means. From the context I understood that ...
- 137
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
1
vote
1
answer
182
views
Pronunciation: the Erdős–Rado partition notation
The Erdős–Rado notation $a \rightarrow (b)^c_d$ is common in partition calculus / combinatorial set theory, as well as its negation $a \not\rightarrow (b)^c_d$. In that field, is there a standard way ...
- 11
6
votes
1
answer
222
views
Reference request: Different definitions of Big O notation
This question might sound strange, but I would like to settle this problem once and for all.
For as long as I can remember, I was introduced to the Big O notation by this definition:
Def. 1: Let $f, g$...
- 61
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
1
vote
0
answers
294
views
What does square bracket superscript star mean in basic group theory typically?
I'm reading some paper where they haven't really defined their notation very well (or I've missed something). You can see the image below.
What does the square bracket and star mean precisely? The ...
- 322
2
votes
1
answer
128
views
Notation for H is isomorphic to a subgraph of G
Is there a notation for the statement $H$ is isomorphic to a subgraph of $G$? I was thinking of using $H<G$, but I'd like to use standard notation if possible.
- 23
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
5
votes
1
answer
208
views
Seven Bridges of Königsberg for hypergraphs
I am teaching a course involving hypergraphs. I would like to have a physical analogy/motivating problem for hypergraphs similarly to how the Seven Bridges of Königsberg motivate graphs. Can you help ...
- 51
5
votes
2
answers
377
views
What is meant by this notation of the real forms of $E_6$?
There are five real forms of the exceptional Lie group, $E_6$. Four of them are notated as in the following:
The split form as EI or $E_{6(6)}$
The quasi-split form as EII or $E_{6(2)}$
EIII or $E_{...
- 2,369
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
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
44
votes
10
answers
11k
views
What kid-friendly math riddles are too often spoiled for mathematicians?
Some math riddles tend to be spoiled for mathematicians before they get a chance to solve them. Three examples:
What is $1+2+\cdots+100$?
Is it possible to tile a mutilated chess board with dominoes?...
3
votes
1
answer
244
views
Finitely-generated conjugation action on a subgroup that is not normal... what is that?
If $H \lhd G$, then $G$ acts on $H$ by conjugation. I need to talk about this action but in a situation where $H$ is not (necessarily) normal. When $H \leq G$, there is a "partial action" of ...
- 6,652
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
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
3
votes
1
answer
243
views
Temporal generalization of graphs: density vs $n$ and $m$?
In short: we generalize graphs to the temporal case, but fail to fully preserve the usual relation between density, number of vertices, and number of edges; how to make better?
Context.
We propose a ...
- 1,444
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
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
-1
votes
1
answer
187
views
Typesetting of symbols and "operators" denoting sets [closed]
Question:
what are the conventions for typesetting sets of certain objects, especially the vertices and edges of a graph or those adjacent to an edge or vertex.
For vectors and matrices there is the ...
- 13.2k
-4
votes
2
answers
228
views
An elementary-looking integral inequality
This might seem a bit easy but I still like to ask it for pedagogical reasons.
QUESTION. Is this inequality true for non-negative integers $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
- 43.2k
0
votes
1
answer
259
views
Explanation of a formula to calculate the zenith distance of sun and moon [closed]
I am studying tidal accelerations and referring to a well known paper by I M Longman :
Formulas for computing.." J Geophys Research 64 (12) Dec 1959.
At Eq 12 he writes a term "1336.rev"...
- 27
9
votes
1
answer
2k
views
Origin of the symbol for the tensor product
I have recently realised that the Paleo-Hebrew (and Phoenician) graph for the Hebrew letter ט (Teth) is $\otimes$. This made me wonder if there is any relation between the choice of the symbol and the ...
- 5,670
20
votes
4
answers
2k
views
PDF readers for presenting Math online
In the current situation it seems especially important to be able to present your mathematical results online in a way that your audience does not fall asleep in front of their screens. But I am ...
23
votes
14
answers
4k
views
Math talk for all ages
I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also ...
9
votes
3
answers
1k
views
Books on the relationship between the Socratic method and mathematics?
Apart from books on heuristics by George Polya.
When trying to engage with and understand mathematical concepts and when applying abstract mathematical concepts to model "continuum" or real ...
- 107
1
vote
0
answers
363
views
Notation for the regular and the adjoint representation of a finite group, in particular the symmetric group
The (left) regular representation of a finite group $G$ is the action on itself by left multiplication, $g\cdot h = gh$.
The adjoint representation of a finite group $G$ is the action on itself by ...
- 5,822
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
4
votes
1
answer
127
views
Question about the notation $N_{\chi}(\alpha, T)$, the number of zeroes of the $L(s, \chi)$ in a rectangle
I am confused with what seems to be a standard notation in analytic number theory and I'd appreciate any clarification. I am interested in the zero density estimates, for example link.springer.com/...
- 3,625
0
votes
1
answer
125
views
Are there search algorithms that are competitive against (gradient based) optimization routines for continuous problems?
Suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a continuous function for which we want to minimize. We may arbitrarily impose good conditions for $f$, such as Lipschitzness, smoothness, convexity, ...
- 479
3
votes
1
answer
323
views
Name and properties of $\mathrm{lcm}(\{1,\,\cdots,\,n\})$ [closed]
one of the most prominent functions of the first $n$ natural numbers is the factorial $n!$ that denotes their product.
Today however I wondered whether the least common multiple $\mathrm{lcm}(n):=\...
- 13.2k
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
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
3
votes
1
answer
202
views
Reference request: Dictionary of the Leibniz notation
Is there any published, somewhat comprehensive, list of (almost?) all the many ways in which the Leibniz notation ($dx,$ $P(dx),$ $d\mu(x),$ $du\wedge dv,$ etc., etc.) gets used in the various areas ...
0
votes
0
answers
290
views
A question about chaining Vinogradov notation
This is not a research question, but I hope it is still legitimate to ask for this platform. Suppose $A(x)$, $B(x)$, $C(x)$, $D(x)$ are positive-valued functions of $x$, and
$A(x) \ll B(x)$ and $ C(x) ...
- 6,224
0
votes
0
answers
39
views
Terminology: Almost stable states
I have a question about fixed points which are almost stable.
I have an increasing transition function $f:[0,1]\rightarrow[0,1]$ where $f(0)>0$ and $f(1)<1$ but I don't necessarily have ...
- 188
118
votes
10
answers
77k
views
What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed to $u^T v$?
In a lot of computational math, operations research, such as algorithm design for optimization problems and the like, authors like to use $$\langle \cdot, \cdot \rangle$$ as opposed to $$(\cdot)^T (\...