All Questions
542 questions
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, ...
CommunityBot
- 1
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 (\...
- 114k
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 ...
- 24.2k
34
votes
6
answers
3k
views
Does seeing beyond the course you teach matter? The case of linear algebra and matrices
This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms ...
- 4,195
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
16
votes
2
answers
889
views
Why are Thompson's groups called $F$, $T$ and $V$?
Why are Thompson's groups called $F$, $T$ and $V$?
I never saw Thompson's unpublished notes, in which he introduces these groups; maybe an explanation can be found there?
- 12.9k
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):=\...
- 64k
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
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) ...
- 12.9k
0
votes
2
answers
542
views
Need help in understanding meaning of a notation and theorem used in research paper due to a reference being in German Language
I thought of utilizing this lockdown period to study research papers in number theory by myself.
I began reading the research paper By T Estermann ->" On Goldbach Problem : Proof that Almost all ...
- 326
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
40
votes
16
answers
11k
views
"Homotopy-first" courses in algebraic topology
A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...
- 35.5k
154
votes
7
answers
85k
views
Where to buy premium white chalk in the U.S., like they have at RIMS? [closed]
While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek.
At RIMS (in Kyoto) in 2005, they had the best white ...
- 45.7k
8
votes
2
answers
693
views
Seeking a combinatorial proof for a binomial identity
Let $n\geq m\geq0$ be two integers. The below binomial identity is provable by other means:
$$\sum_{j=0}^m(-1)^j\binom{n+1}j2^{m-j}
=\sum_{j=0}^m(-1)^j\binom{n-m+j}j.$$
QUESTION. Can you provide a ...
- 43.2k
1
vote
0
answers
102
views
Notation question: bigraded direct sum of graded objects
In some work I'm doing I have two graded modules $M$ and $N$ (graded on $\mathbb Z$, say) and need to take, not the usual direct sum, but the bigraded sum consisting of all $M_p \oplus N_q$ (so graded ...
- 5,429
34
votes
23
answers
29k
views
Textbook recommendations for undergraduate proof-writing class
I am teaching the proof-writing class (for the 3rd time) in the Fall and plan to buck the party line and use a different text than the default Bond and Keane. My parameters are as follows:
Logic, ...
- 33.8k
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 \...
- 6,367
103
votes
13
answers
37k
views
How misleading is it to regard $\frac{dy}{dx}$ as a fraction?
I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as ...
- 161
0
votes
1
answer
158
views
Unknown notation in "Boolean function complexity" by Stasys Jukna [closed]
I am currently reading Boolean Function Complexity - Advances and Frontiers by Stasys Jukna and on page 7 of the latest edition there is a paragraph titled Boolean functions as set systems with the ...
- 873
5
votes
1
answer
1k
views
Why did mathematical notation stay so hard to read? [closed]
One of the first things you learn in a programming 101 course is to write readable code, and to name your variables properly. This notion has seemingly never translated into mathematics. Everywhere ...
- 26.3k
1
vote
0
answers
134
views
What benefits of math can be conveyed to mid/high schoolers? [closed]
I'm teaching mathematical proof writing to a few of math teachers (in the US) this summer. In the beginning of class, I send a survey asking them why they are here. Most of them are here for getting ...
- 5,230
1
vote
0
answers
211
views
Are measures better thought of as densities than differentials?
The standard notation for integrating with respect to a measure $\mu$ is:
$$\int f(x)\,d\mu(x).$$
But I've wondered if it could be better written as:
$$\int f(x)\mu(x)\,dx$$
where $\mu(x)$ is now ...
- 4,943
1
vote
1
answer
7k
views
Maximum/Minimum operator precedence
Is there any standard preceding order for the operators $a \wedge b = \min{(a,b)}$ and $a \vee b = \max{(a,b)}$ with respect to the arithmetic operators.
For example
$$ a \wedge b + c = (a \wedge b)...
- 1
36
votes
3
answers
3k
views
What do we learn from the Wronskian in the theory of linear ODEs?
For a real interval $I$ and a continuous function $A: I \to \mathbb{R}^{d\times d}$, let $(x_1, \dots, x_d)$ denote a basis of the solution space of the non-autonomous ODE
$$
\dot x(t) = A(t) x(t) \...
- 4,713
17
votes
2
answers
2k
views
Notation for "the" left adjoint functor
As far as I know, there is no "official" notation for the left adjoint of a functor $F : \mathcal{C} \to \mathcal{D}$ if it exists. I have seen the notation $F^*$ sometimes, but this looks only nice ...
- 35.5k
6
votes
0
answers
283
views
Interesting things you learned while grading/marking? [closed]
What are some interesting mathematical things you have learned while grading (or marking, if you prefer) student work? For example, clever proofs that students came up with; nice counterexamples or ...
- 6,237
2
votes
2
answers
906
views
Who first discovered the concept corresponding to the symbol of class comprehension?
Who first discovered the concept corresponding to the symbol of class comprehension
$\{x/\varphi\}$ used today in set theory ?
CommunityBot
- 1
5
votes
1
answer
521
views
How to find eigenvalues following Axler?
Preparing my Linear Algebra lecture I like the determinant free approach of Axler because the proof that operators $T$ on an $n$-dimensional complex vector space have eigenvalues is so simple:
Fix ...
- 6,237
6
votes
1
answer
640
views
Why are orthogonal matrices so often denoted $Q$?
I apologize for the stupid question in the title. Of course, we can baptize a particular given matrix as we want but, for example, the QR-decomposition has a fixed meaning.
My humble guess is that ...
- 188k
4
votes
2
answers
813
views
QFT and its notations
I know hardly anything about quantum field theory (QFT) but I'm giving a try to understand some ideas of it. As far as I understand, in QFT one is interested in studying measures such as:
\begin{...
3
votes
1
answer
271
views
Elementary classification of division rings
Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
- 26.5k
8
votes
1
answer
1k
views
Notation for a representable functor
For an object $X$ of a category, $h_X$ is the contravariant functor represented by $X$, i.e. $h_X = Hom(-,X)$.
Question a) Who invented this notation? (My guess: Grothendieck)
b) Is there a special ...
- 9,229
0
votes
1
answer
62
views
Is there a common notation to indicate the final form of a simplified definition? [closed]
I'm trying to become better with using proper terminologies and standard notation when taking notes, which lead me to think:
Similar to the indication of a completed proof by use of the Q.E.D. mark, ...
- 188k
0
votes
0
answers
45
views
Notation of $P^+$-families - bibliography searching
have you ever met with notation of $P^+$-families in other papers than Iian B. Smythe "A local Ramsey theory for block sequences" and his phd?
Thank you in advance
- 5,429
12
votes
1
answer
521
views
Source of a quote by Ferdinand Rudio
I am looking for the source and context of this quote, found e.g. at St Andrews:
Only with the greatest difficulty is one able to follow the writings of any author preceding Euler, because it was ...
- 31.5k
-1
votes
1
answer
118
views
Relative degree of a prime over a number field (notation from Algebraic Number Fields from Gerald J. Janusz) [closed]
I´m working with "Algebraic Number Fields" from Gerald J. Janusz (1. edition from 1973) and I have a question about his notation.
In chapter IV proposition 4.5 he states if K is an algebraic number ...
- 1,322
0
votes
1
answer
60
views
Name for matrix associated to smooth continuation
Is there an established name for the matrices that establish the conditions for a linear combination of $n$ functions $\lbrace f_1(x),\dots,f_n(x)\rbrace$ being the $n$-times smoothly differentiable ...
- 5,429
0
votes
0
answers
82
views
Format of grading Witt Lie Algebra
Let $W(n,m)$ be generalized Jacobson-Witt algebra over a field of characteristic $p>3$. According to the grading of $W(n,m)$, we know that it inherits the grading from $A(n,m)$ as follows: $$W(n,m)...
- 63.9k
15
votes
5
answers
5k
views
How do most people write permutations?
I'd like to know how people prefer to write permutations, or elements of the symmetric group $S_n$ for $n\ge0$.
The most natural way to define a permutation in $S_n$ is as a bijection on the set $\{1,...
- 37.7k
7
votes
3
answers
877
views
Origin of the notation s=\sigma+it in analytic number theory
I was wondering if the standard notation of denoting a complex variable by "$s$" had an interesting origin, or if it dates back to Riemann or Weierstrass. Almost every book in analytic number theory ...
- 63.9k
23
votes
13
answers
7k
views
Pedagogical question about linear algebra
Last semester I taught a linear algebra class that is intended to introduce young students (at a sophmore-junior level) to "abstract mathematics". It seems that a major conceptual hurdle for many of ...
- 50.6k
23
votes
12
answers
15k
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Textbook for undergraduate course in geometry
I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (...
- 2,600
87
votes
2
answers
4k
views
History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$
Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is ...
CommunityBot
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1
vote
0
answers
114
views
Notation for the geometric quotient of a separated Deligne-Mumford stack?
Suppose that $X$ is a separated Deligne-Mumford stack, say over a base scheme.
Is there some standard notation for the geometric quotient of $X$? I've tried using $[X]$ but have had complaints.
- 63.9k
1
vote
0
answers
103
views
Confusion optimal control abuse notation
I'm currently reading this paper describing a numerical scheme for the approximating optimal policy of a stochastic control problem. However, I run into a confusion directly on the first page where ...
- 63.9k
9
votes
4
answers
1k
views
Characterization of the Poisson law
This semester, I teach an introduction to probability course tailored for students with no science background and so with very very little prerequisites. We started with the basics of analytic ...
- 10.9k
10
votes
4
answers
2k
views
Reference for working with the implicit function theorem
I just had a student come to my office hours and ask me a ton of questions, the answer to all of which was "that's a slight variant to the implicit function theorem, which is proved by formal ...
- 5,824