All Questions
542 questions
3
votes
2
answers
550
views
What are some resources discussing mathematical notation?
I'm looking for resources discussing mathematical notation, the theory, the philosophy, the distinct advantages of various notations. Stuff about notation for computer algebra systems is interesting ...
- 31
2
votes
2
answers
465
views
When forcing with a poset, why do we order the poset in the order that we do?
In forcing, we take a collection of forcing conditions and impose a partial order on them. The convention is that if $p$ is stronger than $q$, then we say $p < q$. This is perfectly fine, but it ...
- 1,793
4
votes
1
answer
274
views
Notation for upperbound power sets.
There is a standard notation $\mathrm{ZF}[n]$ for Zermelo Fraenkel set theory with the power set axiom restricted to saying the set of natural numbers has $n$ successive power sets $\beth_0\dots\...
- 11.1k
1
vote
0
answers
77
views
notation for vector product in the space
The notation for vector (a.k.a. cross) product in $\mathbb{R}^3$ I usually see is $\times$.
However, some places use $\wedge$ instead, which IMHO creates a lot of confusion, as $\wedge$ usually is ...
- 14k
1
vote
0
answers
187
views
Default Orientation of Vectors [closed]
When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to column-...
- 13.2k
1
vote
1
answer
152
views
Formula for the Ordinal Number of k-Sets of Positive Integers
Background of my question is, that I would like to store flags indicating the relation between a pairs of non-adjacent edges of a graph (that relation could for example be, whether the edges cross, i....
- 13.2k
7
votes
2
answers
2k
views
Vinogradov's Elements of Number Theory
I can't be the only person here who has fond memories of the problems in Vinogradov's Elements of Number Theory. (For people who have not read it - the text itself is just a concise basic number ...
0
votes
1
answer
2k
views
Dual of Zorn's Lemma? [closed]
It seems to me that the dual of Zorn's Lemma should be true: if $S$ is a non-empty partially ordered set and every chain of $S$ has a lower bound in $S$, then $S$ has at least one minimal element.
...
- 1
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
1
vote
1
answer
2k
views
Is there a notation for natural isomorphism?
There's clearly a notation for isomorphism. It's just $\cong$. But is there notation to indicate that an isomorphism is natural? And in general, for morphisms which are natural?
I ask because one of ...
- 15.4k
12
votes
1
answer
775
views
Teaching Methods and Evaluating them
Hey,
As a lowly graduate student, I'm on a committee (I'm not sure how important my role really is) trying to evaluate how effective different approaches teaching undergraduates. We are looking at ...
5
votes
1
answer
461
views
Is there a standard notation for a "shift space" in functional analysis?
I'm writing up some notes on the nLab about things like embedding spaces and infinite spheres and similar things (can't link to them yet as I haven't put them up yet). One aspect that crops up time ...
- 26.8k
3
votes
1
answer
977
views
Notation for "the inclusion map is a homotopy equivalence"
It's sometimes convenient to have different notations for "$A$ is a subset of $B$" depending on what the inclusion map does:
If it's non-surjective, $A\subsetneq B$ or $A\subset B$, depending on your ...
1
vote
0
answers
3k
views
Notation for space of Lipschitz continuous functions
The Lipschitz norm of a function over a domain $D \subseteq \mathbb R^n$ is easy to define: $$\|f\|_{\mathrm{Lip}} = \sup_{x,y \in D} \frac{|f(x) - f(y)|}{|x-y|}.$$ Is there a standard notation for ...
- 8,512
1
vote
1
answer
742
views
proofs of stochastic boundedness
I'm looking at some statistical literature and trying to compare the results given there in probabilistic big-Oh notation with statements I'm more familiar with.
In particular, I'm trying to ...
- 922
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
2
votes
1
answer
897
views
Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis
Greetings,
I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...
1
vote
1
answer
186
views
What's the name of "twisted semidirect products"?
Let $V$ be an $n$-dimensional real vector space, $\Lambda\subseteq V$ a lattice, and $K$ a subgroup of $Aut_{\mathbb{Z}}(\Lambda)\cong GL(n,\mathbb{Z})$. Let also $\sigma \in Z^1(K,V/\Lambda)$, $\...
- 23.4k
2
votes
1
answer
244
views
How many flavors should a notational system offer for rank-1 tensors?
The notation for tensors is like the plumbing in a very old Vermont farmhouse. It may once have been intentionally designed, but after that it just evolved. As an example, it seems that depending on ...
user21349
4
votes
1
answer
690
views
What does $L^\infty_\varepsilon$ mean?
In Volume 4 of Reed and Simon on page 83 the authors refer to the space $(L^\infty(\mathbb{R}^3))_\varepsilon$,
and later on page 119 they use $L^\\infty_\varepsilon$.
Are these two spaces the same? ...
1
vote
0
answers
33
views
Notation to denote substitution of vector elements [duplicate]
I'm looking for notation to denote vector substitution and elimination of elements. This is possible using set notation, but I am looking for shorthand notation that is perhaps already in use.
...
- 111
0
votes
0
answers
678
views
Notation for isometric spaces?
Metric spaces are isometric if there exists a bijective isometry between them.
Is there a standard notation for this, along the same lines as $X\approx Y$ for homeomorphic spaces and $X\simeq Y$ for ...
- 35.9k
14
votes
1
answer
961
views
Founding of homological without quite involving derived categories
I am looking at the foundations of homological algebra, e.g. the introduction
of Ext and Tor, and am unsatisfied. The references I look at start with
"this is called a projective module, this is ...
- 27.9k
4
votes
1
answer
222
views
Why is there a discrepancy between the normalizations of the central terms for the commutation relations of the Virasoro versus Neveu-Schwarz Lie algebras?
Following the standard conventions in the literature, the commutation relations of the Virasoro Lie algebra are given by
$$[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}\frac1{12}(m^3-m)c,$$
$$[c,L_n]=0.$$
...
- 43.2k
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
1
vote
0
answers
430
views
Professional skills advising for math jobs [closed]
Hi,
I am a postdoc at the University of Nottingham (UK) and I am beginning to apply for Assistant Professor positions in US.
I would like to receive a feedback on the material that I am sending (...
1
vote
0
answers
185
views
Notation for the subobject classifier
Does anyone know why in books on category theory the notation for the subobject classifier is almost everywhere the capital greek letter $\Omega$?
Gérard Lang
- 2,655
2
votes
1
answer
193
views
Terminology for system of equations and...
I am looking for the standard term for a system that consists of things of the form
$p_i(x_1,\ldots ,x_n)=0$ and of the form $q_j(x_1,\ldots,x_n)\neq 0$ with the $p_i$ and $q_j$ polynomials. I have ...
0
votes
0
answers
379
views
Terminology for the image of the diagonal embedding.
Let $X$ be a topological space equipped with maps into two spaces $\bar X_1$ and $\bar X_2$. Is there a standard notation/terminology for the closure $\bar X$ in $\bar X_1 \times \bar X_2$ of the ...
- 5,359
3
votes
1
answer
507
views
What are some interesting grading/curving systems you have seen for a course? [closed]
It seems like every math course has something unique in how things are graded.
1) What are some interesting grading systems you have seen/used? (include curving types, etc.)
2) What are some pros ...
0
votes
0
answers
142
views
Notation for substructure, especially for permutations?
Is there a standard notation that expresses substructure?
The specific case that I care about is the following:
Suppose $\sigma,\tau$ are permutations such that $$\sigma(x)\not=x\implies \sigma(x)=\...
- 1,329
3
votes
0
answers
431
views
Concrete questions that turn into math problems [closed]
I'm writing an article about the way we teach math, trying to find out why so many people are discouraged from learning, and have no interest for math and logic.
At some point, I want to show that ...
- 131
1
vote
1
answer
286
views
Notation for growth $a_n \le c (n!)^\epsilon$
This is just a stupid question about a good terminology. I'm interested in sequences $a_n$ with a growth that can be bounded by an arbitrarily small positive power of $n!$, i.e. for every $\epsilon &...
3
votes
0
answers
311
views
Tensor power- Notation question
Hi everyone
I have a notational question, which is written usually in papers, but I can not figure it out what could be. Let $M$ be an $A$-module. I have seen this notation
$$M^{\otimes -n}$$
I ...
- 2,508
0
votes
1
answer
269
views
What does the *-operation signify in the binary context of two automata?
Hi I have a notation question.
I've recently come across the '*-operation' (star-operation) in the context of a binary operation on two automata e.g., A*B and I'm not sure exactly what it means (and ...
- 141
2
votes
0
answers
179
views
Notation for a canonical quotient of an abelian variety in positive characteristic
This is a light question about notation, but I received no answer in Stackexchange.
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian ...
- 797
0
votes
2
answers
383
views
"X \in \cdot" in Probability Measure [closed]
My question is quite simple, but I was unable to find an answer by googling, since you can't exactly google syntax. What does the $\in \cdot$ mean in:
$$\lim_{n\to\inf}||P(S_n\in\cdot)-P(S_n+k\in\cdot)...
- 3
1
vote
0
answers
396
views
Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922
1
vote
1
answer
311
views
Permutation with repetitions of a vocabulary
Dear all;
Let $\Sigma$={a,b,c,d}, and $\delta$ be a function that returns a string $S$ of infinite length over $\Sigma$, where each character $s \in S$ has been chosen uniformly at random. My ...
- 13
0
votes
0
answers
166
views
Is $\{x_{zt}\}_{Z\times~ T}$ a good notation for specifying the indexed family of entities $x_{zt}$ with $z\in Z,\, t\in T$?
I have a model with lots of variables indexed over a few sets.
After having introduced the model, i.e. having already said that $x_{zt}$ has indexes $z\in Z$ and $t\in T$, instead of writing
"we ...
- 151
2
votes
0
answers
526
views
How much of math could be taught without using mathematical notation? [closed]
Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...
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