All Questions
542 questions
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
23
votes
3
answers
5k
views
History of the notation for substitution
One of the very common notations for syntactic substitution is $[\ /\ ]$.
However, there seems to be an inconsistency in the literature about its usage.
Many write $[t/x]$ for "substitute $t$ for $x$...
- 5,502
1
vote
0
answers
149
views
Name for the Quotient $SU(m+1)/(SU(k) \times SU(m-k))$
The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of ...
- 459
0
votes
0
answers
645
views
Notation for iterated summation
Is there a more compact way to write
$$
\sum_{i_1=0}^{N}
\sum_{i_2=0}^{N-i_1}
\sum_{i_3=0}^{N-i_1-i_2}
\cdots
\sum_{i_{K}=0}^{N-i_1-i_2-i_3-\ldots-i_{K-1}}
a_{i_1i_2i_3\ldots i_K}
$$
as something like
...
- 113
1
vote
2
answers
1k
views
Use of ternary operator in formal writing
I would like to write
$$
f(x) = \begin{cases}1&\mbox{if }x = 1\\ 0&\mbox{otherwise.}\end{cases}
$$
However, this eats up a lot of vertical space for a very simple statement. Is there agreed ...
- 399
1
vote
1
answer
387
views
proof without words for logarithms [closed]
Does anyone know of any PROOF WITHOUT WORDS for logarithmic functions?
The only one I've seen in calculus based and I need one for high school math kids in MATH 1,2,3.
Any suggestions would be ...
0
votes
1
answer
148
views
Comparing vectors with numbers? [closed]
My question pertains to the paper "A Simplified Proof of the Divergence Theorem" by Djairo Guedes de Figueiredo.
It's not a big question, actually, but it's confusing me a lot: In the statement of ...
- 183
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
17
votes
4
answers
3k
views
Languages beyond enumerable
A language is a set of finite-length strings from some finite alphabet $\Sigma$.
It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings.
...
- 151k
1
vote
0
answers
159
views
Notation clash between a representation and spectral radius
I am currently writing a paper where I need talk both about a representation of a semisimple Lie group (usually denoted by $\rho$), and about spectral radii of linear maps (also usually denoted by $\...
- 1,574
2
votes
0
answers
118
views
What does the square root sign tells us in the wave equation? [closed]
I have been reading the paper on wave equations, and I have some confusion in notations.
Consider the initial value problem(IVP)(Wave equation):
$\frac{\partial ^2 u } {\partial t^2}(x,t) = \...
- 1,051
49
votes
5
answers
4k
views
How do you mentor undergraduate research?
Lets say you had an undergraduate who wanted to do some advanced work and some research, possibly for a thesis, or things like that.
There are two slightly more specific groups of questions I have ...
-2
votes
1
answer
514
views
Correction symbols used for mathematical texts [closed]
When proof reading and correcting a mathematical text, I sometimes see people use special notation symbols in the margin to indicate correction, deletion, replacement and so on. Is there any standard ...
- 177
7
votes
1
answer
468
views
What does the notation $[b_1,b_2]$ in M. Hochster's "Prime Ideal Structure in Commutative Rings" mean?
I'm reading the article
M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43--60. Freely available here on the journal's website.
But, I can not find the ...
user89541
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 $...
- 347
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 ...
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....
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 ...
22
votes
4
answers
2k
views
Technical issue in the approach to Lie groups taken in a book
I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer), which I've enjoyed using. I'm confused about ...
- 28.1k
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
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 ...
- 534
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
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 ...
- 109k
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
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
26
votes
4
answers
3k
views
What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$?
This is an embarrassingly simple question, but I was not able to find a definitive answer from literature search.
Suppose one has some collection of functions $f_1: X \to Y_1, \dots, f_n: X \to Y_n$ ...
- 114k
58
votes
4
answers
5k
views
Advice for PhD Supervisors
My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for ...
2
votes
1
answer
243
views
Notation: $Sigma$ and $Pi$ of intersections
In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly.
For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic ...
- 29
-2
votes
1
answer
5k
views
Looking for the name of a mathematical symbol that looks remotely like 1 (answer: indicator function) [closed]
Original question:
The symbol looks like a numeral 1 written like an R in $\mathbb{R}$. It has a double vertical line and a serif at the bottom. It represents a function of a parameter: $1_{\{0,1\}}(x)...
- 121
0
votes
1
answer
328
views
Meaning of $[A,B]$ when $A$, $B$ are self-adjoint
This is just a question about notation, but it got no useful answers on math.stackexchange.
Let $L$ be the Lie algebra of $n\times n$ Hermitian matrices, with Lie bracket $(A,B)\mapsto i(AB-BA)$.
...
- 23k
24
votes
1
answer
1k
views
What is $\infty^6$?
The title of this question may make you want to close it immediately, but bear with me a moment. In several older mathematics papers (early 20th century) I have seen statements such as
The motions ...
- 66.8k
4
votes
1
answer
441
views
How to teach generalizing the induction hypothesis? [closed]
I just finished teaching a class on using proof assistants (in this case, Agda) to write provably correct programs. Reflecting on how it went, the biggest difficulty I noticed the students having was ...
- 9,201
5
votes
4
answers
957
views
Notation for $\log \log \cdots \log n$? [closed]
Is there some accepted, more concise notation for expressions like $\log \log \log n$?
I just noticed an arXiv posting that quotes the bound
$$
\frac{\log X \log \log X \log \log \log \log X}
{ \log \...
- 151k
2
votes
0
answers
812
views
Products between metrics in a product of manifolds
In the "Einstein Manifold" book written by Arthur Besse, chapter 16, there is a notation of a manifold composed by the Cartesian product between two others:
$(M_1\times M_2, f^p(g_1 \times g_2))$
...
- 21
4
votes
2
answers
399
views
Terminology for metrics?
For some reason, I'm currently interested in the following relation - let $d,\delta$ be two metrics on some space $X$. We call the metrics _______ if there are some constants $C,E>0$ such that for ...
- 689
33
votes
15
answers
3k
views
Historical (personal) examples of teaching-based research
The phrase "teaching-based research" brings to mind research about teaching, though important, it is not what I mean. Unfortunately, I couldn't come up with a better phrase, thus please bear with me ...
4
votes
1
answer
609
views
Notation: Categories of measur(abl)e spaces
Is there a common notation in the literature for
the category of measurable spaces and measurable maps?
the category of measure spaces and measure-preserving maps?
The nlab suggests $\mathsf{Measble}...
- 63.1k
3
votes
1
answer
316
views
Was $\Sigma x$ used as quantifier?
Kurt Gödel in 1931 used $x\Pi a$ where we in contemporary notation would use $(\forall x) A$ or $(x)A$, and $Ex a$ where we would use $(\exists x) A$. I believe that I remember that $\Sigma xA$ has ...
- 3,574
87
votes
2
answers
4k
views
History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$
Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is ...
- 31.5k
11
votes
3
answers
729
views
Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? [closed]
In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense.
On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or $\...
- 802
0
votes
0
answers
82
views
Format of grading Witt Lie Algebra
Let $W(n,m)$ be generalized Jacobson-Witt algebra over a field of characteristic $p>3$. According to the grading of $W(n,m)$, we know that it inherits the grading from $A(n,m)$ as follows: $$W(n,m)...
- 367
0
votes
1
answer
155
views
Help with notation for the state of a dynamical system defined by a PDE
Before my question let me briefly describe a simplified version of the dynamical system I'm working with. Suppose that I have a density function $m(\boldsymbol{x},t)$, that describes the abundance of ...
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 ...
- 23.4k
14
votes
1
answer
515
views
Contexts and notations for composing asymmetric simplices
Imagine the elements of a group-like structure as puzzle pieces with essential two sides, an IN-side and an OUT-side.
You can compose two such pieces in two obvious ways:
Now consider triangular ...
- 9,720
1
vote
1
answer
224
views
Lefschetz fixed notation
If $f\colon X\to X$ is a self-map of a nice space with isolated fixed points, then the Lefschetz fixed point theorem relates a global number to local numbers. Some write: $L(f)=\sum_{x\in \mathrm{Fix}(...
- 8,727
8
votes
2
answers
2k
views
Which universities teach true infinitesimal calculus? [closed]
My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
20
votes
2
answers
4k
views
Teaching stochastic calculus to students who know no measure theory (or PDE, or...)
I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)).
I'm to teach the ...
- 61.9k
13
votes
1
answer
605
views
A funny factorization of the Jacobian coming from the lines on the Fermat cubic
Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...
- 156k
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
25
votes
19
answers
20k
views
Math books for advanced high school students
I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...