All Questions
542 questions
4
votes
4
answers
4k
views
Variation on the Sobolev space $H^1_0$
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let
$$
C^1_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|_{\partial\Omega}=0\},
$$
and let $C^1_c(\Omega)$ be the space of ...
- 2,834
2
votes
4
answers
4k
views
Best way to introduce the Chinese Remainder Theorem (to a high school student)
What do you think to be the most effective way to teach the Chinese remainder theorem to a smart high school student, which is supposed to only have a soft idea about how modular arithmetic works, and ...
CommunityBot
- 1
8
votes
1
answer
1k
views
Where does the notation $\pi_1(X,x)$ for the fundamental group first appear?
I've spent the last half hour browsing Stillwell's translation of Poincaré's Analysis Situs and Dieudonné's History of Algebraic and Differential Topology, and I haven't found the source of this ...
- 11.1k
2
votes
0
answers
1k
views
Linear Algebra Text Book [closed]
In our department we do not like our current linear algebra book and so we would want to find a better book. This is for the first course in linear algebra and the title of the course is
Elementary ...
- 44.4k
21
votes
5
answers
38k
views
What does ! above = mean [closed]
Can someone please explain what the symbol $\stackrel{!}{=}$, consisting of an exclamation mark (!) above an equals sign (=) means? Below is the example I'm trying to decipher:
The normalization ...
- 236k
6
votes
4
answers
2k
views
notation for finite sequence with one element is removed [closed]
Often you need a notation for a finite sequence with one element is removed;
i.e. $$(x_1,\dots,x_{i-1},x_{i+1}\dots, x_n).$$
I know one notation
$$(x_1,\dots,\hat x_i,\dots, x_n)$$
and I hate it. It ...
- 60.6k
5
votes
5
answers
2k
views
Topics for a matrix analysis course
I recently taught a new (to my department) course titled "Matrix Analysis". For various reasons that I won't go into here, I was dissatisfied with the textbook I (loosely) followed, and with every ...
- 3,699
5
votes
2
answers
441
views
Name and notation for a binary operation
Is there a standard name or standard symbol for the binary operation that combines $x$ and $y$ to give $xy/(x+y)$, or equivalently $1/(1/x+1/y)$? (At least the expressions are equivalent if we ignore ...
CommunityBot
- 1
18
votes
14
answers
3k
views
Teaching a pedagogy course
At my institution incoming graduate students must take a semester long course on pedagogy taught by current grad students. I may soon be in the position of having to teach this course and I'm looking ...
CommunityBot
- 1
24
votes
1
answer
4k
views
What does the σ in σ-algebra stand for?
I was tutoring someone in analysis and realized I have no idea where this notation comes from (or analogous terms: σ-additive, σ-ring, etc). I would like to know why the letter σ was chosen. I can't ...
CommunityBot
- 1
17
votes
10
answers
109k
views
What are the qualities of a good (math) teacher? [closed]
In forming your answer you may treat the qualifier math or maths as optional, since part of the question is whether there is anything peculiar to the subject of mathematics that demands anything ...
CommunityBot
- 1
13
votes
3
answers
627
views
Certain notations in Cayley's work
Two quick questions on notation, motivated by my being reading Cayley at the moment (I stumbled across a random volume of his Collected Works and now I am unable to do anything else but read it ...
- 44.4k
2
votes
2
answers
1k
views
Decomposition of $K_{10}$ in copies of the Petersen graph
It is a well-known and cute exercise in algebraic graph theory to show that $K_{10}$ cannot be written as the edge-disjoint union of three copies of the Petersen graph $P$. Indeed, the graph $G$ whose ...
- 10.9k
2
votes
4
answers
1k
views
Eigenvalues of powers of linear mappings
Let $\tau$ be a linear map on a finite dimensional complex vector space. Clearly, if $\lambda$ is an eigenvalue of $\tau$ then $\lambda^n$ is an eigenvalue of $\tau^n$, for any natural (integer, on ...
CommunityBot
- 1
21
votes
7
answers
2k
views
Pros and cons of math teaching using smartboards
Currently, there is some talk in my university concerning a change in our lecture rooms from blackboards to smartboards (or other alternatives, such as a smart podium). For that reason, I'm interested ...
- 345
30
votes
3
answers
4k
views
Nearly all math classes are lecture+problem set based; this seems particularly true at the graduate level. What are some concrete examples of techniques other than the "standard math class" used at the *Graduate* level?
In the fall, I am teaching one undergraduate and one graduate course, and in planning these courses I have been thinking about alternatives to the "standard math class". I have found it much easier ...
1
vote
0
answers
2k
views
What does this notation mean: matrix norm with a two-number subscript
I recently came across this notation, without explanation, in a paper:
$||\mathbf{W}||_{2,1}$
From the context, I know that $\mathbf{W}$ is a matrix, which could be any size, and that $||\mathbf{W}||...
- 111
1
vote
1
answer
265
views
Notation of a pregallery
I'm transcribing parts of Harm van der Lek's thesis 'The homotopy type of complex hyperplane complements' and due to it being written in 1983 the typesetting isn't very detailed. In latex, how should ...
- 73.2k
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
13
votes
3
answers
2k
views
History surrounding Gauss Theorema Egregium and differential geometry
I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian ...
- 8,842
0
votes
1
answer
860
views
Sierpinski Triangle and the Chaos Game
The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...
CommunityBot
- 1
28
votes
2
answers
5k
views
The Origin of the Musical Isomorphisms
In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^*M$ and the "raising indices" operator by $\sharp:T^*M \to TM$. These isomorphisms are ...
CommunityBot
- 1
17
votes
6
answers
7k
views
Explaining the concept of projective space: notes for students
This is a question on teaching.
I am teaching at this moment a course in algebraic geometry for master students on a very basic level. Today (this was the fourth lecture) I discovered that only four ...
- 2,281
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 $\sigma \in Z^1(K,V/\Lambda)$, $\...
- 15.4k
7
votes
8
answers
4k
views
Mathematical Advice for Interested Highschool Students
This may not be a research level math question, but I believe it is still relevant to Math Overflow.
What general resources exist for students in highschool who are very interested in Mathematics?...
1
vote
1
answer
994
views
What exactly does \gg and \ll mean?
For example,
$f(T)\ll_T 1$ where $T$ is a positive number.
CommunityBot
- 1
19
votes
3
answers
2k
views
Research level applications of "row rank = column rank"?
No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...
- 17.6k
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\...
- 17.7k
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
- 63.1k
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 ...
- 23k
2
votes
1
answer
439
views
Notation arb(x)
Suppose we have extended $ZF$ by adding to $ZF$ an unary function symbol $arb$ (an arbitrary element of a set) and a corresponding axiom "For every non-empty set $S$, $arb(S)$ is in $S$".
Will be the ...
- 897
4
votes
3
answers
674
views
Is there a (standard) name for $\bar{A}\setminus A$?
This is a notation question:
If $A$ is a set in a topological space and $\bar{A}$ is its closure, is there a (standard) name for $\bar{A}\setminus A$?
- 31
11
votes
1
answer
1k
views
Teaching stacks to differential geometry students
Does anyone have any experience teaching stacks over the category of manifolds to students whose background is, say, a semester-long course on manifolds? Does anyone know of any publicly available ...
- 341
0
votes
2
answers
562
views
Lines on degree 2n-3 Fermat hypersufaces
It is well known that a generic hypersurface of degree $2n-3$ in $\mathbb CP^n$ has finite number of lines. I would like to ask a couple of questions about lines on Fermat hypersurfaces and their ...
- 1,936
12
votes
9
answers
6k
views
Topics for an Undergraduate Expository Paper in Number Theory
I am teaching an undergraduate course in number theory and am looking for topics that students could take on to write an expository paper (~10 pages). No new results are expected of them. Many of the ...
CommunityBot
- 1
9
votes
4
answers
1k
views
Notation for eventually less than
Is there some existing notation for
\[f(n)\leq g(n)\] for sufficiently large n
Apart from just writing that itself?
I'm thinking of something compact like the ...
CommunityBot
- 1
1
vote
1
answer
598
views
Set Exponentiation: Is Y always disjoint from Y^X? [closed]
If $y \in Y$ and $g \in Y^X$, we often write $y+g$ as shorthand for the map $x \mapsto y+ g(x)$. Similarly if $f \in Y^X$ then $f+g = x \mapsto f(x)+g(x)$. However this presupposes that we can ...
- 3,793
5
votes
3
answers
2k
views
Continuous change of basis (and on the definition of determinant) [closed]
Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...
- 23k
13
votes
1
answer
2k
views
conditional equality symbol
Is there a standard notation (perhaps $A \stackrel{\leftarrow}{=} B$) meaning "in all situations where $B$ is defined, $A$ is defined and equals $B$"?
The kind of situation in which such a notation ...
- 3,793
6
votes
2
answers
935
views
Surface Laplace-Beltrami without coordinates, exterior calculus?
Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator $\...
- 27.5k
9
votes
3
answers
1k
views
Where can I find questions motivating important ideas in math?
I would like questions that demonstrate why a mathematical tool or technique is useful, and which can be used to introduce that idea. Ideally, this would be a compilation of problems organized by the ...
CommunityBot
- 1
8
votes
4
answers
1k
views
Multivariable Calculus Lecture Ideas
I am teaching a course in multivariable calculus this semester. We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces. After that ...
CommunityBot
- 1
4
votes
2
answers
3k
views
Standard notation/symbol for an embedding function
Hello everyone,
Suppose that I am defining a function which embeds a surface (manifold) in $\mathbb{R}^3$.
Is there a standard symbol or letter that is used for this function?
Additionally, is ...
- 888
9
votes
3
answers
3k
views
notation for formal Laurent series
I've found a few articles that write the ring of formal Laurent series in $t$ as $R((1/t))$, but what's the underlying meaning of $\cdot ((\cdot))$?
A mathematician of my acquaintance swears that $R((...
- 3,738
2
votes
3
answers
410
views
Pedagogical notes on line bundles on complex projective manifolds
I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems ...
- 15.4k
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
$$[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}\frac1{12}(m^3-m)c,$$
$$[c,L_n]=0.$$
...
- 45.7k
16
votes
6
answers
3k
views
How to mentor an exceptional high school student?
I have a unique and, quite truthfully, humbling opportunity. The parents of an exceptionally talented high school freshman have reached out to me and asked if I might be able to help.
This kid is ...
CommunityBot
- 1
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 ...
- 20.5k
27
votes
5
answers
7k
views
References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
- 13.2k
9
votes
7
answers
1k
views
Mathematics seminar for "non-mathematicians"
Next term I am leading a seminar for students, who will become teachers for elementary school i.e. for kids of age 6-10. The students in the seminar will have no mathematical background beyond the "...
- 12.3k