All Questions
542 questions
2
votes
0
answers
115
views
Name for generalization of property: $f^n(x) \ne x$ for all $n > 0$
I am curious about how to specify with standard terminology that a certain function is non-periodic, in the following sense:
In the simple case of a unary operation $f: X \to X$, this property would ...
- 193
15
votes
4
answers
3k
views
How does one motivates the method of separation of variables when teaching PDE's?
I'm not sure if this question is appropriate for MO. Add comments if it is not. Thanks.
How to explain/motivate the method of separation of variables for PDEs to undergraduates? What's the real math ...
- 5,052
23
votes
4
answers
4k
views
Curriculum reform success stories at an "average" research university
Greetings all,
There's a never-ending story that many of us have sunk our teeth into. How do we go about teaching subjects like calculus and analysis "well?" Most universities that I'm familiar ...
1
vote
1
answer
211
views
Notation for the restriction map in Galois cohomology
My coauthors and I are writing a paper based on MO questions and answers:
Friedrich Knop's answer,
my answer 1
and
my answer 2.
For a linear algebraic group $G$ over a perfect field $k$, I consider a ...
- 14.2k
6
votes
3
answers
691
views
Meaning of historical fluxion notation
I've noticed that in 18th century English books on calculus writers would say that 'the fluxion of $ax$ is $a\dot{x}$' and 'the fluxion of $x^n$ is $n x^{n-1} \dot{x}$'. What does this extra '$\dot{x}$...
- 863
2
votes
1
answer
215
views
Notation for the automorphisms of a $S$-scheme over automorphisms of $S$
Here is a slightly anecdotical notational question.
Let $S$ be a scheme and let $X$ be a scheme over $S$, with structural morphism $s\colon X\to S$. Is there a good suggestive notation for the group $...
- 1,017
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
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
votes
1
answer
771
views
Stochastic Process Notation
Note: I'm not an expert on stochastic processes. Please use small words and speak real slow.
I'm reading a paper [1], which uses a notation for stochastic processes that doesn't seem to be standard.
...
- 245
18
votes
4
answers
2k
views
Origin of symbol *l* for a prime different from a fixed prime?
I've never seen an authoritative explanation for the choice of the lower case letter $\ell$ or $l$ to denote an arbitrary prime different from a given prime $p$. This now has its own LaTeX command \...
- 52.9k
6
votes
1
answer
462
views
How to talk about certain "free" categories?
Given two categories $\mathcal{C}$ and $\mathcal{D}$, we can describe the following category $\mathcal{E}$. It is the initial category whose object set contains $\mathrm{Obj}(\mathcal{C}) \times \...
- 595
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 ...
0
votes
1
answer
114
views
Name of a matrix with one column and row removed [closed]
I am looking for the exact name of a matrix where the i-th column and rows have been removed.
I cannot remember how it is called in linear algebra, does anyone got an idea?
Thanks!
- 111
3
votes
6
answers
2k
views
Teach a course in 1 month
I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?
14
votes
3
answers
3k
views
Open source LaTeX lecture notes/slides/books [closed]
In the mathematics community it's quite common for professors to write their own notes for the classes they are teaching. The notes are then usually published in both PDF and PS form on the course ...
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
4
votes
1
answer
784
views
Notation diversity
This morning I had a brief discussion about different notations of trigonometric functions in Europe, so I looked for an online resource dealing with these diversities in mathematical notation. I ...
- 1,524
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
12
votes
5
answers
2k
views
Introducing Cryptology to Undergraduates
This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures on,...
- 4,842
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 ...
- 1,269
6
votes
0
answers
622
views
How necessary is the knowledge of Lebesgue integral for non-analysts? [closed]
Recently I have learned that at some math department the introductory course to Lebesgue integration not obligatory. Thus in another course on introduction to Hilbert spaces the $L^2(0,1)$ space is ...
- 21.8k
8
votes
3
answers
2k
views
The harmonic (series) beetle: live illustrations of mathematical theorems
In my analysis class I use the following problem to illustrate the divergence
of the harmonic series (consider this as a hint for solving it).
Exercise.
A beetle creeps along a 1-meter infinitely ...
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
0
votes
5
answers
2k
views
How to teach addition of negative numbers? [closed]
I have a friend with dyscalculia and was teaching her some some mathematics (namely, solving a linear equation, simplifying certain expressions, and what (affine linear) functions are).
She ...
- 648
24
votes
2
answers
2k
views
Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$
Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, ...
- 156k
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
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
4
votes
5
answers
12k
views
What is the difference between the biconditional iff. and equality = ?
Hello,
I've been used to writing logical transformations using equality, but the other day it struck me that perhaps I should be using the biconditional $\iff$?
So my question is:
What is the ...
11
votes
3
answers
729
views
Calculus Teaching: Is it possible or desirable to give a severely abbreviated treatment of series convergence tests?
I will be teaching Calculus 2 this fall at a large U.S. state university. Our incoming students tend to have a limited or inconsistent background, which limits the amount of material we can cover.
...
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 ...
- 1,785
5
votes
1
answer
409
views
What countable ordinals are called $\kappa_\alpha$?
Jervell has a notation for countable ordinals up to the small Veblen ordinal using trees:
• Herman Ruge Jervell, How to wellorder finite trees
and get good ordinal notations, Berkeley Logic ...
- 22.3k
8
votes
4
answers
4k
views
How to teach introductory statistic course to students with little math background?
Next semester I will teach an elementary statistic course for the first time (which I am actually quite excited about). A brief description can be found here. I am told to expect very little math ...
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 ...
- 822
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 ...
- 47.3k
-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
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
2
votes
0
answers
240
views
What does the $\pi_1(\mathsf{C})$ really mean?
Assume that $\mathsf{C}$ is a small category (in my case with finitely many objects but this is probably irrelevant). In a paper I'm studying at the moment there is a notion used constantly, this of $\...
- 441
1
vote
4
answers
1k
views
Kro-necker versus Kron-ecker: which hyphenation is preferred? [closed]
Synopsis and concrete practices
Everyone is thanked for their comments, and in view of the diversity of views expressed, I have converted this question to a community wiki.
Here is a working ...
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 ...
- 2,372
2
votes
0
answers
323
views
Is there standard notation for restriction partial functions?
Given a partial function $f : A \rightarrow B$, and a subset $S \subseteq A$, we get a new partial function $$f \restriction_S : A \rightarrow B$$ by restriction. However, I prefer to analyse $f \...
- 3,793
2
votes
1
answer
293
views
Notation and reference for polynomials with coefficients not commuting with the indeterminates
Let $R$ be a noncommutative ring (with unit). Then a "fully noncommutative" (for a lack of better wording) monomial over $R$ in the single noncommutative indeterminate $X$ of degree $d$ is given by a ...
- 7,127
5
votes
1
answer
393
views
Not quite adjoint functors
What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
- 47.3k
9
votes
4
answers
2k
views
Applications of Math: Theory vs. Practice
I have a problem: I learned about a lot of the applications of mathematics from academics. Neither they nor I have had much contact with the "real world" to go and see for ourselves how mathematics ...
3
votes
1
answer
793
views
Are traditional notations for elliptic integrals/functions in Latin or Greek letters?
I am doing some calculation involving elliptic integrals/functions, and find the notations confusing.
In Wittaker-Watson, the "Jacobi's earlier notation" H(u) is called the Eta-function, so the "H" ...
- 133
6
votes
0
answers
466
views
What is the "permanence relation" really?
I have come across the words "permanence relation" in a 1969 paper by Keith Hannabuss The Dirac equation in de Sitter space. The only other similar google hit for this phrase appears in ...
- 33.3k
7
votes
0
answers
214
views
Notation: Why Ω for the based loop functor?
This is just a question about notation - probably useless, but it's always baffled me:
Why was $\Omega$ chosen to denote the based loop functor?
I once heard someone speculate: "It's because $\Omega$...
- 113
8
votes
1
answer
4k
views
Who is this guy : Z.A. Melzak (wrote Companion to Concrete Mathematics) ? [closed]
Author : Z.A. Melzak
Book Title : Companion to Concrete Mathematics.
Publication : Dover renewed 2004 2 volumes in one. Copyright 1972/1976.
I found this book extremely nice.
To whet your appetite ...
- 1,306
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
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 ...
6
votes
3
answers
1k
views
Publishing with Undergraduates
Is doing research with a student considered to be good for a dossier? Is it okay to have few research publications but a lot of student projects? I am finishing up a grad program and am looking at ...
- 61