All Questions
542 questions
7
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Where can I find resources for creating a mathematics "bridge course"?
My department is in the very early stages of developing a "bridge course" or "introduction to proofs" course, motivated by our lower-level courses not currently doing a good job of preparing our ...
- 5,544
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 ...
7
votes
0
answers
367
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
7
votes
0
answers
1k
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Conventions for Riemann curvature tensor
I am aware of two conventions for the Riemann curvature tensor, namely the expression
$$\langle\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W\rangle$$
is either declared to be $R(X,Y,Z,W)$ or $...
- 18.7k
7
votes
0
answers
214
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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
6
votes
8
answers
2k
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Mathematical objects whose name is a single letter
(Not research-level, but perhaps not easily answered elsewhere — you decide if MO can afford the innocent fun. If so, it should likely be “community-wiki” i.e. one object per answer.)
I am seeking ...
6
votes
3
answers
1k
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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
6
votes
1
answer
4k
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Examples of separable ordinary differential equations in economics
I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic ...
- 1,665
6
votes
3
answers
691
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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
6
votes
5
answers
656
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Is there a name for the involution on Laurent polynomials?
This is a simple terminology question: I want to know if the involution $z \mapsto z^{-1}$ on Laurent polynomials (over some ring, I happen to be working over $\mathbb{Z}$ but that's not important) ...
6
votes
2
answers
1k
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Pages from a known textbook on Euclidean geometry?
Do you recall having seen the attached pages in a textbook once? If so, would you be so kind as to share its bibliographic record (or the main items in it) with me below?
A teacher provided us xerox ...
- 8,842
6
votes
4
answers
2k
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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
6
votes
3
answers
1k
views
An application of Maschke's theorem
I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like ...
- 1,472
6
votes
2
answers
945
views
Notation/name for "Artin-Schreier roots"?
If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x.
Of course ...
- 65.4k
6
votes
2
answers
2k
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Notation: Exponent of a group
The exponent of a group $G$ is the least positive $n$ such that $g^n = e$ for all $g \in G$. This is obviously a sensible name for the concept.
A notational awkwardness arises however when the group $...
- 1,793
6
votes
2
answers
2k
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Why are the Dynkin diagrams E6, E7 and E8 always drawn the way they are drawn?
The Dynkin diagrams of type ADE are ubiquitous in mathematics as solutions of various classification problems. The diagram E6 is usually drawn by five dots in a row with a sixth dot above the third, ...
- 61
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 $\...
- 3,059
6
votes
1
answer
877
views
Is $O(10^{-6})$ an acceptable notation in numerical analysis? [closed]
The following question has been on math.SE for several days. Without having a satisfying answer, I'd like to ask the experts here.
In mathematics, the big $O$ notation is used to describe the ...
user14319
6
votes
3
answers
1k
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Names of noncompact riemannian symmetric spaces?
Irreducible riemannian symmetric spaces come in pairs: one compact and one not compact, usally called the noncompact dual.
The compact symmetric spaces include spheres, complex and quaternionic ...
- 33.3k
6
votes
1
answer
181
views
Does $\mathsf{SVC}^\ast$ exist?
$\mathsf{SVC}(S)$ is the assertion that for all sets $X$ there is an ordinal $\eta$ and a surjection $f\colon\eta\times S\to X$. I would like to denote by $\mathsf{SVC}^\ast(S)$ the same assertion but ...
- 2,206
6
votes
2
answers
588
views
Applications of isotropic quadratic forms
I will soon be teaching an introductory course on bilinear algebra and quadratic forms. I will likely spend most of the time and effort on positive definite quadratic forms and euclidean spaces. These ...
6
votes
1
answer
641
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 ...
- 16.5k
6
votes
1
answer
749
views
Origin of symbols used for half-sum of positive roots in Lie theory?
The Weyl character formula is a central result in the finite dimensional representation theory of semisimple Lie groups, algebraic groups, Lie algebras. Related questions on MO include these here ...
- 52.9k
6
votes
1
answer
222
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Reference request: Different definitions of Big O notation
This question might sound strange, but I would like to settle this problem once and for all.
For as long as I can remember, I was introduced to the Big O notation by this definition:
Def. 1: Let $f, g$...
- 61
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
6
votes
0
answers
283
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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
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
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
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
5
votes
9
answers
2k
views
Suggestions for teaching advanced high school students
Hi all,
I'm a grad student and just joined a mentoring program in which I will visit a group of advanced year ten high school students (around 16 years old) from a group of schools in the area. I don'...
5
votes
5
answers
2k
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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 ...
5
votes
2
answers
377
views
What is meant by this notation of the real forms of $E_6$?
There are five real forms of the exceptional Lie group, $E_6$. Four of them are notated as in the following:
The split form as EI or $E_{6(6)}$
The quasi-split form as EII or $E_{6(2)}$
EIII or $E_{...
- 2,369
5
votes
3
answers
799
views
Euclidean function of Euclidean domain defined at 0
In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things as ...
5
votes
4
answers
1k
views
Lecture on Fractals for Middle School Students
I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject.
I'll show some fractal images and a few ...
- 87
5
votes
1
answer
393
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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
5
votes
3
answers
647
views
Looking for ideas concerning the teaching of lower-division differential equation courses...
I'm looking for problems/lessons plans that could be used in a lower-division differential equations course that involve discerning properties of solutions of an equation, IVP, or BVP, without looking ...
5
votes
3
answers
2k
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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 ...
- 1,174
5
votes
1
answer
589
views
what do empty parens symbol mean?
Quick easy question: what is the meaning of the symbol $(\space\space )$. I've seen it now in two papers, one of which is Milgram's Group Representations and the Adams Spectral Sequence, available at ...
- 1,147
5
votes
2
answers
445
views
About the maximum number of leaves adjacent to a vertex in a tree
Let $T$ be a finite tree graph with the set of vertices $V(T)$. For an arbitrary vertex $ v \in V(T)$, I define $l(v)$ to be the number of leaves connected to $v$.
In my study, I need to define the ...
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 ...
5
votes
1
answer
331
views
Meaning of $g_d^r$ in algebraic geometry
As an editor I often encounter the symbol $g_d^r$ as a noun. I tried googling but I only get papers where the symbol is used without a definition. Can someone supply a reference to a definition? ...
- 483
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 ...
- 19.7k
5
votes
1
answer
208
views
Seven Bridges of Königsberg for hypergraphs
I am teaching a course involving hypergraphs. I would like to have a physical analogy/motivating problem for hypergraphs similarly to how the Seven Bridges of Königsberg motivate graphs. Can you help ...
- 51
5
votes
1
answer
548
views
Question about denoting/designating of algebraic structures
I saw this image on Wikipedia (Template:Group-like structures, current revision):
Since there are five "properties" that we can have (in this context), namely: totality, associativity, identity, ...
user114642
5
votes
1
answer
1k
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Generalizing Big O notation to arbitrary vector spaces
I'm constructing a Coq library for Big-O notation. Naturally, I'd like it to be as general as possible. The Wikipedia page on Big-O notation says
The generalization to functions taking values in ...
- 153
5
votes
2
answers
2k
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Any suggestions for a course in Mathematical Logic?
I am teaching a topics course for Mathematics majors (at Temple), and am considering Logic as the topic. I was wondering if people (a) have suggestions for an appropriate text and (b) how much might ...
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 ...
- 16.5k
5
votes
1
answer
1k
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Is Diagonalization worth to be taught? [closed]
When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a ...
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
5
votes
1
answer
461
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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