All Questions
542 questions
4
votes
0
answers
283
views
Pairing in Group Cohomology [closed]
I am following Ararat Babakhanian's Cohomological Methods in Group theory.
Let $A,B,C$ be $G$ modules then we have a $G$ module structre on $\text{Hom}_{\mathbb{Z}}(B,C)$ with $$\sigma.f(x)=\sigma(f\...
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33
votes
20
answers
5k
views
Do names given to math concepts have a role in common mistakes by students?
Perhaps this question overlaps with similar ones, ... but I want to focus on a particular possible cause of confusion. I notice that students are often confused by the concepts of "infinite" and "...
- 6,541
1
vote
0
answers
59
views
Notation for largest universal subclass and class of arrows "locally in" a given class of arrows
Let $\mathcal M$ be a class of arrows in a category $\mathsf C$. I would like suggestions for good notation for the following two classes.
The smallest universal (pullback stable) subclass $\mathcal ...
- 10.5k
0
votes
1
answer
179
views
Theory of integration of Kernel in çinlar probability and stochastic
I'm reading the probabilistic book write by çinlar, but I don't understand the Kernel theory, in details:
$ (E,\mathcal{E}),(F,\mathcal{F})$ are two measurable space
$$K:E \times \mathcal{F} \...
- 29.8k
16
votes
9
answers
4k
views
How to motivate the skein relations?
I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this skein relation to compute all these ...
- 118k
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
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
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 ...
- 26.3k
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 ...
- 41.1k
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 ...
- 16.5k
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 ...
- 3,911
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.
...
CommunityBot
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22
votes
1
answer
33k
views
vector to diagonal matrix [closed]
For any column vector we can easily create a corresponding diagonal matrix, whose elements along the diagonal are the elements of the column vector.
Is there a simple way to write this transformation ...
- 19.6k
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 ...
CommunityBot
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52
votes
9
answers
26k
views
Is Galois theory necessary (in a basic graduate algebra course)?
By definition, a basic graduate algebra course in a U.S. (or similar) university with
a Ph.D. program in mathematics lasts part or all of an academic year and is taken
by first (sometimes second) ...
-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 ...
- 19.6k
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 ...
- 31.2k
4
votes
4
answers
3k
views
Is there a standard notation for binary relations in category theory?
In set theory, I learned that a binary relation is simply a subset of a Cartesian product. Moving on to category theory, it seems that this definition is not enough. Just as a function is no longer ...
- 131
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 ...
CommunityBot
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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
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
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
0
votes
2
answers
1k
views
What is a maximal set in the context of argumentation in AI [closed]
I am computer scientist, not a mathematician, I've been reading some papers on argumentation in AI that uses the term 'maximal' set without defining it. I think it's left undefined because it's a ...
- 32.5k
21
votes
10
answers
6k
views
Not especially famous, long-open problems which higher mathematics beginners can understand
This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...
CommunityBot
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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 ...
CommunityBot
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52
votes
22
answers
19k
views
Interesting Calculus Questions/Exercises
I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...
- 39.9k
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 $f_1: X \to Y_1, \dots, f_n: X \to Y_n$ ...
- 15.5k
9
votes
3
answers
3k
views
Math History Question about the exponential function
While tutoring a student recently, I have come across the situation of explain logarithms by first introducing functions of the form $$f(x)= a^x$$ where $a \ge 0,x\in \mathbb{R}$. My student then ...
- 1,481
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 $(A,B)\mapsto i(AB-BA)$.
...
CommunityBot
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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 ...
- 73.2k
-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
11
votes
2
answers
3k
views
Good examples of random variables whose image is not a measurable set?
Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued ...
- 41.1k
27
votes
3
answers
4k
views
Why is the identity element of a group denoted by $e$?
The question was asked by a student, and I did not have a ready answer. I can think of the German word ``Einheit'', but since in German that is not how the identity element of a group is called, I ...
- 101
35
votes
19
answers
9k
views
Interesting applications (in pure mathematics) of first-year calculus
What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...
CommunityBot
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8
votes
2
answers
2k
views
What is the best *general triangle*?
During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...
CommunityBot
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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 ...
- 82.7k
16
votes
12
answers
10k
views
How seriously should a graduate student take teaching evaluations? [closed]
Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...
- 50.6k
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 \...
- 1,375
27
votes
3
answers
3k
views
Is “problem solving” a subject to be taught?
I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...
CommunityBot
- 1
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:
$(M_1\times M_2, f^p(g_1 \times g_2))$
...
- 690
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 ...
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 ...
- 61
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
7
votes
4
answers
2k
views
Help me find good math questions for my students [closed]
I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad.
Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...
CommunityBot
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16
votes
5
answers
1k
views
Permission to use Online Notes
I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by ...
CommunityBot
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28
votes
6
answers
2k
views
Means of Promoting Mathematics in Young Countries!
We all know mathematics is life, this question is for Mankind. It's mathoverflow here when some parts of the world we have mathunderflow! I think we can do something through ideas. A similar ...
CommunityBot
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