All Questions
542 questions
10
votes
3
answers
1k
views
About the classification of commutative and of cocommutative, fin. dim. Hopf algebras
I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....
- 7,746
10
votes
4
answers
2k
views
Reference for working with the implicit function theorem
I just had a student come to my office hours and ask me a ton of questions, the answer to all of which was "that's a slight variant to the implicit function theorem, which is proved by formal ...
- 156k
10
votes
2
answers
2k
views
What is a convenient shorthand notation for a category
Set theory has a very convenient and well established curly brace notation to specify a set by its elements: $\{2,3,4,6\}$ or $\{\text{finite subgroups of }SU(2)\}$ are simple examples.
There should ...
- 491
9
votes
3
answers
1k
views
Books on the relationship between the Socratic method and mathematics?
Apart from books on heuristics by George Polya.
When trying to engage with and understand mathematical concepts and when applying abstract mathematical concepts to model "continuum" or real ...
- 107
9
votes
5
answers
3k
views
Assessing effectiveness of (epsilon, delta) definitions [closed]
There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...
- 16.6k
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 "...
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 ...
- 297
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 ...
- 7,013
9
votes
4
answers
1k
views
Characterization of the Poisson law
This semester, I teach an introduction to probability course tailored for students with no science background and so with very very little prerequisites. We started with the basics of analytic ...
- 10.9k
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((...
- 19.7k
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 ...
9
votes
1
answer
2k
views
Origin of the symbol for the tensor product
I have recently realised that the Paleo-Hebrew (and Phoenician) graph for the Hebrew letter ט (Teth) is $\otimes$. This made me wonder if there is any relation between the choice of the symbol and the ...
- 5,670
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 ...
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
9
votes
0
answers
887
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.9k
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 ...
user112109
8
votes
5
answers
6k
views
Advantages of the sequence definition of limits
I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of ...
8
votes
4
answers
788
views
Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$
Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...
- 43.2k
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
2
answers
1k
views
Mathematics of sustainable development and energy sobriety in the classroom
Faculty members are encouraged to highlight the connection between the courses we teach and climate change, and raise awareness of the issue in our lectures, across subjects in my university. I am ...
8
votes
2
answers
6k
views
Equal signs with fancy marks
Some people use $\stackrel{\mathrm{def}}{=}$, $:=$ or $\stackrel{\Delta}{=}$ for definitions.
In more informal contexts, I have also seen $\stackrel{?}{=}$, for "I wish to prove this equality, which ...
- 20.2k
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
8
votes
1
answer
1k
views
Why aren‘t op and co switched?
When reading through Loregian and Riehl - Categorical notions of fibration, on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^...
- 355
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 ...
8
votes
2
answers
3k
views
What is the standard notation for reversing the order of vector's components? [closed]
If we have a vector $x=(x_1,x_2,\ldots,x_n)$, is there any standard way to denote the vector $(x_n,x_{n-1},\ldots,x_1)$?.
I think that $x^{-1}$ could be a good option.
- 89
8
votes
2
answers
2k
views
Examples of analytic functions to motivate a first course in complex variables
[Changed title as a plea to re-open the question.]
If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an ...
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
8
votes
1
answer
1k
views
Notation for a representable functor
For an object $X$ of a category, $h_X$ is the contravariant functor represented by $X$, i.e. $h_X = Hom(-,X)$.
Question a) Who invented this notation? (My guess: Grothendieck)
b) Is there a special ...
- 63.1k
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
8
votes
2
answers
693
views
Seeking a combinatorial proof for a binomial identity
Let $n\geq m\geq0$ be two integers. The below binomial identity is provable by other means:
$$\sum_{j=0}^m(-1)^j\binom{n+1}j2^{m-j}
=\sum_{j=0}^m(-1)^j\binom{n-m+j}j.$$
QUESTION. Can you provide a ...
- 43.2k
8
votes
2
answers
447
views
Big ideas and big ways of thinking in statistics?
I'm moving to a new university for the fall semester, and I'll be teaching a statistics class for the first time. I'm familiar enough with doing statistics (my dissertation in math ed was a mixed-...
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 ...
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
8
votes
0
answers
554
views
Lower semicontinuity of naive fiber size
I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
- 156k
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?...
7
votes
4
answers
841
views
Easy to state applications of dimension theory in algebraic geometry
Dimension theory is quite a sophisticated topic (at least for me), it is fully settled in Shafarevich's book on the first 100 pages.
Shafarevich gives two nice applications of the theory. 1) A proof ...
- 14.3k
7
votes
5
answers
2k
views
Commutative algebra final project
I'm looking for a topic for a final project in commutative/homological algebra, for first year master's students (in a decent European university). During the course, they will cover the following ...
7
votes
3
answers
3k
views
The etale fundamental group of a field
Background and motivation:
I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch ...
- 7,338
7
votes
6
answers
1k
views
Another chicken or egg: sequence or series
This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...
7
votes
3
answers
1k
views
Higher dimensional Bezout via Hilbert polynomials: a reference
For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...
- 14.3k
7
votes
2
answers
1k
views
Maximal Ellipsoid
John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...
- 1,588
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, ...
7
votes
3
answers
3k
views
Problems reducing to a graph-theory algorithm
This is essentially a question in pedagogy -- the answers could be useful to teach (or rather, motivate) graph theory, and especially the algorithmic side of it.
I have been very impressed with this ...
- 2,287
7
votes
2
answers
1k
views
How should you respond to a student who asks whether a very nice physical example constitutes a proof? [closed]
"Is this really a proof?" is the exact question e-mailed to me today from an undergraduate mathematics student whom I know as a highly competent student. The one sentence question was accompanied with ...
- 2,437
7
votes
2
answers
1k
views
Two different kinds of definitions of $C^k(\overline{\Omega})$ — extension and restriction
This is cross-posted in MSE.
I have seen two different kinds of definitions of the notation $C^k(\overline{\Omega})$ — by "extension" of functions on $\Omega$ or by "restriction" of functions on $\...
user14319
7
votes
1
answer
677
views
What does the t in t-category stand for?
To my knowledge the notion of a t-category was first introduced Beilinson, Bernstein and Deligne's Faiseaux Pervers. But while they explain the name "perverse sheaf", they don't give any indication ...
7
votes
3
answers
877
views
Origin of the notation s=\sigma+it in analytic number theory
I was wondering if the standard notation of denoting a complex variable by "$s$" had an interesting origin, or if it dates back to Riemann or Weierstrass. Almost every book in analytic number theory ...
- 899
7
votes
1
answer
372
views
Theory of surfaces in $\mathbb{R}^3$ as level sets
Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...
- 7,197
7
votes
1
answer
723
views
Alternate algorithms for Chinese remainder theorem
I was teaching Discrete this semester and set the students loose on a system of linear congruences. One of them came up with this solution. Say $$ x \equiv 1 \textrm{ mod } 3 $$ $$ x \equiv 3 \textrm{ ...
- 525
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