All Questions
542 questions
3
votes
3
answers
1k
views
Pedagogical question concerning $\Gamma(z)$
Pedagogically speaking, I see two problems with defining
$\Gamma(z)$ (at least for real $z$) by the limit
$$\Gamma(z)=\lim_{m\to\infty}\frac{m! m^z}{\prod_{i=0}^m (z+i)}$$
as compared with the formula
...
- 2,762
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
11
votes
2
answers
1k
views
Social Reading Platform for Math or LaTeX texts
Social reading is considered to be one of the big trends that could be catalysing learning by reading. Features could include:
Highlighting or annotating paragraphs or single steps in a proof for ...
0
votes
1
answer
269
views
What does the *-operation signify in the binary context of two automata?
Hi I have a notation question.
I've recently come across the '*-operation' (star-operation) in the context of a binary operation on two automata e.g., A*B and I'm not sure exactly what it means (and ...
CommunityBot
- 1
3
votes
1
answer
507
views
What are some interesting grading/curving systems you have seen for a course? [closed]
It seems like every math course has something unique in how things are graded.
1) What are some interesting grading systems you have seen/used? (include curving types, etc.)
2) What are some pros ...
- 30.1k
6
votes
5
answers
656
views
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) ...
CommunityBot
- 1
71
votes
11
answers
9k
views
How to introduce notions of flat, projective and free modules?
In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this ...
- 12.4k
1
vote
0
answers
396
views
Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922
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 ...
- 798
1
vote
1
answer
2k
views
Is there a notation for natural isomorphism?
There's clearly a notation for isomorphism. It's just $\cong$. But is there notation to indicate that an isomorphism is natural? And in general, for morphisms which are natural?
I ask because one of ...
- 21
14
votes
2
answers
7k
views
What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first?
When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator $\{f\in V^*: f(u)=0\quad \forall u\in U\}$ of a ...
- 491
19
votes
1
answer
2k
views
Resources for teaching arithmetic to calculus students
Every time we teach calculus we discover that a significant portion of our students never understood arithmetic. I don't mean that they can't multiply numbers, but rather that they don't know ...
- 12.1k
27
votes
5
answers
5k
views
Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties
This really is two questions, but they are kind of related so I would like to ask them at the same time.
Question 1:
In a question asked by Amitesh Datta, BCnrd commented that it is important to ...
CommunityBot
- 1
3
votes
4
answers
3k
views
two sequences whose difference converges to zero
Is there a name for the relationship between sequences $A_n$ and $B_n$ which means that the sequence $A_n - B_n$ converges to zero? I want to say something like "sequence $A$ converges to sequence $B$...
- 1,795
4
votes
2
answers
3k
views
Is the notation $f(x)$ "deprecated by professional mathematicians" (as claimed by Wolfram)? [closed]
Wolfram's MathWorld website, at the page on functions, makes the following claim about the notation $f(x)$ for a function:
While this notation is deprecated by professional mathematicians, it is ...
- 24.8k
10
votes
7
answers
2k
views
Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
CommunityBot
- 1
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 ...
- 41
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?
- 2,368
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 ...
CommunityBot
- 1
0
votes
1
answer
2k
views
Dual of Zorn's Lemma? [closed]
It seems to me that the dual of Zorn's Lemma should be true: if $S$ is a non-empty partially ordered set and every chain of $S$ has a lower bound in $S$, then $S$ has at least one minimal element.
...
- 39.8k
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 \...
- 32.6k
1
vote
1
answer
311
views
Permutation with repetitions of a vocabulary
Dear all;
Let $\Sigma$={a,b,c,d}, and $\delta$ be a function that returns a string $S$ of infinite length over $\Sigma$, where each character $s \in S$ has been chosen uniformly at random. My ...
- 1,318
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 ...
- 7,442
0
votes
1
answer
630
views
Useless question on rank
What is the rank of $A^{n}$ if A is the zero ring? It's clearly not $n$ as many careless authors claim, since it's not even invariant. I don't think it's 0 either because it does have a linearly ...
- 33.8k
1
vote
0
answers
3k
views
Notation for space of Lipschitz continuous functions
The Lipschitz norm of a function over a domain $D \subseteq \mathbb R^n$ is easy to define: $$\|f\|_{\mathrm{Lip}} = \sup_{x,y \in D} \frac{|f(x) - f(y)|}{|x-y|}.$$ Is there a standard notation for ...
- 8,512
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 ...
- 1,882
6
votes
2
answers
2k
views
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, ...
- 2,992
4
votes
1
answer
3k
views
Notation for algebras
Is there standard notation for
(1) exterior algebras
(2) free graded commutative algebras
(3) divided polynomial algebras ?
I've seen (and used) $\Lambda$, $\Gamma$, $\Delta$ etc. used for ...
- 20.8k
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 ...
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 ...
CommunityBot
- 1
32
votes
22
answers
9k
views
Origins of Mathematical Symbols/Names
I'm not sure if this has been asked. I'll explain the question by an example.
Fields are often denoted by the letter k, which comes from the German word Körper, meaning body (like corpse, corporeal).
...
CommunityBot
- 1
2
votes
2
answers
465
views
When forcing with a poset, why do we order the poset in the order that we do?
In forcing, we take a collection of forcing conditions and impose a partial order on them. The convention is that if $p$ is stronger than $q$, then we say $p < q$. This is perfectly fine, but it ...
- 255
14
votes
3
answers
4k
views
Mathematical symbols, their pronunciations, and what they denote: Does a comprehensive ordered list exist?
Often, certain symbols in mathematics denote different things in different fields. Is there any sort of ordered list that will tell you what a certain symbol means in alphabetical order by the symbol'...
- 4,994
2
votes
1
answer
897
views
Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis
Greetings,
I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...
- 50.6k
12
votes
1
answer
775
views
Teaching Methods and Evaluating them
Hey,
As a lowly graduate student, I'm on a committee (I'm not sure how important my role really is) trying to evaluate how effective different approaches teaching undergraduates. We are looking at ...
- 6,734
28
votes
3
answers
3k
views
Why is "h" the notation for class numbers?
A student asked me why $\mathcal{O}_K$ is the notation used for the ring of integers in a number field $K$ and why $h$ is the notation for class numbers. I was able to tell him the origin of $\...
- 4,279
7
votes
2
answers
1k
views
Maximal Ellipsoid
John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...
- 31.5k
3
votes
1
answer
977
views
Notation for "the inclusion map is a homotopy equivalence"
It's sometimes convenient to have different notations for "$A$ is a subset of $B$" depending on what the inclusion map does:
If it's non-surjective, $A\subsetneq B$ or $A\subset B$, depending on your ...
- 47.3k
14
votes
1
answer
961
views
Founding of homological without quite involving derived categories
I am looking at the foundations of homological algebra, e.g. the introduction
of Ext and Tor, and am unsatisfied. The references I look at start with
"this is called a projective module, this is ...
CommunityBot
- 1
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 ...
- 403
2
votes
2
answers
6k
views
Examples of random variables
I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent ...
- 11.4k
2
votes
0
answers
526
views
How much of math could be taught without using mathematical notation? [closed]
Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...
