All Questions
495 questions
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
4
votes
3
answers
507
views
Defining negation
I'm currently coauthoring a book intended to teach first-year students basic proof techniques. One of the chapters, written by my coauthor, is about basic logic. In that chapter the negation of a ...
- 18.7k
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 ...
-8
votes
1
answer
378
views
Why is it impossible to find work of John Tate online? [closed]
The work of John Tate belongs to mankind. Why is not online in pdf´s? Who is dirty enough to earn money on HIS work?
- 13
0
votes
0
answers
148
views
About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
13
votes
17
answers
3k
views
Short Course Suggestions For High School Students
I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are ...
13
votes
3
answers
1k
views
Teaching polarisation formula
When teaching about Hilbert spaces, one begins with a polarisation formula, which allows us to reconstruct the scalar product from the norm:
$$\langle u,v\rangle=\frac14(\|u+v\|^2-\|u-v\|^2+\imath\|u+\...
- 52.4k
11
votes
4
answers
2k
views
Why do mathematicians prefer one definition over the other when they both define the same concept?
Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...
- 2,437
18
votes
12
answers
10k
views
Theorems in Euclidean geometry with attractive proofs using more advanced methods
The butterfly theorem is notoriously tricky to prove using only "high-school geometry" but it can be proved elegantly once you think in terms of projective geometry, as explained in Ruelle's book The ...
15
votes
13
answers
23k
views
Math journal for high school students?
I recently discovered The College Mathematics Journal and enjoyed reading through some of the articles on fun applications of mathematics. I'd like to send some of the articles to my younger sister, a ...
- 169
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
14
votes
11
answers
35k
views
Why does undergraduate discrete math require calculus?
Often undergraduate discrete math classes in the US have a calculus prerequisite.
Here is the description of the discrete math course from my undergrad:
A general introduction to basic
...
11
votes
4
answers
6k
views
Place of Analytic geometry in modern undergraduate curriculum
I am a freshmen student in mathematics at Moscow State University (in Russia) and I'm confused with placing the subject called "analytic geometry" into the system of mathematical knowledge (if you ...
- 111
12
votes
10
answers
2k
views
A place to find original papers
I currently use scholar.google.com to find papers in cases like Sophus Lie's original papers on "Transformation Groups". Does anyone know of other places that collect original works like this, i.e. ...
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 151k
39
votes
6
answers
5k
views
What is the simplest, most elementary proof that a particular number is transcendental?
I teach, among many other things, a class of wonderful and inquisitive 7th graders. We've recently been studying and discussing various number systems (N, Z, Q, R, C, algebraic numbers, and even ...
- 533
14
votes
3
answers
1k
views
Where can I read reviews of mathematical theories? [closed]
I'm really enjoying the AMS column "What is ..." (http://arminstraub.com/math/what-is-column) and The Princeton Companion to Mathematics.
I am looking for something similar. I'd like to acquire some ...
4
votes
4
answers
2k
views
When did you "meet Polya"? [closed]
I guess most of us didn't meet Polya in person (this is the answer to the title)! Perhaps, it is much easier to guess that most of us have met one of his writings (or alike) on problem solving, and ...
16
votes
10
answers
6k
views
Undergraduate Topology
I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than ...
9
votes
2
answers
637
views
Constructivist defininition of linear subspaces of $\mathbb{Q}^n$?
Let me preface this by saying I'm not someone who has every studied mathematical logic or philosophy of math, so I may be mangling terminology here (and the title is a little tongue in cheek).
I (and ...
- 2,478
29
votes
2
answers
2k
views
Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?
In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before:
quoted from Leo Corry, Modern algebra, German original:
Why did Dedekind doubt that $(\...
- 9,720
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 ...
- 30.6k
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" ...
20
votes
4
answers
2k
views
Problems for developing mathematical visualization expertise
Einstein stated that he often explored and reasoned visually and spatially, and only after achieving understanding cast his insights into algebraic form. He could just "see" the answer. There are ...
3
votes
0
answers
873
views
Hard problems solving tricks
This question is motivated by this one that I posted on math.stackexchange.
When I fail to solve a hard math problem (like the ones I presented in the linked post), I read a solution and I noticed ...
- 161
14
votes
9
answers
2k
views
math circles video lectures for school children?
Hello,
I am from India. I find the mathoverflow amazing.
I have a question: Are there any good quality video lectures on school math topics?
There are a lot of high quality lectures available on ...
5
votes
0
answers
186
views
Examples of partial adjoints
Recall that a functor $$R: D \to C$$ is said to have a partial left adjoint $L$ defined at an object $X \in C$ if the functor
$$D \to Sets, Y \mapsto Hom_C(X, R(Y))$$
is corepresentable by some object ...
- 2,040
16
votes
13
answers
4k
views
Do you find your students are less competent in basic algebra and arithmetic, and, if so, do you believe that this is due to overuse of calculators at an early level? [closed]
So first I gave my class the quiz problem: Compute $$\lim_{h\rightarrow 0} \frac{\frac{1}{3+h} - \frac{1}{3}}{h}.$$ Upon finding that they could not do that (no real surprize) I asked them to compute $...
16
votes
2
answers
1k
views
Teaching Steenrod Operations
I am teaching a class on topology and want to introduce Steenrod Operations. I have talked about simplicial sets and classifying spaces of groups but have not talked about Eilenberg–MacLane spaces. ...
- 161
16
votes
1
answer
2k
views
A conjecture in which both "if" and "only if" are near misses
[Migrated from Math Stack Exchange]
More than a year ago, I posted the following on the Math Stack Exchange.
Consider $2^n-1$. Based on checking a few small numbers for $n$ (in
fact, the first ...
- 2,437
18
votes
14
answers
3k
views
Teaching a pedagogy course
At my institution incoming graduate students must take a semester long course on pedagogy taught by current grad students. I may soon be in the position of having to teach this course and I'm looking ...
19
votes
10
answers
6k
views
Research Experience for Undergraduates: Summer Programs
Some time ago, I found this list of REU programs held in 2009.
The main aspects that characterize such programs are: (a) a great deal of lectures on specific topics; and, admittedly more importantly,...
20
votes
2
answers
2k
views
Bitcoin Research
I have recently been assigned to advise a student on a senior thesis. She has taken linear algebra, introductory real analysis, and abstract algebra. Her interest is in cryptography. And she has a ...
- 822
15
votes
7
answers
6k
views
Freshman's definition of sin(x)?
I would like to know how you would rigorously introduce the trigonometric functions ($\sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ ...
- 23.4k
21
votes
7
answers
3k
views
What should be taught in a 1st course on Riemann Surfaces?
I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...
- 3,284
13
votes
7
answers
35k
views
Real analysis has no applications?
I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications ...
6
votes
2
answers
1k
views
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
3
votes
1
answer
271
views
Elementary classification of division rings
Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
- 26.5k
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
9
votes
3
answers
12k
views
An image of the hierarchy of algebraic structures
Hello! Does anybody know an image of a graph featuring the hierarchy of algebraic structures? Something rather complete.
So far I've found similar images describing the hierarchies of classes/...
- 441
8
votes
12
answers
13k
views
How do I explain the number e to a ten year old? [closed]
Hardly a research level question, but interesting nonetheless, I hope. Pi is easy, but not e. Where could I start?
3
votes
1
answer
806
views
What are some problems for research in functional analysis that can possibly be solved by someone with basic knowledge of the subject? [closed]
I wanted to know are there any problems in Functional Analysis (FA) that can possibly be successfully tackled by someone like me who does not have any expertise in this area but is only familiar with ...
- 167
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
25
votes
6
answers
25k
views
What are the advantages and disadvantages of the Moore method?
Describe your experiences with the Moore method. What are its advantages and disadvantages?
50
votes
4
answers
4k
views
What algorithm in algebraic geometry should I work on implementing?
This summer my wife and one of my friends (who are both programmers and undergraduate math majors, but have not learned any algebraic geometry) want to learn some algebraic geometry from me, and I ...
6
votes
8
answers
1k
views
Reference for elementary and "cool" statistics or financial math
I signed up for a Math Mentorship Program (for high school students) this term, but one of the students assigned to me is more interested in Statistics and Finance - something that would help him to ...
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
5
votes
2
answers
707
views
Books on the History of math research at European universities
Are there good books that cover the history of math and mathematical science (ex. physics, chemistry, computer science) PhD programs in the Occident? My primary motivation is to figure out how the PhD ...
- 3,871
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 ...
- 28.1k
14
votes
1
answer
3k
views
An elementary proof that the degree of a map of spheres determines its homotopy type
I'm helping to teach an undergraduate algebraic topology course (out of Hatcher's textbook). We have recently defined the degree of a map of spheres using homology, and the professor and I thought it ...
- 7,338