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49
votes
5answers
5k views

Changing field of study post-PhD

I am doing my PhD in algebraic graph theory, for not much more reason than that was what was available. However, I love deep structure and theory in mathematics, and I do not particularly want to be ...
6
votes
1answer
687 views

Stylistic question

I'm writing up a paper now where I'm the only author and have a stylistic question. Should I write ''I'' or ''we'' as in ''I/we recall the definition...'' etc. I think this simple example will make ...
2
votes
0answers
172 views

Finite topological dimension x local compactness

Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance: A topological vector space is finite dimensional ...
29
votes
9answers
3k views

Dimensional Analysis in Mathematics

Is there a sensible and useful definition of units in mathematics? In other words, is there a theory of dimensional analysis for mathematics? In physics, an extremely useful tool is the Buckingham Pi ...
13
votes
14answers
3k views

What are some examples of “chimeras” in mathematics?

The best example I can think of at the moment is Conway's surreal number system, which combines 2-adic behavior in-the-small with $\infty$-adic behavior in the large. The surreally simplest element ...
19
votes
5answers
2k views

Where do surreal numbers come from and what do they mean?

I know about Conway's original discovery of the surreal numbers by way of games, as well as Kruskal's way of viewing surreal numbers in terms of asymptotic behavior of real-valued functions, leading ...
23
votes
3answers
2k views

An elementary problem in Euclidean geometry [closed]

This problem was first put to me by Luke Pebody (who did not know the answer at the time) and after some work I am yet to find a proof or counterexample. I would be grateful of any insights. Call a ...
8
votes
2answers
1k views

Naturally occurring orderings

The are many orderings that naturally occur in interesting but seemingly unrelated circumstances. Here are some examples: The volume spectrum of orientable hyperbolic 3-manifolds has order type ...
24
votes
4answers
3k views

Overview of the interplay of Harmonic Analysis and Number Theory

I'm kind of disappointed that the question here was never sharpened. The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...
6
votes
10answers
887 views

Examples of “Unusual” Classifications

When one says "classification" in math, usually one of a handful of examples springs to mind: -Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...
37
votes
14answers
3k views

A set for which it is hard to determine whether or not it is countable.

I got thinking recently, while trying to come up with a problem, that I did not know of any sets which were reasonable to define but for which it was very difficult to determine whether or not they ...
8
votes
3answers
878 views

P vs. NP resistant problems

According to Stephen Cook on wikipedia, http://en.wikipedia.org/wiki/P_versus_NP_problem ...it would transform mathematics by allowing a computer to find a formal proof of any theorem which has a ...
106
votes
15answers
14k views

When should a supervisor be a co-author?

What are people's views on this? To be specific: suppose a PhD student has produced a piece of original mathematical research. Suppose that student's supervisor suggested the problem, and gave a few ...
3
votes
0answers
625 views

How many projects do you work on concurrently? [closed]

I was wondering how many concurrent research projects a typical math researcher works on at a given time. I ask because I currently have the oppertunity to start a second project on something I'm ...
2
votes
1answer
312 views

A question on a special type of function

Suppose I have a function $f$, and positive integers $x$ and $y$ such that $x$ is a square, $x \ne y$ and $y \ne \sqrt{x}$. Now, assume that: $|\frac{f(x)}{y} - \frac{f(y)}{x}| > 2$ ...
3
votes
0answers
1k views

Where does Aphex Twin's “windowlicker” equation come from? [closed]

$\Delta M_i^{-1} = -\alpha \sum\limits_{n=1}^N D_i [n] \left[\sum\limits_{j \in C[i]} F_{ji} [n-1] + Fext_i [n^{-1}]\right]$ This is the name of the second song on Aphex Twin's album "Windowlicker". ...
9
votes
3answers
984 views

Spaces with a quasi triangle inequality

How do you call a space with a function which is symmetric, non negative, positive definite and which satisfies a quasi-triangle inequality: $d(x,z) \leq C( d(x,y)+d(y,z) )$ for all $x,y,z$ and some ...
14
votes
3answers
1k views

Card game / options pricing / Brownian bridge question

We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black ...
14
votes
12answers
2k views

Why semigroups could be important?

There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in ...
7
votes
2answers
922 views

Are there uncountably many essentially inequivalent versions of Mathematics?

Hi everyone, Disclaimer 1: logic and set theory are a long way from my field, so apologies in advance if I demonstrate extreme ignorance or stupidity, and please correct me if (when?) I write stupid ...
1
vote
0answers
284 views

Why semigroups are important? [closed]

There is known a lot about semigroups, mostly about their inner structure etc. But is there any use of semigroups in the general Maths -- like that of groups?
1
vote
1answer
456 views

Is there a conjunction bias?

This is slightly related to question The unprecedented success of the “intersection” operator . Apart from a set of maths books of null measure, most have the following property: Objects ...
2
votes
2answers
1k views

Advice on Giving a Talk [closed]

What advice do you have for giving a talk on a mathematical research paper to people in other fields in science (not physics nor astronomy) but without lot of math background? Thanks.
4
votes
1answer
2k 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 ...
4
votes
3answers
672 views

Is the componentwise square-root of a positive-definite matrix also pos.-def.?

Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a_{ij} = a_{ji} \geq 0$ and $B=(b_{ij})$ with $b_{ij} = \sqrt{a_{ij}}$. Is $B$ positive-definite whenever $A$ is? In other words: ...
-2
votes
1answer
1k views

Unpopular “elementary” theorems/identities to impress an audience of mathematicians. [closed]

This question grew out of my recent job interview. Since the interviewers were math professors, I had a hard time searching for interesting elementary theorems in case I got asked for one. I thought ...
63
votes
6answers
2k views

Good ways to engage in mathematics outreach?

Greetings all, I have often heard that it would be good if we as a community did more in the way of mathematics outreach: more to explain what it is we do to the community at large, more to expose ...
83
votes
90answers
11k views

What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...
7
votes
1answer
1k views

Technical trends quietly aimed at big open problems? [closed]

When I was an undergraduate 35 years ago, I made the mistake of asking some of my mathematics professors what well-known open problems they liked to think about. I got the message that this was ...
9
votes
11answers
3k views

What advanced Area of Mathematics can be delved into with only basic Calculus and Linear Algebra

Hello Mathoverflow Community, I would really appreciate some advice on this: All I know is Basic Calculus and Basic Linear Algebra, I want to start learning more advanced material on my own while ...
12
votes
17answers
2k views

Individual mathematical objects whose study amounts to a (sub)discipline? [closed]

Certain mathematical objects have a theory so rich that their study alone arguably constitutes a distinct (sub)discipline. My own list would begin with 1) the absolute Galois group of the rationals; ...
1
vote
0answers
165 views

Regarding a Feature of Multivariate Real Function

Any real function can be expressed as a function of the sum of two monotonic real functions? More precisely, for real function p(x, y), there exist continuous real functions P(x), h(x,y), g(x) such ...
1
vote
0answers
530 views

What is the name for(a^2 + b^2 + c^2 +…)/(a + b + c +…)? [closed]

That is, the sum of squares of some numbers divided by the sum of the numbers. The term "anti-harmonic mean" has been coined for this quantity. I'm hoping there is a better name.
50
votes
6answers
6k views

Still Difficult After All These Years

I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were ...
54
votes
26answers
5k views

Proof synopsis collection

I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself! Definition (Fraleigh): A proof synopsis ...
15
votes
35answers
2k views

Basic results with three or more hypotheses

Consider the following statement of the Arzela-Ascoli theorem. Theorem. Let K be a compact topological space and let S be a subset of C(K). Then S is relatively compact if and only if S is uniformly ...
13
votes
2answers
844 views

Is there a name for sets for which it is easier to test membership than to find members---and vice versa?

This is a question my son Bob asked me. For some sets it is relatively easy to test for membership but a lot more difficult to find members, and for others the reverse is true. Here is an elementary ...
3
votes
4answers
3k views

Theorems true but wrong. [closed]

Many theorems have the form : Premise(es) implies Conclusion(s) Example A of wrongness: There are many examples in which a theorem is stated without mentioning that part of the premise is not ...
46
votes
6answers
5k views

Does a referee have to check carefully the proof ?

I have always checked very carefully the papers I was refereeing when I wanted to suggest "accept". Actually I spend almost as much time checking the maths of a paper I referee than checking the maths ...
30
votes
6answers
4k views

Why is the Gaussian so pervasive in mathematics?

This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...
2
votes
2answers
877 views

Example of a function that behaves like another function

I need a function $f(x)$ with the following properties - It should be monotonically non-decreasing. For $x \geq 1$, $x + \frac{1}{x} - f(x) < \epsilon$ where $\epsilon$ is an extremely small ...
7
votes
3answers
576 views

What is the strategy for “all words valid” scrabble?

The rules for "all words valid" scrabble are exactly the same as ordinary scrabble, except that every single combination of letters is in the dictionary. To make the game deterministic, we will also ...
124
votes
36answers
22k views

Demonstrating that rigour is important

Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...
7
votes
1answer
608 views

Sum of subset of geometric series: a^2^n

The formula for 1 + a + a^2 + .... where 0 < a < 1 is $\frac{1}{1-a}$, but what if you wanted to sum only those where the exponent is a power of 2? That is, $S = a + a^2 + a^4 + a^8 + \cdots$ ...
40
votes
8answers
3k views

Published results: when to take them for granted?

Two kinds of papers. There are two kinds of papers: self-contained ones, and those relying on published results (which I believe are the vast majority). Checking the result. Of course, one should ...
1
vote
6answers
3k views

Proofs by induction [closed]

Background I'm interested in the issue of "explanatory" mathematical proofs and would like to try to find out what intuitions mathematicians have about induction, because there seems to be some ...
2
votes
7answers
2k views

Books about polynomials [closed]

Hi,have you a good reference (books) for the study of polynomials with one variable or many variables ? Thanks for your help. Don't hesitate to correct my English.