MathOverflow Stack Exchange
MathOverflow Stack Exchange

Most Popular

more tags
1500 questions
52
votes
7 answers

On an example of an eventually oscillating function

For $x\in(0,1)$, put $$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$ This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate around the value $1/2$ on a left neighborhood of…
Twi
  • 2,188
52
votes
5 answers

When exactly and why did matrix multiplication become a part of the undergraduate curriculum?

The story about Heisenberg inventing matrices and matrix multiplication in 1925 is very well known and well documented. A few weeks later, Born and Jordan read this work and recognized matrix multiplication, because one of them happened to take a…
Alexandre Eremenko
  • 88,753
52
votes
7 answers

Triangulating surfaces

I've had a few undergraduate students ask me for references for the classical fact (due to Rado) that closed topological surfaces can be triangulated. I know two sources for this, namely Ahlfors's book on Riemann surfaces and Moise's book…
Andy Putman
  • 43,430
52
votes
1 answer

Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes. Let $\mathrm{Li}(x)$ be the offset logarithmic integral, let $\pi(x)$ be the prime counting…
Jim Belk
  • 8,433
52
votes
5 answers

What is the best graph editor to use in your articles?

Here is the criteria for a "perfect" graph editor: it should be able to perform an automated, but controllable layout one is able to make "manual" enforcements to nodes and edges locations when you need it (or at least such fine automated layout so…
Leonid Dworzanski
  • 371
52
votes
3 answers

Function extensionality: does it make a difference? why would one keep it out of the axioms?

Yesterday I was shocked to discover that function extensionality (the statement that if two functions $f$ and $g$ on the same domain satisfy $f\left(x\right) = g\left(x\right)$ for all $x$ in the domain, then $f = g$) is not an axiom in the standard…
darij grinberg
  • 33,095
52
votes
1 answer

There's something strange about $\sqrt{\big(j(\tau)-1728\big)d}$

Given discriminant $d$ and j-function $j(\tau)$, I was looking at, $$F(\tau) = \sqrt{\big(j(\tau)-1728\big)d}$$ which appears in Ramanujan-type pi formulas. Let $C_d$ be the odd prime factors of the constant term of the minimal polynomial for…
Tito Piezas III
  • 12,044
52
votes
7 answers

"Algebraic" topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact. Question: Are there other categories of algebraic…
Harry Gindi
  • 19,374
52
votes
6 answers

Mathematical explanation of the failure to quantize gravity naively

One often hears in popular explanations of the failure to find a "Grand Unified Theory" that "Gravity goes off to infinity, but cutting off the edges gives us wrong answers", and other similar mathematically vague statements. Clearly, this issue…
Harry Gindi
  • 19,374
52
votes
11 answers

Does the exponential function have a (compositional) square root?

(asked by Nathaniel Hellerstein on the Q&A board at JMM) Is there a "half-exponential" function $h(x)$ such that $h(h(x))=e^x$? Is it unique? Is it analytic? Related question: Is there an invertible smooth function $E$ such that $E(x+1)=e^{E(x)}$?…
2010 Joint Meetings
  • 1,119
52
votes
8 answers

Publishing a bad paper?

First, I apologize if mathoverflow is a bad fit for this question, but it is the only place where I can think to get advice from professionals given my circumstance. I'm also sorry about any vagueness in my post since I need to make sure I maintain…
anonymous
  • 171
52
votes
10 answers

Intuition behind Thom class

The Thom class and Thom isomorphism theorem for oriented vector bundles are proven ( at least to my knowledge) by induction on the open covers and some manipulation with Mayer-Vietoris sequences. What is the "actual reason" behind the existence of…
Axel
  • 1,297
52
votes
0 answers

Class function counting solutions of equation in finite group: when is it a virtual character?

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by $$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,x_n)=g \}|. $$ The question is: Can we…
Frieder Ladisch
  • 6,869
52
votes
1 answer

D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over a sheaf of differential operators, but for…
Ben Webster
  • 43,949
52
votes
2 answers

How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.807})$ for the multiplication of two $n \times n$ matrices (the exponent is $\frac{\log7}{\log2}$). However, the constant is so large that this…
Vidit Nanda
  • 15,397
1 2 3
…
99 100