All Questions
12,823 questions
85
votes
12
answers
90k
views
Why is the gradient normal? [closed]
This is a somewhat long discussion so please bear with me. There is a theorem that I have always been curious about from an intuitive standpoint and that has been glossed over in most textbooks I ...
- 3,613
4
votes
5
answers
2k
views
Isodiametric Inequality
I was reading the proof of Isodiametric inequality in the book by Evans and Gariepy. As a remark they said that the set, say A, is not necessarily contained in a ball of diameter diamA. I thought ...
- 41
3
votes
1
answer
914
views
Range of a Certain Linear Operator
Consider the following hermitian form on the sobolev space H^1(I), of an interval I:
g(u,v):= \int_I (du/dt dv/dt - \rho(t) u v)dt, where \rho is a nice bounded function on I.
Riesz representation ...
- 31
1
vote
2
answers
132
views
Multiplicative stability for integration [closed]
I remember back in undergraduate to ask myself this question :
In the general case, I is an interval,
\int_I fg =! \int_I f \int_I g (*)
But how to describe the egality case, i.e find all couples (f,g)...
- 19
8
votes
1
answer
688
views
Universal covers of domains in complex projective space
The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
- 4,485
5
votes
3
answers
1k
views
Asymptotics of a hypergeometric series/Taylor series coefficient.
I was planning on figuring this problem out for myself, but I also wanted to try out mathoverflow. Here goes:
I wanted to know the asymptotics of the sum of the absolute values of the Fourier-Walsh ...
- 6,694
26
votes
8
answers
4k
views
Euclidean volume of the unit ball of matrices under the matrix norm
The matrix norm for an $n$-by-$n$ matrix $A$ is defined as $$|A| := \max_{|x|=1} |Ax|$$ where the vector norm is the usual Euclidean one. This is also called the induced (matrix) norm, the operator ...
- 365
40
votes
5
answers
10k
views
Is there a natural measures on the space of measurable functions?
Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
- 1,475
21
votes
5
answers
4k
views
Isomorphisms of Banach Spaces
Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...
- 629
7
votes
1
answer
1k
views
Explicit convergence of Baker-Campbell-Hausdorff
Let g be a finite dimensional simple Lie algebra over C. The Baker-Campbell-Hausdorff series defines a (multivariable) analytic function from a neighborhood of 0 in g \times g \to g. What is the ...
- 71
8
votes
3
answers
698
views
L_p norm balls for 1<p<2 - is it always similar to an L_q norm ball for some q>2?
The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same ...
- 101
11
votes
3
answers
4k
views
erfc lower bound
I've seen the following lower bound for the complementary error function (erfc) but I haven't been able to prove it. Does anyone know how to establish the following?
$$erfc(x) > \frac{ x \exp(-x^...
- 5,227
23
votes
6
answers
2k
views
Is there a topological description of combinatorial Euler characteristic?
There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...
- 54.6k
56
votes
17
answers
13k
views
Atiyah-Singer index theorem
Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of ...
- 44.8k
10
votes
4
answers
966
views
What m minimizes E(|m-X|^3) for a random variable X?
Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x.
A couple weeks ago in a technical ...
- 14k
2
votes
2
answers
372
views
Limit of sequence involving gamma functions
Let G be the gamma function, and b be a constant in (-2,inf). Let
H(n, i) = G(i+1+b) * G(n-i+1+b) / [G(i+1) * G(n-i+1)]
for integers n > i > 0. Let
S(n) = \sum_{i=1}^{i=n-1} H(n, i).
Let x_ n = H(...
- 23
3
votes
2
answers
1k
views
roots of analytic functions
Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $\mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic ...
- 4,265
2
votes
1
answer
493
views
Convergence of Affine Transformations
Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:
...
- 690
10
votes
4
answers
2k
views
Reading for finite Fourier analysis
Can anyone recommend some good reading for Fourier analysis (and the Fourier transform) over finite abelian groups? I've found it given brief descriptions in both books on representation theory and on ...
- 7,013
18
votes
2
answers
3k
views
Zeta-function regularization of determinants and traces
The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.
Let A be an operator (on an infinite-dimensional ...
- 54.6k
5
votes
3
answers
1k
views
Does the "continuous locus" of a function have any nice properties?
Suppose $f:\mathbf{R}\to\mathbf{R}$ is a function. Let $S=\{x\in \mathbf{R}|f\text{ is continuous at }x\}$. Does $S$ have any nice properties?
Here are some observations about what $S$ could be:
$S$ ...
12
votes
4
answers
877
views
Can you describe the image of the exponential map $B(H)\to B(H)$?
James Tener asks at the 20-questions seminar:
The exponential map $\exp:B(H)\to B(H)$ is just defined by its Taylor series. Can you describe its image?
- 1,059
0
votes
0
answers
2k
views
Ignore this question [closed]
This question is a hacky way to create some tags for you to use. Move along.