All Questions
12,780 questions
2
votes
1
answer
722
views
How to determine bounds on the extremal length around annuli?
I wish to determine bounds for the sum of moduli of a family of topological annuli in the complex plane. Towards that end I would like to ask a question about the closely related concept of extremal ...
- 367
2
votes
3
answers
672
views
Do approximately the same polynomials have approximately the same roots? [closed]
"If $U$ is an open subset of the complex plane, then matrices $X\in\textrm{M}(n,\mathbb C)$ all of whose eigenvalues belong to $U$ make up an open subset of $\textrm{M}(n,\mathbb C)$." Trying to prove ...
- 486
1
vote
1
answer
3k
views
How to show this Holder bound?
Define the seminorm on the space $S=[0,1]\times[0,T]$
$$\mid u\mid_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$
Define the norms on the same space
$$\lVert u \...
- 67
0
votes
0
answers
73
views
A constrained prolongement
Let $\Omega$ be a domain of $R^n$, let $\omega$ be open subset of $\Omega$ and let $\theta \in W^{2,\infty}(\omega).$
I am wondering about the existence of a function $\tilde{\theta} \in W^{2,\infty}...
- 25
3
votes
2
answers
503
views
Holomorphic image of a strip in complex plane
I'm reading a proof of Arnold's theorem about analytic linearization of analytic circle diffeomorphisms. The following result is used in the proof and I don't see why it should be true.
Let $S_\sigma=...
- 315
1
vote
1
answer
2k
views
The zeros of the digamma function
I wonder what work have been done on the zeros of the digamma function and on the values of the gamma function at such points (on the negative real axis). Any help please :)
- 19
4
votes
1
answer
262
views
Are faces of a compact, convex body "opposed" iff their extreme points are pairwise "opposed"?
Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W_1$, $W_2$ $\subset P$ are opposed if there ...
- 95
16
votes
4
answers
4k
views
Geometric invariant theory for geometers
I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry.
So ...
- 161
0
votes
1
answer
320
views
Derivable functions & Sobolev spaces [closed]
Is a C^1-function in a bounded domain $\Omega\subset R^n;$ an element of the Sobolev space $W^{2,\infty}(\Omega)$ ?
- 25
1
vote
2
answers
660
views
A function which belongs on a concrete Besov Space
Please, anyone of you know a simple example of a function which belongs to the Besov Space with $p=q=\infty$ and $s=0$ (over $\mathbb{R}$ or $I\subset\mathbb{R}$ where $I$ is a closed interval). I ...
- 11
3
votes
1
answer
294
views
Automorphisms of bounded symmetric domains
Let $D \subset \mathbb{C}^n$ be a bounded symmetric domain. It is known that $D$ can be realized as the unit ball of some complex norm $||\cdot||$. Using the Bergman metric on $D$, one can define a ...
- 1,169
2
votes
1
answer
301
views
finite generation of $G$-equivariant holomorphic maps by polynomials?
Let $V$ and $W$ be two complex vector spaces with an action of a finite group $G$. The $G$-equivariant polynomial maps from $V$ to $W$ are finitely generated as a module over the ring of $G$-invariant ...
- 483
6
votes
0
answers
457
views
Jet differentials and hyperbolicity: possible mistake in the literature?
I was reading this note by Jingzhou Sun http://arxiv.org/abs/1109.1329
about Demailly's approach to hyperbolicity using jet differentials. The author seems to claim that there is a mistake in one of ...
- 61
2
votes
2
answers
330
views
Computability of finding roots in holomorphic functions.
Consider a holomorphic function $f: S \to \mathbb{C}$ where $S$ is a path connected open subset of $\mathbb{C}$ (not necessarily simply connected). Is it then possible to determine if $f$ contains a ...
- 123
3
votes
0
answers
396
views
Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$
Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range.
That ...
- 5,327
13
votes
0
answers
385
views
Are the zeros of Tutte polynomials dense in $\mathbb C^2$?
For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\...
- 85.6k
7
votes
3
answers
2k
views
Why do we distinguish the continuous spectrum and the residual spectrum?
As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension.
If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...
- 391
6
votes
1
answer
538
views
Growth rate of the infinity norm of Discrete Fourier Transform of +1,-1 vectors
Let $f=(f_0,\ldots,f_{n-1})$ be a vector in $V_n=\{\pm 1\}^n$. Let
$F=(F_0,\ldots,F_{n-1})$
be its (discrete) Fourier transform defined by
$$
F_k=\sum_{x=0}^{n-1} f_x \omega_n^{x k}
$$
where $\...
- 10.4k
3
votes
0
answers
302
views
Dense subalgebras of von Neumann algebras and increasing nets
[Question previously asked on Math.SE]
Let $N$ be a von Neumann algebra, and $A$ be a dense $∗$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that:
For any $x∈N^+$, there ...
- 33
5
votes
1
answer
1k
views
When is a Banach Algebra $C^\star$
I know that if there are enough Hermitian elements in a Banach algebra, then the Banach algebra is stellar. In particular, I'm interested in the two spaces $B(L^1(S^1,\Sigma,\mu))$ the space of ...
- 53
1
vote
1
answer
357
views
semi group of contractions
Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative, and let $B$ is a monotone linear operator such that $D(A)\subset D(B)$.
...
- 11
0
votes
1
answer
193
views
Dissipative operator
Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative.
is it true that : if $\|y\|\leq \|z\|$ then $\|Ay\|\leq \|Az\|$?
Thank ...
- 11
8
votes
2
answers
1k
views
What is the simplest oscillatory integral for which sharp bounds are unknown?
I have either heard or read that sharp asymptotics and bounds for oscillatory integrals of the form
$ \int e^{i \lambda \Phi(x)} \psi(x) dx \quad \lambda \to \infty $
are unknown when the critical ...
- 2,243
1
vote
1
answer
511
views
Heat equation of spatial complex variable
Suppose that $v(t, z)$ is analytic with respect to the complex variable $z$ and differentiable with respect to the real variable $t$ and satisfies the partial differential equation
$$\frac{\partial ...
- 19
4
votes
0
answers
374
views
generalizations of the interplay between Cauchy Riemann equations and complex-linearity
A function $f: \mathbb{C} \rightarrow \mathbb{C}$ is naturally viewed as mapping $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose $f = (u,v)$ is continuously differentiable on $D \subseteq \mathbb{...
- 453
3
votes
2
answers
466
views
Question on a Basel-like sum
Hello all,
I have happened upon the following sum:
$ 1^2 + \Big(1 \times \frac{1}{3} + \frac{1}{3} \times 1 \Big)^2 + \Big(1 \times \frac{1}{5} + \frac{1}{3} \times \frac{1}{3} + \frac{1}{5} \times ...
- 237
8
votes
2
answers
1k
views
Does infinite-dimensional Brownian motion live in hyperplanes?
I'll begin this question with the finite-dimensional case, as a
warmup.
Let me say a continuous path $\omega : [0,1] \to \mathbb{R}^d$ is
hyperplanar if there exists a nonzero $x \in \mathbb{R}^d$ ...
- 29.8k
4
votes
1
answer
2k
views
Composition of $C^{k, \alpha}$ function with $C^\infty$ function on a compact domain
(I asked this question on MSE but I did not receive an answer so I hope I can post here.)
Let $S$ be a compact set in $\mathbb{R}^2$ and let $C^{k, \alpha}(S)$ denote the usual Holder space with $k$ ...
- 67
4
votes
2
answers
845
views
Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard)
What are modular forms or cusps forms, resp. ?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The ...
- 85
-2
votes
1
answer
578
views
Simply-Connected Regions and Phragmen-Lindelöf Theorem
It's easy to see that the Phargmen-Lindelöf theorem from complex analysis can be generalized to non-simply-connected regions. Namely to regions $G$ with the property that for each $z \in \partial_\...
- 253
2
votes
1
answer
199
views
A Function with Exactly $k$ Minima in a Bounded Space
Is it possible to have a function with the following properties?
(i) The function maps a bounded $n$-dimensional space $\mathcal{X}$ (say $\left[0,1\right]^n$) onto a bounded interval $\mathcal{Y}$ (...
- 23
14
votes
2
answers
2k
views
Determining rational functions by their critical points
Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...
- 1,199
2
votes
1
answer
687
views
Solutions to Heat Equations with Obstacles!
Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = \...
2
votes
0
answers
202
views
Frames and reproducing kernels
Hello MathOverFlow
I have some questions about frames and reproducing kernels and here they are:
For a Hilbert space $H$ spanned by a frame $\lbrace f_n \rbrace$ there exists a reproducing kernel $K(...
- 21
0
votes
1
answer
338
views
Ultraweak closure inside a closed ball
Let $H$ be a Hilbert space, and $S\subseteq \mathcal{B}(H)$. We denote
$\bar S$ the ultraweak closure of $S$, and $B_r$ the closed ball of center 0
and radius $r>0$ of the normed space $\mathcal{...
- 33
1
vote
1
answer
3k
views
In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives
I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem.
"Bochner's theorem states that a
positive ...
- 11
1
vote
2
answers
452
views
Showing $\int f_{n+1} dx / \int f_n dx\to 0$
I have formally derived a solution to a PDE as a power series
$$u = \sum_{n=0}^\infty \epsilon^n u_n.$$
I would like to show that the radius of convergence for is $\mathbb{R}$. I assume that the ...
- 351
3
votes
1
answer
380
views
Special Morse function on a Riemann surface
Let $f: S \to \Bbb R$ be a Morse function on a Riemann surface. Let $x_0$ be a saddle point of $f$. Since $x_0$ is a critical point of $f$, it makes sense to talk about the bilinear forms $f_{z\...
- 1,211
2
votes
2
answers
393
views
Globally generation of $\Omega_{\mathbb{P}^n}(2H)$
I have an elementary question about globally generation of a vector bundle. I would like to see why $\Omega_{\mathbb{P}^n}(2H)$ is globally generated (it seems this is well-known among experts). Here $...
- 63
0
votes
0
answers
143
views
description of a convex set of functions
Hi everyone,
I have a question about the characterization of a set of functions.
Let $\Phi$ a set containing all the functions $\phi(x): \mathbb{R}_+\rightarrow \mathbb{R}_{+}$ that satisfy the ...
- 69
0
votes
1
answer
251
views
Schrodinger Operators with diverging Potential
Is it well known that if $ H = -\bar{\Delta} + V$ (which is defined over $ L^2( \mathbb{R} ^n $ ) and $ lim_{|x| \to \infty } = + \infty $, then $ H$ has compact resolvent?
Does someone know of any ...
- 253
2
votes
1
answer
382
views
Function extension in a Sobolev space
Let $\Omega$ be a domain of $R^n$ and let $H^2(\Omega)$ be the usual Sobolev space.
Let $\emptyset\ne \omega_1\subset\omega_2$ be open subsets of $\Omega$, and let $\theta \in H^2(\omega_1)$.
I ...
4
votes
1
answer
4k
views
Weak compactness and weak sequential compactness in Banach spaces
If $E$ is a Banach space, $A$ is a subset of $E$ and is compact with the weak topology $\sigma(E,E')$, that is the most coarse topology which make every $f\in E'$ continuous, is it true that $A$ is ...
4
votes
2
answers
543
views
Zeros of incomplete exponential functions
Let
$$ f_N(z) = \sum_{n=N}^\infty \frac{z^n}{n!},$$
where $N$ is a positive integer.
Where are the (complex) zeros of those functions located?
It would be sufficient for me to know what the ...
- 577
8
votes
4
answers
2k
views
Manifold-Valued Sobolev Spaces
I have the following basic question about Sobolev-spaces which take their values in a
Riemannian manifold $(M,g)$, i.e.
functions $u:\Omega \to M$, $\Omega \subset \mathbb{R}^n$ bounded, such that ...
- 233
0
votes
1
answer
156
views
Does homeomorphism preserves the family of cones?
Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 +...
- 165
6
votes
1
answer
363
views
von Neumann automorphisms: does convergence on a dense algebra imply $u$-convergence?
Let $M$ be a separable von Neumann algebra and let $A$ be a (von Neumann-)dense *-subalgebra.
Suppose that $\alpha,\alpha_1,\alpha_2,\dots$ are automorphisms of $M$, such that for every $a \in A$,
$$ \...
- 1,786
5
votes
1
answer
513
views
Adding segments to an annulus - a question regarding the conformal modulus
Let $A \subset \mathbb{C}$ be an topological annulus, i.e. a region of $\mathbb{C}$ bounded by two disjoint Jordan curves.
Let $B \subset \mathbb{C}$ be a quadrilateral, i.e. a topological disc with ...
- 367
7
votes
2
answers
622
views
Is there a tropical analogue of a reproducing kernel Hilbert space?
In classical functional analysis, one can construct a reproducing kernel Hilbert space by starting with a positive definite kernel, say
$K: [0,1]\times [0,1] \rightarrow \mathbb{R}$.
One then creates ...
- 1,666
19
votes
1
answer
5k
views
A Fourier-analytic inequality used by Jean Bourgain
I am currently reading Jean Bourgain's 1986 paper A Szemerédi type theorem for sets of positive density in $R^k$ and would appreciate some help in understanding a Fourier-analytic estimate used in ...
- 6,206