All Questions
12,780 questions
2
votes
1
answer
253
views
A question on Schwartz distributions
I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak* topology. Is is a Barreled space, say, a space whose convex, balanced, ...
- 23
3
votes
3
answers
406
views
When can Hodge filtrations (decompositions?) be described explicitly in terms of periods?
It seems that there is no chance to explain the Hodge theory (to students) in an hour or so. Yet do there exist any cases when the Hodge filtration (or the Hodge decomposition) of the cohomology of a ...
- 16.9k
11
votes
1
answer
633
views
Inequivalent complete norms and the axiom of choice
Hi,
I've been wondering about the following :
Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space?
All the examples of inequivalent complete norms ...
- 2,154
0
votes
1
answer
455
views
Sequences satisfying $x_{n+1} \geq \alpha (x_1+\ldots{} +x_{n})$ for $\alpha<1$
Consider a positive sequence $x_n >0$ that satisfy the condition that there exists a constant $0<\alpha<1$ such that $x_{n+1} \geq \alpha (x_1+\ldots{} +x_{n})$.
What can be said about the ...
- 81
2
votes
1
answer
224
views
Subalgebras of $B(E)$
Let me fix an infinite-dimensional (complex) Banach space $E$. There is a cute result of Bracic and Kuzma which says that every maximal abelian subalgebra of $\mathscr{B}(E)$, the algebra of bounded ...
- 11.3k
4
votes
1
answer
586
views
Upper bounds for the solution of an elliptic PDE depending on a parameter.
Suppose I have the following PDE on $[0,1]^n$
$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$
with periodic boundary conditions and $\int f(x) dx =0$ . ...
- 617
6
votes
2
answers
605
views
$\beta\mathbb{N}$ vs $\beta\mathbb{Z}$
Just started learning the Stone-Cech compactification of discrete groups this week. My motivation comes from a question on $\beta\mathbb{Z}$. Surprisingly, I realized there are muchhhh more literature ...
7
votes
1
answer
659
views
Compactness of Sobolev embedding for domains of finite measure
Let $\Omega \subset \mathbb{R}^d$ be a domain of finite Lebesgue measure, not assumed to be smooth or bounded. Is it true that the embedding of, say, $W^{1,p}_0(\Omega)$ (Sobolev functions with zero ...
- 4,010
2
votes
1
answer
400
views
Existence of a measure under certain condition
Hi everyone,
my problem seems quite simple: I have a set $\Gamma$ along with a nice $\sigma$-algebra $\mathscr{B}$. Then I have a vector space of bounded measurable functions $A \subset \mathscr{B}_{\...
2
votes
1
answer
267
views
Fourier transform and spectrum of PDOs in $L^p$
Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ?
Motivation: If $K$ is a ...
- 1,644
10
votes
0
answers
508
views
Tensorial decomposition of $B(H)$
Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
- 101
5
votes
1
answer
306
views
Periodic Holomorphic ODE
Suppose I have an annulus $U\subset \mathbb{C}$ and a single-valued holomorphic function $V:U\to \mathbb{C}$.
I would like to know if there are (tractable) conditions on $V$ that ensure that the ...
- 2,299
1
vote
1
answer
426
views
Continuous embedding of Hardy space in Lebesgue space
I would like to have a reference to the following statement which I think is true:
$$h^1 \hookrightarrow L^1.$$
The closest I came to this is in D. Goldberg's paper, "A local version of real Hardy ...
11
votes
0
answers
758
views
A basic question on Stone-Cech compactification of $\mathbb{Z}$
Can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? ...
- 895
6
votes
0
answers
324
views
Ricci-flat metrics on Cotangent bundles in adapted complex structure
greetings,
Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...
- 61
5
votes
1
answer
225
views
Extending Jordan loops
I encountered this issue recently, but do not know of any general results to deal with it, so I would appreciate any pointers.
Let $\mathbb T=\{z\in\mathbb C\mid |z|=1\}$, and let $f:\mathbb T\to\...
- 32.5k
-2
votes
1
answer
1k
views
holomorphic extension of a function [closed]
hi,
I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{...
- 29
0
votes
1
answer
1k
views
A continuous linear functional on $L^\infty(R)$ that vanishes on $C(R)$.
Intuitively, I want to construct the functional F in this way:
$$F(f)=\lim_{x\rightarrow 0+}f(x)-\lim_{x\rightarrow 0-}f(x)$$
for $f\in L^\infty$. I know this is not well defined so I'd like to find ...
3
votes
1
answer
510
views
A more direct proof of Dedekind Reciprocity
Question: Let $s(p,q)=\sum_{i=1}^{q-1}((i/q))((pi/q))$ where (p,q)=1 and $((x))=x-[x]-1/2$ for $x\notin Z$. I want to prove that $s(p,q)+s(q,p)=(p/q+\frac{1}{pq}+q/p)/12-1/4$ using at least one of ...
- 124
8
votes
5
answers
545
views
Reference for : a Fréchet nuclear space is Montel
I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact"
Thank you in advance for the help!
- 5,428
0
votes
1
answer
488
views
Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \...
- 17
8
votes
4
answers
2k
views
Is there an explicit formula for the modulus of an annulus given a parameterization of the inner and outer boundries?
Every open set in the complex plane homeomorphic to an annulus is biholomorphic to exactly one annulus whose inner radius is 1 and whose out radius is $r>1$. Each value of $r$ gives a different ...
- 12.1k
1
vote
2
answers
437
views
Analytical predicate for integers over complex numbers
A complex number $z$ is an integer if and only if $\sin(\pi z)=0$.
It follows that a complex number $z$ is an integer if and only $\sin^2(\pi z) = 0$. So for a real analytic function $f$ and any real ...
- 123
28
votes
2
answers
2k
views
A 14th and 26th-power Dedekind eta function identity?
Given the Dedekind eta function $\eta(\tau)$. Define $m = (p-1)/2$ and a $24$th root of unity $\zeta = e^{2\pi i/24}$.
Let p be a prime of form $p = 12v+5$. Then for $n = 2,4,8,14$:
$$\sum_{k=0}^{p-...
- 12.6k
4
votes
1
answer
532
views
why do we need to study entire curves?
Good afternoon,
I'm just curious about this question, because I see that there are a lot of papers which study the value distribution of an entire curve $f\colon \mathbb{C}\to X,$ with X a complex ...
- 758
2
votes
0
answers
427
views
How to produce a biholomorphism
If one deals with a simply-connected domain in the complex plane which is not the whole plane then it is easy to construct the biholomorphism mapping it to the unit disc. This can be done by means of ...
- 141
0
votes
1
answer
223
views
Relation between the wave front set and the semiclassical frequency set
I need to prove that the wave front set of a distribution (as defined in Hormander's "The analysis of linear partial differential operators I") is equal to the semiclassical frequency set of an h-...
1
vote
2
answers
622
views
Kähler manifold with Ricci-flat Kähler form
hallo,
I have the following problem: Let $X$ be a $n-$dim Kähler manifold with Ricci-flat Kähler form $\omega$. There is a known fact that then there exists a holomorphic (n,0)-form $\Omega$ such ...
- 29
1
vote
1
answer
111
views
Log-nonexpansive functions: terminology and references
During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important.
(Defn.). Let $h: (0,\infty) \to (0,\infty)$ be ...
- 28.6k
6
votes
0
answers
411
views
Birth-Death Process associated with Orthogonal Polynomials
I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...
- 22.8k
1
vote
1
answer
340
views
Reflexive Besov spaces
I don't know whether the Besov space $B^1_{1,1}$ on a one dimensional torus is reflexive or not? Can someone help me please?
- 13
1
vote
2
answers
1k
views
Existence of solution of a Non-linear PDE via Fixed point theorem
Hi all
I've the following non-linear PDE
$-\Delta Y + Y^3 =U$ on $\Omega \subset R^n $, open, bounded, Lipschitz boundary domain
$Y=0 , $ on $\partial\Omega$
1.Let $Y\in H_0^1 $ and as $H_0^1 \...
- 17
1
vote
0
answers
302
views
Dedekind eta function identity involving two complex variables
Given the Dedekind eta function $\eta(\tau)$ and complex numbers a,b with imaginary part > 0, anybody knows how to prove the proposed identity,
$$\sum_{k=0}^{p-1} e^{2\pi i k/4}\eta^3\big(\tfrac{a+k}{...
- 12.6k
10
votes
1
answer
930
views
Non-probabilistic proof of the Johnson–Lindenstrauss lemma
The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...
- 225
1
vote
1
answer
414
views
Fourier inversion formula for complex-valued random variables?
The characteristic function of a complex-valued random variable $X$ with pdf $\mu$ is given by
$$
\phi(t) = \int \exp[i \Re(\bar{t} X)] \; d\mu
$$
(or, so says Wikipedia). How does one recover the ...
3
votes
1
answer
189
views
Sufficient (and concrete) condition for a function to satisfy some measure theoretic property
I'm interested in the following property, for a positive and locally bounded function $\omega:\mathbb{R}\to\mathbb{R}^d$, $d\ge 1$: there exists a countable sequence of open and pairwise disjoint sets ...
- 381
7
votes
2
answers
315
views
Duality between extremal points and extremal maps
Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set
...
- 421
2
votes
1
answer
202
views
tensor product with projective topology
There are two Banach spaces $X,Y$. These spaces have unconditional Schauder bases $\{e_i\}$ and $\{f_i\}$ respectively.
Is this right that $e_i\otimes f_j$ is the unconditional Schauder basis in $X\...
- 21
24
votes
12
answers
4k
views
2D problems which are easier to solve in 3D
It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex ...
0
votes
1
answer
382
views
Double duals characteristic [closed]
Recall that (for $1\le p<\infty$), $\ell^p = \{\{a_n\}_{n=1}^\infty:\sum\limits_{i=1}^\infty|a_i|\lt\infty\}$, with norm $||\{a_n\}||=(\sum\limits_{i=1}^\infty|a_i|^p)^{\frac{1}{p}}$.
It is well ...
- 13
6
votes
1
answer
751
views
left- and right- Folner sets
Given an amenable group, it is a standard trick to turn a left-invariant mean ( i.e. a continuous positive normalised linear functional $m:\ell_\infty(G) \to \mathbb{R}$ such that $\forall g \in G, m \...
- 4,432
0
votes
1
answer
352
views
If Fatou set has a Multiply connected Fatou component implies every component of F(f) is bounded
Recently when I read a paper about Fatou component, I met the following theorem which cited in Professor
Eremenko's paper "on the iteration of entire functions"
If Fatou set has a Multiply connected ...
- 1,706
0
votes
1
answer
142
views
A special Integral Kernel
Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\...
- 17
1
vote
1
answer
1k
views
Almost analytic continuation
Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
- 1,644
0
votes
0
answers
186
views
Properties of Eigenfunctions of a Kernel
I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references.
I've and Kernel function $K(x,y)$
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$...
- 17
2
votes
1
answer
823
views
Lambert $W_{-1}(x)$ as $x\rightarrow 0^-$: Asymptotic behavior
There are well known bounds for $W_0$, the "principal" real-valued branch of the Lambert-W function. For example, $W_0(x)$ lies between $\log x - \log\log x$ and $\log x - \frac{1}{2}\log \log x$, ...
- 23
4
votes
2
answers
821
views
Elliptic regularity in $L^1$
Dear all,
I am looking for a good reference for elliptic regularity in $L^1$. To be more precise
Let $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain, let $A$ be a properly elliptic ...
- 103
2
votes
1
answer
403
views
The set of Upper semi-continuous functions as a ring.
I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R} $} as it's base.
If $X$ is a topological space, an upper ...
- 1,788
2
votes
1
answer
261
views
Is C^{k+1}(X) compactly contained in C^{k}(X) for a closed manifold X?
Hi all,
I apologize if this question is too low level for mathoverflow. I'm happy to move it to math.stackexchange if so.
Let $X$ be a closed manifold, let $k$ be a nonnegative integer and let $C^k(...
- 31
5
votes
2
answers
291
views
Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group
Hi All,
I am new to this (though I seem to be a latecomer); so forgive me if this is not your most favorite question:
I am trying to understand the structure (e.g., decomposition) of the unitary ...
- 955