All Questions
12,779 questions
1
vote
1
answer
559
views
Sum of a Gaussian and an independent second moment constrained random variable
I am studying the (asymptotic) behavior of the p.d.f of the random variable $Y = X + Z$, where $X$ is an r.v. with any distribution function $F(x)$ such that $\int_{-\infty}^{\infty} x^2 \mathrm{d}F(x)...
- 51
0
votes
1
answer
130
views
Maximal length vector under constraints
Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
- 800
7
votes
2
answers
1k
views
Yang Mills gradient/heat flow on 4-torus
The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,
$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta ...
- 362
2
votes
2
answers
411
views
Functional Minimization: When is this heuristic rigorous?
I'm trying to solve a functional minimization problem of the following form:
$$\arg\min_{f:\mathbb{R}\rightarrow [0,1]} h(f)$$
where $h$ is some expression in terms of several integrals over $f$.
I ...
- 21
1
vote
1
answer
219
views
fourier transform of cumulative function
Hi
I've encountered a test that uses the cumulative value of a finite time series to deterime the data set's stationarity.
I would like to know the characteristics of this test in frequency space,...
- 11
27
votes
2
answers
8k
views
Compact embeddings of Sobolev spaces: a counterexample showing the Rellich-Kondrachov theorem is sharp
Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...
- 309
18
votes
1
answer
564
views
Is the space of Hankel operators complemented in B(H)?
Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$.
Let ...
- 25.8k
4
votes
2
answers
2k
views
mean value theorem for operators
This might be a trivial question but I am not very familiar with the subject matter. I was wondering if some sort of mean value theorem works for operators on function spaces. Say $F: \mathcal{S_1} \...
- 85
4
votes
3
answers
2k
views
looking for a book on banach manifolds
Hi,
I am looking for a book on Banach manifolds. Can somebody recommend me something.
Thanks in advance.
leo
- 41
9
votes
2
answers
902
views
Subtlety in the definition of the Kobayashi metric
When defining the Kobayashi metric on a connected complex analytic space $X$, one makes the following auxiliary definition:
A holomorphic chain from $x\in X$ to $y\in X$ is a finite sequence of ...
- 1,288
15
votes
3
answers
8k
views
What is an isomorphism of Banach spaces?
The nLab page on Banach spaces (http://ncatlab.org/nlab/show/Banach%20space) was recently criticised as being, in effect, too heavily biased to category theory (not of the Baire kind) and not enough ...
- 26.8k
8
votes
0
answers
751
views
The log kernel and Bochner Theorem
I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that
$$
L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)
$$
for every $x\in [0,1/2]$.
On a structural ground, this ...
- 2,135
3
votes
0
answers
481
views
Characterizing essential singularities
In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...
- 757
4
votes
2
answers
1k
views
How to use DFT to solve this minimization problem?
This is a problem when I'm reading a paper.
Equation:
$min\{\sum_p(S_p-I_p)^2+\beta((\partial_xS_p-h_p)^2+(\partial_yS_p-v_p)^2) \} $
where $S,I,h,v$ are all $M*N$ matrices and p stands for every ...
- 43
5
votes
2
answers
1k
views
General form of Schwarz reflection principle
Hello all,
It is easy to find results on reflecting holomorphic functions over circles and lines, but I am wondering what there is for reflecting over smooth curves, or analytic arcs, etc. In ...
- 237
7
votes
2
answers
1k
views
Analog of residue for meromorphic quadratic differentials
Hi I had asked this already on math.stackexchange.com but got no answers.
I was wondering if there was any sort of (natural) analog of the residue of a meromorphic one form that made sense for a ...
- 2,299
3
votes
1
answer
394
views
Topological weak mixing and $\omega$-linearly-independent sequences generated by composition operators
A research problem on which I am currently working requires a construction in topological dynamics of the following type:
Let $T \colon X \to X$ be a continuous transformation of a compact metric ...
- 6,206
8
votes
0
answers
196
views
Parametrizing derivations from the algebra of smooth functions on a manifold to its dual
$\newcommand{\Der}{\operatorname{Der}}$
$\newcommand{\Real}{{\mathbb R}}$
(Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential ...
- 25.8k
4
votes
2
answers
419
views
Proper morphisms
Suppose that $f:X\to S$ is a holomorphic morphism of Hausdorff complex manifolds and that $s\in S$ such that $f^{-1}(s)$ is compact (and maybe singular). Then is it true that there is an open ...
- 3,137
5
votes
1
answer
331
views
Entire calculus and clmc algebras
If $\mathcal{A}$ is a complete locally convex (Hausdorff) associative unital algebra (over $\mathbb{C}$) one is interested in defining "transcendental" functions of a given algebra element $a \in \...
- 8,098
4
votes
1
answer
426
views
Analytic continuation of ordinary Dirichlet series
Consider an ordinary Dirichlet series which is absolutely converge in some half plane Re s>c.
Question:Suppose it can be extended meromorphically to the whole complex plane with finite many poles.is ...
- 3,337
0
votes
0
answers
624
views
zeros of a holomorphic function in several variable
Let $f$ be a holomorphic function in $n$ complex variables on a domain $D\subset \mathbb C^n$. Let $S$ be a subset of $D$ such that for a polynomial $P$ in $n$ variable, $P(S)=(a,b)$ for an interval $...
- 415
5
votes
2
answers
642
views
Is the Hausdorff metric on sub-$\sigma$-fields separable?
Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ ...
- 6,287
2
votes
1
answer
901
views
Geometry of the Hilbert sphere
Let $X$ be the unit sphere in $\ell^2$, i.e. $X=\{x\in\ell^2: \|x\|=1\}$. Let the metric on $X$ be the geodesic metric, i.e. $d(x,y)=\cos^{-1}\langle x,y\rangle$. Call a set a ball-intersection if ...
- 744
43
votes
1
answer
5k
views
Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?
Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
- 559
1
vote
3
answers
849
views
How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?
$a \in \mathbb{R}$
$f:\mathbb{R} \rightarrow \mathbb{R}$
$g:\mathbb{R} \rightarrow \mathbb{R}$
For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below?
$f(x+a)=f(x)+a\times g(...
- 175
3
votes
2
answers
461
views
Complemented subspaces isomorphic to $c_0$ in $\mathcal{B}(E)$ [closed]
It is well known that neither
1) $c_0$ is isomorphic to a complemented subspace of $\mathcal{B}(H)$
nor
2) $c_0$ is a quotient of $\mathcal{B}(H)$
for a Hilbert space $H$. Can we replace $H$ above ...
2
votes
2
answers
966
views
Converse to a theorem of Landau on Dirichlet series
Landau's Theorem for Dirichlet series with real coefficients ($c_n$) states that if the coefficients are of fixed sign for all sufficiently large $n$, then the point $\sigma_0$ on the abscissa of ...
- 2,480
2
votes
2
answers
560
views
Will the eigenvalue of the dirac operater tend to negative infinity?
Question: If M is a spin manifold. Condider the dirac operator on a spinor bundle. Can the eigenvalue of this operater tend to negative infinity? If it can, can we choose a riemann matrix such that ...
- 283
5
votes
1
answer
419
views
positive hermitian elements in $M_n(\mathbb{C})$
Elements of the set $P$ of positive hermitian $n×n$ matrices over complex numbers
have some special properties:
(i) they are closed under sum,
(ii) they are closed under multiplication by positive ...
- 179
2
votes
1
answer
2k
views
linear bijective operator
Let $X$, $Y$ be Banach spaces, and $T\colon X\to Y$ be linear and bijective ($D(T)=X, R(T)=Y)$. Can one infer directly that $T$ is continuous? If not, is there a counterexample?
- 225
4
votes
1
answer
461
views
Bounded spherical derivative implies finite order
Hi,
Let $f$ be an entire function. The spherical derivative $\rho(f)$ is defined by
$$\rho(f)(z):= \frac{|f'(z)|}{1+|f(z)|^2}.$$
A result from Clunie and Hayman states that if $\rho(f)$ is bounded, ...
- 2,154
1
vote
1
answer
220
views
How to compute Kobayashi distance of compact Kaehler manifolds with postive Ricci curvature?
Recently I just learned the Kobayashi distance on complex manifolds and wants to get some feeling of how it looks like on exmaples of manifolds with positive Ricci curvature. I have a feeling that the ...
- 637
1
vote
1
answer
283
views
$L^2$ boundeness of a sequence
Let $f_n \in C^2(\bar{\Omega})$ be a sequence satisfying
$\Delta f_n - f_n^3 \to 0 \ \ {\rm in} \ \ L^2(\Omega)$
where $\Omega \subset {\mathbb R}^2$ is bounded and open with a smooth boundary. Is ...
- 595
0
votes
1
answer
611
views
Weak star separable and separable quotient problem
My first question is the following:
Q1: Let $X$ be a Banach space. If its dual $X^\ast$ is weak* separable, does $X$ admit an infinite-dimensional and separable quotient $X/M$?
To the best of my ...
- 515
1
vote
0
answers
109
views
Bounding a q-expansion on a bounded open subset of the complex upper-half plane
Let $f:\mathbf{H}\to \mathbf{C}$ be a holomorphic function on the complex upper-half plane and let $q:\tau\mapsto \exp(\pi i \tau)$ be the nome on $\mathbf{H}$. Suppose that there are integers $a_j$ ...
- 11
1
vote
1
answer
280
views
Sheaf without embedded associated points
Hi. I want to know if the following is true:
Let $f:X\rightarrow S$ be a flat morphism of complex spaces without embedded components. Let $F$ be a $O_{X}$-coherent $S$-flat sheaf. Then the following ...
- 11
2
votes
2
answers
386
views
Reversed disc algebra?
Take $U=\mathbb{D}_2\setminus \overline{\mathbb{D}_1}$ ($\mathbb{D}_r$ is the open disc centered at 0 with radius $r$) and consider the space $A(U)$ of all functions on $\overline{U}$ which are ...
- 23
2
votes
1
answer
452
views
What do we get from an euclidian affine structure ?
Imagine you investigate a set of objects $\mathcal{E}$, and you just realize this that $\mathcal{E}$ possesses an affine structure with respect to some real vector space $\mathcal{V}$ having a scalar ...
- 2,135
4
votes
1
answer
581
views
Are all quantum cellular automata invertible & representable?
A little background: As far as I know there is no standard definition of a quantum cellular automaton in the literature. Different authors use different definitions. Here I propose my own definition (...
- 1,368
6
votes
1
answer
1k
views
Symmetric basis of harmonic homogeneous polynomials
Recently, a question about the beautiful theory of harmonic polynomials made me aware there is something
I've wanted to know for a long time.
As is well known, for any number of variables $n$ and any ...
- 60.5k
8
votes
1
answer
267
views
Is the class of elementary integrals "small" ?
This I read in a paper:
"The class of integrals that are elementary is very
small compared with nonelementary integrals."
What is the precise meaning of this sentence? E.g., does that mean that the ...
- 83
10
votes
1
answer
607
views
Properties of a matrix-valued generalization of the $\Gamma$ function
I am interested in the following function (Mellin transform of matrix exponential):
$$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$
Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ 0$....
- 1,243
1
vote
1
answer
1k
views
Convergence of Dirichlet series
I have arrived at an elementary-looking "result" via a sketchy argument. Having unsuccessfully searched for the statement and its "proof" in the literature, I would like to hear if anyone knows ...
- 2,480
0
votes
2
answers
225
views
Codimension of $J(\omega_1)$ in its bidual
I am reading the paper
G. A. Edgar, A long James space, in: Measure Theory, Oberwolfach 1979, Lectures Notes in Math. 794, Springer-Verlag (1980) pp. 31-37.
and I am pretty confused by the remarks ...
- 3
9
votes
0
answers
462
views
$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$
Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...
- 3,076
2
votes
1
answer
2k
views
Modified Lebesgue differentiation theorem
Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
- 2,270
9
votes
1
answer
596
views
Classical analogue of the Stone-von Neumann Theorem?
Let $U_s$, $V_t$ be a pair of continuous $n$-parameter groups ($n < \infty$) of unitary operators on a complex Hilbert space $\mathcal{H}$. The Stone-von Neumann Theorem establishes that any such ...
- 657
2
votes
2
answers
387
views
holomorphic function with special decreasing property
If you consider $f=\frac{P}{Q}$ the quotient of two polynomial function (i.e. $P,Q\in \mathbb{C} [z]$) then $\frac{f'}{1+|f|^2}$ decrease like $\frac{1}{z}$. My question is, is the converse true? is ...
- 71
20
votes
2
answers
1k
views
P-adic C* algebras
I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C*-...
- 373