All Questions
12,780 questions
1
vote
0
answers
347
views
HyperKaehler manifolds are Ricci-flat
Hi,
I have the following question: Let $M$ be a Hyperkaehler manifold with complex structures $I,J,K$ and Hyperkaehler metric $g$. Let $\omega_{I} = g(I *, *), \omega_{J} = g(J *, *), \omega_{K} = g(...
- 93
4
votes
1
answer
838
views
Finding the spectrum of the composition of a projection with a multiplication operator
In reading a paper on numerical quadrature I've come across a result that is proved in a manner that is very clever:
Let $X \subset \mathbb{C}$ be a compact, convex set. If $U$ is a finite-...
12
votes
2
answers
3k
views
Direct proof of injectivity of $L_\infty$
I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result.
$L_\infty$ as ...
- 1,697
0
votes
1
answer
177
views
Laurent series with analytic coefficients
Let $A=H(D(0,1))$ the ring of holomorphic functions on the open unity disc.
I consider the function $f$:
$$f (t)=\sum f_{i}t^{i} \in A[[t]]$$
I suppose that the $t$-adic valuation of it is less or ...
- 3,472
3
votes
1
answer
474
views
Nondegenerate involutions on the complex numbers -- always just conjugation?
Suppose you have a ring homomorphism $(-)':\mathbb{C} \to \mathbb{C}$, which is an involution such that $\sum_i a_i a_i' = 0 \Leftrightarrow \forall i \ a_i=0$, where this sum is finite.
Must this ...
- 2,513
4
votes
0
answers
1k
views
Resonance of Schrödinger operator
Consider the dispersive estimates for the Schrödinger flow
$$
e^{itH}P_{c},\quad H=-\Delta+V \quad \text{on}\quad \mathbb{R}^n,n\ge 1
$$
where $P_{c}$ is the projection onto the continuous spectrum ...
- 1,644
2
votes
0
answers
484
views
Mellin inverse of the Hadamard product rep. of the Riemann zeta function?
The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely :
$$\left \lfloor x \right \rfloor=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta(s)\frac{x^{s}}{...
- 181
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 3,245
75
votes
3
answers
9k
views
Does a power series converging everywhere on its circle of convergence define a continuous function?
Consider a complex power series $\sum a_n z^n \in \mathbb C[[z]]$ with radius of convergence $0\lt r\lt\infty$ and suppose that for every $w$ with $\mid w\mid =r$ the series $\sum a_n w^n $ converges ....
- 47.5k
0
votes
2
answers
400
views
functional equation
I have the following question:
Let $f$ be an analytic function satisfying the functional equation: $f(z)=u(z)f(a-z)$ where $a$ is a real constant. Let $g$ be another function satisfying the same ...
- 1,197
2
votes
2
answers
514
views
Isometric embedding of a Kaehler manifold as a special Lagrangian in a Calabi-Yau manifold
Hallo,
I am reading the paper "Hyperkaehler structures on the total space of holomorphic cotangent bundles" by D.Kaledin and I am asking if it is possible to embedd a real-analytic Kähler manifold, ...
- 23
1
vote
2
answers
488
views
Sobolev-type inequality.
Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then:
$ \left \| \int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha} } \right\|_{L^q(\mathbb{R}^n)...
- 13
0
votes
0
answers
183
views
Continuity of the Shadow of a Nondecreasing Function
So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
- 485
3
votes
0
answers
163
views
Isometric automorphism of $c_0$ different than coordinate permutation
Does there exist an isometric automorphism of $c_0$ which is not a permutation of coordinates?
- 311
7
votes
1
answer
1k
views
Is a simply connected Ricci-flat Kaehler manifold a Calabi-Yau manifold?
Hi,
I have the following question: Let $(M,\omega, J)$ be a simply connected Kaehler manifold with Ricci-flat Kaehler metric. How can one show that $M$ is a Calabi-Yau manifold. By Calabi-Yau ...
- 71
4
votes
2
answers
229
views
Dimension of the full automorphism
Let $\mathbb P_1$ be the one dimensional complex projective space.
What is the connected component of the full automorphism of
$\mathbb C^*\times \mathbb P_1$.
Is it a complex Lie group? I mean is it ...
- 645
3
votes
2
answers
599
views
Weak topology of WOT
Let $E$ be a reflexive Banach space and let $B(E)$ be the space of bounded operators on $E$ endowed with the weak operator topology. In particular, the unit ball of $B(E)$ is then WOT-compact. $(B(E), ...
- 115
1
vote
2
answers
522
views
Variational problems whose lagrangian density depends on derivatives higher than 1.
The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...
- 469
4
votes
2
answers
670
views
Limits of von Neumann algebras
Consider (abstract) von Neumann algebras topologised by their weak*-topology arising from the unique predual.
In the theory of topological vector spaces, there is a natural notion of an inductive ...
- 115
1
vote
1
answer
3k
views
The conormal derivative of a function
Hi!
I was wondering about the definition of the conormal derivative of a function
$u$ which is given on a domain $\Omega$. It is known that if $-\Delta u = f$, considered
as functionals on $H^1_0(\...
- 13
6
votes
0
answers
322
views
Terminology for notion dual to "support"
If $X$ is a set (feel free to think of it as finite, but it doesn't have to be) and $f$ a real function on $X$, call the support $\operatorname{supp} f$ the subset of $X$ consisting of all elements $i\...
- 21.6k
9
votes
2
answers
524
views
Projections onto $n$-codimensional subspaces of a Banach space: norms.
Hello, I'd like some help to find an answer I've been looking for since this morning.
Let $X$ be a Banach space and let $Y$ be an $n$-codimensional subspace of $X$. Let $P$ be a projection from $X$ ...
- 93
7
votes
1
answer
4k
views
Functional/variational derivative and the Leibniz rule
I am currently trying to understand the BV-formalism, which makes heavy use of the functional derivative.
Let us consider the functional derivative, as defined in for example its Wikipedia article.
...
- 515
4
votes
6
answers
2k
views
Real functions with finitely many zeroes
I am looking for as general a class as possible of real functions defined on $\mathbb{R}^+$ that are guaranteed to have a finite number of zeroes - no, polynomials are not enough :).
Specifically, ...
- 392
8
votes
2
answers
540
views
Maximum entropy priors in infinite dimensional spaces
Is there an extension of maximum entropy probability distributions for function spaces?
For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as "non-...
- 1,160
0
votes
1
answer
308
views
Linear Riemann-Hilbert problems in multiply-connected domains
Hello,
I am looking for a reference giving the solution of the linear Riemann-Hilbert problem in multiply-connected domains of C. Many authors refer to Gakhov's book, but the presentation is quite ...
- 41
12
votes
4
answers
2k
views
Seeking a Geometric Proof of a Generalized Alternating Series' Convergence
Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges:
$$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$
Note that $S(...
- 7,831
8
votes
4
answers
13k
views
Eigenvalues of infinite matrices [closed]
I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can ...
- 431
2
votes
1
answer
812
views
About Sobolev embedding theorem of the case $W^{s,2}$.
Let $s>n/2, \; f \in W^{s,2}(\Bbb R^n)$ . Then how can I show that there is an embedding into the space of uniformly bounded, continuous functions, that is, $$ |f(x)| \leqslant C\| f \|_{W^{s,2}}$$ ...
- 43
5
votes
1
answer
445
views
A sequence that tell us if a holomorphic function of several variables is identically zero
Is there any sequence $ \{ Z_{\nu} \}_{\nu \in \mathbb{N}}$ in $\mathbb{C}^{n}$, $Z_{\nu} \rightarrow 0$, such that any holomorphic function in $\mathbb{C}^{n}$ which vanishes in $Z_{\nu}$ for all $\...
- 283
0
votes
1
answer
279
views
Normal Family for several complex variable (from $C^n$ to $C^n$ \ $U$)
Recently I faced a problem, which I realized has a close connection with the following problem.
$\{ f_{n} \}_{n=1}^{\infty}$ is analytic map from $C^{n}$ to $C^{n} $\ $U$ where $U$ is open ...
- 1,706
6
votes
2
answers
1k
views
Commuting Linear Operators In Hilbert Spaces
Let $V$ be a finite dimensional vector space over the complex field $\mathbb C$. Let $L:V\rightarrow V$ be a linear operator. Using the matrix of $L$ and the Jordan canonical form it is easy to find ...
- 545
10
votes
3
answers
1k
views
ordered exponential of unbounded operators
Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
\...
- 43.2k
7
votes
2
answers
2k
views
Arbitrary union of meagre open sets
Let $X$ be a topological space. A subset $M$ of $X$ is called meagre (or of first category) if it is covered by the union of a countable family of closed subsets of $X$ with empty interior.
Can you ...
- 73
8
votes
1
answer
668
views
Hahn-Banach theorem with real extended valued function
Hello to everyone,
My problem is the following: I have this version of the Hahn-Banach theorem:
Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any
convex function. Let $W$ be a vector ...
- 83
20
votes
1
answer
2k
views
Any algebraic substitute for Morse theory (and homology) in arbitrary characteristic?
As far as I know, Morse theory yields much information on the topology of smooth manifolds; in particular, it can be used to prove Artin's vanishing (that the singular cohomology of smooth complex ...
- 16.9k
0
votes
1
answer
276
views
Positive definite Hermitian matrices of countable rank
Say that a $\omega\times \omega$ Hermitian matrix $A$ is positive semidefinite of rank $n$ if there exists a $\omega\times n$ complex matrix $B$ such that $A=B B^\dagger$ where $^\dagger$ denotes the ...
user2529
18
votes
6
answers
3k
views
What's the use of Malgrange preparation theorem?
The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
- 1,644
3
votes
2
answers
1k
views
Infinite dimensional manifold
In Hamiltonian mechanics, one essentially work with $\mathbb{R}^{2n}$. However, this is only a local description of our configuration manifold $M$. More precisely, the mechanical system is regarded as ...
- 49
2
votes
1
answer
240
views
BM-distances between $B(\ell_p^n)$ and $\ell^{n^2}_p$
Let $\ell_p^n$ be the $n$-dimensional real or complex $\ell_p$-space and let $\mathscr{B}(\ell_p^n)$ be the space of matrices on $\ell_p^n$ endowed with the operator norm. I am looking for any ...
- 11.3k
1
vote
1
answer
534
views
Unique symplectic form in an adapted complex structure
I have the following question: Due to Stenzel, Lempert, Szöke etc. we know that a Riemannian manifold $(M,g)$ admits a complex structure on a neighbourhood of the zero section of the cotangent bundle. ...
- 13
4
votes
1
answer
427
views
Smooth functions tangent to the leaves of a foliation
Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space
$$T_f C^\infty(M,N) = \...
- 1,829
3
votes
1
answer
2k
views
A question about a formal power series manipulation
I want to find a function $f(x,y)$ which can satisfy the following equation,
$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = exp [ \sum _{n=1} ^\infty \frac{f(x^n,y^n)}{...
- 3,541
8
votes
1
answer
894
views
Certain compact subset of $L_1$
Let $(\Omega,\Sigma, \mu)$ be a probability measure and $X$ a Banach space. I am interested in subsets $F\subseteq L_\infty (\mu,X)$ that satisfy these two compactness conditions:
$F$ is a norm-...
- 900
1
vote
0
answers
221
views
Distance between probability amplitude functions
Suppose we have two probability measures $P_1$ and $P_2$ on some Riemannian manifold $(\Sigma,g)$. There are many potential distance measures between $P_1$ and $P_2$:
The Wasserstein distance
For $...
- 705
7
votes
0
answers
300
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
3
votes
1
answer
399
views
Smoothness of solution to a PDE
Let $X$ be a Riemann surface and let $E$ be a smooth complex vector bundle on $X$ with a connection $D$. We can write the connection $D$ as the sum $D'+D''$ where $D'$ is the (1,0) part and $D''$ is ...
- 1,563
4
votes
2
answers
904
views
Nash inequality on a compact domain?
I have come across a few papers that make use of the Nash inequality for functions on a compact domain. Unfortunately, nobody cites a reference for the proof of this result. Is going from the ...
- 617
16
votes
2
answers
1k
views
Maximum of a function of one variable
Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs
of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie ...
- 91.8k
2
votes
0
answers
82
views
Description of the norm of certain interpolation space
Dear all,
I suspect that there should be some detailed description of the norm (or of the unit ball) of the following complex interpolation space (for any $0< \theta < 1$): $$\Big(B(\ell_1^n, \...
- 769