All Questions
12,780 questions
3
votes
0
answers
498
views
PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field
Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,...
- 800
2
votes
1
answer
90
views
How a matrix of C^1 functions on a domain Ω in Cn generates a C^1 distribution in Ω
I am reading the paper Complex foliation generated by (1,1)-forms by M. Klimek. I have a problem understanding why is true a detail in one of the theorems:
Let $\Omega$ be an open connected subset ...
- 601
1
vote
2
answers
504
views
Do all graphs of C1 functions have Hausdorff dimension 1?
Suppose f is a real-valued function of one variable, and suppose f is of differentiability class C1. My question is, if $\Gamma$ is the graph of f, then must $\dim_H(\Gamma)=1$? If anyone knows of a ...
1
vote
0
answers
84
views
Parabolic partial differential equation, initial conditions
Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
- 11
0
votes
1
answer
503
views
When are operators extended by linearity bounded?
Greetings.
Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite
dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
- 129
1
vote
1
answer
318
views
Does the Bergman kernel always arise as the Jacobian of a biholomorphism?
Let $0 \in D$ be a bounded domain. Is it true that we can always find a injective holomorphic map $F: D \to \mathbb{C}^n$ such that $JF=K(z,0)$? Here, $K$ denotes the Bergman kernel of $D$, and $JF$ ...
- 1,169
0
votes
0
answers
83
views
Comparison between operators
I have found the following two concepts:
$\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The
operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$
and for any $\varepsilon>0$,...
2
votes
1
answer
240
views
Are there polynomials (almost) all of whose intersection numbers are divisible by some integer?
I've been playing around with some basic intersection theory, and I've wondered the following:
For every two integers $n$ and $m$, and complex numbers $a_1,...,a_n$, are there polynomials $f_1(x),...,...
- 5,929
1
vote
1
answer
367
views
An integral which is related to Biharmonic extension
In my research, I need to evaluate an integral:
$$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}\log(|\xi^{2}|+\frac{1}{4})d\xi$$
where $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it ...
- 11
2
votes
1
answer
259
views
What does non-levi flat point mean geometrically
Hello,
$CR$ manifold for example $S^1\times C^{n-1}$ is every where levi flat. Can I have example of $CR$ manifold which has at least one non levi flat point.
I can't see what the happening in Non-...
- 541
1
vote
0
answers
66
views
singularity of the solution to an integral equation
I consider a function $x\mapsto f(x)$ which is the positive solution to the integral equation
$$\int_{f(x)}^\infty \frac{du}{u^\gamma-x}=1, \quad x\ge 0,$$
where $\gamma\in (1, 2]$ is some parameter....
- 11
1
vote
0
answers
327
views
Are there any known bounds on this function?
For a sequence of functions $f_{k}(z,s)=\frac{1}{k} \sqrt[s]{Li_s(z^k)}$ with $s>2$ and $Li_{s}(s)$ is the Polylogarithm, I am trying to show
If $\Re f_{1}(e^{\frac{2\pi i}{3}},s) > \Re f_3(1,...
- 1,306
5
votes
1
answer
514
views
Request for reference: Banach-type spaces as algebraic theories.
Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight ...
- 26.8k
1
vote
0
answers
153
views
Functional Analysis Generalizations: indeterminated inner product and functions over manifolds
There are books or articles that deal with generalizations of functional analysis in the sense that the inner product need not be positive-definite or that works with functions over manifolds?
- 11
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
1
vote
0
answers
149
views
(localized) L^2 norm of quasimode for Laplacian
Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq k^{0.99}}e^{ikx}e^{ily}=\frac{1}{\sqrt{2\...
- 11
3
votes
1
answer
330
views
Is a compact subset of a Stein space admitting a fundamental system of Stein neighbourhoods necessarily holomorphically convex?
Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a ...
- 169
3
votes
0
answers
496
views
When checking if a harmonic function is continuous on its boundary, is a dense subset enough?
Let $U$ be an open connected subset of $\mathbb{C}$ and let $u:U\rightarrow \mathbb{R}$ be harmonic and bounded on $U$.
Let $f:\partial_\infty U \rightarrow \mathbb{R}$ be a continuous function, ...
- 1,601
0
votes
2
answers
337
views
Is there a general notion of entropy for the states of a C*algebra?
I've seen some definition of the relative entropy between two states of a C*algebra. However this definitions work only for finite dimensional C*algebras and I don't know if there is a correspondent ...
2
votes
0
answers
175
views
Grauert's theorem for infinite dimensional Frechet Lie groups
Stein manifolds are complex analytic submanifolds of some $\mathbb{ C}^N.$
(A version of) Grauert's theorem states that on a Stein manifold $X$ every continuous map
$g\colon X\to G$ to a complex Lie ...
- 6,825
6
votes
0
answers
291
views
What is the status of the subadditivity problem for analytic capacity?
Hi,
Here is another question that concerns analytic capacity. For a compact set $K$ in the plane, define the analytic capacity of $K$ by
$$\gamma(K):=\sup|f'(\infty)|,$$
where the supremum is taken ...
- 2,154
1
vote
1
answer
448
views
Cohen macaulay morphism
Hi.
I have a doubt about this fact:
Let f:XS be a flat, proper and surjective morphism of complex spaces (or locally noetherian, excellent schemes) with n-pure dimensional fibers. Then f is Cohen-...
- 435
0
votes
0
answers
160
views
Is this function in the weighted Sobolev space $H^{2,-s}$?
I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order $...
- 1
3
votes
1
answer
235
views
Odd element of L^1 group algebra of the integers
Giving some motivation is hard here, so I'll just ask the question. I want an element $a=(a_n)\in\ell^1(\mathbb Z)$ such that:
$\|a\|>1$
a is power bounded (turn $\ell^1(\mathbb Z)$ into a Banach ...
- 18.7k
0
votes
0
answers
436
views
cokernels of semi-Fredholm operators
I did not find a reference for the following question, so I will pose it here. I think the answer should be elementary.
Let $F:X\rightarrow Y$ be a semi-Fredholm operator between Banach spaces, i.e. $...
- 2,935
1
vote
1
answer
506
views
Bessel sequence, uniformly minimal, separated
Is every unit norm Bessel sequence in a Hilbert space a finite union of separated ones? Is every unit norm separated sequence a finite union of uniformly minimal (minimal with uniformly bounded ...
- 11
2
votes
0
answers
99
views
analytical behaviour of holomorphic functions on boundary
Is there any survey about criterions for a holomorphic function to be analytical on the boundary of a pseudo convex domain on which it is defined ?
- 19
21
votes
0
answers
876
views
Are the eigenvalues of the Laplacian of a generic Kähler metric simple?
It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
- 6,247
2
votes
1
answer
214
views
union of Stone-Cech remainders
Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [...
- 607
7
votes
1
answer
362
views
Nonexpansive multi-valued maps in $\ell^2$
Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
- 744
1
vote
0
answers
126
views
Is scalarwise measurability determined by the strong dual?
Since this question has not received an answer so far, I try to reformulate the question in a simpler manner as follows: Do there exist $E,F,\ell,f$ such that
$E$ and $F$ are separable (real) Banach ...
- 3,584
2
votes
1
answer
137
views
Discriminant on boundary of semi-algebraic surface
Let
$P(t)$ be a polynomial in $t$ of degree $n$,
with some contiguous coefficients (not the first or last) being $x_1,\dots,x_k$
and the rest of the coefficients are fixed.
(E.g. $p(t)=1+2t+x_1t^2+...
- 15.8k
1
vote
2
answers
515
views
continuity of extension of maps along curves
Let $a\le b$ and $k\ge 0$ be given and fixed. Let furthermore $x$ and $y$ denote two different elements of a Hilbert space $H$. Suppose $u:\mathbb{R}\rightarrow H$ is a $C^k$-embedding connecting $x$ ...
- 2,935
1
vote
0
answers
720
views
Does the tangent bundle of this fiber product split?
Let $\mathcal X \to S$ be the local universal family of an elliptic curve, and let $E \to S$ be a vector bundle over $S$. Then we can form the fiber product $\mathcal Y = \mathcal X \times_S E$, which ...
- 4,343
-2
votes
1
answer
314
views
holomorphic equation
hi,
i am working for some time on a problem and at some point i cant go further. here the critical part: Let $U \subset \mathbb{C}^{n}$ be a open set and consider $c : U \rightarrow \mathbb{R}$ a ...
- 11
1
vote
1
answer
706
views
Plancherel-Polya Type Inequality for non-compactly Fourier-supported Functions??
Hi!
The Plancerel-Polya inequality can be stated as follows:
Let $0 < p\le \infty$ and $ \nu \in \mathbb{Z}$. Suppose that $g$ is a (smooth) function satisfying $\mbox{supp }\hat g \subset \...
- 979
4
votes
0
answers
87
views
Question concerning Mellin transforms
I've recently come across a result I've been trying to generalize.
Say that $\phi(\sigma \pm iy) < Ce^{\frac{\pi}{2}|y|}$ in the strip $a < \sigma < b$
then then the following integral is ...
- 181
1
vote
1
answer
663
views
What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)?
[Background:]
Looking at the powerseries for the gamma-function
$ \Gamma(1+x) = 1 + a_1 x + a_2 x^2 - a_3 * x^3 + ... $
then we can arrive at a decomposition
$ \Gamma(1+x) = r(x) + g(x) $
...
- 5,342
1
vote
2
answers
534
views
Local representation of an analytic sets
Let V be a analytic set of $C^n$, $I(V)$ is the sheaf of ideals of V (the sheaf whose stalks are ideals defining germs of V at its points). Since $I(V)$ is a coherent analytic sheaf, we see that in a ...
- 750
2
votes
1
answer
230
views
Completing The Space Sections in a Vectorbundle
Hi there.
Assume $(M,g)$ is a Riemanian manifold and $E\to M$ is a
vector bundle with a bundle metric $\langle\cdot,\cdot\rangle$. We then have the pre-Hilbert space $H_0:=\Gamma_c^\infty(E)$ of ...
- 23
3
votes
1
answer
575
views
Kahler forms on Cohen Macaulay spaces
Hi.
Can anyone answer the two following questions:
For $n$-dimensional $X$ Cohen-Macaulay complex space, is it true that the sheaf of top degree homolorphic forms $\Omega^{n}_{X}$ has no ...
- 435
2
votes
1
answer
262
views
Little Picard for (open) complex manifolds?
"Little Picard" states that if a complex function $f(z)$ is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. The ...
- 3,806
6
votes
0
answers
179
views
Perturbations of zero-dimensional algebraic varieties
Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C$ its restriction to the real torus. Assume generic situation, so by ...
- 959
4
votes
1
answer
474
views
Are these operators defined on 2D surfaces self-adjoint?
My research group finds/proposes a fundamental operator in quantum mechanics, the Cartesian momentum as I called (I think for mathematician the ref. 2007 is sufficient). However, I do not know whether ...
- 199
4
votes
1
answer
313
views
Maximally symmetric smooth projective varieties in CP^2
Let P(X,Y,Z) be a homogeneous polynomial in ℂ[X,Y,Z] whose locus M in ℂℙ2 is a nonsingular curve of genus ≥ 2.
Define M to be maximally symmetric if the following is not true:
...
- 2,858
1
vote
0
answers
305
views
Strong minimum principle for maximal plurisubharmonic functions
Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog ...
- 1,211
1
vote
0
answers
336
views
Residue Cancellation
I am trying to understand how to apply the residue theorem to solve
$\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) \left(\frac{b}{c}\right)^s\,\...
- 447
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
2
votes
0
answers
520
views
Eigenvector of infinite matrix
I consider the system of reaction-diffusion PDEs in a ball
with Robin boundary condition.
It is a Steklov eigenvalue problem
(see G Auchmuty (2004) "Steklov eigenproblems and the representation
of ...
- 31
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