All Questions
12,780 questions
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
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
3
votes
0
answers
564
views
analytic continuation of an integral involving the mittag-leffler function
greetings. we have the following integral :
$$I(s)=s\int_{0}^{\infty} \frac{dx}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)-\left(E_{s/2}(( x)^{s/2})-1\right)e^{-x}$$
where :
$E_{\alpha}(z)$ ...
0
votes
1
answer
864
views
Sequence of smooth functions converging to sgn(x)
I'm looking for a sequence of smooth functions $f_i(x)$ converging to Sign$(x)$, each of which additionally have the following property:
\begin{equation}
f_i(x_1+x_2) = g_i(x_1, f_i(x_2))
\end{...
- 561
25
votes
16
answers
4k
views
functions satisfying "one-one iff onto"
Hello Everybody.
I need some more examples for the following really interesting phenomenon:
A function from the class ... is one-one iff it is onto.
Some ...
16
votes
10
answers
5k
views
Good book on Riemann surfaces and Galois theory?
I'm supervising an undergraduate project on Galois theory and covering spaces. I want to have him read about the fact that from a branched cover of a Riemann surface you get an extension of its field ...
- 683
4
votes
1
answer
315
views
Spectral Properties of $A(I-A)^{-1}$
I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...
- 8,512
1
vote
0
answers
202
views
Weak solution of a certain pde with integral term
Let us consider the following pde on the domain $(0,T)\times(0,1)$
$
\dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0
$
with initial data $p(0,x)=p_{0}(x)$ and boundary data $...
- 11
8
votes
2
answers
819
views
Decomposition of an integral operator into a composition
I've been musing about the following question for a while now. Given an integral operator $G$ defined by
$$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$
Is it possible to decompose this into two separate "...
- 757
0
votes
0
answers
184
views
Extension of closed linear functionals...
If f is a closed linear functional defined on a dense subspace of a Banach space X, and, consider f1 which is an extension of f to X, is there a way to show that f1 is also closed without invoking the ...
- 55
1
vote
0
answers
114
views
Mappings preserving convex compactness
Let $H$ be a Hilbert space.
How can one describe continuous mappings $F:H \to H$
that satisfy the following condition:
There exist two elements $c$, $F(c) \neq c$
and a convex compact $M$ containing ...
- 11
11
votes
5
answers
5k
views
A criterion for the sum of two closed sets to be closed ?
Let $V$ and $I$ be two closed subsets of a Banach space $A$.
The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$.
I would like to know whether $I+V$ ...
- 452
0
votes
0
answers
173
views
Implications of complex solutions of Matiyasevich / Chaitin diophantine polynomials.
This is a shot in the dark: In twf:202, an isomorphism $T\cong T^{7}$ between binary trees $T$ and seven tuples of binary trees T^{7} is mentioned. The argument for this isomorphism starts with the ...
- 975
1
vote
0
answers
393
views
Unambiguous "weak" vector valued $L^{+\infty}$ spaces?
For some time, I have been stuck to the problem to be described as follows. The (perhaps not so commonly known) facts given here are taken from R. E. Edwards' Functional Analysis (Holt, Rinehart and ...
- 3,584
2
votes
3
answers
604
views
Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes?
The problem is that we want to approximate domain-bounded functions (that is, functions restricted to a domain such as [0,10]) as (probably infinite) series of other functions. We know that, for ...
- 23
1
vote
1
answer
294
views
Reference request: Rate of convergence of sequence of functions
Suppose you are given a sequence of functions $f_n \rightarrow F$ with a certain notion of convergence. Suppose that in your setting where this implies that $f_n^{'} \rightarrow F^{'}$ with the same ...
- 81
26
votes
1
answer
820
views
The maximal "nearly convex" function
The following problem is only tangentially related to my present work, and I do
not have any applications. However, I am curious to know the solution -- or
even to see a lack thereof, indicating that ...
- 23k
10
votes
1
answer
869
views
Completeness of Borel measure
Let $X$ be a compact Hausdorff space and $\mu$ a finite Borel measure without atoms which is outer regular with respect to open sets and inner regular with respect to compact sets. Can such measure be ...
- 277
3
votes
1
answer
1k
views
A corollary to Stone-Weierstrass theorem
Can i get the answer to the following problem. I am having a proof, i feel there is something wrong here..Can you please point out!
Let $D\subset \mathbb C$ be a simply connected domain, and $\gamma:...
- 541
5
votes
0
answers
200
views
Diffusion processes in wide generality
It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.
Hard question: What are the most general structures on which one may define something ...
- 8,512
32
votes
2
answers
2k
views
Example of a compact Kähler manifold with non-finitely generated canonical ring?
A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely ...
- 6,871
6
votes
2
answers
4k
views
Is there dual space of the distributions $\mathcal{D}'(R)$?
Dear MOs,
Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
- 1,649
5
votes
1
answer
318
views
What's the link between the Toeplitz operators on H^2 and those used to define Cuntz-Pimsner algebras?
An alternate way to phrase this question might be, "How did the Toeplitz operators used in the definition of the Cuntz-Pimsner algebra come by their name?" or, "What's the relationship between the ...
- 151
0
votes
1
answer
261
views
Norm functionals of $B(H)$ restricted to sub ven-Neumann algebras [closed]
Let $H$ be a Hilbert space, we know that weak topology over $B(H)$, operator algebra of bounded linear operators from $H$ into $H$, is the topology generated by
$\{\langle \cdot \xi,\eta\rangle:\; \...
6
votes
3
answers
1k
views
Topological vector spaces that are isomorphic to their duals
After reviewing the (locally convex)
topological vector spaces that I know,
the only examples I could find where there is an isomorphism from the
space to its (anti)dual, are Hilbert spaces.
So my ...
- 357
3
votes
0
answers
217
views
Is this integral operator about Stokes' Flow compact?
Consider the following integral operator $\mathcal{A}$ on [EDITED: continuous vector function $f=(f_i):\partial S\to{\mathbb R}^3$]:
$$
({\mathcal A}f)_j(x_0):=\int_{\partial S}\sum_{i=1}^3 f_i(x)G_{...
user14319
12
votes
3
answers
2k
views
Reference request: Simple facts about vector-valued Sobolev space
Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...
- 29.8k
1
vote
1
answer
254
views
references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
- 1,222
12
votes
2
answers
878
views
The ground state is signed and symmetric
Background
In Berestycki and Lions it is asserted that (on page 316), if I am not misreading, that the "ground state", i.e. action minimizer among nontrivial solutions, corresponding to the action
$$...
- 39.1k
7
votes
1
answer
2k
views
Real integral which cannot be evaluated without complex analysis?
One motivation for studying contour integrals in complex analysis is that they can be used to give elegant evaluations of certain real integrals, which tend to have more direct physical and ...
- 3,813
4
votes
1
answer
2k
views
On the nascent delta 'function'
I have two questions regarding the sinc function in the weak limit, where it can be used as a nascent delta function.
The definition:
$\lim_{\varepsilon \rightarrow 0}\frac{1}{\pi }\int_{-\infty}^{\...
- 181
3
votes
1
answer
515
views
How to extend evaluation at a point from continuous maps to square-integrable ones?
Consider the Hilbert space $L^2[0, 1]$ of square integrable functions on $[0, 1]$.
Saying more carefully, there is one subtle detail: this space is defined as factor space by the functions which are ...
- 24.9k
6
votes
2
answers
742
views
Symmetric Feller processes and Dirichlet forms
Let $(G, \mathcal D)$ be a densely defined operator on $C_0$ (continuous functions vanishing at infinity on some nice topological space) whose closure $\bar G$ generates a Feller semigroup and let $X$ ...
- 448
15
votes
1
answer
2k
views
Matrices with entries in a $C^*$-algebra
Let $\mathcal{A}$ be a $C^\ast$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution ...
- 1,697
9
votes
1
answer
450
views
A question on infinite dimensional Gaussian measure and affine tranformations.
Let $\gamma_\infty$ denote the product Gaussian measure on $\mathbb{R}^\mathbb{N}$. Which $a,b \geq 0$ satisfy that for every Borel set $K\subseteq \mathbb{R}^\mathbb{N}$ of positive measure, $a K + ...
- 3,547
10
votes
2
answers
3k
views
Absolute continuity on $R^{n}$
I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$.
I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
- 399
4
votes
2
answers
312
views
Algebraic function with extra condition, what can it be?
I have an unknown function $\psi(\xi_1,\dots,\xi_n)$, such that
$\psi$ satisfy an (unknown) polynomial equation with coefficients polynomials in the $\xi_i$.
The function is homogeneous, that is, $\...
- 15.8k
10
votes
4
answers
1k
views
References: Infinite dimensional Lie algebras
What I really want are properties (if it is abelian, nilpotent, solvable, simple, or semisimple; Cartan subalgebras...) of the Lie algebra of smooth functions on a symplectic manifold $(M,\omega)$; ...
- 641
3
votes
0
answers
188
views
Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
This question is related to the following question
Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
A couple of authors have observed that composing a ...
- 265
3
votes
1
answer
908
views
The version of Montel's theorem used in the proof of Jenkins-Strebel differential
Hello,
I am afraid that my main question might be a bit too elementary, but still I ask :
In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an ...
- 1,471
2
votes
0
answers
524
views
What essential property justifies the name "derivative"?
Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...
- 643
4
votes
2
answers
976
views
Wightman fields vs local functionals vs operators
In QFT literature one wants to look at $n-$point correlation functions of "operators" inserted at $x$ say, $\cal{O}(x)$ and if $\phi_i$ are the fields then the quantity one has in mind is written as, $...
- 3,541
2
votes
1
answer
188
views
quadrature domains from circles?
If $h(z)$ is analytic on the disk centered at 0 of radius r, by the Cauchy Residue formula
\[ \int \int_D h(z)\, dx dy = \pi r^2 h(0) \]
The disk is the simplest example of a quadrature domain since ...
- 22.8k
4
votes
3
answers
644
views
Traceless GUE : Four Centered Fermions
The proof of the Wigner Semicircle Law comes from studying the GUE Kernel
$$ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...
- 22.8k
5
votes
3
answers
519
views
Is there a PDE for this phenomenon?
At a point on a surface an incompressible fluid begins to up well at a constant rate and spread across the surface.
Is there a physical law - like the heat equation - that describes the flow?
Will ...
- 439
3
votes
1
answer
332
views
Continuity of a weight on its definition domain in a von Neumann algebra
Let $M$ be a von Neumann algebra and $\varphi$ be a normal weight on it,
and let $A$ be its definition subalgebra. We still denote $\varphi$
the extension to $A$ as a linear positive functional.
It ...
- 357
4
votes
1
answer
293
views
Convergence of the summation 1/p^(1+iy) (over all primes p with y a nonzero real number)
For $z\in\mathbb{C}$ with real part greater than $1$ the sum $$\sum_{p}{\frac{1}{p^z}},$$ where the sum is taken over all primes $p$, converges absolutely. It is also well known that the same sum with ...
- 43
7
votes
1
answer
390
views
What are the relations in the unbounded model of K-homology?
I have posed this question to some experts at my university who would probably know the answer if there were a complete one, so my expectations are limited. It's possible that the question deserves ...
- 29.2k
0
votes
1
answer
328
views
Direct limit of sheaves over paracompact spaces
Let $\mathcal F_i, i\in I$, be a directed family of sheaves of abelian groups on a paracompact Hausdorff space $X$. Let $\mathcal F=\varinjlim F_i$ denote the direct limit sheaf. Is it true that $\...
- 741
6
votes
0
answers
387
views
Local minimum from directional derivatives in the space of convex bodies
I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...
- 5,971