All Questions
10,050 questions
5
votes
1
answer
1k
views
why is it called the diamagnetic inequality?
How is the Diamagnetic inequality born? Why is it call this name?
Diamagnetic inequality： $\big|\nabla|u|(x)\big|\leq \big|(\nabla+iA)u(x)\big|$.
- 53
11
votes
4
answers
2k
views
problems from the scottish book
Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from ...
3
votes
2
answers
236
views
Reconstructing a Lie group Banach representation from the Lie algebra rep. on analytic vectors
Dear all,
I have some difficulties with the following assertion in the book of Kirillov.
Let $G$ be a connected Lie group, and T a given (!) representation of G on a Banach space V.
Let $V^\omega$ ...
- 399
10
votes
2
answers
6k
views
Characterizing the Dual of $W_0^{s,p}$
I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and $...
- 882
10
votes
3
answers
1k
views
Compact subgroups of the unitary group of operators in a hilbert space
Is there a characterization for the compact subgroups of the unitary operators in a Hilbert space, where the unitaries are furnished with the norm topology? What about other topologies?
- 101
3
votes
2
answers
534
views
On exponential formula for $C_0$ semigroups
Let $T(t),t\geq 0$, be a $C_0$-semigroup on a Banach space $X$. If $A$ is the infinitesimal generator of $T(t),t\geq 0$, then
$$T(t)x=\lim_{n\infty}(I-\frac{t}{n}A)^{-n}x$$
for every $x \in X, t\geq ...
9
votes
3
answers
2k
views
2-Wasserstein (optimal transport) and extension to the set of all signed measures
Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
...
- 1,731
1
vote
1
answer
283
views
Cotangent bundle in the category of locally convex spaces
I'm trying to understand the definition of a differential form on $M$ in the context of Fréchet spaces or, more generally, locally convex spaces. The standard procedure defines a k-form as a map $\...
- 5,824
5
votes
1
answer
219
views
Do there exist (almost surely) $C^{\infty}$-smooth Gaussian random fields?
Let $d \ge 1$. Do there exist Gaussian random fields on $\mathbb R^d$ which are (almost surely) $C^{\infty}$-smooth, but which are not analytic?
If so, what are necessary and sufficient conditions ...
- 8,512
11
votes
2
answers
2k
views
Complexifying a real Banach space and its dual
A standard way to define the "complexification" $E_\mathbb{C}$ of a real Banach space $E$ is to define a complex linear structure on $E\times E$ by (1) $(x,y)+(u,v)=(x+u, y+v)$, (2) $(a+ib)(x,y)=(ax-...
- 1,704
5
votes
3
answers
700
views
When is a sequentially closed cone, closed?
The following question I also posed here, but still got no answer. Let $X$ be a locally convex, Hausdorff topological vector space and $C\subseteq X$ a convex cone, which is sequentially closed. What ...
- 215
35
votes
4
answers
6k
views
How are infinite-dimensional manifolds most commonly treated?
I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for ...
- 655
1
vote
1
answer
226
views
How are real-analytic functions encoded in computer algebra?
I would like to know how are encoded the real-analytic functions on the interval by the computers. When I think in a real-analytic function I just think in a composition of the ''typical'' analytic ...
- 105
4
votes
2
answers
360
views
On the descent homomorphsim of Kasparov equivariant KK theory
Hello,
I have recently read about the construction of the descent map in Kasparov KK theory, which, for a group $G$ and two $G$-equivariant $C^*$ algebra $A$ and $B$ send $KK_i^G(A,B)$ to $KK_i(A \...
- 42.4k
1
vote
0
answers
119
views
Boundedness of Riesz transforms.
The Riesz $R_i$ transform on $\mathbb{R}^n$ is defined by
$$ R_if(x)= \int_{\mathbb{R}^n} \frac{t_i-x_i}{\vert x-t \vert^{n+1}}f(t) dt$$
for a Schwartz function $f$ on $\mathbb{R}^n$. Can you please ...
- 583
6
votes
1
answer
754
views
Banach Manifold
Let $M$ and $N$ be closed manifolds. Is it true that
$C^{k}(N,M)$, which is the space of functions $f: N\to M$ such that $f\in C^{k}$, is a $C^{\infty}$ Banach manifold? If so, can you help me to ...
- 225
3
votes
1
answer
200
views
Dense subspaces in primitive ideals of C-star algebras
Let $G$ be a unimodular locally compact group (my main examples are algebraic groups over local fields. Thefore we can assume $G$ is Type I, if necessary). Then there are at least three group algebras ...
- 955
1
vote
0
answers
52
views
Extension of $S_+$ type operators
Let $X$ be a reflexive Banach space and $G\subset X$ a open bounded set. Let $F:\overline{G}\rightarrow X^\star$ be a $S_+$ operator, i.e., if for any sequence $x_n$ in $G$ for which $x_n\...
- 71
7
votes
0
answers
199
views
Central Extension of Continuous Loop Group
For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the ...
- 1,040
1
vote
1
answer
318
views
Integration by parts wrt. a Morse function on its basin of attraction
Let $f:\mathbb R^n\to\mathbb R$ be a Morse function with uniform nondegenerate Hessian at critical points, i.e. for some $\delta>0$
$$
\forall x \in \{\nabla f=0\}\;\forall \xi\in\mathbb R^n: \quad ...
- 53
2
votes
1
answer
247
views
Factorization of bivariate polynomial
Let $q(y, z) = u_1 + u_2y + u_3 z + u_4y^2 + u_5yz + u_6z^2 + u_7y^3 + u_8y^2z + u_9yz^2 +$ $\hspace{2.55cm}u_{10}y^3z + u_{11}y^2z^2 + u_{12}y^3z^2$
Can $q(y, z)$ be factorized as
\begin{...
- 23
3
votes
0
answers
145
views
Growth of inner functions on the disk
Recall that an inner function on the disk $D$ is a bounded analytic function on $D$ having radial limits of modulus one almost everywhere.
There has been many works on the growth of the inner ...
- 769
1
vote
1
answer
651
views
Riesz representation for an infinite-dimensional space
Suppose $X$ is an infinite-dimensional Banach algebra (hence not locally compact).
Does there exist any sort of Riesz representation theorem that says something about elements of $C(X)^*$?
- 29
4
votes
0
answers
454
views
Binomial Expectation of Convex Function
Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate
$$\frac{\partial}{\partial \alpha} \...
- 41
6
votes
0
answers
262
views
Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
- 8,512
1
vote
0
answers
237
views
bivariate polynomial
Hello,
Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex.
If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where $|...
- 9
10
votes
2
answers
804
views
General recipe for building C*-algebras out of combinatorial object
I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out ...
- 493
3
votes
1
answer
292
views
Generator of a generated $C_0$ semigroup
Consider a $C_0$-semigroup $S_t:\mathscr{B(H)} \to \mathscr{B(H)}$ with generator $U$. Now define $P_t:\mathscr{B_1(H)} \to \mathscr{B_1(H)}$ where $P_t(\rho)=S_t\rho S_t^*$. How can I prove $P_t$ ...
- 81
20
votes
3
answers
3k
views
Realizing universal $C^*$-algebras as concrete $C^*$-algebras
How do I in general realize a universal C*-algebra generated by some generators and relation as concrete C*-algebras? For example, I know that universal C*-algebra generated by a single unitary is $C(\...
- 493
1
vote
1
answer
4k
views
Showing a Banach space is reflexive
I need to know if a certain Banach space I stumbled upon is reflexive or not. I need to know what are the state of the art techniques to determine if a Banach space is reflexive or not. For example, ...
2
votes
1
answer
352
views
Cyclic vectors for C* algebras
Let A be a C* algebra of operators on a Hilbert space H. Can it happen that for some x in H the set Ax is dense in H but it is not the whole H?
- 51
3
votes
1
answer
785
views
on an inequality of Brezis-Lieb
In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...
- 3,388
2
votes
1
answer
179
views
Second quantization of partial isometry
If we have a unitary map from Hilbert space $H$ to $H$, we get a unitary map from $e^{H}$ to
$e^{H}$, where $e^{H}$ is the symmetric Fock space of $H$. But if we replace the unitary with partial ...
- 95
16
votes
4
answers
2k
views
Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?
This question arose a few years back when I was an assistant teacher on a course of basic (Lebesgue) measure theory, but I didn't find an answer or anyone able to solve the problem. The setting of the ...
- 588
2
votes
0
answers
272
views
Continuity of multiplicative character
Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ...
- 61
6
votes
2
answers
469
views
Orthonormal basis for $L^2(G/H)$.
Let $G$ be a locally compact group and $H$ be a closed subgroup of $G$. Is there any way to define a reasonable orthonormal basis for $L^2(G/H)$? By "reasonable" I mean elements of the orthonormal ...
user23860
1
vote
0
answers
316
views
"Integration by parts" formula for functionals
We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$
then
$$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$
where the $...
- 29
3
votes
1
answer
2k
views
Norm of differential operator between Sobolev spaces
It is easy to check that the differential operator $\partial^a$ (where $\alpha\in \mathbb{N}_0^n$) is continuous between the Sobolev spaces (with usual norms)
$W^{m,p}(U)\to W^{m-|\alpha|,p}(U)$, ...
- 31
3
votes
0
answers
183
views
Is the construction of ring C*-algebra functorial?
Cunz and Li defined defined C*-algebras for arbitrary rings with of course some condition. One can look at their article (http://arxiv.org/abs/0905.4861). My question: is the construction functorial? ...
- 95
3
votes
2
answers
581
views
Banach lattice subspace of $C([0,1])$ not a sublattice
This is probably easy, but I did not see it in standard texts. Describe a closed subspace $V$ of $C([0,1])$ such that $V$ is a Banach lattice (in the pointwise ordering), but $V$ is not a sublattice ...
- 1,704
3
votes
2
answers
416
views
Stabilization in Banach algebras
In $C^\ast$-algebras we use $K(H)$, the algebra of compact operators on a separable Hilbert space, for stabilization of a $C^\ast$-algebra, i.e. $S(A):=A\otimes K(H)$. Is there any similar ...
user23860
11
votes
1
answer
645
views
Subspaces of $l_p$ and Banach-Mazur distance
This is a question I posted on SE, and I have been advised to post it here.
https://math.stackexchange.com/questions/146427/subspaces-of-l-p-and-banach-mazur-distance
It is well-known that every ...
- 113
9
votes
2
answers
1k
views
Rescaling positive definite matrices to force a unit eigenvector
Hello,
Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones.
I'm hoping to construct a positive, diagonal matrix $W$ such that
$$(W X'X W) \mathbf{1} = \mathbf{1}$$...
- 193
2
votes
1
answer
545
views
Characters separating points on Maximal Torus modulo Weyl group?
Let G be a compact Lie group, for example, SU(n). Let T be its maximal torus. Let W be its Weyl group.
Every finite-dimensional representation of G has a character, which is a function on G, T and T/...
- 585
2
votes
1
answer
386
views
Decomposing bilinear forms in Hilbert spaces
You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...
- 1,211
7
votes
2
answers
2k
views
Uniform bound on the eigenfunctions of the Laplacian
Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...
- 73
0
votes
2
answers
182
views
RFC for definite integral connection to second derivative
Hi,
During my research I found an interesting fact, and I'd like to know if it's interesting for others as well.
Find a function $g(x,t):[0,T]\times[0,T]\rightarrow[0,T]$ such that for any twice ...
- 310
5
votes
1
answer
418
views
Robin-Laplacian in unbounded domains
Let $\Omega\subset \mathbb R^n$ be an open domain and $\tau>0$. Consider the following boundary value problem
$-\Delta v=f $ in $\Omega$, $\partial_\nu v+\tau v=g$ on $\partial\Omega$.
If $\Omega$...
0
votes
1
answer
861
views
Norms agreeing on dense subspace [closed]
Suppose $(B,\|\cdot\|)$ is a Banach space, $V\subset B$ a dense subspace, and $V$ is equipped with a norm $\|\cdot\|_V$ such that $\|x\|_V = \|x\|$ for all $x\in V$.
Is $(B,\|\cdot\|)$ a completion ...
- 143
3
votes
1
answer
354
views
Solvability for constant-coefficient partial differential operators
Let $\mathcal{S}$ denote the space of Schwartz functions on $\mathbb{R}^n$, and $\mathcal{S}'$ the space of tempered distributions. Let $L$ denote a linear, constant-coefficient, partial differential ...
- 111