All Questions
947 questions
4
votes
0
answers
126
views
Relationship between three different definitions of solutions for ODE with irregular coefficient
What is the difference between the notions of
Regular Lagrangian flow
Filippov solution
Caratheodory solution
of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
- 839
4
votes
1
answer
92
views
Approximate a one-form on the disk with nowhere vanishing one-forms satisfying an asymptotic vanishing of some derivatives
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk, and let $g:\mathbb{D}^2 \to \mathbb{R}$ be a non-constant harmonic function (smooth up to the boundary).
Does there exist a sequence of ...
- 6,741
4
votes
0
answers
281
views
Dual space of ${\rm Lip}_0(\mathbb R^d)$
This question comes to me when I read this paper : https://arxiv.org/pdf/1702.06049.pdf
Let ${\rm Lip}_0(\mathbb R^d)$ be the space of Lipschitz functions $F$ on $\mathbb R^d$ with $F(0)=0$. Then is $...
user128095
4
votes
1
answer
352
views
Minimality properties of James' space
I am interested in the following question about James' quasi-reflexive Banach space $\mathcal{J}$:
Does there exists a non-Hilbertian subspace $X$ of $\mathcal{J}$ such that $X$ isomorphically ...
- 1,005
3
votes
0
answers
124
views
Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
- 969
3
votes
2
answers
280
views
Question about the Bessel operator
For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by
\begin{equation*}
L_\...
- 421
3
votes
0
answers
87
views
Instances of c-concavity outside of optimal transport?
Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
- 232
3
votes
2
answers
457
views
Integrality of complex infinite series
Let $(a_n)$ be a (double-sided) sequence of complex numbers satisfying
$$\sum_{\mathbb{Z}}\vert n\vert\,\vert a_n\vert^2<\infty, \tag1$$
$$\sum_{\mathbb{Z}}a_n\bar{a}_{n+k}=\delta_0(k), \qquad \...
- 43.2k
3
votes
0
answers
306
views
Metric analogues of bounded variation
A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if
$$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$
for some finite $V>0$, where the supremum is over all finite partitions
$...
- 6,541
3
votes
1
answer
233
views
A special approximation of the Heaviside function
Is there a $C^m$ approximation $f_\epsilon$ of the Heaviside function such that
$$f_\epsilon(x) = f_1(x/\epsilon) = \begin{cases} 0 & \text{ if } x < 0 \\
1 & \text{ if } x/\epsilon \ge 1
\...
- 131
3
votes
4
answers
2k
views
Suitable references for the the Stone-von Neumann Theorem
Hi all,
I am working on a mathematical physics project now and I need to understand the Stone-von Neumann Theorem properly. Wikipedia says that it is any one of a number of different formulations of ...
- 1,719
3
votes
1
answer
845
views
Moser estimates?
Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
- 31
3
votes
1
answer
476
views
Kernel of the Radon transform
Consider the following generalized version of the Radon transform. Let $X,Y,Z$ be compact smooth manifolds. Let $p\colon Z\to X$, $q\colon Z\to Y$ be smooth maps. Let $m$ be a fixed smooth density (...
- 21.8k
3
votes
1
answer
463
views
"Strongly mutually singular" families of measures, and the set of ergodic measures
Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish].
Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume ...
- 708
3
votes
1
answer
362
views
Solution of the fractional Laplace equation on a ball
What is the expression of the (non $u \equiv 0$) solutions to
\begin{align*}
(-\Delta)^s u &= 0 && x \in B_r(0) \\
u&=0 && x \in \mathbb R^N \setminus B_r(0),
\end{align*}
...
- 99
3
votes
1
answer
746
views
Is inverse Laplace Transform of a power of $s$ a positive function?
It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace ...
- 123
3
votes
1
answer
169
views
Copy of $\ell_\infty$ inside $\ell_\infty(\Gamma)$ containing given subspace
To complete a proof I need to know if the following is true:
Given a non-empty set $\Gamma$ and a separable subspace $Y$ of $\ell_\infty(\Gamma)$, there exists a subspace $A$ of $\ell_\infty(\Gamma)$ ...
- 271
3
votes
1
answer
207
views
Embedding of a Banach space in $\ell_\infty(\Gamma)$ with a subspace embedded in a copy of $\ell_\infty$
In this question (which I think may be interesting in its own) I was asking if we can find a copy of $\ell_\infty$ between a separable subspace $Y$ contained in $\ell_\infty(\Gamma)$ and the whole ...
- 271
3
votes
2
answers
1k
views
Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...
- 3,738
3
votes
1
answer
229
views
Existence of topologically transitive map on Euclidean space
I was reading this post and wondered. Does there exist a topologically transitive (TT) map $f:\mathbb{R}^n\to\mathbb{R}^n$ when $n\geq 2$? I know that post asks for compactness and topological ...
- 5,405
3
votes
1
answer
182
views
How to choose some $h$ so its Fourier transform supported in some set?
Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
- 325
3
votes
1
answer
740
views
Centralizers in C*-algebra
Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, $\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$. Can we say anything about the correspondence between $a$ and $b$?
For ...
- 179
3
votes
0
answers
160
views
Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras
I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...
3
votes
1
answer
322
views
Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
- 159
3
votes
1
answer
420
views
Transform Riesz basis $\{e^{i \lambda_n t}\}_{n\in\mathbb Z}$ to an orthogonal basis
It is well-known that $\{e^{i n t}\}_{n\in\mathbb Z}$ is an orthonormal basis for $L^2(-\pi,\pi)$. A theorem by Kadec (Kadec $1/4$ theorem) studies the perturbed exponential system:
If $\{\lambda_n\...
- 297
3
votes
1
answer
769
views
Strichartz estimates for the heat equation
Consider the heat flow $e^{t\Delta}$, $t>0$, on an Euclidean domain (say $\mathbb{R}^3$). I expect, in analogy with the Strichartz estimates for the Schrodinger equation, that the following ...
- 943
3
votes
1
answer
406
views
Exactness of injective tensor products
For (algebraic) tensor products, it is well-known that the functor $A\otimes_R \cdot:Mod_R\rightarrow Mod_R$ is only (left-) exact when $A$ is a flat $R$-module. In particular, all vector spaces are ...
- 5,405
3
votes
0
answers
233
views
Is it possible to reconstruct the compact space $X$ from the space of measures $M(X)$?
Let $X$ be a compact Hausdorff topological space and $C(X)$ the Banach algebra of continuous functions $u:X\to\mathbb C$ (with the usual $\sup$-norm). It is well-known that the structure of Banach ...
- 7,426
3
votes
1
answer
949
views
ODE continuous dependence on parameters to PDE
I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the ...
- 55
3
votes
2
answers
571
views
Inverted harmonic oscillator
I am looking for the spectrum of th inverted oscillator $H=-\frac{d^2}{dx^2}-x^2$.
Thanks in advance.
- 135
3
votes
1
answer
328
views
A numerical radius inequality
Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{B}(E)$ the algebra of all bounded linear operators from $E$ to $E$.
...
- 1,154
3
votes
1
answer
225
views
Quotient of weak amenable Banach algebras
Let $A$ be a weak amenable Banach algebra and $I$ be a closed (two-sided) ideal of $A$. In general $\frac{A}{I}$ is not weakly amenable. Is there an example of this type of weak amenable Banach ...
- 565
3
votes
1
answer
476
views
Strict topology on the multiplier algebra
Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by
$$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ...
- 175
3
votes
1
answer
428
views
Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?
Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions.
In this case, I ...
- 3,477
3
votes
1
answer
182
views
Perimeter continuity of $BV$ sets on any sequence from $W^{1,1}$
In the book of S.Osher & R.Fedkiw - Level set methods and dynamic implicit surfaces, at page 15, is stated without proof, a formula like that:
$$\text{Per}_{\Omega}(\omega)=\lim_{\varepsilon\to 0} ...
- 1,759
3
votes
1
answer
775
views
Is there some quantitative version of interlacing of eigenvalues of a matrix under rank-1 update?
Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$.
But do we have any quantitative ...
- 1,893
3
votes
1
answer
1k
views
A calculus question related to the nonnegative definite functions
I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that
$$
\int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$...
- 1,649
3
votes
1
answer
327
views
$L^1_{\mu}$ as limit
Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. Does there exist a countable set of finite measures $\{\mu_n\}_{n \in \mathbb{N}}$ on $(X,\Sigma)$ such that $L^1_{\mu}(\Sigma)$ can be ...
- 5,405
3
votes
1
answer
695
views
Continuous embedding between parabolic Sobolev spaces
I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct?
Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider ...
- 503
3
votes
1
answer
274
views
Function square-integrable
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$
where $x_0$ is an ...
3
votes
1
answer
250
views
Characterization of a subset of [0,1] $II$
My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...
- 1,835
3
votes
0
answers
512
views
Domain of the adjoint of the Laplacian
Given a domain $\Omega \subset\Bbb R^n$, consider the set $D := \{u \in L^2(\Omega)| \Delta u \in L^2(\Omega)\}$, where $-\Delta$ is the Laplacian.
I think this is the domain of the adjoint of $-\...
- 31
3
votes
2
answers
869
views
How do functions operate in a Sobolev space $H^{s}$?
Let $s>\frac{1}{2};$ and define a Sobolev space as follows:
$$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$
Fact: Let $m$ ...
- 1,051
3
votes
1
answer
233
views
Extending linear isometries from subspaces of $\ell_p^n$
Take $p\in (1,\infty)\setminus \{2\}$. Let $X$ be a subspace of $\ell_p^n$ and let $U\colon X\to \ell_p^m$ ($m\geqslant n$) be a linear isometry. Is it possible to extend $U$ to a (non-surjective) ...
- 331
3
votes
0
answers
411
views
Bounded functions dense in Sobolev Spaces
Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...
- 9,853
3
votes
1
answer
788
views
Higher order functional derivatives
Let $E, F$ be Banach spaces. A continuous bilinear functional ${\langle \cdot\,, \cdot \rangle }: E \times F \to \mathbb{R}$ is called $E$-non-degenerate if $\langle x,y\rangle = 0$ for all $y \in F$ ...
- 3,535
3
votes
1
answer
299
views
Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)
Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality
\begin{align*}
|f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n.
\end{align*}
For $n \ge 2$, can we ...
- 1,731
3
votes
2
answers
849
views
Integral of fractional Laplacian is zero
Is it true that $$\int_{\mathbb{R}^N}(-\Delta)^su(x) dx = 0,$$
where $(-\Delta)^s$ is the fractional Laplacian?
user60665
3
votes
0
answers
160
views
Elements of vector-valued $L^1$-spaces
Let $E$ be a complete locally convex space and let $(X, \Sigma, \mu)$ be a measure space where $\mu$ is a Radon measure. Then the space $L^{1}(X,E)$ is defined as a the completion of the space $S(X,E)$...
- 799
3
votes
2
answers
207
views
Commutative direct summands of C*-algebras
I have a question about commutative direct summands of $C$*-algebras.
Let $A$ be a $C$*-algebra (with unit) and suppose that its bidual $A^{**}$ has a commutative direct summand, that is, $A^{**}=B\...
- 633