All Questions
1,003 questions
2
votes
1
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586
views
Inverse Problem for Pullback
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) ...
- 61
2
votes
1
answer
755
views
Existence of a solution to an infinite dimensional Stratonovich SDE
Let
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace
$U_0:=Q^{1/2}U$
$(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ ...
- 167
2
votes
1
answer
452
views
Can we choose an element from a class?
Let $H$ be a complex Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$.
Set $H_0:=H_1\cap H_2\cap...\cap H_n$ and let
$P_i$ be the orthogonal projection onto $H_i$, $i=0,1,2,...,n$.
I study ...
- 935
2
votes
0
answers
96
views
Is the Leray projection continuous with respect to the Frechet topology of smooth periodic vector fields in $3$ dimensions?
Let $\mathbb{T}^3:=(\mathbb{R}/\mathbb{Z})^3$ be the $3$-torus and $C^\infty(\mathbb{T}^3,\mathbb{R}^3)$ be the Frechet space of smooth periodic vector fields on $\mathbb{T}^3$.
By Helmholtz ...
- 3,477
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
2
votes
2
answers
243
views
Given a specific function $f$, how to compute the left-inverse of $f$ in the sense of $\approx$?
For a non-negative function $\varphi$ defined on $[0,\infty)$, the left-inverse $\varphi^{-1}$ of $\varphi$ is defined by setting, $\forall t\geq 0$,
$$\varphi^{-1}(t):=\inf\{u\geq0:\varphi(u)\geq t\}....
- 51
2
votes
0
answers
73
views
Alberti rank-one theorem and irregular jump discontinuities
Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...
- 839
2
votes
2
answers
336
views
Metrization of a topological vector space
Let $C(\mathbb R^d)$ be the space of continuous functions on $\mathbb R^d$, and $C_{lip}(\mathbb R^d)\subset C(\mathbb R^d)$ be the subspace of Lipschitz functions. We endow $C_{lip}(\mathbb R^d)$ ...
user128095
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
2
votes
2
answers
485
views
Dual space of the completion of the space of Lipschitz functions
This question is a continuation of this post : Metrization of a topological vector space
Let $C_{lip}(\mathbb R^d)$ be the space of Lipschitz functions on $\mathbb R^d$. We endow $C_{lip}(\mathbb R^...
user128095
2
votes
2
answers
447
views
Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$
Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
- 5,405
2
votes
0
answers
216
views
Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?
Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a ...
- 369
2
votes
0
answers
250
views
Dense property of intersection of Sobolev space
I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the ...
- 197
2
votes
1
answer
996
views
Derivative and Jacobian determinant of solution of ODE [closed]
Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
...
user124345
2
votes
1
answer
233
views
${\rm II}_1$-factors with finite commutant: $\mathcal{A} \cap \mathcal{B} = \mathbb{C} \Rightarrow \mathcal{A}' \cap \mathcal{B}'$ hyperfinite?
Let $\mathcal{A} , \mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}', \mathcal{B}' $ are also a ${\rm II}_1$-factors.
Question: $\mathcal{A} \cap \mathcal{B} = \mathbb{...
1
vote
0
answers
75
views
Derivation of the vortex filament equation from Euler equation
How can the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
be derived from the Euler equation
$$\partial_t \...
- 277
1
vote
2
answers
234
views
Find $\inf_{P_{X_1,X_2}}P_{X_1,X_2}(\|X_1-X_2\| > 2\alpha)$ , where $\alpha > 0$ and inf is over couplings
Let $\mathcal X$ be a seperable Banach space with norm $\|\cdot\|$, and let $X_1$ and $X_2$ be random vectors on $\mathcal X$ with finite means.
Question. Given $\alpha > 0$, what is value of, ...
- 6,853
1
vote
1
answer
654
views
Properties of the trace term in the Itō formula
Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where
$U,H$ are separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ is nonnegative ...
- 167
1
vote
2
answers
535
views
Non-closed range space of Laplace operators?
Set $ -\Delta: H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3) $. Then $ \mathcal{R}(-\Delta) $ is non-closed?
Sorry if this question is trivial. I am not familiar with theory of ...
- 269
1
vote
1
answer
154
views
BV function with absolutely continuous divergence
Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$.
Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and ...
- 839
1
vote
1
answer
247
views
Elliptic interface problem without conditions on the interface
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...
user60665
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 463
1
vote
1
answer
148
views
How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?
Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$
How can we calculate the generalized gradient $\partial_Cf(x)$ of ...
- 167
1
vote
3
answers
345
views
Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mid \pi: X \to A\right\}$ closed?
Let $X$ be a compact subset of $\mathbb R^n$ and let $A$ be a compact subset of $\mathbb R^k$. Let $P$ be a probability distribution on $X$ and $v$ be a $P$-measurable function from $X$ to $\mathbb R^{...
- 6,853
0
votes
2
answers
494
views
Semifinite measure and spectral theorem
Let $H$ be a complex Hilbert space (not necessary separable).
Spectral Theorem: Let $A_1$ and $A_2$ be two commuting normal operators, then there exists a measure space $(X,\mathcal{E},\mu)$,
two ...
- 1,154
0
votes
2
answers
403
views
Application of uniform boundedness principle
$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
user128095
0
votes
2
answers
125
views
Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 835
0
votes
2
answers
1k
views
Question on Hartogs's Extension Theorem
Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)?
For Hartogs's Extension Theorem see here:
http://en.wikipedia.org/wiki/...
- 53
94
votes
1
answer
11k
views
The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
- 23.5k
71
votes
2
answers
6k
views
Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used?
I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, ...
- 26.8k
65
votes
9
answers
12k
views
Polish spaces in probability
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
- 651
59
votes
4
answers
15k
views
Group theory in machine learning
I'm a Machine Learning researcher who would like to research applications of group theory in ML.
There is a term "Partially Observed Groups" in machine learning theory which has been ...
- 607
59
votes
9
answers
10k
views
Motivation for and history of pseudo-differential operators
Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...
- 5,327
45
votes
7
answers
16k
views
What is an intuitive view of adjoints? (version 2: functional analysis)
After realising that I don't have an intuitive understanding of adjoint functors, I then realised that I don't have an intuitive understanding of adjoint linear transformations!
Again, I can use 'em, ...
- 26.8k
43
votes
0
answers
820
views
A kaleidoscopic coloring of the plane
Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
- 41.8k
37
votes
2
answers
2k
views
Moving one family of commuting self-adjoint operators to another without losing commutativity on the way
This is actually not a question of mine, so I'll be short on motivation and say nothing beyond that if this were true, a few fancy harmonic analysis techniques that a colleague of mine used in proving ...
- 61.9k
34
votes
2
answers
3k
views
Can we recover a von Neumann algebra from its predual?
By definition, a von Neumann algebra is a C*‑algebra A
that admits a predual, i.e., a Banach space Z such that
Z* is isomorphic to the underlying Banach space of A.
(We require that isomorphisms in ...
- 37.8k
33
votes
1
answer
2k
views
Stone-Weierstrass theorem for holomorphic functions?
The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called
Naсhbin's theorem: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth ...
- 7,426
33
votes
1
answer
2k
views
For which maps $S^1\to S^1$ is the winding number defined?
There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to S^...
- 43.2k
31
votes
3
answers
5k
views
When is an integral transform trace class?
Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator
$$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$
the operator $K$ is Hilbert ...
- 11.2k
31
votes
1
answer
2k
views
Topology on space of hyperfunctions
This is a reference request, coming from someone with little knowledge of hyperfunctions:
Which methods have been used to endow the space of hyperfunctions $\mathcal B(\mathbb R)$ with something like ...
- 21.3k
30
votes
3
answers
3k
views
Surjectivity of operators on $\ell^\infty$
Can anyone give me an example of an bounded and linear operator $T:\ell^\infty\to \ell^\infty$ (the space of bounded sequences with the usual sup-norm), such that T has dense range, but is not ...
- 301
29
votes
1
answer
4k
views
Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...
- 25.5k
29
votes
6
answers
9k
views
Nonseparable Hilbert spaces
Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
- 9,330
28
votes
2
answers
1k
views
Can an operator have Exp(z) as its characteristic "polynomial"?
Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define
$$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$
the ...
- 5,898
27
votes
2
answers
5k
views
What can be said about the Fourier transforms of characteristic functions?
What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular,
What properties are common to ...
- 2,719
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
27
votes
1
answer
1k
views
The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
- 683
26
votes
3
answers
3k
views
Sum of Gaussian pdfs
I learned from a colleague that if one sums translates of the Gaussian density $f(x)=(2\pi)^{-1/2}e^{-x^2/2}$ translated by the integers (i.e. one considers $F(x)=\sum_{n\in\mathbb Z}f(x+n)$), the ...
- 23.2k