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1 vote
0 answers
146 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
0 votes
1 answer
239 views

Are ALL linear functionals on $C[0,1]$ generated by measures? [closed]

Consider derivative of the convolution of a given function $f(\cdot)$ with a fixed $C^\infty$ function $s(\cdot)$, evaluated say at $1/2$. Is there a measure which generates the functional so defined?
2 votes
0 answers
63 views

Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$

In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
4 votes
0 answers
101 views

There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions

Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$. Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
1 vote
0 answers
48 views

What makes the generalized projection different than metric on a Banach space?

I have came across the notion of generalized projection in Banach spaces, introduced by Ya. Alber and has seen many iterative algorithms being solved by using this projection. It helps in finding the ...
2 votes
1 answer
61 views

$k$-subset with minimal Hausdorff distance to the whole set

Let $(\mathcal{M}, d)$ be a metric space. Let $k \in \mathbb{N}$. Let $[\mathcal{M}]^k$ be the set of $k$-subsets of $\mathcal{M}$. Consider the following problem: $$ \operatorname*{argmin}_{\mathcal{...
2 votes
0 answers
155 views

Inscribed square and convexity

Let $b(X)$ be the boundary of any $X$ subset of the plane. Does there exist $A,B$ convex compact sets of the plane, such that $C:=A\setminus B$ is simply connected and not empty, and such that ...
3 votes
2 answers
354 views

General version of Weyl's lemma

The classical Weyl's lemma say, suppose $u \in L^1_{loc}(\Omega­)$ satisfies $$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$ then $u$ is harmonic in $\Omega.$ What I want ...
0 votes
0 answers
53 views

Spectral theory of compact operator for quasi-Banach spaces

Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
8 votes
1 answer
422 views

Why $(\mathrm{Lip}([0,1]^2))^*$ is finitely representable in 1-Wasserstein space over the plane?

In "Snowflake universality of Wasserstein spaces"" by Alexandr Andoni, Assaf Naor, and Ofer Neiman, they have the following notation: For a metric space X they write $\mathcal{P}_1(X)$ ...
0 votes
0 answers
77 views

Nice formula for powers of modified Bessel function

Let $K_\nu(z)$ be the modified Bessel function of second kind. I am looking the geometric series $$1+aK_v+(aK_v)^2+(aK_v)^3...$$ I know there are formula for product of two such functions. I would ...
3 votes
2 answers
247 views

Morse approximation with bounded number of critical points

Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
11 votes
1 answer
652 views

How to correctly state Cauchy's rigidity theorem?

Cauchy's rigidity theorem is often stated briefly as Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent. As a more formal generalization to general ...
3 votes
0 answers
158 views

Gowers' dichotomy for quotients

Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable. A ...
1 vote
0 answers
84 views

Does sets of positive capacity rule out constant functions?

Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by \begin{align*} \text{Cap}_{p}(K, U) := \inf \left\{ \int_U |\...
1 vote
1 answer
142 views

Operator norm of some type of discrete Fourier matrix

Let $N$ be a natural number and let $w$ be a complex number. We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows, $$ a_{k,l}=\begin{cases}1 & l=k+1\\ w &...
3 votes
1 answer
128 views

Comparing two different principles of premeasure-to-measure extension

It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for ...
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
0 votes
1 answer
170 views

Is the evolution family self-adjoint?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading Roland Schnaubelt's survey ...
1 vote
1 answer
59 views

Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains

On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$. I'm interested ...
2 votes
0 answers
52 views

On distributions and kernels

Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
3 votes
0 answers
155 views

Colimits in commutative Banach algebras?

Let $K$ be a complete non-Archimedean field. It is known that the category $\mathrm{Ban}_K$ of $K$-Banach spaces with bounded linear maps does not have infinite colimits. The usual argument for $\...
5 votes
1 answer
537 views

Twin circles in a quadrilateral

The circumcenters of the four triangles of a complete quadrilateral along with the two points of completion form two congruent circles (in black). Surely this must've been done before - what's the ...
1 vote
0 answers
67 views

Quasi-geodesics in Alexandrov spaces

I am trying to understand the notion of quasi-geodesic in Alexandrov space with curvature bounded below following the Perelman-Petrunin paper. I have two questions: Is it true that the shortest ...
3 votes
1 answer
158 views

Upper and lower bounds for a Rademacher-type expectation

Suppose that $\varepsilon_i$ are independent Rademacher random variables (that is, $ \mathbb{P}(\varepsilon_i=-1) = \mathbb{P}(\varepsilon_i=1) =1/2 $. Fix an $a\in\mathbb{R}^n$ and define the random ...
5 votes
1 answer
206 views

Compactness in trace class operators space

Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$. Are there easy ...
0 votes
0 answers
55 views

Compactness and Leray-Schauder degree

What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
0 votes
1 answer
188 views

Does the second Bourgain–Delbaen space belong to C_p?

The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$. An operator $T:X\to Y$ ...
3 votes
0 answers
130 views

A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)

As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
0 votes
1 answer
65 views

Strict positive definite function gradient tuple

I have a (Gaussian) random function (aka "stochastic process" or "random field") $(f(t))_{t\in \mathbb{R}^d}$. I now want to consider the vector valued random function $g=(f, \...
25 votes
2 answers
2k views

Functional approach vs jet approach to Lagrangian field theory

Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...
0 votes
0 answers
113 views

Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?

Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$. We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if $$\forall_{u \in A(\...
2 votes
1 answer
383 views

Geometry in $\mathbb{R}^n$: angle between projections of a rectangle

Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$. Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$. For ...
3 votes
1 answer
189 views

Ribe's Theorem: finitely representability between two uniformly homeomorphic Banach spaces

An infinite-dimensional Banach space $X$ is said to be crudely finitely representable (with constant $\lambda$) in an infinite-dimensional Banach space $Y$ if there is a constant $\lambda>1$ such ...
2 votes
0 answers
70 views

Is the hypothesis "uniformly equivalent" needed?

I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows: Assume $\mathscr{H}$ is a Hilbert space of ...
5 votes
1 answer
630 views

Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution

Examples of infinite dimensional involutions Edit 2/25/23, as suggested by YCOR below: (Start) The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
1 vote
0 answers
111 views

References on the partial trace

For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows : $$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(...
0 votes
1 answer
119 views

Nonstationary phase method for oscillatory integral

I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth. The stationary phase method says that if $t_0\in [a,b]$ is such that ...
2 votes
1 answer
168 views

Ratio of inscribed/circumscribed ellipsoids: geometrical proof?

Let $K$ be a convex subset of ${\mathbb R}^n$, with non-void interior. The Löwner-John theorem states that there are a minimal volume ellipsoid $\cal E$ containing $K$, a maximal one $\cal F$ ...
3 votes
0 answers
95 views

Commutator of $A\otimes I$ and $I \otimes B$ vanishes?

Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
6 votes
0 answers
213 views

Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck

In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
10 votes
1 answer
2k views

Chain rule for distributional derivative

Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$). Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
0 votes
1 answer
231 views

Divide angles by coefficients relate to Fibonacci sequence

In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle $(...
2 votes
0 answers
43 views

Distributions and time-kernels

Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
-1 votes
0 answers
53 views

convergence of convolution in Bochner space

I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$ let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and ...
-1 votes
0 answers
94 views

Why define Schwartz by supremum rather than limit?

The Schwartz space is defined as the set of all indefinitely differentiable functions such that the supremum over the free variable of any (order) derivative times any (order) power is finite. However,...
1 vote
0 answers
38 views

Metric entropy of an ellipsoid

Let $B^d_2$ denote the unit ball of $\ell_2^d$ and let $T$ be an invertible linear map. Consider the function $$ H(T) := \log M(TB_2^d, B_2^d), $$ which is the packing entropy for $TB_2^d$ by $B_2^d$....
9 votes
5 answers
2k views

Convexity of distance-to-boundary function

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded convex domain. Denote $d_{\Omega}:\Omega\rightarrow\mathbb{R}$ the distance-to-boundary function, that is, $$ d_{\Omega}\left(x\right):=\inf\left\...
11 votes
7 answers
1k views

What are some interesting ways of making new metrics out of old metrics?

If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics. If $d_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum_{i=1}^n{d_i^2(x,y)}}$ Are ...
1 vote
1 answer
90 views

Sobolev inequality with weight in the case $1<n\leq p$

Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...