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16 votes
2 answers
1k views

New series for $\pi$ from string theory

This is a direct followup to the post Possible new series for $\pi$ by Timothy Chow and its numerous answers and comments. Using another formula in the same string theory paper by Saha and Sinha one ...
Henri Cohen's user avatar
  • 13.1k
0 votes
0 answers
60 views

On an oscillatory property of the Riemann Xi-function

In their paper "The Integral of the Riemann Xi-Function", Lagarias and Montague mention Wintner's 1947 proof that $$ \Xi^{(-1)}(t) > 0 \quad \text{when} \quad t > 0. $$ This result ...
Tokita Ohma's user avatar
5 votes
2 answers
258 views

Boundary value of Sobolev space

Let $D$ be a regular domain in $\mathbb R^2$. Suppose that $u \in H_0^1(D) \cap C(D)$. Does this imply $u \in C(\overline D)$ and $u|_{\partial D} = 0$?
Focus's user avatar
  • 177
5 votes
1 answer
173 views

Properties of Poisson Integral and Traces of functions in Sobolev space $W^{1,2} (\mathbb D)$

Let $W^{1,2}(\mathbb D)$ be the complex valued Sobolev space on $\mathbb D$ where $\mathbb D $ is the open unit disk of the complex plane. By definition, $W^{1,2} (\mathbb D)$ is the set of all ...
ash's user avatar
  • 151
0 votes
0 answers
118 views

Find the maximum of an expression under the logconcave assumption

Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
Ningjingzhiyuan's user avatar
1 vote
0 answers
30 views

Generalization of subadditivity analogous to quasiconvexity, and variants

I am curious if there are natural generalizations of subadditivity which have been studied in the past or have been stated in the literature? I (and people that I have talked to) have not had much ...
Alex Rutar's user avatar
2 votes
1 answer
118 views

Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius

This is a follow up from this question. I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
Ryan Hendricks's user avatar
5 votes
2 answers
364 views

Euler-Lagrange equations for minimizer of energy with indicator function

I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for $$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
BBB's user avatar
  • 93
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 &...
ABB's user avatar
  • 4,058
1 vote
1 answer
76 views

Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius

I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
Ryan Hendricks's user avatar
1 vote
1 answer
115 views

Block-diagonal embedding of $U(n)$ into $U(mn)$

What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding $$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$ for $\alpha$ appearing $m$ times? For ...
Alonso Perez-Lona's user avatar
2 votes
0 answers
94 views

Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces

Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
Bogdan's user avatar
  • 1,759
7 votes
1 answer
253 views

Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?

In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed. Does a Banach algebra ...
Qingping Zeng's user avatar
8 votes
0 answers
177 views

Understanding spaces of negative regularity

I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here. Let $C^k(\mathbb{R}^n$) be the ...
CBBAM's user avatar
  • 721
0 votes
0 answers
85 views

Measurable selection for the mean value theorem

When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that: Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
Bogdan's user avatar
  • 1,759
2 votes
0 answers
84 views

Question about the Nemytsky operator on $L^p$ space

Let $\Omega\subset\mathbb{R}^N$ be a bounded open set, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function, i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is ...
Bogdan's user avatar
  • 1,759
3 votes
0 answers
147 views

Embeddings of Bochner-Sobolev spaces with second time derivative

NOTE: I also asked this question here in MSE. In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
MathsGoose's user avatar
0 votes
1 answer
123 views

Proving a Fourier transform inequality for functions with mixed variable bounded support

I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide. Let $\gamma\...
Julian Bejarano's user avatar
1 vote
0 answers
175 views

Solution of recurrence relation with summation

I have the following recurrence relation: $$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
Cardstdani's user avatar
-2 votes
2 answers
322 views

Is there a term for a countour integral that disregards direction?

Is there a name for integration of the form $\oint_\gamma f(z) |dz|$? In other words, the integral that only takes into account the length of the contour and the values of the function but not the ...
Anixx's user avatar
  • 10.1k
0 votes
0 answers
79 views

Is the Bures metric equivalent to the Euclidean one?

Let $K=\mathbb R$ (reall numbers) or $K=\mathbb C$ (complex numbers). Define $\mathcal M_n$ to be the space of $n\times n$ matrices $A=(a_{i,j})_{1\le i,j\le n}$, with $a_{i,j}\in K$. Let $\|\cdot\|$ ...
GJC20's user avatar
  • 1,334
1 vote
2 answers
156 views

Numerical evaluation of monomial divided differences

Suppose $f(x)=x^{n+1}$ for some $n\in\mathbb{N}$, and define the divided difference $$f[a,b]=\frac{a^{n+1}-b^{n+1}}{a-b}.$$ I am wondering about the best way to numerically evaluate $f[a,b]$ to high ...
Stephen Berg's user avatar
3 votes
1 answer
227 views

Algebraic and continuous duals of an inverse limit of finite dimensional vector spaces

I have been trying to understand the following section of a paper "Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique" by Gabriel Dospinescu and Arthur-César Le ...
Tom Adams's user avatar
  • 117
5 votes
1 answer
220 views

How big is the class of all closed range bounded linear operator?

Let $X$ and $Y$ be Banach spaces and let $CR(X,Y)$ denote the set $B(X,Y)$ of all bounded linear maps from $X$ to $Y$ with $T(X)$ closed in $Y$. Certainly $CR(X,Y)$ is not open in $B(X,Y)$ as given ...
Anupam's user avatar
  • 585
2 votes
1 answer
236 views

A sensible topology on the space of continuous linear maps between Fréchet spaces

Let $V_1$ and $V_2$ be Fréchet spaces. Let $\{ \lVert \cdot \rVert_{1,n} \}_{n \in \mathbb{N}}$ be a family of seminorms for $V_1$ and similarly $\{ \lVert \cdot \rVert_{2,n} \}_{n \in \mathbb{N}}$ ...
Isaac's user avatar
  • 3,477
2 votes
0 answers
60 views

Semigroup property in SPDEs

In fact, we know that a bounded linear operators on a Banach space $X$ satisfies the semigroup property, i.e. $$S(t+s)=S(t)S(s), \text{for every}\ t,s\geq 0.$$ However, in various literatures, I ...
Y. Li's user avatar
  • 57
1 vote
0 answers
98 views

$(\lambda I-A)^{-1}-(\lambda I-B)^{-1}$ compact implies $\sigma_\text{ess}(A)=\sigma_\text{ess}(B)$

Suppose $H$ is a Hilbert space and $A$, $B$ are two adjoint operators on it (not necessarily bounded), satisfying $D(A)=D(B)$. Question: If $\exists \lambda\in \rho(A)\cap\rho(B)$ such that $(\lambda ...
Richard's user avatar
  • 775
1 vote
1 answer
170 views

Mean of probability distribution

I have a probability distribution defined by the following density function: $f(k,j,n,m)=\frac{(m n)! \mathcal{S}_k^{(j)}}{(m n)^k (m n-j)!}$ (With $\mathcal{S}_k^{(j)}$ being the Stirling number of ...
Cardstdani's user avatar
1 vote
1 answer
132 views

Deriving a specific bound for functions in Hardy Space

Reading some article a while ago I read the following: (here $H^2$ represents the Hardy space) Let $f\in H^2$ be such that $f(0)=1$, and let $0<\lvert\lambda\rvert<1$, then $$\lVert f(\lambda z)\...
Tomas smith Smith's user avatar
1 vote
0 answers
66 views

The derivative of semigroup in the weak sense imply strong sense

Suppose $X$ is a Banach space, and $T(t)$ $t\ge0$ is a strongly continuous semigroup with generator $A$. Assume $\frac{T(t)-I}{t}x$ weakly converges to $y\in X$ when $t\to 0$, then I need to prove $x\...
Richard's user avatar
  • 775
7 votes
1 answer
184 views

Functional calculus on the Schwartz space instead of $L^2$?

As far as I know, functional calculus is typically carried out on Hilbert spaces with (possibly unbounded) self-adjoint operators. However, I wonder if there is a way to do it on the space of test ...
Isaac's user avatar
  • 3,477
1 vote
0 answers
63 views

Is there any other norms besides cut norm defined on graphon?

Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions $W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
tom jerry's user avatar
  • 349
5 votes
2 answers
432 views

Does closedness of the image of unit sphere imply the closed range of the operator

Let $X$ and $Y$ be Banach spaces and let $T:X\to Y$ be a bounded linear operator such that $T(S_X)$ is closed in $Y$. Does it imply that $T(X)$ is closed? Any hint is appreciated.
Anupam's user avatar
  • 585
0 votes
0 answers
66 views

convolution of the fundamental solution with the homogeneous solution

I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero? Let $U$ and $E$ ...
Alucard-o Ming's user avatar
-1 votes
2 answers
251 views

$p$-norm of random variables and weighted $L^p$ space resemblance

I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
Mark Ren's user avatar
7 votes
0 answers
131 views

Approximation of a continuous curve on commuting matrices

I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that $[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$....
Ayman Moussa's user avatar
  • 3,425
2 votes
0 answers
96 views

Isometric Schröder-Bernstein theorem for injective Banach spaces?

It's known that every injective Banach space is of the form $C(M)$ where $M$ is a compact, Hausdorff, extremally disconnected topological space. Let $X$, $Y$ be two injective Banach spaces such that, ...
Onur Oktay's user avatar
  • 2,605
6 votes
0 answers
207 views

Partial fraction expansions of meromorphic functions

Sorry if this question (inspired by the recent flurry of activity around a "new" formula for $\pi$) is too naive. Imitating what one does with Hadamard products, one can try to do the same ...
Henri Cohen's user avatar
  • 13.1k
2 votes
0 answers
102 views

Existence of unique-up-to-shift solution of a Volterra equation

Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\...
e.lipnowski's user avatar
2 votes
0 answers
71 views

How to naturally define an output space with certain properties

Consider the following regression problem $v=A(u) + \varepsilon$ for some operator $A:\mathcal{U} \rightarrow \mathcal{V}$ and some function spaces $\mathcal{U},\mathcal{V}$, mapping from $\mathcal{X}$...
emma bernd's user avatar
7 votes
2 answers
350 views

Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?

This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
David Gao's user avatar
  • 2,830
1 vote
1 answer
87 views

Convergence in $H^{-2}$ of $L^2$-functions with limit in $L^2$

Assume a sequence $f_n$ in $L^2(\mathbb{R}^d)$ converges in $H^{-2}$ (w.r.t. its norm topology) to a limit $f \in L^2(\mathbb{R}^d)$. In this case, can one improve the convergence, for instance to ...
PDEprobabilist's user avatar
5 votes
2 answers
517 views

Functions whose product with every $L^1$ function is $L^1$

Let $\mu$ be a probability measure and $f$ a measurable function whose product with any integrable function is integrable: $$ \int|g|\,{\rm{d}}\mu<\infty\implies \int|fg|\,{\rm{d}}\mu<\infty. $$ ...
KhashF's user avatar
  • 3,599
2 votes
0 answers
66 views

interchange of integrals and semigroup without the semigroup being an integral operator

In Cazenave's book: BREZIS, HAIM.; CAZENAVE, T. Nonlinear evolution equations. IM-UFRJ, Rio, v. 1, p. 994, 1994. The following corollary appears The formula (1.5.2) is Duhamel formula: $$u(t) = T(t)u(...
Ilovemath's user avatar
  • 677
2 votes
1 answer
211 views

Hölder continuity in time of heat semigroup for regular initial distribution

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e., $$ p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
Akira's user avatar
  • 835
3 votes
0 answers
104 views

Comparing unitaries which are perturbatively close

Let $\mathcal{H}$ be a Hilbert space and let $H_0$ and $H_1$ be two Hermitian operators on $\mathcal{H}$. Thinking of $H_1$ as a perturbation of $H_0$, the Duhamel formula allows us to write $e^{-...
felipeh's user avatar
  • 452
5 votes
1 answer
483 views

Can you always extend an isometry of a subset of a Hilbert Space to the whole space?

I remember that I read somewhere that the following theorem is true: Let $A\subseteq H$ be a subset of a real Hilbert space $H$ and let $f : A \to A$ be a distance-preserving bijection, i.e. a ...
Cosine's user avatar
  • 609
3 votes
1 answer
263 views

Hölder continuity in time of heat semigroup

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that $$ \|\ell\|...
Akira's user avatar
  • 835
2 votes
0 answers
137 views

Why a function induced by the infimum of the arclength of curves is Lipschitz?

Recently I have read a paper "Weighted Trudinger-type Inequalities" written by Stephen M. Buckley and Julann O'Shea and published by Indiana University Mathematics Journal in 1999, MR1722194,...
Javier's user avatar
  • 69
1 vote
0 answers
105 views

Can a uniform weak null set of $c_0$ be uniformly embedded into a Hilbert space?

Remark that a uniform weak null set $A$ of $c_0$ satisfied that for any $\epsilon>0$ there exists a positive number $N(\epsilon)$ such that for every $f$ in $B(\ell_1)$, the unit ball of the ...
yu tianfeng's user avatar

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