All Questions
1,778 questions
8
votes
3
answers
2k
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References request: constructive quantum field theory
I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman ...
- 663
8
votes
3
answers
3k
views
The mean of points on a unit n-sphere $S^n$
A unit n-sphere is defined as $$\mathcal{S}^n = \{\mathbf{p} \in \mathbb{R}^{n+1}: \|\mathbf{p}\| = 1\}$$
The distance between two points $\mathbf{p}$, $\mathbf{q}$ on $\mathcal{S}^n$ is the great-...
- 147
8
votes
1
answer
355
views
Proving a certain $ C^{*} $-algebraic inequality
Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality
$$
|\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} \|...
- 1,396
8
votes
1
answer
1k
views
Conditional law as a random measure and convergence of random measures
I'm looking for a reference book or article for the following two facts. In both statements, a Polish space $E$ and an ambient probability space $(\Omega, {\cal A}, \Pr)$ are given, and I consider ...
- 2,560
8
votes
1
answer
455
views
Closure of $L(\ell^2,\ell^2)$ in $L(\ell^2,\ell^\infty)$
Let $\ell^2$, $\ell^\infty$ denote the usual sequence spaces and let $L(\ell^2,\ell^2)$ the Banach space of bounded linear operators from $\ell^2$ to $\ell^2$ as well as $L(\ell^2,\ell^\infty)$ the ...
- 261
8
votes
2
answers
3k
views
$L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$
It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in ...
- 176
8
votes
2
answers
1k
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Does Multiplicative Version of Azuma's Inequality Hold?
It is known that there are multiplicative version concentration inequalities for
sums of independent random variables. For example, the following
multiplicative version Chernoff bound.
Chernoff bound:...
- 175
8
votes
1
answer
726
views
continuity of the Boltzmann entropy in the Wasserstein metric
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 5,380
8
votes
3
answers
1k
views
Dual Banach space of $B(X,Y)$ when $X$ is finite dimensional
Denote $B(X,Y)$ the Banach space of bounded operators between Banach spaces $X$ and $Y$.
When $X$ and $Y$ are both finite dimensional, it follows from the formula
$$\|u\|_{B(X,Y)} = \sup_{\|x\|_X <...
- 9,486
8
votes
2
answers
577
views
Ultracoproducts and Cartesian products
Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...
- 1,786
8
votes
1
answer
5k
views
integration by parts for the fractional Laplacian
Is there an integration by parts formula for fractional laplacians in $L^p(\mathbb{R}^N)$, something like
$$
s\in(0,1),\qquad\int\limits_{\mathbb{R}^N}f[(-\Delta)^sg] =\int\limits_{\mathbb{R}^...
- 5,380
8
votes
2
answers
785
views
Is taking the product of signed measures weakly continuous?
For a Polish space $X$, let $C_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped ...
- 29.8k
8
votes
0
answers
6k
views
Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
- 8,512
8
votes
1
answer
434
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Self-dual finite-dimensional complex normed spaces
Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space?
Remarks: There are easy counterexamples in the real case, and in ...
- 11.4k
8
votes
1
answer
421
views
$C^k$ one-parameter family of metrics
Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
- 1,407
7
votes
2
answers
605
views
Gaussian and the convex hull of moment curves
Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...
- 1,127
7
votes
1
answer
861
views
Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold
I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
- 159
7
votes
1
answer
3k
views
The question about Kolmogorov tightness criterion
We know about Kolmogorov Criterion for the tightness of a stochastic process $X_n(t)$
1.The sequence $(X_{n}(0))_{n\geq0}$ is tight.
2.There exist constants $\gamma\geq0$,$\alpha>1$, $K>0$ and ...
- 71
7
votes
5
answers
682
views
Bound on sum of complex summands involving binomial coefficients
I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have $|x|...
- 93
7
votes
0
answers
927
views
What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?
I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi \...
- 1,051
7
votes
1
answer
222
views
Bound on queries to a tree with unusual probabilities
Consider a tree $\mathcal{T}(r) = (V,E)$ rooted at $r \in V$. Let $\kappa_r: V \longrightarrow [0,1]$ such that $\sum_{v \in V} \kappa_r(v)^2 = 1$. Furthermore, for any given vertex $v \neq r$, $\...
- 209
7
votes
0
answers
304
views
Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
- 167
7
votes
1
answer
545
views
Is the fundamental group of $II_{1}$ factors invariant under a relation?
In order to define the equivalence relation, let's first recall the Tomita-Takesaki modular theory and conditional expectation for von Neumann algebras.
Let $H$ be a separable Hilbert space and $B(H)$...
7
votes
2
answers
1k
views
Edge probability for connected Erdős–Rényi model
Consider the Erdős–Rényi model $G_{n,p}$ with corresponding probability measure $\mathbb{P}_{n,p}$. For any two vertices $x,y$, $\mathbb{P}_{n,p}[E_{x,y}]=p$, where $E_{x,y}$ is the event that there ...
- 71
7
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2
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639
views
Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?
Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...
7
votes
3
answers
3k
views
Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space
I previously posted this question on Math.SE but didn't receive an answer. It is perhaps a little vague; part of what I want to know is what question I should ask.
First, consider the following form ...
- 29.8k
7
votes
1
answer
337
views
First Collision Time for k Random Walkers on a Torus
I consider $k$ random walkers on $\mathbb{Z}^{d}/n \mathbb{Z}^{d}$, the $d$-dimensional torus of side length $n$. More precisely, I will define a Markov chain $Z_{t} = (X_{t}[1], \ldots, X_{t}[k])$ ...
- 71
7
votes
1
answer
414
views
Criteria for operators to have infinitely many eigenvalues
Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.
For non-normal operators this no longer has to be true.
There ...
- 536
7
votes
4
answers
476
views
What does $\mathbb E_V \max_{x \in V,\,\|x\|=1} x^T Ax$ evaluate to when $V$ is random $k$-dim suspace of $\mathbb R^n$ and $A$ is fixed psd matrix?
Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by
$...
- 6,853
7
votes
1
answer
439
views
series representation in injective tensor products
All books on tensor products of Banach spaces contain the well-known theorem of Grothendieck that every element of the completed projective tensor product
$X \tilde{\otimes}_ \pi Y$ has a ...
- 16.4k
7
votes
1
answer
757
views
Length of nearest neighbor path in travel salesman problem
Given $n$ nodes uniformly distributed in $[0,1]^2$, consider the nearest neighbor algorithm to solve traveling salesman problem, i.e., each time I select the nearest neighbor not visited so far as the ...
- 367
7
votes
0
answers
519
views
Squaring random Schwartz distributions
Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance
$$
\mathbb{E}
[\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)}
\widehat{g}(\xi)}{|\xi|^{d-2[\...
7
votes
1
answer
499
views
Moments of a random variable and of its conditional expectation
Let $X$ be a bounded random variable with $\mathbb{E}X=0$. Since $X$ is bounded, all its moments exist. Let $\mathcal{G}$ be any $\sigma$-field and let $Y:=\mathbb{E}[X|\mathcal{G}].$ I am interested ...
- 1,269
7
votes
1
answer
592
views
topologies on U(H)
There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space:
the weak, strong, ultraweak (also called σ-weak), ultrastrong (also called σ-strong), and some more......
- 43.2k
7
votes
1
answer
220
views
Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bundle $E\to M$ with compact fiber?
The question is a special case of a previous question.
Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection ...
- 9,019
7
votes
1
answer
609
views
$H^s$ norm of a solution of a nonlinear Schrödinger equation
I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...
- 71
7
votes
3
answers
330
views
Quantifying the noninvertibility of a function
Given a function $f$ from a finite set $X$ to itself, it seems natural to consider $\kappa_f := (\sum_{x \in X} |f^{-1}(x)|^2)/|X|$ as a measure of the non-invertibility of $f$: it equals 1 if $f$ is ...
- 19.7k
7
votes
1
answer
857
views
Trace of inverse of random positive-definite matrix in high dimension?
Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
- 2,306
7
votes
2
answers
321
views
Random suborbits of a rotation
Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...
- 2,560
7
votes
0
answers
245
views
Distribution of trivial subset sums
Suppose I have a set $S$ of $n$ integers picked independently, uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the ...
- 96.4k
7
votes
1
answer
261
views
Comparison of several topologies for probability measures
Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
- 243
7
votes
2
answers
984
views
Brownian motion in $n$ dimensions
Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in \...
- 73
7
votes
1
answer
429
views
Open projections and Murray-von Neumann equivalence
Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
- 619
7
votes
1
answer
572
views
What is $\hat{A}=\{[\pi]:\pi$ is a irreducible representation of $A$} ( $A$ is a $C^*$-algebra)?
Let $A=\{f:[0,1]\to M_2(\mathbb{C}): $f continuous and $ f(0)=\begin{pmatrix} f_{11}(0) & 0 \\ 0 & f_{22}(0) \end{pmatrix}\}$ be a $C^*$-algebra with pointwise multiplication, involutions and ...
- 1,188
7
votes
2
answers
920
views
Exotic spectrum of Laplace operator
Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator,
it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
- 101
7
votes
1
answer
390
views
Combinatorial/probabilistic statements having $F_{\text{un}}$/$F_q$ geometric interpetation
$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in recent years.
One of the points is that it provides bridge between geometrical and ...
- 24.9k
7
votes
1
answer
789
views
Subadditivity of the square root for matrices
For positive numbers $a$ and $b$ we have the inequality $\sqrt{a+b} \leqslant \sqrt{a} + \sqrt{b}$. Is it true that the same holds if we take $a$ and $b$ to be positive semidefinite matrices?
If not, ...
- 5,005
7
votes
2
answers
3k
views
Arzelà-Ascoli theorem and Hölder spaces
Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist ...
- 21.8k
7
votes
1
answer
814
views
An equivalent condition for separability of $X^*$
Let $X$ be a Banach space. By the weak operator topology on $B(X)$, we mean the locally convex topology implemented by the following semi-norms:
$$B(X)\to[0,\infty) : T\to|\langle Tx,x^*\rangle|$$
...
- 4,058
7
votes
0
answers
497
views
Extreme unitary minimal models of conformal field theory
Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge
$$
c=1-\frac{6}{m(m+1)}\ .
$$
I ...