All Questions
18,181 questions
0
votes
0
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114
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Some stability and estimate of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54
2
votes
0
answers
143
views
Convergence of eigenfunctions
In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
3
votes
1
answer
199
views
Can gradient zero implies that a function is constant with Hörmander vector fields
Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by
$$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
- 561
1
vote
1
answer
152
views
Points in the Stone Cech compactification are intersection of open sets
Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...
4
votes
0
answers
73
views
Find reasonable definition for endpoint Lorentz function spaces $L^{\infty,q}$ via the idea from endpoint Triebel-Lizorkin ${\scr F}_{\infty,q}^s$
On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
- 938
3
votes
1
answer
169
views
Can homomorphisms be Borel in some weaker topology?
Let $X$ be a Banach space with separable dual and let $A$ be a Banach algebra. Consider a norm continuous homomorphism $h$ from $L(X)$, the Banach algebra of bounded operators on $X$ onto $A$. In $L(X)...
- 31
2
votes
1
answer
119
views
Deriving the distribution of standardized variables with empirical mean and standard deviation
I'm working with a set of independent and identically distributed random variables $\{ x_i \}_{i=1}^N$, where each $x_i$ follows a Gaussian distribution $P_X(x) = \mathcal{N}(x; \mu, \sigma^2)$. This ...
- 305
-1
votes
1
answer
504
views
Minimum of exponential distribution
Consider $n$ independent random variables $𝑋_𝑖\sim\exp(𝜆_𝑖)$ for $I=1\ldots,n$. Let $\lambda = \sum_{i=1}^n\lambda_i$. Of course, the minimum of these exponential distributions has distribution:
$...
- 9
1
vote
2
answers
498
views
Show convergence result
Consider the following sets:
$$
A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\},
$$
and
$$
A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)...
- 108
2
votes
0
answers
111
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
- 825
2
votes
0
answers
114
views
Asymptotic Independence of random walks from increments?
Suppose we have two random walks $(S_n:n\geq 1)$ and $(T_n:n\geq 1)$ building from independent identically distributed increment vectors $\{(X_k,Y_k):k\geq 1\}$, i.e. $S_n=\sum_{k=1}^n X_k, T_n=\sum_{...
- 715
0
votes
0
answers
82
views
Conditional distributions of random orthogonal projection matrix
I have encountered a rather curious question.
Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n$-dimensional ...
- 319
1
vote
0
answers
62
views
A small lemma on cache resets (Bloom filters in particular)
Assume a fixed set of message $D$ and an associated distribution for selecting each message $d_i$ such that the total probability $\sum_{i \in D} d_i = 1$. We create a cache with $M$ bits and $k$ ...
3
votes
1
answer
114
views
Dimension of spectral projection subspaces under local convergence
I'm interested in estimates on dimension of spectral projection subspaces of some limit operator. I recently asked a related question in the thread Dimension of spectral projection subspaces under ...
- 633
2
votes
1
answer
172
views
Does there exists an example of a Banach space that is compactly LUR; but not LUR
We know that a Banach space $X$ is locally uniformly rotund(LUR) if for $x, x_n \in S_X$ with $\Vert x+x_n \Vert \to 2$, we have $x_n \to x$. In the same case, if $(x_n)$ has a convergent subsequence, ...
- 85
1
vote
0
answers
78
views
Moment method / genus expansion for random matrices with i.i.d. entries
Given a (say real) random matrix $M=(M_{i,j})_{1\leq i, j \leq N}$, the moments method consists in computing (the limits in $N$ of) the quantities $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$
...
- 31
4
votes
1
answer
227
views
Probability problem in Sheehan's conjecture
As my first math project, I have been working on Sheehan's Conjecture
and am stuck for weeks. I wonder if I am at a dead end.
Sheehan's Conjecture states that every Hamiltonian 4-regular simple
graph ...
- 43
0
votes
0
answers
126
views
Building representation of an arbitrary umbral calculus
Consider a set of integrable functions on the interval $(0,1)$.
Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function).
In such system the ...
- 10.1k
0
votes
0
answers
211
views
Gauss transformation in fractional Sobolev space
Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
- 101
3
votes
2
answers
183
views
Dimension of spectral projection subspaces under strong convergence of operators
I have a possibly simple question regarding estimating bounds on spectral projection subspace.
Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$ converging in the ...
- 633
5
votes
1
answer
267
views
Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions
Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...
- 63
1
vote
1
answer
143
views
Projection of an element of the $n$-simplex onto subset
Let $\mathbb{S}^{n}$ denote the $n$-dimensional probability simplex and let $\{e_1,...,e_{n+1}\}$ be the canonical basis of $\mathbb{R}^{n+1}$. Consider the subset $\mathbb{S}^{n}(K) \subset \mathbb{S}...
3
votes
0
answers
122
views
Slow points of diffusion processes
Let $W$ be a standard $d$-dimensional Brownian motion, and $X$ the solution to the SDE
$$dX_t = \mu(X_t) dt + \sigma(X_t) \, dW_t,$$
with $\mu$ and $\sigma$ Lipschitz continuous.
Given a (...
- 6,233
4
votes
0
answers
116
views
Weakly null sequences in projective tensor products II
The question in this post is the question below from an article by Rodriguez & Rueda Zoca [1].
Below is a complimentary salad/side dish that accompanies the main course.
Let $B^2(X,Y)$ denote ...
- 2,605
7
votes
1
answer
524
views
What happens when the diffusion term in an SDE becomes zero?
Consider this time-homogeneous SDE, in the Ito sense:
$$dX_t= -(X_t-a)\,dt+\sigma(X_t)\,dW_t,$$
where $W_t$ is standard Brownian motion, $a<b\in\mathbb{R}$, $X_0\leq b$ a.s., and $\sigma(b)=0$. ...
- 103
2
votes
2
answers
317
views
How does pairwise independence restrict dependence to a third variable?
Let $X,Y,Z$ be random variables such that
$X,Y,Z$ have mean $0$ and variance $1$.
$X$ and $Y$ are pairwise independent.
$Z$ can be arbitrarily dependent on $X$ and $Y$.
My question is: What can we ...
- 73
1
vote
1
answer
120
views
Characterization of an integral operator with a Bessel kernel
I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$
I am ...
- 141
3
votes
0
answers
122
views
Multi-scale 3- and 5-arm exponents for critical planar percolation
Consider critical site percolation on the planar triangular lattice. Denote by $A_j(m,n)$ the event that there are $j$ arms (paths from the inner boundary to the outer boundary) of alternating colour ...
- 311
2
votes
1
answer
185
views
Koopman operators on $L^p(X)$
On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
0
votes
0
answers
21
views
Proof that Component-wise MH algorithm is invariant w.r.t. target measure
consider a standard situation in Bayesian modelling,
given real vector parameter $\theta=(\theta_1,\dotsc,\theta_n)$ and observations $x$ we derive a posterior distribution $\pi$ with posterior ...
- 31
2
votes
0
answers
114
views
Poincare inequality on the hemisphere
Background:
Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
- 537
5
votes
1
answer
297
views
Expected number of coin flips before you see a $k$-term arithmetic progression of heads
Let $\{X_i\}_{i \in \mathbb Z_+} $ be independent fair coin flips. Write $S := \{i \in \mathbb Z_+\, | \, X_i \text{ is heads}\}$, and define, for an integer $k \geq 3$,
$$Y := \inf \{n \in \mathbb N \...
- 6,233
6
votes
1
answer
207
views
Hopf monads in categorical probability theory
1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...
- 764
0
votes
1
answer
102
views
Lower bounds for truncated moments of Gaussian measures on Hilbert space
Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
- 505
2
votes
0
answers
109
views
The fluctuations of a random path
Suppose I have a $n \times n$ square grid and for each square, I assign 1 with probability $\frac{1}{2}$ and 0 with probability $\frac{1}{2}$. On the boundary, I put 1s on the lower half and 0s on the ...
- 1,054
1
vote
0
answers
108
views
Infinite tensor product of Hilbert spaces [duplicate]
Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...
- 11
4
votes
1
answer
287
views
Reference request: Gaussian measures on duals of nuclear spaces
I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcal{S}'(\mathbb{R}^n)$) play a key role. ...
- 721
1
vote
0
answers
72
views
Dimension-free sample complexity for the inverse of Gaussian sample covariance?
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in ...
- 517
3
votes
1
answer
128
views
Fréchet-valued symbols
Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...
- 395
1
vote
0
answers
89
views
Definition of second quantization
The standard textbook for second quantization is Reed & Simon. However, I am a bit confused with their notation. They write:
Let $\mathscr{H}$ be a Hilbert space, $\mathcal{F}(\mathscr{H})$ the ...
- 1,305
6
votes
1
answer
319
views
How are coordinate charts constructed in noncommutative geometry?
In noncommutative geometry, one is given a triple $(A,D,H)$, where $A$ is a commutative C* algebra, $H$ is a Hilbert space, and $D$ is an operator. There is a somewhat long list of conditions that ...
- 593
4
votes
1
answer
175
views
Large JN-sets in Banach spaces
For every infinite-dimensional Banach space $X$ there is a weak*-null sequence in the unit sphere of $X^\ast$. Does this extend under suitable circumstances to the non-separable setting?
Say that $X$ ...
- 41
0
votes
1
answer
192
views
A continuous injection from the Hilbert cube to the real line?
Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question:
Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
- 3,104
0
votes
0
answers
71
views
Weak limit of pushforward measures with finite second moments is also a pushforward measure with finite second moment
Suppose $\mu \in P(\mathbb{R}^d)$ and for each $n$, $T_n:\mathbb{R^d} \rightarrow \mathbb{R^d}$ is such that the pushforward $T_n \# \mu$ has a finite second moment. If $\{T_n \# \mu\}$ converges ...
- 69
3
votes
0
answers
153
views
Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 131
1
vote
1
answer
109
views
Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex
Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
1
vote
1
answer
162
views
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
It seems too good to be possible, but:
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
- 3,104
3
votes
0
answers
132
views
Takesaki's duality in representation theory of $C^*$-algebras
In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting.
...
- 139
6
votes
1
answer
228
views
Non-adjacent permutations
Suppose we have an $N$ by $M$ table. Suppose that $x=(a,b)$ and $y=(c,d)$ are two locations in the table, specified by their row and column indexes. We say that (x,y) is horizontally adjacent if $c=...
- 3,979
0
votes
1
answer
150
views
Property of $p$-norm in the $n$-simplex
Let $\mathbb{S}^{n}$ be the canonical simplex of $\mathbb{R}^{n}$ and let $u = (1/n,\dotsc,1/n)$. Is it true that
$$\lVert x - u \rVert_p \leq \lVert y - u \rVert_p$$
implies that
$$\lVert x\rVert_p \...