All Questions
13,944 questions
4
votes
1
answer
297
views
Oscillation of monotone real-analytic function
Let $f:(a,\infty)\rightarrow \mathbb{R}$ be a real-analytic and strictly monotone function. I have been wondering how much this function can "oscillate". Namely, can we always find a ...
- 1,045
-1
votes
1
answer
98
views
Spectrum of sum of positive and negative operators
Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
- 21
0
votes
0
answers
78
views
What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?
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
$$\forall_{u \in A(\...
- 63
1
vote
1
answer
124
views
$d(x,y) = \min\{|x_1−y_1|+|x_2−y_2|, 1−|x_1−y_1|+|x_2−(1−y_2)|\}$ defines a metric on $[0,1)\times[0,1]$? [closed]
For $x,y \in [0,1)\times[0,1]$, let $d(x,y)$ be the minimum of $|x_1−y_1|+|x_2−y_2|$ and $1−|x_1−y_1|+|x_2−(1−y_2)|$. Prove or disprove that $d$ is a metric.
I was unable to find a counterexample to ...
- 21
4
votes
1
answer
214
views
Characterisation of Sobolev spaces using their Lipschitz approximations
Let $f \in W^{1, p} (\mathbb R^n)$. A classical approximation theorem (see for instance, the book by Evans and Gariepy) says that we can approximate $f$ by Lipschitz functions, in the sense that for ...
- 6,165
5
votes
2
answers
149
views
Showing an operator is (or not) closed on $L^2(\mathbb{R})$
I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$.
Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
3
votes
0
answers
100
views
How to compute the partial derivatives of this function?
For any probability measure $\mu$ on $\mathbb R^2$ and $\theta\in [0,2\pi]$, denote by $\mu_\theta$ its projection along $v:=(\cos\theta,\sin\theta)$. Namely, if $X$ is a random variable distributed ...
- 1,389
1
vote
0
answers
98
views
Equivalence of Sobolev norms for smooth functions with compact support
Let $f\in C^\infty_c([0,1]^n)$, then we can extend it to a $1$-periodic smooth function $\tilde f$. We define the fourier transform (series) of $f$ ($\tilde f$):$$
\hat f(\xi):=\int e^{2\pi i x\cdot \...
- 435
1
vote
1
answer
157
views
Is finding the CDF from the Laplace transform well-posed?
In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
- 654
6
votes
1
answer
194
views
The most even partition of $\mathbb R$ into measure dense sets
Notation: $\mu$ denotes the Lebesgue measure. Let $\mathcal D$ be the set of Lebesgue measurable subsets of $\mathbb R$ such that itself and its complement have nonzero Lebesgue measure in every ...
- 6,165
3
votes
1
answer
344
views
Asymptotic behavior of a recursion
Let $x_n(0)=1$,
$$
x_n(N+1) = \frac{1}{N+1}\sum_{k=0}^N \sum_{j=1}^n x_j(k)x_{n+1-j}(N-k) + \frac{10}{N+1} x_{n+1}(N) , \quad\quad N\ge 0 .
$$
So the recursion is on $N$, and at each level, we compute ...
- 24.2k
2
votes
0
answers
189
views
Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation}
G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
- 185
2
votes
1
answer
149
views
Proof that superlinearly convergent sequence converges faster than linearly convergent sequence
Given real sequences $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$, both converging to the same limit $A$ and such that $|a_n-A|\neq 0$ and $|b_n-A|\neq 0$ for every $n$ sufficiently large, we ...
- 131
1
vote
0
answers
60
views
Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
- 745
3
votes
2
answers
118
views
Does the derivative of the antiderivative of a BV function $f$ agree with $f$ at all but countably many points of differentiability?
Let $f: (a, b) \to \mathbb R$ be a function of bounded variation, and write
$$F(x) := \int_a^x f(t) \, dt$$
for the antiderivative. Is it true that at all but countably points of differentiability of $...
- 6,165
3
votes
1
answer
247
views
Is the derivative of a Lipschitz function continuous a.e.?
Let $f:(a,b) \to \mathbb R$ be Lipschitz.
The derivative $f'$ exists on some set $D \subset (a,b)$ of full measure and is bounded (by Rademacher).
Is $f'$ continuous (or some representative) on the ...
0
votes
1
answer
231
views
Questions on the compactness of $L_1([0,1]^2)$'s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
- 349
2
votes
1
answer
276
views
Estimating a sum over set partitions
Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$.
I would like to estimate the following alternating sum.
QUESTION. Is this true?
...
- 43.2k
0
votes
1
answer
101
views
Limit sequence of regular function in $L_1$‘s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
- 349
4
votes
0
answers
147
views
Weakly compact sets forced to contain $0$
Let $E$ be an infinite-dimensional real normed space and let $K\subset E$ be a weakly compact set such that, for each $\varphi\in E^*\setminus \{0\}$, there exists a unique $\tilde x\in K$ such that
$$...
- 283
0
votes
0
answers
121
views
How to find the inverse of this linear integral operator?
Let $f(x): \mathbb{R}^d \rightarrow \mathbb{R}$ be a function that decays ``fastly enough'' at infinity.
We can define the following linear operator
$$L[f](x):= \int_{\mathbb{R}^d} d^d y \, \frac{f(y)}...
- 1
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 ...
- 6,541
3
votes
1
answer
309
views
Extremizing sequence consists of two elements
Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,...
5
votes
0
answers
160
views
Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
- 929
-6
votes
1
answer
180
views
An analog of Anderson's result in C* algebra setting [closed]
Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.
For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$
It's known that $...
- 307
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
5
votes
0
answers
156
views
What is the Hausdorff dimension of the set on which this exponential sum is bounded?
This is a direct follow up to For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
What is the Hausdorff dimension of the ...
- 6,165
1
vote
1
answer
102
views
Does $C^{k,s-k}$ function with lipschitz lower order derivatives give a certain bound on the Taylor remainder?
Let $\Omega \subseteq \mathbb{R}$ be open (not necessarily an interval). Let $ s > 0$ and $k \in \mathbb{N}_0$ be such that $s \in (k, k+1]$. Suppose that
$f \colon \Omega \to \mathbb{R}$ is an ...
- 820
2
votes
1
answer
152
views
Co-locating slowly increasing smooth functions in two different ways
This question is subsequent from my previous one.
I will write everything in detail for the sake of completeness.
Let $g_1$ and $g_2$ be smooth functions on $\mathbb{R}$, whose derivatives of all ...
- 3,477
5
votes
2
answers
256
views
On the closed convex hull of a weakly compact set
Let $H$ be an infinite-dimensional real Hilbert space and let $B$ be the closed unit ball of $H$. Let $K\subset B$ be a weakly compact set whose closed convex hull agrees with $B$. Question: does $K$ ...
- 283
4
votes
1
answer
204
views
Stationary phase formula for a complex valued phase
I'd be interested in computing an asymptotic expansion when $h \rightarrow 0$, of an integral of the form
$$
I_h = \int_{\mathbb{R}}{e^{\frac{i}{h}\varphi(x)}dx}
$$
where $\varphi : \mathbb{R} \...
- 2,696
2
votes
0
answers
102
views
Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential
Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
- 1,551
5
votes
2
answers
373
views
Weak Archimedean property instead of Archimedean property
We say that a sequence $(z_n)$ of real numbers is a modulated Cauchy sequence, whenever there exists a function $\alpha:\mathbb{N} \rightarrow \mathbb{N}$ such that:
$$
|z_i-z_j| \le \frac{1}{k} \quad ...
0
votes
0
answers
121
views
Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?
Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
- 7,127
3
votes
1
answer
182
views
Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product
Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing).
For any pair of ...
- 3,477
12
votes
2
answers
866
views
Sets that project to zero measure on all lines except one
It is a (difficult) exercise to show that there exists a measurable set $E \subset [0,1]^2$ (necessarily with zero 2-dimensional Lebesgue measure) such that the projection on every line passing ...
- 355
-1
votes
1
answer
167
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349
1
vote
1
answer
40
views
Envelopes of functions with respect to some convex cone $\mathcal{F}$
Let's say we are given a function $f:\mathbb R ^d\to \mathbb R$ continuous. Assume that $\mathcal F$ is a convex cone of continuous functions ($\mathbb R^d$ to $\mathbb R$) closed under maxima. I am ...
- 291
4
votes
0
answers
80
views
Interpolation-extrapolation scales of H. Amann
I am currently reading H. Amann's notes titled "Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems", and I have a question regarding the abstract ...
- 145
8
votes
1
answer
342
views
How large can the set of turbulent points be?
This question resisted attempts on MSE.
Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:
$$\...
- 6,165
1
vote
1
answer
50
views
Increasing function of $\theta$ for the Ali-Mikhail-Haq Survival Copula
I have been trying to solve the following function is non-increasing (non-decreasing) with respect $\theta$ where $\theta \in (0,1)$ (resp. $\theta \in (-1,0)$)
\begin{equation}
f(\theta)= \frac{h(t,\...
- 23
5
votes
1
answer
183
views
What is a natural interpretation of the commutator of the conditional expectation operator?
Notation: We denote by $\mathbb E_{\mathcal F} X$ the conditional expectation of the random variable $X$ with respect to the $\sigma$-algebra $\mathcal F$.
Given two $\sigma$-algebras $\mathcal G, \...
- 6,165
1
vote
1
answer
330
views
Does $\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$ converge?
I am trying to study the converge of the series
$$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$
But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? ...
- 287
5
votes
1
answer
164
views
Does quadratic asymptotic growth imply log-Sobolev inequality?
Let $f : \mathbb{R}^n \rightarrow [0,\infty)$ be a smooth function and consider $h$ s.t $h(\vec{x}) = f(\vec{x}) + \lambda \Vert \vec{x} \Vert^2$.
Does this imply that irrespective of any other ...
- 617
10
votes
1
answer
314
views
Weakly metrizable sets in normed spaces
A similar question was asked on MSE without getting an answer.
In the proof of lemma 1.2 of Asplund operators and holomorphic maps the author (my attempt to contact him failed because the only e-mail ...
- 16.4k
5
votes
0
answers
285
views
How do you go about making ranges (for integer variables) independent?
Basic question: say you have a sum
$$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$
where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
- 20.2k
0
votes
0
answers
60
views
Spectral analysis of Dirac operators coupled to gauge potential on $\mathbb{R}^n$
Dirac operators on compact manifolds seem to have been studied well, such as in this book and also this one.
However, I cannot easily find comprehensive treatment of Dirac operators coupled to gauge ...
- 3,477
1
vote
2
answers
102
views
About the recursive inequality $w_p \geq (1-\frac {\pi}n)w_{p-2n} + 2\pi + o(1)$
Suppose we have a non-decreasing sequence of positive real numbers that tend to infinity: $0<w_1\leq w_2\leq w_3\leq...$ It is known that:
For every $n$ and $p\geq 2n$, we have $w_p \geq (1-\frac {...
- 525
0
votes
1
answer
139
views
Proving negativeness of function involving $-\log t$
I have been trying to solve the following function is non-increasing with respect $\theta$
\begin{equation}
h(t,\beta) = \frac{1-t-\frac{\beta(-\log t)^{\theta}}{\theta(-\log \beta)^{\theta -1}}}{1-\...
- 23
3
votes
0
answers
196
views
Parabolic smoothing for semilinear PDE
Consider the semilinear energy-critical parabolic PDE in $\mathbb{R}^3$
\begin{align}
\partial_t u &= \Delta u + |u|^{4/(n-2)}u = \Delta u + u^5\\
u(0,x) &= u_0\in \smash{\dot{H}}^1(\mathbb{R}^...
- 537