All Questions
5,850 questions
2
votes
1
answer
255
views
A lower estimate of the derivative of a distance function
I have a question about the derivative of a distance function.
Let $D \subset \mathbb{R}^{d}$ be a connected and unbounded open subset with smooth boundary. $B(z,r)$ denotes the closed (not open) ...
- 721
4
votes
2
answers
243
views
Summability of iterates of analytic function
This question, although appearing deceptively easy, has resisted many attacks against it. The question, being simple to state, is something rather non-trivial that is rather crucial towards more ...
user78249
1
vote
1
answer
2k
views
On the derivative of a distance function
I have a question about the derivative of a distance function.
Let $D \subset \mathbb{R}^{d}$ be a connected and unbounded open subset with smooth boundary. $B(z,r)$ denotes the open ball of radius $...
- 721
4
votes
1
answer
222
views
Is every regular Borel outer measure topologically additive?
If $m$ is a regular Borel outer measure is it true that $m$ is topologically additive?
If so what is a proof or a counterexample?
Definitions:
Topologically Additive: $X$ is a topological space, $m$ ...
- 41
6
votes
1
answer
239
views
Positive semidefinite ordering for covariance matrices
Suppose that X and Z are matrices with the same number of rows. Let
$$ D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & ...
- 1,068
3
votes
1
answer
670
views
A specific mollified functions in the Sobolev space H^1(R)
Let $u>0$ be in $H^{1}(\mathbb{R})=W^{1,2}(\mathbb{R})$, we know that the set of $C^{\infty}$ functions with compact support are dense in the Sobolev space $H^{1}(\mathbb{R})$. Hence, we have a ...
- 31
1
vote
1
answer
100
views
Can this equality hold for a nonzero $b$?
Please may you kindly assist me on this integration exercise: For real $a, b$ with $a \neq 0$, consider the equality
$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\...
- 19
2
votes
1
answer
349
views
Polynomial with subset of critical points and values prescribed
Motivated by this question I wish to pose the following question:
Given $k$ points $(x_1, y_1), \ldots (x_k, y_k)$ with (WLOG) $x_i < x_{i+1}$, can we find a polynomial $p(x)\in\Bbb R[x]$ ...
- 1,049
1
vote
1
answer
133
views
Every $W^{1,p}$ has a representative in ACL
Let $\Omega:=(0,1)^n$ and define $ACL_i(\Omega)$ as the set of all Borel functions $u:\Omega\to\mathbb{R}$ such that
$$ t\mapsto u(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n) $$
is $AC$ for a.e. $(x_1,\...
- 3,146
2
votes
1
answer
1k
views
Doubling metrics, doubling measures, Lebesgue density
As stated in this question,
Lebesgue differentiation theorem holds on locally doubling space?
and proved here,
http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf
the Lebesgue differentiation theorem (...
- 6,541
2
votes
1
answer
887
views
A uniform Lebesgue density theorem
The Lebesgue density theorem in $\mathbb{R}^n$ may be stated as follows. For a Lebesgue-measurable $A\subseteq\mathbb{R}$ and $r>0, x\in\mathbb{R}^n$, define
$$ \chi_{A,r}(x)=\frac{\mu(A\cap B_r(x))...
- 6,541
0
votes
1
answer
160
views
Global Poincaré type estimate
For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x ...
- 4,115
2
votes
0
answers
81
views
Convolution of decaying polynomials [closed]
I conjecture that if the functions $f$, $g$ defined on $\mathbb{R}^n$ satisfying
$$|f(x)| ≤ A(1+|x|)^{−M}, \quad |g(x)| ≤ B(1+|x|)^{−N}$$for some
$M$, $N > n$, then$$|(f * g)(x)| ≤ ABC(1+|x|)^{−L},$...
- 355
1
vote
0
answers
117
views
The eigenfunction of modified $1$-laplace equation?
Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
- 679
15
votes
1
answer
602
views
Integrability property of polynomials in several variables
This might be very trivial, or not.
Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...
- 151
1
vote
0
answers
161
views
level sets portrait near a critical point
Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a smooth ($C^{\infty}$) function and $O$ be an
isolated critical point of $f$. I am looking at the local level sets diagram
near $O$ from topological ...
- 45
2
votes
1
answer
389
views
An irresistible inequality
The following occurred while working on some research project. Since the methods of proof I used were lengthy, I wish to see a skillful or insightful approach (perhaps even conceptual). Anyhow, here ...
- 43.2k
1
vote
0
answers
167
views
Expected amount of linearly dependent random vectors? [closed]
Given a random Matrix $A\in \mathbb{F}_2^{n\times n}$ what is the expectation value of the amount of linearly dependent row-vectors of $A$?
EDIT: As said in the comments, I'm looking for the ...
- 123
5
votes
0
answers
116
views
For $f$ a polynomial, does strict convexity of $\log f(e^s)$ imply that the second derviative of $\log f(e^s)$ has no zeros?
Let $f(t)$ be a monic real polynomial such that $f(t) > 0$ for all $t \ge 0$. Suppose that $\log f(e^x)$ is strictly convex on $\mathbb{R}$, i.e.
$f(s^2) \cdot f(t^2) > f(st)^2$
for all $s, t \...
user94803
2
votes
0
answers
254
views
Prove this function is increasing
I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$:
\begin{eqnarray}
\Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...
- 21
2
votes
0
answers
110
views
If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$
Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$.
Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
user76025
2
votes
1
answer
113
views
estimation of a vector-function
Let $x(t)\in C^1(\mathbb{R}_+,\mathbb{R}^m)$ be a vector-function such that
1) $\|x(t)\|+\|\dot x(t)\|\to 0$ as $t\to\infty$ and
2) for some real $c_1>0$ and all $t>0$ one has $\|x(t)\|\le c_1\...
- 21
1
vote
0
answers
197
views
A certain measure on Banach algebras
According to the comments of Nate Eldredge I did revise the question. In particular I change "$C^{*}$ algebras" to "Banach algebras".
Is there a reference who introduce the following measure on ...
- 366
1
vote
0
answers
71
views
The jump set of $SBV$ function for different value of parameter in image denoising problem
The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary,
$$
u_\...
- 679
4
votes
2
answers
436
views
Variation of Radon transform for probability measures on $\mathbb C$
Let $\mu$ be a probability measure on $\mathbb C$. For $z \in \mathbb C$, let $$f^z \colon \mathbb C \to \mathbb R_{\geq 0}$$ be the function $f^z(\lambda) = |\lambda - z|$. Consider now the family $(\...
- 25.5k
2
votes
0
answers
86
views
I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection
I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
user74319
9
votes
1
answer
966
views
A question about composition of functions
Recently, I heard this question: are there two functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ such that $f\circ g$ is strictly increasing on $\mathbb{R}$ and $g\circ f$ is ...
- 343
1
vote
0
answers
111
views
Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?
I also put this question on MSE here
Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable).
Let $\...
- 679
3
votes
1
answer
378
views
What is the mathematics behind the random experiment which produces the data with this strange property?
I have a following scenario. there is a huge collection of data resulting from a random experiment $E$ (I do not say random variable yet, for reasons that you will need to explain in your answer). Let ...
- 698
2
votes
1
answer
266
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 121
2
votes
1
answer
228
views
Convergence of Discrete Geodesic
Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$.
Suppose $f^{-1}:U_p \mapsto U$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence:
\...
- 5,405
2
votes
0
answers
55
views
Does there exist $\lambda_{\sigma(1)}$ such that $\mu(A\cap\{\lambda_{\sigma(1)}\neq0\})>0$?
Let $(\mathcal F,\Omega,\mu)$ be a measure space and $A\subseteq\Omega$ such that $\mu(A)>0$. Let $L^0$ be the space of all measurable functions.
We say $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$...
- 85
5
votes
1
answer
309
views
Density of convolution
Let $\{X_i\}$ be i.i.d random variables uniform on a measurable, symmetric set $A$ contained in $[-1,1]$. Let $g_{n}$ be density of $X_1+\ldots + X_n$.
Question (general): Is there any non-trivial ...
- 1,491
1
vote
1
answer
115
views
Well-definedness for a singular integral
Let $T_\alpha$ be a singular integral operator defined by
$$
T_{\alpha}[f](t):=\int_{0}^{t}\frac{f(t)-f(s)}{(t-s)^{\alpha+1}}ds
$$
for continuous functions $f$ on $[0,\infty)$ and $0<\alpha<1$.
...
- 201
2
votes
2
answers
636
views
Continuous upper envelope of upper semicontinuous function
Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by
$$A = \{\phi \in C(K): \phi \ge u\}.$$
[Q.] Is the following ...
- 1,399
0
votes
1
answer
217
views
Reproducing Kernel Hilbert Spaces with positive kernels
In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...
- 11
10
votes
1
answer
439
views
Interpolation between $L_1^0$ and $L_2^0$
Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...
- 31.5k
1
vote
1
answer
156
views
Step 2 of The Strichartz's Estimates in Cazenave's Book
My question is from Cazenave's book "Semilinear Schrödinger Equation", page 35. I am stuck with Step 2 of the Strichartz's estimates.
The book says that $||\Phi_f(t)||_{L^2}^2=\left(\int_0^t \...
- 13
2
votes
0
answers
79
views
Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127
6
votes
1
answer
2k
views
Weak convergence in $H_0^1$ and strong convergence in $L^2$
I'm reading a hand-waving argument in a proof of Chapter 7 of Navier–Stokes Equations by Constantin and Foias. I would like to know if I understand it correctly.
Let $\Omega\subset{\mathbb{R}^n}$ be ...
user14319
3
votes
1
answer
189
views
Equivalent Definitions of the Gaussian Surface Measure for Regular Sets
I wonder if the following definitions of the Gaussian surface measure are equivalent.
First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....
- 1,127
5
votes
3
answers
2k
views
Morrey's inequality for Sobolev spaces of fractional order
Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions (or distributions), $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that
$$
\|u\|_{H^s}^2=\sum_{k\...
- 2,933
6
votes
0
answers
210
views
Generalized singular numbers and the Haagerup $L^p$ spaces
Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...
- 361
2
votes
1
answer
5k
views
Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$
Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...
- 1,127
1
vote
1
answer
221
views
Methods to tackle this series and get to the limit?
Take a look at the averaging sum
$$\frac{\pi}{n}\sum_{k=1}^n\;\exp{(-\sin\theta_k)}\cdot \sin(\theta_k +\cos\theta_k)\, \qquad\text{where }\;\theta_k=(2k-1)\frac{\pi}{2n}$$
depending on $n\in\...
- 489
46
votes
2
answers
6k
views
Is the following identity true?
Calculation suggests the following identity:
$$
\lim_{n\to \infty}\sum_{k=1}^{n}\frac{(-1)^k}{k}\sum_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}.
$$
I have verified this identity for $n$ up to $5000$ ...
- 2,183
6
votes
1
answer
260
views
bounding derivative of a sequence
I've been banging my head against the wall on this one ... define a sequence of polynomials $q_n$ by $q_0 = 0$ and $$q_{n+1} = q_n + .5(t^2 - q_n^2).$$ If $q_n \leq t$ on $[0,1]$ then $$.5(t^2 - q_n^2)...
- 42.8k
9
votes
1
answer
299
views
Sequence of nested sets in $[0, 1]$ with bound on gaps
What is the best possible $\epsilon$ and sequence $(a_n)_{n = 1}^\infty \subset [0, 1]$ we can find such that
$$
d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in ...
user78817
3
votes
1
answer
289
views
Domain of Laplacian
Let $L$ be an operator on $C^2(\mathbb R)$, defined by
$$L \phi (x) = \int_{|y|<1} (\phi(x+y) - \phi(x) - \phi'(x) \ y)\ \nu(dy), \text{ for all } x\in \mathbb R$$
for a measure $\nu(dy) = |y|^{-2} ...
- 133
4
votes
1
answer
1k
views
Density argument with Schwartz functions?
I was wondering whether the Schwartz functions are also dense in
$$\{f \in L^2(\mathbb{R}^n); \int_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$
...
- 129