All Questions
1,549 questions with no upvoted or accepted answers
5
votes
0
answers
428
views
Is there an appropriate weighted Sobolev space to include exponential map and projection map?
Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: \...
- 3,245
5
votes
0
answers
596
views
Literature on Exponential of a Quadratic Form
Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials
\begin{align}
f(\mathbf{x})=\sum_{i=1}^{L}\operatorname{exp}(-{\mathbf{x}^T\mathbf{A}_i\...
- 1,421
5
votes
0
answers
270
views
Differential operators that preserve real-rootedness
Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of $\...
- 156k
5
votes
0
answers
760
views
two versions of the nested interval property
There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (...
- 19.7k
5
votes
0
answers
583
views
Cohomology of Real algebraic Varieities
I understand Serre's GAGA theorem as saying that equations over algebraically closed fields can be studied equally from the algebraic and analytic points of view, at least with respect to cohomology.
...
user2529
5
votes
0
answers
369
views
Independent Events Inducing Probability Measures
Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...
- 4,952
5
votes
0
answers
558
views
continuous selection of a multivalued function?
The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...
- 1,839
5
votes
1
answer
829
views
Clarke generalized Jacobian of an inverse function
For a Lipschitz function $f: X \rightarrow X$, Clarke's generalized Jacobian at $x$ is defined as the convex hull of the following set:
$$\delta f (x) = \text{convex hull} \left \{\lim_{x_i \...
- 301
4
votes
0
answers
140
views
Condition under a function is uniquely identifiable by the supremum values
Let $f(x),g(x)$ be two real-valued functions on $\mathbb{R}$ and $h(x,y)$ be a real-valued function on the plane. We can assume continuity (maybe piecewise differentiability also) of these functions. ...
- 271
4
votes
0
answers
198
views
When a null uncountable set can be image of some increasing function with discontinuities on a dense countable set
Consider the following result:
A: Let $f:D \to \mathbb R$ be an increasing function with discontinuities on a dense countable subset of $D$ such that the jump values sum to $\mu(D)$, where $D$ is a ...
- 303
4
votes
0
answers
88
views
A question concerning regularly varying functions
In my work I need some results about regulary varying functions, which I only have a very vague understanding.
A strongly related reference I found is "On the Existence of a Regularly Varying ...
- 119
4
votes
0
answers
238
views
Conjectural values of some determinants involving Legendre symbols (I)
$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants
$$\det\left[\Legendre{i+j}p\right]_{...
- 15.6k
4
votes
0
answers
158
views
Measurability of $L^{p}(L^{q})$ integrable functions
Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that
$
\int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty
$
In addition we ...
- 41
4
votes
0
answers
140
views
Does an instance of this generalisation of the determinant exist?
Let $n$ be composite, $d$ a divisor greater than $1$ and $m=n/d$. Does anybody know if there is a general mapping $T$ from $n×n$ matrices to $m×m$ matrices that preserves the determinant? Over a field ...
- 1,245
4
votes
0
answers
116
views
Lipschitz extension of a flow can still be a flow?
Consider a map $\Phi: [0,T] \times \mathbb{R}^d \to \mathbb{R}^d$, and assume that there exists a set $U \subset \mathbb{R}^d$ such that $\Phi\rvert_{[0,T] \times U}$ is $L$-Lipschitz. It is well ...
- 697
4
votes
0
answers
208
views
Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
- 43.2k
4
votes
0
answers
181
views
Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.2k
4
votes
0
answers
114
views
Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,375
4
votes
1
answer
287
views
Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
- 1,467
4
votes
0
answers
130
views
A "counterbalancing" trigonometric sum inequality
Is it true that
$$s_{n,k}:=\sum_{j=1}^{n-1} r_{n,k,j}
<0$$
for all natural $n\ge2$ and all natural $k\in\{1,\dots,n-1\}$,
where
$$\text{$r_{n,k,j}:=\frac{x_{n,2j}}{y_{n,k,j}\;y_{n,k+1,j}},\quad$
$...
- 128k
4
votes
0
answers
140
views
Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
- 1,075
4
votes
0
answers
179
views
Approximation by gaussian mollification in Sobolev spaces
I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify)
$$\label{0}\tag{0}
\|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
- 230
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
- 61
4
votes
0
answers
481
views
Generalized Jensen's inequality for positively homogeneous functions
The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
- 519
4
votes
0
answers
82
views
Maximal function in Orlicz space
Consider the maximal operator defined for a function $f\in L^1_{loc}$:
$$
Mf : x\mapsto \sup_{r>0} \frac{1}{|B(x,r)|} \int\limits_{B(x,r)} f.
$$
It is well know that $M : L^1 \to L^{(1,\infty)}$ ...
- 363
4
votes
0
answers
81
views
Does this sequence of functions converge in a distributional sense?
Let $f\in W^{1,12/5}(\mathbb{R}^3)$ (time-independent), let $K^{\epsilon}$ be a uniformly in $\epsilon$ bounded sequence in $L^{1}\cap L^{7/5}(\mathbb{R}^3)$ and let
$$\tilde{K}^{\epsilon} := K^{\...
- 469
4
votes
0
answers
191
views
Approximating a uniformly elliptic function of bounded variation by Lipschitz functions
Let $\sigma: \mathbb R^d \to \mathbb R^{d \times d}$ be of bounded variation.
Suppose $\sigma$ is uniformly elliptic, in the sense that there exists some constant $C > 0$ such that $\xi^{T} \sigma(...
- 6,155
4
votes
0
answers
140
views
Is the existence of Banach limits independent of ZF+DC?
Is the existence of Banach limits independent of ZF+DC?
Assuming this is known, where can I find a proof?
- 6,453
4
votes
0
answers
188
views
Branch cuts, inverse Fourier transform and large time asymptotics
Let the Fourier transform of $f(t)$ be defined as $F(\omega) = \int_{-\infty}^\infty dt f(t) e^{i\omega t}$ for values of $\omega$ where the integral exists. What are the precise conditions on $F(\...
- 1,150
4
votes
0
answers
253
views
Legendre-Fenchel transform
Suppose $F:\mathbb R^n\to \mathbb R$ is a convex continuous function.
Moreover, for any $x\in \mathbb R^n$,
$$
\limsup_{\lambda\to\infty} \frac {|F(\lambda x)|}{\lambda}<\infty.
$$
I would like to ...
- 650
4
votes
0
answers
169
views
Questions on a cone defined by elementary symmetric functions
Let $1\le k\le n$ be given integers. Define the following cone
$$\Gamma_k=\{\lambda\in\mathbb{R}^n| S_j(\lambda)>0, j=1, ..., k\},$$
where $S_j(\lambda)$ is the $j$th elementary symmetric function ...
- 69
4
votes
0
answers
656
views
Eigenvalues of Matérn covariance function
Recall that Matérn covariance function $C_\nu(d)$ is defined as
$$
C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...
- 397
4
votes
0
answers
826
views
Showing that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$
Prove, without evaluating the integrals that:
$$\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$$
Originally I posted this here on MSE, however it's ...
- 215
4
votes
0
answers
115
views
Delta distributions that are smooth on strata of a singular manifold
This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\...
- 7,952
4
votes
0
answers
194
views
What are the "local degrees of freedom" in the space of smooth functions?
Let $C^k$ be the set of $k$th-order smooth real functions $f:\mathbb{R}\to\mathbb{R}$, and $C^\infty$ the set of smooth real functions. One can specify an $f\in C^k$ by specifying all its derivatives ...
- 49
4
votes
0
answers
116
views
Is every analytic set the projection of a set with sections large in some sense?
Is every analytic set $A$ in, say, $I=[0,1]$, the projection of a Borel set $B$ in, say, $[0,1] \times [0,1]$, $A = \pi_1(B)$, with the following property: For every regular Borel probability measure $...
- 119
4
votes
0
answers
318
views
Integral representation of solution of an elliptic PDE in divergence form
Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
- 261
4
votes
0
answers
300
views
Is the set of approximating sequences for irrationals dominating?
Let $r\in [0,1]\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\{|r-\frac{...
- 48.9k
4
votes
0
answers
135
views
Zygmund class, Schwartz class and Littlewood-Paley projection operators
I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions:
Consider the Zygmund class of functions defined as ...
- 349
4
votes
0
answers
176
views
Finding roots of equation with gamma functions
Encountered this function in one of my research problems
$$\frac{\Gamma \left(1-\dfrac{i c}{a}-\gamma \right) \Gamma \left(1+\dfrac{i c}{a}+\dfrac N 2-\gamma \right)}{\Gamma \left(1+\dfrac{i c}{a}-\...
- 199
4
votes
0
answers
112
views
Properness of real analytic maps?
Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, ...
- 10.5k
4
votes
0
answers
151
views
Estimating the size of $\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}$
Let $\Omega$ be a bounded domain in $\Bbb R^n$. Define
$$
\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \},
$$
i.e. it the ring of thickness $r$ at the boundary of $\Omega$. Intuitively, ...
- 1,245
4
votes
0
answers
204
views
Harmonic functions in upper half plane
Let $\mathbb H^+$ denote the upper half plane in $\mathbb R^2$. Consider the following equation
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
\Delta u=0\,\quad &\text{on $\mathbb H^+$},...
- 4,135
4
votes
0
answers
179
views
Condition on kernel convolution operator
I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with
$$
\Phi_i(2t)\approx \Phi_i(t), \quad i=1,2
$$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
- 101
4
votes
0
answers
84
views
Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms
Let $F$ be the set of all convex functions $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0=f'_+(0)$ and $f_+(\infty-)=\infty$, where $f'_+$ is the right derivative of $f$. For any function $f\in F$, its ...
- 128k
4
votes
0
answers
141
views
What is the completion of $L^\infty$ in the dual of BV?
Every $f \in L^\infty([0,1])$ induces a continuous linear functional on BV via $g \mapsto \int f g \mathrm{d}x$. I believe $L^\infty([0,1])$ is also separable in BV$^\ast$, while BV$^\ast$ is not ...
- 403
4
votes
0
answers
150
views
A maximum of an integral
It seems that the following functions $$G(R,s)=(1-R^2)\int_0^1\int_0^{2\pi}
Adr da,$$ where $$A=\frac{ \sqrt{\left(\left(1+r^2\right) \cos(a+s)-2 r R \cos s\right)^2+\left(1-r^2\right)^2 \sin^2(a+s)}}{...
- 719
4
votes
0
answers
97
views
Smoothing continuous functions in metric space
Let $(X,\rho)$ be a metric space.
For any $f:X\to\mathbb{R}$, define the local Lipschitz constant of $f$ at $x$ by
$$ \Lambda_f(x) := \sup_{x'\in X\setminus\{x\}} \frac{|f(x)-f(x')|}{\rho(x,x,')}
.
$$...
- 6,541
4
votes
0
answers
220
views
A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic
I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...
- 91
4
votes
0
answers
272
views
An inequality in harmonic analysis with the BMO flavour
I am asking myself this question (which seems to be a natural generalization of Remark 4.4 of these lecture notes).
Question. Let $I_s, s \in \mathcal{S}$ be a collection of intervals included in
$[0,...
- 489