All Questions
1,491 questions with no upvoted or accepted answers
1
vote
0
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122
views
Series and solution of $-\Delta u + \lambda u = f(x)$
Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of
\begin{align*}
-\Delta u(x) + \lambda u(x) &= f(x), & x \in \...
- 131
1
vote
0
answers
33
views
Existence theory for first order scalar discontinuous ODE
Consider the scalar i.v.p. in ${\mathbb R}$
$$
x'=f(t,x), \; t\in[0,T], \; x(0)=x_0,
$$
where $T\in {\mathbb R}$, $T>0$, $x_0\in {\mathbb R} $, and $f:[0,T] \times {\mathbb R}\mapsto {\mathbb R}$...
- 31
1
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0
answers
76
views
Second question on a real sequence
I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
- 903
1
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0
answers
188
views
Best approximation of a Lipschitz function with a piecewise polynomial Lipschitz function
Let $g : [-1, 1] \to R$ be a $1$-Lipschitz function and $f_{k,d} : [-1, 1] \to R$ a $1$-Lipschitz function whose restriction to any subinterval $[h_i, h_{i+1}] \subset [-1, 1]$, $i = 0 ... (k-1)$ with ...
- 31
1
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0
answers
157
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Does a sequence of Jacobians converge to the 'correct' continuous part plus some controlled singular part?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \in W^{1,...
- 6,741
1
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0
answers
81
views
Compact imbedding for weight space
We begin with some definitions. Let $\gamma \geqslant 1,\,p \in \left[ {1,\infty } \right)$, we define
$$L_\gamma ^p\left( {0,1} \right) = \left\{ {v:\left( {0,1} \right) \to \mathbb{R}:{{\left\| v \...
- 183
1
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0
answers
81
views
Hardy maximal function on the torus
A few years ago I asked a reference about the Hardy maximal function on the flat torus. Mateusz Kwaśnicki kindly answered in a comment, and confirmed my conviction that basically everything which is ...
- 3,425
1
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0
answers
119
views
Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
$\DeclareMathOperator\rad{rad}$
Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ be compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
In $H_{\rad}^1(\mathbb{R}^3)$, by Struass estimate $|f(x)| \lesssim |x|^{-1} ...
- 429
1
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0
answers
42
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Energy estimate for $\theta_t + H(\theta)_x = 0$ in $t>0, x >0$?
Consider the IBVP for $$\theta_t + H(\theta)_x = 0, \qquad t>0, \ x>0$$ with $$H(\theta) = \frac{1}{\pi} \text{pv}\int_{0}^\infty \frac{\theta(y)}{y-x} dy$$
with Dirichlet boundary conditions. ...
- 161
1
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0
answers
79
views
Bounds on the inverse multivariate beta function
Can i find some constants $\mathbf a$ and $\mathbf b$ in $\mathbb{R}_+^d$ such that for all $\mathbf{x} \in \mathbb{R}_{+}^{d}$, the inverse beta function :
$$IB(\mathbf x) = \frac{\Gamma\left(\lvert \...
- 686
1
vote
1
answer
493
views
Sufficient and necessary conditions for decomposing the sum of random variables
Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and
$\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...
- 550
1
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0
answers
47
views
Convergence of a certain serie
I came cross the following serie :
$$\sum\limits_{\mathbf k \in \mathbb N^d} e^{\langle \mathbf r, \mathbf k \rangle}$$
What would be the conditions on the d-dimensional real vector $\mathbf r$ for ...
- 686
1
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0
answers
47
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Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder
Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there ...
- 339
1
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0
answers
52
views
Functions that vanish weakly to $\infty$ and a uniqueness problem
I am reading the article "User’s guide to the fractional Laplacian and the method of semigroups" by P.R. Stinga, there is a link. At page 17, in theorem 7, the author state that, for a given ...
- 339
1
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0
answers
210
views
The translation is continuous in $L^1(\mathbb{R}^n,d\mu)$, $d\mu=\frac{1}{1+|y|^{n+a}}dy$,$ a>0$
For any function $f\colon\mathbb{R}^n\to\mathbb{R}$, set: $\tau_hf(x):=f(x+h)$, $x,h\in\mathbb{R}^n$. Consider the following finite measure on $\mathbb{R}^n$:
$$\mu(A):=\int_A\frac{1}{1+|y|^{n+a}}\,dy$...
- 339
1
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0
answers
139
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A lower-bound on matrix-function with vector product
I am currently trying to show that a sequence of homeomorphisms converges to some limiting homeomorphism using Anderson's the inductive convergence criterion. However I can't explicitly compute the ...
- 5,405
1
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0
answers
74
views
Fourier transform of a Sobolev function dependent on a "parameter"
Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...
- 339
1
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0
answers
42
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Extension problem of fractional laplacian and Fourier transform of $L^1_\text{loc}$ function?
I have understand the proof oh the lemma 4.1.9 "SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY" by C.Bocur, there is a link. In this paper, we define, for $u\in L^1_\text{loc}(\...
- 339
1
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0
answers
68
views
wave equation with $H^{-1}$ source
Let $\Omega$ be a bounded domain with smooth boundary and given $f \in H^{-1}((0,T)\times \Omega)$, consider the wave equation
$$ \Box u =f\quad \text{on $(0,T)\times \Omega$},$$
with $u|_{(0,T)\times ...
- 4,143
1
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0
answers
213
views
Fractional Laplacian extension problem and uniqueness question
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. Consider the following problem:
$$ \Delta_xu+\frac{a}{y}u_y+u_{yy}=0, $...
- 339
1
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0
answers
78
views
A question about extension problem related to fractional laplacian
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. link. At page 2, for a function $f\colon\mathbb{R}^n\to\mathbb{R}$, we ...
- 339
1
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0
answers
55
views
Projection of a real analytic manifold onto subspace is union of real analytic submanifolds
Let $M$ be a compact connected real analytic submanifold of the Euclidean space $\mathbb{R}^{n} \times \mathbb{R}$ and denote by $\pi : \mathbb{R}^{n} \times \mathbb{R} \rightarrow \mathbb{R}^{n}$ the ...
- 19
1
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0
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151
views
Fourier transforms exhibiting symmetries about their critical points
Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
- 113
1
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0
answers
91
views
A bilinear estimates involving critical Sobolev norms
Given $q>1$, consider the critical Sobolev space $W^{n/q,q}(\mathbb{R}^n)$, which fails to embed in $L^{\infty}(\mathbb{R}^n)$. I'm wondering if we can recover some critical estimate by considering ...
- 943
1
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0
answers
218
views
Asymptotic inverses and de Bruijn conjugates (etc.) for complex-valued functions
I recently got my hands on a copy of Regular Variation by Bingham, Goldie, and Teugels ("BGT"), and it's been an absolute revelation for my research. The thing is, my current work centers ...
- 1,284
1
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0
answers
74
views
Well-posedness for a wave equation with degenerate coefficients
Let $\Omega$ be a bounded domain with smooth boundary and consider the following initial boundary value problem:
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
t\partial_t(t\partial_t u)-\...
- 4,143
1
vote
0
answers
52
views
Non constant delay differential equations
Given $\varphi:[0,1] \to [0,1]$ a continuous function, let $(E)$ be the delay differential equation (I am not sure about the terminology, as the delay is non constant): $y'(t) = y(\varphi(t))$. It is ...
- 111
1
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0
answers
84
views
A Riemann Hilbert problem on the unit square
Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
- 4,143
1
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0
answers
52
views
A local base for space of probability measures with Prohorov metric
Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
- 111
1
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0
answers
302
views
Transforming a continuous function into a differentiable function
Given a continuous function $f(x)$ when does there exist a non-constant continuous function $g(x)$ such that $f(g(x))$ is differentiable what about $g(f(x))$?
Does there exist any examples of $f(x)$ ...
- 19
1
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0
answers
56
views
Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
1
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0
answers
44
views
Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system
Consider the initial value problem
\begin{equation}\label{fainait ve}
\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; t \geq 0, \; \;\boldsymbol{f}(\boldsymbol{0}_n) = \boldsymbol{0}...
- 81
1
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0
answers
148
views
Convergence of infinite linear programming
Suppose we have the following linear program (LP1),
$$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{...
- 53
1
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0
answers
79
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Conditions on triangle inequality for integral kernel
Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$.
Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$
A(t,v)=\int_0^{1/v}L(1/t,s)ds,
$$
which is decreasing with $v$ and ...
- 343
1
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0
answers
244
views
Möbius function and polynomials
Let $\mu$ be the Möbius function. It is well known that $\sum_{n|k} \mu(n) = 0$ for $k>1$. What could be said about the polynomials $R_k = \sum_{n|k} \mu(n) x^n$ for $x \in [0,1]$? There does not ...
- 433
1
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0
answers
60
views
About an estimate of an oblique derivative problem of Laplace's equation
Suppose $n\geq 2$, set $B=B_r(0)\subset \mathbb{R}^n$, $B^+=\{x\in B|x_n>0\}$, $H=\{x\in B|x_n=0\}$, Let $u\in C^\infty(B^+)\cap C^1(B^+\cup H)$ be a solution of the following oblique derivative ...
- 11
1
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0
answers
151
views
Log-concavity inequality
Let $x,y,$ and $t$ be fixed real numbers, $1<x<y$, $0<t<1$. Does the following inequality hold for some $c$
$$\frac{\log{(tx+(1-t)y)}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{(sw+(1-s)z)}}{...
- 3,209
1
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0
answers
297
views
Minkowski (box-counting) dimension of generalized Cantor set
I'm trying to solve this problem.
For $0<\alpha, \beta<1,$ let $K_{\alpha, \beta}$ be the Cantor set obtained as an intersection of the following nested compact sets. $K_{\alpha, \beta}^{0}=[0,...
- 111
1
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0
answers
107
views
Comparison of two Fourier transforms
I am looking for $\delta>0$, such that
$$
\delta \int_{-\infty}^{\infty} \exp(its)
{ \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\
\int_{-\infty}^{\infty} \exp(its)
{ \Gamma (it+1)\over \...
- 93
1
vote
0
answers
353
views
Eigenvalues of convolution matrices
Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
- 315
1
vote
0
answers
79
views
Conditions for a function to vanish almost nowhere on its support?
Let $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function and $\mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\}$ its support.
Under which conditions is it true that $f≠0$ (...
- 463
1
vote
0
answers
71
views
Reference request : Convergence of radial basis function interpolation or spline interpolation as points become dense, for a continuous function
Is there any proof for this. Kindly request a reference in case available or any related documents towards this.
PS : I am specifically interested in the case of scattered data (irregularly placed), ...
- 698
1
vote
0
answers
120
views
Natural candidates for super-half-exponential which limit to half-exponential function from above
There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth.
However super-half-exponentials (functions whose composition grows ...
- 1,826
1
vote
0
answers
66
views
Path connectedness of a certain subspace of measurable functions
Note: Functions that differ on a null set are not identified.
Consider the space of measurable functions $[0, 1] \to [0, 1]$ that are continuous exactly on a set of Lebesgue measure $r$ , $0 < r &...
- 2,069
1
vote
0
answers
126
views
Continuity of Helmholtz-Hodge projection in $H^1(\Omega)$
Let $\Omega \subset \mathbb{R}^d$ (for simplicity $d = 2$ or $3$) be a bounded Lipschitz domain. For any vector-valued function $\mathbf{f} \in \mathbf{L}^2(\Omega):= \left ( L^2(\Omega) \right )^d$, ...
- 23
1
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0
answers
60
views
Maximum value of $\int (aF^2(x)g(x)+G^2(x)f(x))dx$ over all $f,g$ densities satisfying $\int F(x)g(x)dx=1/2$
I want to maximise $$I(f,g):=\int_{-\infty}^\infty (aF^2(x)g(x)+G^2(x)f(x))dx$$ where $a>0$ is a given constant, over all possible probability densities $f,g$ satisfying $$\int_{-\infty}^\infty F(x)...
- 319
1
vote
0
answers
203
views
Construction of weight function to satisfy condition on given functional
Consider the following function :
$$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$
Here, $\omega(z)$ is a weight we are going to consider
The following two conditions should meet for $\...
- 375
1
vote
0
answers
47
views
Preservation of $(C,1)$ summability of a function, when multiplied by a bounded strictly decreasing function
Let $f(x)$ be a $(C,1)$ summable function defined on $[1,∞)$:
$$ \lim_{R \rightarrow \infty} \int\limits_1^R \left(1−\frac{x}{R}\right)f(x)\,\mathrm{d}x= L, $$
where $L$ is a finite number.
Let $g(x)...
- 71
1
vote
0
answers
83
views
Gradient descent in $U(n)^r$
I have a function $f:U(n)^r\rightarrow \mathbb{R}$ which I would like to minimize. Here, $U(n)$ is the set of unitary matrices, and $r$ should be considered to be much bigger than $n$. For instance, $...
- 111
1
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0
answers
46
views
Independence of variables in curvilinear coordinate systems
Let $U$ be a connected open subset of $\Bbb{R}^n$, and let $(\xi_1,\dots,\xi_n)$ be a curvilinear smooth ($C^\infty$) coordinate system on $U$. Suppose $1\leq k<n$. A smooth function $f:U\...
- 3,599