All Questions
1,533 questions with no upvoted or accepted answers
1
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229
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How to prove a concentration isoperimetric inequality for a non-Lipschitz function
Definition $1$. A probability measure $\mu$ on $\mathbb{R}^{d}$ satisfies c-isoperimetry if for any bounded L-Lipschitz $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, and any $t \geq 0$,
\begin{align}
\...
- 79
1
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0
answers
182
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Hardy's inequality proof using Doob's inequalities
Consider a probability space $([0,1],\mathcal{B}([0,1],\lambda),p>1$ and $f \in L^p(]0,\infty[).$
We want to prove Hardy's inequality using martingale theory and Doob's maximal inequalities.
Let $\...
- 573
1
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0
answers
194
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Uniform norm bounds for linear approximation of 1-Lipschitz functions
This problem seems like it should be quite easy/standard, but I've not found a solution written down anywhere.
Consider the set of 1-Lipschitz functions on the $[0,d]$ interval. Define the linear ...
- 181
1
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123
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On Riesz decomposition of Volterra operator
Let $T:L^2((0,1)) \to L^2((0,1))$ be the Volterra operator defined by
$$ Tf(x) = \int_0^x f(t)\,dt.$$
Given any $\lambda\neq 0$ it is well known that there exists some positive integer $n=n(\lambda)$ ...
- 4,143
1
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0
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96
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Limit of alternating sum of factorial moments which diverge
Consider the non-negative, integer valued random variable $X$, and its $i^{\text{th}}$ factorial moment $E_{i}[X]$. Then we have that
$$
P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!}
$$
...
- 640
1
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0
answers
96
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Building random homeomorphisms of the circle
Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as
\...
- 101
1
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1
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125
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Approximation of two densities with a single transformation
Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
- 63
1
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103
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Real analytic map with connected fibers
Let $X,Y$ be compact real analytic varieties. Suppose $Y$ is connected and there is a surjective analytic map $f:X\to Y$ such that each fiber of $f$ is connected. How to prove that $X$ is connected as ...
- 14.3k
1
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71
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Control of solutions to nonlinear elliptic equations away from boundary
Let $\Omega$ be a bounded domain in $\mathbb R^3$ with a smooth boundary. Consider a smooth real valued function $F:\overline\Omega \times \mathbb R \to \mathbb R$ with the property that $\partial_s F(...
- 4,143
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59
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Identification of a limit point of a sequence of solution of ODE
Let $v^0$ and $v^1$ be the following vector fields over $\big(\mathbb{R}_+^*\big)^3$: for $x\in\big(\mathbb{R}_+^*\big)^3$ and $1\leq i\leq 3$,
\begin{align*}
& v^0_i(x)=x_i(x_{i-1}-x_{i+1}) \\
&...
- 449
1
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0
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155
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Study of the class of functions satisfying null-IVP
$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$.
Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property :
$\...
- 307
1
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48
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Optimal regularity of polynomial interpolators
Definitions
We define the "complexity" of any polynomial function $p:\mathbb{R}^n\rightarrow \mathbb{R}^m$ as $m\binom{n+\deg(p)}{n}$ (i.e the dimension of $\oplus_{i=1}^m\,\mathbb{R}[X_1,\...
- 5,405
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75
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finite (non-disjoint) open covering a finite set
A collection of sets $O_{\lambda}$ open covers a set $A$ in $\bf R$ that is bounded and measurable. Assume that $A$ can not be open covered by a finite subcollection of $O_{\lambda}$, let $\epsilon>...
- 11
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48
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Notation for dominating (or uniformly bounded) function
While I developing a new statistical estimator, I wondered is there any good notation for dominating (or uniformly bounded) function.
A situation like this. For some true function $f:\mathbb{R} \to \...
- 284
1
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0
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56
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Differentiability of functions given as integral of some singular kernel
Let $A: \mathbb R_+\to [0,1]$ be $1/2$-Holder continuous and $k: \{(s,t): 0\le s\le t\}\to\mathbb R$ be continuous. Define $f:\mathbb R_+\to\mathbb R$ by
$$f(t):=\int_0^t\frac{k(s,t)}{\sqrt{t-s}}\big(...
- 1,334
1
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0
answers
142
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Tiling a rectangle with squares
Recently, the German science journal Spektrum put online a riddle about squares being tiled to a rectangle:
The task was to determine the area of the rectangle tiled with $8$ squares, of which the ...
- 48.9k
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73
views
Straightening a function supported on a strip
Given a positive smooth function $b:\mathbb{R}^{N+M}\to \mathbb{R}$ which vanishes exactly outisde of some $U\times\mathbb{R}^M$ ($U\subset \mathbb{R}^N$ open), is there another positive smooth ...
- 151
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144
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Zeroes of Mellin transform
There exist a "standard" or canonical way to construct a real valued function whose Mellin transform has a prescribed set of zeroes? Clearly for some set of zeroes this could be impossible ...
- 131
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202
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Function uniquely determined by its values at integer arguments
A smooth enough, slow growing real-valued function $f(x)$, is uniquely determined by its values at integer arguments. I don't remember the name of the theorem and the conditions for this to be true. ...
- 3,098
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46
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Help with a surface of delay differential equations
This question is difficult for me to phrase, as it's very much outside of my mathematical purview. This is a question which intersects directly with my research, but as I work predominantly in ...
- 388
1
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0
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37
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Reference for a general theory of spaces of one-directional rays?
There is a lot of work done on projective spaces, over real, complex numbers or over an abstract field. But I do not find a reference for similar theory where the vectors are projected to the same &...
- 330
1
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165
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distance between two orthogonal projection matrices and its covering number
Let $X, Y \in \mathbb{R}^{n\times p}$ such that $\Vert X- Y \Vert_{HS} \leq \delta$ (Hilbert-Schmidt norm). Also, assume that both $X, Y$ have full column rank. Let the orthogonal prpjection operator ...
- 399
1
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158
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examples of function difficult to prove to be $\geq0$?
I have often wondered whether there has ever come a point in your research,
when you were confronted with an explicit real function $f(x_1,x_2,\ldots,x_n)$ and an explicitly defined compact set $S\...
- 211
1
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35
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How to relate this integration with the integral expansion of special functions?
I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
- 11
1
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198
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Complexity of singular value decomposition using matrix multiplication oracles
Suppose I have an $n\times m$ real matrix $A$, $n\ll m$ with full row rank $(\mathrm{rank}(A) = n)$. I have an oracle that can compute $Ax$ or $A^T y$ for any $x\in \mathbb{R}^m, y\in \mathbb{R}^n$. ...
- 662
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113
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Maximizing a parametric integral over the unit sphere
I am trying to compute the nonnegative quantity
$$
\underset{y\in\mathbb{S}^{d-1}}{\sup}\int_{0}^{t}(\Vert A(\tau)y\Vert_{1}- \Vert A(\tau)y\Vert_{q})d\tau, \quad 1 < q < \infty
$$
where $\...
- 1,538
1
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0
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159
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Generalized functional for solution of PDEs
Asked this on Math Stack Exchange awhile ago but it got ignored then deleted.
To solve a differential equation of one variable, you need constraints equal to the number of derivatives.
For a partial ...
- 131
1
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0
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596
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What is $T T^*$ argument?
During my studying of many papers, some authors used what so-called $T T^*$ argument. I have no clue about this concept (or mathematical tool). Could you please enlighten me with some explanations or/...
- 159
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76
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Geometric series involving the Laguerre polynomials
Let put $\alpha=5$ and $x=3$. Consider the following set given by
$$M=\lbrace \; n \in N, \; \; 0 < |L_{n}^{5}(3)| < 1 \; \rbrace$$
Where $L_{n}^{\alpha}(x)$ is the generalized Laguerre ...
1
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0
answers
134
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Number of solutions to a diophantine equation
Given a positive integer $n$, consider the diophantine equation $4x^2+y^2+4x+y=2n$ with solutions in non-negative integers $x$ and $y$.
Define the proportion
$$\delta_n=\frac{\#\{(x,y)\in\mathbb{Z}^2_{...
- 43.2k
1
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0
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70
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Examples of reaction-diffusion systems with analytical solutions
I want to study how some numerical schemes work on $2$-dimensional reaction-diffusion systems on rectangles with Neumann Boundary conditions and I search for a while for a problem of the form:
$$\...
- 1,759
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110
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Zeroth homology of the complement of a closed set
Suppose $F$ is a closed set in $\mathbb{R}^n$ with $n>1$.
Are there some known conditions that must be imposed on $F$ so that its complement in $\mathbb{R}^n$ has a finite number of components? ...
- 411
1
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0
answers
72
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Compute surface Sobolev norm using local coordinate
For a bounded $\Omega\subset \mathbb{R}^n$ with Lipschitz boundary, there are various definitions of fractional Sobolev spaces (a.k.a. Sobolev-Slobodeckij spaces) on $\partial \Omega$, either by using ...
- 503
1
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0
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72
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Elliptic systems with two dimensions
Let $ \Omega\subset\mathbb{R}^2 $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq 2 $ and $ 1\leq\...
- 1,219
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0
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75
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Existence of solutions to $\alpha(s)=\mathbb P[Y_s>0] + \int_0^s \dot{\alpha}(t)\mathbb P[Y^{t,0}_s>0] dt$
Let $\alpha:\mathbb R_+\to\mathbb R_+$ be a "nice" function with $\alpha(0)=1$. Define the process
$$Y_t=Y_0+t+\int_0^t\frac{dW_u}{1+\alpha(u)},\quad \forall t\ge 0,$$
where $Y_0>0$ has a ...
- 1,334
1
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0
answers
85
views
Boundary estimates for elliptic systems
Let $ \Omega\subset\mathbb{R}^d $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq d $ and $ 1\leq\...
- 1,219
1
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0
answers
99
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Estimate on integral with logarithmic weight
Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, ...
- 839
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0
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76
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Representing a function in terms of higher order differences
I want to write a function in terms of its mollification and higher order
forward differences. Given a function $u:\mathbb{R}\rightarrow\mathbb{R}$ and
$h>0$, we set $u_{h}(x):=\frac{1}{h}u\left( \...
- 411
1
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0
answers
76
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Error estimates for orthogonal polynomial approximation
tl;dr: Are there explicit bounds for the approximation error by orthogonal polynomials?
There are various ways to formulate this question more precisely, so want I emphasize up front that this is a ...
- 19
1
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0
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259
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Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics
Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$?
One ...
- 969
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100
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Exponential decay of a random matrix falling into a ball
Let $A=U\Sigma V^T\in\mathbb{R}^{n\times n}$ be a random matrix defined in the following way: $U,V$ are uniformly distributed on the orthogonal group $O(n)$, $\Sigma$ is a diagonal matrix such that ...
- 1,495
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67
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Solution to recurrence relation from integro-differential dynamical system?
Consider the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=2\int_0^t J_1(x_t-x_s)e^{-\epsilon(t-s)}ds.\tag{1}
\end{equation}
such that $\kappa,\epsilon\in\mathbb{R}$, $t\in\...
- 79
1
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0
answers
98
views
Two definitions of Sobolev spaces and the trace theorem
Let $M=[0,\infty) \times S^2$. We have the regular regular Sobolev space $H^1(M)$.
We also have the space $H^1\bigg([0,\infty); H^1(S^2)\bigg)$. Are those two spaces the same? Does one contain the ...
- 969
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260
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A sufficient condition for multiple differentiability of a function of several variables
While working on some properties of partial derivatives and multiply differentiable functions of several variables, I came across the following Hypothesis 1:
Let $f: \mathbb{R}^n\to\mathbb{R}$, $\...
1
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0
answers
34
views
$L^p$-continuity for discrete linear causal systems
Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by:
\begin{...
- 41
1
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0
answers
142
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Fourier transform of the Bochner-Riesz multipliers
How to obtain the decay of Fourier transform of the Bochner-Riesz multipliers? For $\lambda>0$ define:
$$
\hat{m_{\lambda}}(x)=\int\limits_{\mathbb{R}^d} (1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \...
- 351
1
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0
answers
38
views
Solving an equation containing Laplace transform
Consider the equation
\begin{equation}
\frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}%
(y)(s_{2})=\mathcal{L(}y)\mathbf{(}p),
\end{equation}
where $\mathcal{L}$ is the ...
- 47
1
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0
answers
87
views
Computation of the trace of a convolution operator
I try to understand a particular step on page two from the research paper "A multidimensional analog of a limit theorem of G. Szegö".
https://iopscience.iop.org/article/10.1070/...
- 197
1
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0
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77
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Divergence between random variables after transformation
Let $X$ and $Y$ be random variables with laws $\mu_X$, $\mu_Y$ and $d$ be some $f$-divergence (e.g. KL, total variation, Hellinger). Writing $d(X,Y)$ for the divergence between $\mu_X$ and $\mu_Y$, ...
- 19
1
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0
answers
87
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Estimating the impact of replacing a negative exponential by a truncation of its Taylor series in an integral
Let $f(x)$ be a smooth function that takes both positive and negative values and suppose there exists an increasing sequence of positive numbers $R_i$ diverging to $\infty$ such that
$$\lim_{i \...
- 81