All Questions
739 questions
2
votes
0
answers
470
views
Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?
If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
- 10.1k
2
votes
1
answer
301
views
A question about pushforward measures and continuous Borel isomorphisms
It is fairly well known that if $\mu$ and $\nu$ are nonatomic measures on the standard Borel spaces $(X,B)$ and $(Y,C)$ such that $\mu(X)=\nu(Y)$. If $X$ and $Y$ are uncountable, then there exists a ...
- 33
2
votes
2
answers
667
views
Power series of ratio of Gamma functions
Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...
- 39
2
votes
1
answer
150
views
When is it true that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?
First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...
- 870
2
votes
1
answer
206
views
Bound for zero-crossings of heat equation
I am considering the following problem.
Let $\mathcal{P}$ the classical heat-diffusion problem:
$$\mathcal{P} : \left(\partial_t u (t,x)=\frac{1}{2}\partial_{xx}^2u(t,x)\text{ with }u(0,\cdot) = f(x)\...
- 393
2
votes
1
answer
143
views
Proving convexity of the expected logarithm of binomial distribution
I would like to prove that the following function, for an arbitrary integer $n$:
\begin{equation}
\begin{split}
f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\
& = x \cdot \sum_{k=0}^{n} \...
- 23
2
votes
1
answer
260
views
Non-Fourier complete orthogonal basis?
The Fourier Transform (FT)
Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero
Is invertible: info-preserving, has inverse function
Is energy-...
2
votes
1
answer
157
views
Inequality with decreasing rearrangement and non-decreasing function
This question is a continuation of the question here.
Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$...
- 459
2
votes
1
answer
186
views
Local equality of functions implies global equality?
The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
- 1,117
2
votes
1
answer
307
views
Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 459
2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
2
votes
1
answer
118
views
Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius
This is a follow up from this question.
I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
- 765
2
votes
1
answer
211
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 825
2
votes
1
answer
105
views
Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$
For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define
$$
\begin{split}
K_n(a,t) &:= \inf_{x \in \mathbb R^n} \|x-a\|_2 + t\|x\|_1,\\
M_n(a,t) &:= ...
- 6,853
2
votes
1
answer
168
views
Validity of formula $u(x)=\frac{1}{4\pi}\int_G \nabla_y \frac{1}{\lvert x-y \rvert} \times \omega(y) \, d^3y +A(x)$ for periodic boundary case
I think it is better to provide context in which the previous question Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? has been raised.
The motivation is the ...
- 3,477
2
votes
1
answer
143
views
Roots of rational function
Sorry, I asked a similar question yesterday which contained a mistake in the question posed, here is the real question.
Let $(x_n)_{n=1}^N$ be a sequence taking values in $[1,2]$ with the property ...
- 73
2
votes
1
answer
154
views
Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$
Let $\alpha \in (0, \infty)$ and $\beta \in (0, 1]$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that
$$
f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(...
- 825
2
votes
1
answer
61
views
$K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?
This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised.
As before, let us ...
- 3,477
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
2
votes
2
answers
218
views
Convergence for a non-linear second order difference equation
In my work, I need to study the convergence of sequence defined by the non-linear recurrence relation
$$
u_0,u_1>0, \qquad \forall n\in \mathbb N, \; u_{n+2}=a\ln(1+u_n)+b\ln(1+u_{n+1})
$$
with ...
- 1,503
2
votes
0
answers
190
views
What is the smallest dimension that allows finding $n$ points at distances $|x_i-x_j|^{\delta/2}$, where $0<\delta<1$, and $x \in \mathbb{R}^n$?
Let $x_1,\cdots,x_n \in \mathbb{R}$, are there $\xi_1,\cdots,\xi_n \in \mathbb{R}^s$, such that
$|x_i-x_j|^{\delta}=||\xi_i-\xi_j||^2$, $0<\delta<1$, what is the smallest $s$ to guarantee the ...
- 178
2
votes
1
answer
154
views
Is the optimum of this problem convex in the constraint parameter?
Let $f:\mathbb R^+ \to \mathbb R$ be a smooth function, satisfying $f(1)=0$, and suppose that
$|f|$ grows with the distance from $1$: $|f(x)|$ is strictly increasing when $x \ge 1$, and strictly ...
- 6,741
2
votes
2
answers
953
views
Differentiability of Nemytskii operator on Sobolev space
I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
- 201
2
votes
1
answer
144
views
Do we have independence if we let the indices of the events increase?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
- 247
2
votes
1
answer
451
views
Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$
$g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
- 167
2
votes
1
answer
433
views
bounding the absolute value of a trigonometric polynomial
Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$
\begin{equation*}
f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})}
\...
- 859
2
votes
1
answer
193
views
A question on the partial sum of infinite doubly stochastic matrix
Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Is the following statement true ?
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^na_{ij} >0
$$
Any reference or comment on this is ...
- 171
2
votes
1
answer
249
views
linear recurrence inequality
Given two real analytic functions, $g(x)$ and $f(x)$, on an open interval $I\subset \mathbb{R}$, it is obvious that $g(x) \leq f(x)$ does not imply $g_n \leq f_n$ (here $g_n = [x^n] g(x)$ denotes the $...
- 145
2
votes
1
answer
273
views
Is it always possible to partition $[a,b]\times[c,d]$ into disjoint blocks $D_{ij}$ s.t. $\left.f\right|_{D_{ij}}$ is bijective?
Consider the function given by $f:[a,b]\times[c,d]\to[0,1]^{2}$ such that $0\leq a < b \leq 1$, $0 \leq c < d \leq 1$.
Moreover, we do also have that $f\in C^{1}([a,b]\times[c,d],[0,1]^{2})$ and ...
user127351
2
votes
1
answer
160
views
Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?
The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form.
Is there possible an extension of real/complex numbers in which logarithms and ...
- 10.1k
2
votes
0
answers
77
views
Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 5,405
2
votes
2
answers
255
views
Do we have a name for this space?
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Consider the class
$$
\mathcal{F}=\{f\in L^{1}(\Omega):\exists C>0 \text{ s.t. } \int_{U}|f|\leq C\sqrt{|U|},\text{ for any }U\subset \Omega.\...
- 21
2
votes
1
answer
389
views
Intersections of algebraic surfaces with hypercubes of a $d$-dimensional grid
This is a follow-up question, to a question I asked earlier.
See Algebraic curve intersecting square-grid.
Consider $n^d$ unit hypercubes in $d$-dimensional Euclidean space
tightly packed in the ...
- 479
2
votes
1
answer
289
views
On semi-discrete Wasserstein distance
Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$, where $\nu$ has a bounded support. Consider the $2-$Wasserstein distance below:
$$...
- 345
2
votes
2
answers
1k
views
Approximation of smooth compactly supported functions on $\mathbb{R}^2$ using sums of products of one variable functions
Let $f \in C^{\infty}(\mathbb{R}^2)$ be smooth and compactly supported. Can we approximate $f(x,y)$ by sums of the form $\sum_{i=1}^m g_i(x) h_i (y)$ where $g_i, h_i \in C^{\infty}(\mathbb{R})$ are ...
- 33
2
votes
0
answers
232
views
Is an orthogonal projection of a Lipschitz domain still a Lipschitz domain?
Let $\mathcal{X}\subseteq\mathbf{R}^n$ be a Lipschitz domain, i.e., for each $x\in\partial\mathcal{X}$, there exists a radius $r_x>0$ and a Lipschitz continuous function $F^x:\mathbf{R}^{n-1}\to\...
- 21
2
votes
3
answers
3k
views
dual space of a subspace of the space of bounded measures
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
- 1,835
2
votes
1
answer
157
views
Is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane almost surely?
Let $d = 2$. With probability $1$, is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane?
user75246
2
votes
1
answer
101
views
Convergence of energy of Sobolev functions near the boundary
Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. $h\in W_0^{1,2}(B_0(1))$. For $r\in (0,1)$, define a function $f_r(x):[0,1]\rightarrow \mathbb R$ by
\begin{equation}
f_r(x):=
\begin{cases}
...
- 169
2
votes
0
answers
197
views
Orthogonality relation in $L^2$ implying periodicity
Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties
$$
\int_0^{2\pi} e^{i\theta(t)} dt=0.
$$
Geometrically this means ...
- 405
2
votes
1
answer
162
views
On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$
Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
- 1,209
2
votes
2
answers
4k
views
a limit of the laplace transform and its derivative
If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
2
votes
2
answers
257
views
Reference request on Min-Max theorem
Consider the following min-max problem
$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$
where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
user128095
2
votes
1
answer
437
views
If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set
If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...
- 167
2
votes
1
answer
107
views
Lower bounds on translates of a function over a compact set
Let $f\in L^p(\mathbb{R})$ and define $f_\theta(x)=f(x-\theta)$. Let $K\subset\mathbb{R}$ be a compact set. I would like to compute (or at least lower bound) the following:
$$
\inf_{\theta\ne\theta'\...
- 45
2
votes
1
answer
677
views
Lipschitz continuity of an implicit function
Let $z=F(x,y)$ be a function from $\mathbb R^d\times \mathbb R$ to $\mathbb R$ and $z=F(x,y)$ is Lipschitz continuous. Assume that for any $x\in\mathbb R^d$, there is a unique $y$ such that $F(x,y)=0$....
- 601
2
votes
0
answers
144
views
Does this geometric PDE have a solution?
Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes.
Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
- 6,741
2
votes
0
answers
275
views
Smoothness of coefficients of remainder term in Taylor expansion
Given a $C^{k}$ function $f:\mathbb{R}^d\to\mathbb{R},$ we can use Taylor's theorem to write it as
$$f(x)=\sum_{|\alpha|\le k-1} c_\alpha x^\alpha + R(x),$$
where $R$ is $C^k$ and can be expressed ...
- 203
2
votes
1
answer
131
views
Uniformly Converging Metrization of Uniform Structure
This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...
- 12.5k
2
votes
1
answer
497
views
Truncated Euler products, Dirichlet eta function, and convergence issues
Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as
$$W(\sigma,...
- 3,098