All Questions
5,876 questions
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207
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Functions that satisfy a reverse triangle inequality: do they have a name?
Let $f : \mathbb{R}^n \to \mathbb{R}_+$ satisfy
$$f(a) - f(b) \le C f(a - b)$$
$\forall a, b \in \mathbb{R}^n$ for some $C \ge 0$. Is there a name for such functions? (I would be happy to have a name ...
- 337
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0
answers
95
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When does a potential function with given partial derivatives exist
I am looking for the answer to the following question:
Consider an integrable function $f:X\rightarrow X$ with $X$ being a compact subset of $\mathbb{R}^n$. What are the conditions on $f$ so that a ...
- 37
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votes
0
answers
82
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Integral equality involving fractional laplacian
Let $s\in(0,1)$, let $u\in H^s(\mathbb{R}^n)$. For all $\psi\in\mathcal{S}(\mathbb{R}^n)$, let:
$$ (-\Delta)^s\psi(x)=c(n,s)\lim_{\epsilon\to0^+}
\int_{\mathbb{R}^n\setminus B_\epsilon(0)}\frac{\psi(...
- 339
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votes
0
answers
82
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A question about Fourier transform of a function defined by an integral
I have the function:
$$ G_k(x)_=\frac{1}{(4\pi)^{k/2}\Gamma(k/2)}\int_0^\infty e^{-\pi|x|^2/\delta}e^{-\delta/4\pi}\delta^{-(n-k)/2}\,\frac{d\delta}{\delta}, $$
for all $x\in\mathbb{R}^n$ and $k>0$....
- 339
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0
answers
49
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Non-square multiplication operator matrix
Let $A(x), x\in (0,1)$ an $2\times n$ matrix, with $n\geq 2$.
Consider the multiplication operator $K$ on $[L^2(0,1)]^n$ defined as $$K: f(x) \mapsto Kf(x)=A(x).f(x).$$
Intuitively, $$K: [L^2(0,1)]^n ...
- 617
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votes
1
answer
152
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A simple clarification on Riesz decomposition theorem
Let $D$ be a domain of $\mathbb{R}^{m}$ and let
$K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions&...
- 411
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0
answers
53
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Are the densities of a continuous stochastic process locally positive in time?
Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ a (non-degenerate) interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\...
- 463
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146
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Does the following sequence $\{g_n\}$ converge?
Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where
\begin{eqnarray}\label{eqn:constraint1}
f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
- 550
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votes
1
answer
84
views
Vanishing sequence and subsequence with particular decay [closed]
Assume I have a sequence $\{a_m\}$ that is vanishing and strictly positive:
$$
0<a_{m+1}\leq a_m\leq\ldots\leq a_1<\infty, \quad \lim_{m\to \infty}a_m = 0
$$
Is it true or false that this has a ...
- 379
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votes
0
answers
76
views
Constructing a small Radon-Nikodym derivative
Let $u:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ function. Is it possible to (explicitly construction) a function $h$ such that:
$0<h(x)$.
$\int_{x \in \mathbb{R}^n} |h(x)|<\infty$,
$\sup_{x \in ...
- 5,405
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answers
211
views
Understanding Krantz's proof of Hefer's lemma in $\mathbb{C}^2$
Note: I initially phrased the question in a different way, and it did not receive much attention. In the hope to make it more interesting, I have included a (long) introduction to contextualize and ...
- 11
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0
answers
61
views
Weak topology of Gaussian measures
Let us consider a space of Dirac measures $\delta_{x}$ on a Tychonoff space $X$. I know that this space is homeomorphic to $X$. A space of Gaussian measures (weak topology) on some loсally convex ...
- 11
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1
answer
169
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How to obtain envelope equation for oscillating functon? [closed]
Some oscillating function is given. How can I obtain its envelope? For example, for $ \sin x$ I should get $\pm 1$.
Particularly, I am interested in envelope for $$\begin{equation}\frac{(1-x \cot (2 ...
- 145
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votes
2
answers
156
views
Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$
I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that
If for all $t$, we have $$0=A_{0}(t)+A_{1}(t)W(t),$$ where $(A_{0}(t),\mathcal{F}_{t})$ ...
user157103
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0
answers
132
views
How can GL(n) acts on the determinant polynomial?
I'm reading Landsberg's paper, which provide an introduction to geometric complexity theory. At chapter 2 of this paper, the author defined the following objects:
Let $W = \mathbb {C}^{n^2}$, $det_n \...
- 47
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0
answers
263
views
Existence of the inverse Fourier transform, Carr Madan
I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant.
So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...
- 109
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answers
151
views
Making area/volume calculations that use SIA rigorous
There are some intriguing "proofs" using Smooth Infinitesimal Analysis of theorems concerning areas and volumes. Some examples:
A proof that $\sin'(0) = 1$.
A proof that the surface area of a cone is ...
- 4,943
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votes
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answers
73
views
Sufficient and necessary condition for the continuity of an improper integral
Let $f(\cdot) \in \mathscr{C}\left( \mathcal{D}; \mathbb{R} \right)$ where $\mathcal{D} \subseteq \mathbb{R}$ is open with $0 \in \mathcal{D}$ and
$$ f(0) = 0, \quad \forall x \in \mathcal{D}\...
- 81
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77
views
Construction of (general class of) function(s), which sieves out primes, w.r.t. given conditions:
Consider the function $F(x)$ defined in following manner:
$F(n)$ is finite (likely $F(x)\in[0,1]$) if $n$ is prime and zero otherwise:
It has to satisfy following conditions:
(1) $F(x)$ is ...
- 375
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0
answers
86
views
Let $f$ be periodic with a continuous image and $a_n = cn$ for some $c > 0$. When is $\{f(a_n)\}$ dense in the image of $f$?
Let $f:\mathbb{R}\to\mathbb{R}$ be a periodic function with period $T$ and continuous everywhere except perhaps on a countable set, and have an image that's an uncountable subset of $\mathbb{R}$. Let $...
- 277
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84
views
Relation between two matrices associated with a positive definite function
Let $f:\mathbb{R}^N \to \mathbb{R}$ be a positive definite function. Let $$g(h) = \int_{\mathbb{R}^N}f(x)f(h-x)\mathrm{d}x$$ Due to Bochner's and Parseval's theorems, $g$ is also a positive definite ...
- 698
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147
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Approximation of Inductive Tensor Product $C(X) \bar{\otimes} C(Y)$
The following question is from Banach Algebra Techniques in Operator Theory written by Ronald G. Douglas.
Assume both $X, Y$ are Banach spaces and $X \otimes Y$ is the algebraic tensor product. Let ${...
- 307
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255
views
Span of a nonlinear function
Fix vectors $x,y\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. Define $\phi^d: \mathbb{R}^d \rightarrow \mathbb{R}^d$ as applying $\phi$ entrywise (i.e. $\phi^d(x_1, ...
- 159
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162
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Help to understand a limit $\varepsilon\rightarrow 0$ computation on a fluid mechanic paper
In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if
$$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
- 383
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0
answers
51
views
A set of zero harmonic measure 2
Let 𝑉 be a bounded open set in $\mathbb{R}^{m}$, $m\geq 2$, and 𝑊 the interior of the closure of 𝑉. Let 𝐸 be a subset of ∂𝑉∩𝑊 (∂ means boundary) such that:
1) $E$ has positive ($m-$dimensional) ...
- 411
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0
answers
53
views
Reference for inequality for TV of positive measures
Let $\mu,\nu$ be positive measures on some measurable space $(X,\mathcal{F})$.
Let $||\mu-\nu||$ denote the total variation distance between $\mu$ and $\nu$.
Is the inequality
$$ ||\mu-\nu|| \le 2(|\...
- 6,541
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0
answers
59
views
Gauss lemma for nonsmooth metric
$g_{ij}(x)\in L^\infty(\mathbb{R}^n, M^{n\times n})$ is a metric in $\mathbb{R}^n$ satisfying $\lambda |x|^2\leq g_{ij}x^ix^j\leq \Lambda |x|^2$($\lambda>0$&$\Lambda>0$)
Can we find a ...
- 101
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146
views
Derivatives in unusual support domains
Originally posted on Math.StackExchange, here, but I was advised to post it on MathOverflow as it is a research question. Now two final, great answers have been posted, see on Math.StackExchange.
I ...
- 3,098
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0
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82
views
equivalent of an alternating series
Let $d_n=\mathrm{lcm}(1,\cdots,n)$. By the prime number theorem $d_n=e^{n+o(n)}$.
I look for an equivalent of the function $\sum_{n\ge0}(-1)^n\frac{d_n}{n!}t^n$ when $t\to+\infty$. Unfortunately, the ...
- 3,998
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0
answers
84
views
A question about multivariable calculus and optimization
Consider the function $f(x) :\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x)\geqslant 0\; \forall x\in \mathbb{R}$, and has a set of extremum points at $x_{j}$.
Consider the integral :
$$\int_{\bar{...
- 181
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0
answers
85
views
Could the convex hull of $\operatorname{Lip}_1(\mathbb R)$ be dense in $\operatorname{Lip}_1(\mathbb R^d)$?
$\DeclareMathOperator\Lip{Lip}$My problem is slightly different from the title, but I don't have a more straightforward title. Sorry for that.
For $d\ge 1$, denote $\mathbb S^{d-1}:=\{x\in\mathbb R^...
user128095
0
votes
0
answers
40
views
Derivative of a valuation map for a second order ODE
Let $V:\mathbb{R}^+\to\mathbb{R}$ a smooth potential. Given $\lambda\in\mathbb{R}$, let $\psi_{\lambda}$ be the solution to
$$-\psi_{\lambda}''+\lambda V\psi_{\lambda}=0$$
with initial condition $\...
- 943
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0
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91
views
Does $L^1$ convergence preserve the regularity of this sequence of functions?
Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges
$...
- 143
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0
answers
89
views
Families of Riccati Equations for a closed subspace?
Let $N$ be a positive integer, fix constants $a_i,b_{i,j},c_{k,i},d_{k,i,j}\in \mathbb{R}$,
and let $X$ be the subspace of all $f\in C^2(\mathbb{R};\mathbb{R}^d)$ satisfying:
$f=f_0e_0+\sum_{i=1}^N ...
- 5,405
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0
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135
views
Help showing F is weakly lower semicontinuous
Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\...
- 383
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0
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275
views
Are Bernstein polynomials bounded by their coefficients?
I am representing a function by nth order Bernstein polynomial coefficients. I have bounded the coefficients between some $f_{min}$ and $f_{max}$. From what I can see experimentally, it appears that ...
- 1
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0
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81
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Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article
To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question.
We consider the ...
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0
answers
66
views
Is there a dyadic cube decomposition where edge length is comparable to L^2 averages?
Suppose I have some measurable function $f : B_1(0) \to [0,1)$, which is pointwise very small, i.e. $\|f\|_{\infty} << 1$.
I'm looking to construct some kind of dyadic cube decomposition or ...
- 1,179
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votes
0
answers
237
views
Semicontinuity of the Lebesgue measure of images of a family of functions
Let $\mu$ be the usual Lebesgue measure on $\mathbb{R}^m$. Suppose $f:\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^m$ is uniformly continuous, and let $U$ be a measurable subset of $\mathbb{R}^n$....
- 326
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0
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185
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Express $\zeta(n)$, in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral
The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-...
0
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0
answers
55
views
Smooth compactly supported function with good scaling with respect to the fractional Laplacian
Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
user139845
0
votes
0
answers
479
views
What are the sets on which norm-closedness implies weakly closedness?
Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non ...
- 369
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votes
0
answers
159
views
Limiting property of polylogarithm ratio
I try (without success) to figure out what could be the following limit if any...
For real $s$ strictly $> 1$ and $x \rightarrow +\infty$ (x real) the limit of the polylogarithm ratio $\frac{Li_{s-...
0
votes
0
answers
77
views
Energy-minimizing set of discrete points in a bounded domain
Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain.
Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize
$$
\sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|}
$$
over all ...
- 401
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votes
0
answers
54
views
Existence of a solution for the Laplace equation with sub-linear non-linearity
At first, I do apologize if my question is silly. I know that by variational methods it is possible to prove the existence of a solution for
$$
\begin{cases}
-\Delta u = u^p & \Omega \subset \...
- 371
0
votes
0
answers
221
views
Positivity of an integral
Let $f(x)=x-\theta(x)$, and for $x\ge 2$ let $J(x)=\int_x^{\infty}\frac{f(y)(1+\log y)}{y^2\log^2 y}dy$. Furthermore, $J(2)>0$. Suppose $J(x)>0$ when $f(x)=0$. Does it follow that $J(x)>0$ ...
- 1,018
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votes
0
answers
75
views
Does there exist $\alpha>0, \beta\in (0,1)$ such that $\dfrac{\sum_{k=1}^n a_k}{n}\le \alpha (a_1\cdots a_n)^{1/n} + \beta \max_i(a_i)$ holds?
Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following:
Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...
0
votes
0
answers
63
views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user60665
0
votes
1
answer
246
views
Cross entropy loss is not twice differentiable?
I was reading a recent theory paper in machine learning by Kenji Kawaguchi and Leslie Pack Kaelbling
https://arxiv.org/pdf/1901.00279.pdf
and the authors seem to suggest in section 2.2 that cross-...
- 283
0
votes
0
answers
117
views
Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 1