All Questions
5,848 questions
5
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0
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167
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Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,351
0
votes
0
answers
31
views
What is the Fisher information matrix of the von Mises-Fisher distribution?
Assuming the von Mises-Fisher distribution as
$$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$
where $\kappa \ge 0$,...
- 287
1
vote
2
answers
231
views
A real root of a cubic equation for a stationary point
Let us consider the quartic polynomial in $x$
\begin{equation}
F(x) = (2 a p +2)x^4+ (6a(1-a)p^2+(6-12a)p-6)x^3
+ p(2(a-2)(a-1)a p^2 + 3(5a^2-9a+2)p +12a-18)x^2
- p^2 ((a-2)(4a^2 ...
- 371
0
votes
0
answers
54
views
Inequality between inverses of real functions
Let $s\geq 0$ and
$$
f(x)=-\log(x) \quad\text{an}\quad g(x)= \log(\log(1/x)+1)$$ for all $x\in(0,1)$. Is there exists $C_s>0$ such that for all $x,y\in(0,1)$,
$$
f^{-1}(s g(x)) \cdot f^{-1}(s g(y))...
- 39
54
votes
4
answers
12k
views
Everywhere differentiable function that is nowhere monotonic
It is well known that there are functions $f \colon \mathbb{R} \to \mathbb{R}$ that are everywhere continuous but nowhere monotonic (i.e. the restriction of $f$ to any non-trivial interval $[a,b]$ is ...
- 35.5k
1
vote
2
answers
209
views
Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$
Let $f=\sum_{i=1}^n c_i 1_{\Delta_i}$ be a simple function on $\mathbb{R}^d$, where $c_i\in\mathbb{C}$. Then we can find sequnces of continuous functions $\{f_k^{(i)}\}$ for each $i=1,\ldots,n$ such ...
- 6,367
1
vote
0
answers
100
views
Difference of two completely monotonic functions
We know by the Hausdorff-Bernstein-Widder theorem that any completely monotonic function on the positive half line $[0, \infty)$ is given by the Laplace transform of a positive Borel measure on $[0, \...
CommunityBot
- 1
7
votes
5
answers
513
views
Probability of $\operatorname{Bin}(n,p)=\operatorname{Bin}(n,q)$ is decreasing when $n$ increases
$\newcommand{\Bin}{\operatorname{Bin}}$I would like to show that $\mathbb P(\operatorname{Binomial}(n,p) = \operatorname{Binomial}(n,q))$ decreases when $n$ increases for a fixed pair $(p,q)$. This ...
0
votes
0
answers
237
views
Pair of real functions satisfying some conditions
Consider two functions $\psi$ and $\varphi$ defined on the interval $(0,c)$ where $c\in(0,+\infty)$ and they exhibit the following characteristics:
$\psi$ and $\varphi$ are continuous, positive, and ...
- 39
15
votes
1
answer
1k
views
Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
CommunityBot
- 1
2
votes
0
answers
29
views
Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
1
vote
0
answers
58
views
Asymptotics of Jacobi form
What are the large $x\in\mathbb R$ asymptotics of $f(x)=\theta_3(c_1+c_2 x^3,e^{-x^2})$ where $c_1,c_2$ are a pair of complex numbers (say, $\Re(c_2)>0$ and $\Im(c_2)<0$), and $\theta_3(a,b)=\...
- 11
0
votes
0
answers
63
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Calculating hyperbolic Fourier series
Question:
is it possible to uniquely express functions locally as infinite sums of hyperbolic sines and cosines
$f(x)=\sum\limits_{i=0}^\infty \alpha_i\sinh(i\cdot x)+\beta_i\cosh(i\cdot x)$
or even ...
- 13.2k
8
votes
4
answers
1k
views
For what sets does the Lebesgue Differentiation Theorem hold in one dimension?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
- 34.7k
6
votes
2
answers
225
views
On a trigonometric inequality by Huygens
The following inequality, ascribed to Huygens, appeared in this post:
\begin{equation*}
1-\frac43\,\frac{\sin^3\theta/2}{\theta-\sin\theta}
>(1-\cos\theta/2)\Big(\frac35-\frac3{1400}\frac{\...
- 1
7
votes
1
answer
580
views
Sobolev spaces are smooth? Their dual is strictly convex?
Do you know any reference which says something about the:
Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton.
...
- 2,670
0
votes
0
answers
56
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What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?
I have asked this here. Due to inactivity and no satisfying answers, I am asking here. Hope that's okay.
We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$
is $n$ (thanks to this ...
- 113
4
votes
1
answer
255
views
Asymptotic behavior and of an integral on a d-dimensional torus
I am trying to evaluate the asymptotic behavior of the following integral as $t \to \infty$:
$$
I(t; \mathbf{v}) = \int_{[-\pi, \pi]^d} \frac{\sin(t f(\mathbf{k}))}{\sin(f(\mathbf{k}))} e^{i t \mathbf{...
- 81
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 ...
- 60.5k
1
vote
1
answer
76
views
Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius
I have a smooth 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 curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
- 128k
1
vote
1
answer
204
views
A question on Borel measurability
Let $(X, \mathcal{B}_{X}, \mu)$ be a measure space. Here, $\mu$ is an infinite Borel measure and $\mu$ is not $\sigma$-finite. Let $\pi$ be surjective Borel measurable map form $(X, \mathcal{B}_{X}, \...
- 2,830
0
votes
1
answer
255
views
Carleson's theorem: proof of a lemma
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...
1
vote
1
answer
62
views
Integrability in the product space can follow from a property of the Nemytskii operator?
Let's say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where ...
- 2,670
0
votes
0
answers
116
views
Integral of a measurable function with parameter is measurable?
Say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ is an open set, is a function such that:
$f(x,\cdot)\in L^1_{\text{loc}}(\mathbb{R})$ for a.a. $x\in\Omega$
$f(\...
- 12.9k
1
vote
0
answers
175
views
Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
2
votes
1
answer
246
views
Ramsey type property of the Lipschitz constant
The following problem was proposed by Pietro Majer as an extension of an earlier question of mine on Lipschitz functions.
For $f$ a Lipschitz function on $\mathbb R^n$, we denote by
$$\text{Lip}(f, U) ...
- 60.5k
0
votes
0
answers
79
views
Is the Bures metric equivalent to the Euclidean one?
Let $K=\mathbb R$ (reall numbers) or $K=\mathbb C$ (complex numbers). Define $\mathcal M_n$ to be the space of $n\times n$ matrices $A=(a_{i,j})_{1\le i,j\le n}$, with $a_{i,j}\in K$. Let $\|\cdot\|$ ...
- 1,334
0
votes
0
answers
63
views
Arrangements of fixed $k$-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
- 12.9k
4
votes
1
answer
254
views
On the Lipschitz constant outside the stretch set
Let $f: \mathbb R^n \to \mathbb R^m$ be a Lipschitz map. We define the local Lipschitz constant $Lf$ of $f$ at $x \in \mathbb R^n$ by
$$Lf(x) := \lim_{r \to 0_+} \text{Lip}(f, B_r (x)),$$
where $\text{...
- 6,367
9
votes
1
answer
845
views
Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
I asked this question on MSE here.
Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
- 541
5
votes
1
answer
279
views
Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?
Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $:
$$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
- 91.8k
11
votes
2
answers
8k
views
About the Fourier transform of the logarithm function
I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
- 201
1
vote
1
answer
60
views
Does Monotone (linear) convergence of iterates imply monotone (linear) convergence of function values?
I am considering a proof that would require a certain connection between convergence of iterates and corresponding function values: Consider an algorithm with iterates $\left\{{\mathbf{x}}^k\right\}_{...
- 12.9k
5
votes
1
answer
229
views
Intersection between Lipschitz domains
Let $\Omega\subset\mathbb{R}^N$ be an open, bounded and connected Lipschitz domain. Is it true that we can find some $R>0$ such that any $N$-dimensional open ball $B(x,r)$ with $r\leq R$ that ...
- 1,759
3
votes
0
answers
95
views
Is it true that p-integrable function can be written as a convolution of an integrable function and p-integrable function?
We know that convolution of an integrable function with an $p$-integrable is an $p$-integrable function. This follows from Young's inequality.
My question: Is it true that $L^p(\mathbb{R}^n)\subseteq ...
- 31
2
votes
0
answers
138
views
Sufficient initial conditions for "non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
- 63.9k
2
votes
1
answer
206
views
Deriving an inequality for the integral of maximum indicator functions under measure-preserving transformations
Let's denote the measure space by $(X, \mathcal{B}, \mu)$ and the measure-preserving transformation by $T: X \to X$. Let $A \in \mathcal{B}$ be a measurable set with $0 < \mu(A) < \infty$. Let $...
- 578
7
votes
1
answer
290
views
Equivalence of omniscience principles for natural numbers and analytic omniscience principles for Cauchy real numbers
In constructive mathematics, a proposition $P$ is decidable if $P \vee \neg P$, and a proposition is stable if $\neg \neg P \implies P$. We have the following principles of omniscience for the natural ...
- 32.5k
1
vote
1
answer
187
views
Bound the distance between two vectors on the probability simplex
Let $a,b$ be two vectors with strictly positive elements and $\delta = 1 - \frac{\langle a,b \rangle}{\|a\|\|b\|}$. Bound the following optimization problem as a function of $\delta$
$$\sup_{x>0} \...
CommunityBot
- 1
8
votes
0
answers
414
views
For $f$ Lipschitz with $|\nabla f| = 1$ a.e., what is the supremal Hausdorff dimension of the set on which $\varepsilon< |\nabla f| < 1-\varepsilon$?
Let $f$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere.
Let $\varepsilon \geq 0$. What is the supremal Hausdorff dimension of the set on which $f$ is differentiable with $\varepsilon &...
- 6,165
3
votes
1
answer
248
views
Can any function in $C^\alpha$ be approximated in $C^{\alpha^-}$ by singular functions?
For every positive $\alpha < 1$, we consider the space $C^{\alpha}$ of Holder continuous functions of order $\alpha$ on $[0, 1]$, equipped with the norm
$$\|f\|_{C^\alpha} := \sup|f| + \sup_{x, y \...
- 60.5k
0
votes
0
answers
66
views
convolution of the fundamental solution with the homogeneous solution
I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero?
Let $U$ and $E$ ...
CommunityBot
- 1
6
votes
1
answer
413
views
Analyticity of $f*g$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on $\mathbb{R}^*$
Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ analytic on $\mathbb{R}\setminus\{0\}$ but non-analytic at $x=0$. Is the convolution (...
- 12.9k
1
vote
1
answer
93
views
Bound measure of difference of advected sets by norm of difference of vector fields
Consider two smooth vector fields $v$ and $u$ in $\mathbb{R}^n$, and a smooth set $\Omega$. Consider the flow of $\Omega$ via $v$ and $u$ for a time $T$, namely let
$$ \Omega_v =\{x(T, x_0) | x \text{ ...
- 2,772
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.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 578
3
votes
1
answer
263
views
Hölder continuity in time of heat semigroup
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\|\ell\|...
- 1,190
0
votes
0
answers
54
views
Weyl equidistribution for a periodic $L^2$ function
Let $\alpha $ be a fixed irrational number. For a function $g:\Bbb R\to\Bbb C$, define $$g^*(x)=\sup_{N\geq 1} \frac{1}{N} \sum_{n=1}^N |g(x+\alpha n)| ,$$
and assume that there is a constant $C>0$ ...
- 213
1
vote
1
answer
120
views
Does Gaussian heat kernel ensure $\int_{\mathbb R^d} (1+|x|) \sqrt{\ell_{t_0} (x)} \, \mathrm d x < \infty$?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\int_{\bR^d} (1+|x|) \sqrt{\ell (x)} \diff x < ...
- 578
4
votes
1
answer
259
views
Hausdorff dimension of the zero set of the gradient of an eikonal function
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ is ...
- 6,165
5
votes
2
answers
2k
views
Elementary proof of the uniqueness of smooth structures on $\mathbb{R}$
Is there any 'elementary' proof of the uniqueness of smooth structures on $\mathbb{R}$? By elementary, I mean that the proof does not use any sophisticated topological machinery. In particular, I'm ...
- 63.9k