All Questions
5,848 questions
1
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1
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111
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How to show such result for generalized $ O(|x|^{-1/2}) $ function?
Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
- 24.2k
1
vote
0
answers
67
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Distribution of zeros for arbitrary Bessel functions
Consider the ODE $x^2 y''+x y' + (x^2-\alpha^2)y = 0$, where $\alpha$ is an arbitrary positive irrational number that is less than $ 2 \pi$. Let $J_{\alpha}(x)$ be a solution to the equation and ...
2
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0
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81
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Extension of a tangent vector field
Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
- 63.9k
2
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0
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207
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Seeking alternative elementary proof instead of applying Lojaseiwicz's inequality for $f(x,y) \geq c (x^2+y^2)^{\frac{M}{2}}$
Let $B\subseteq \Bbb{R}^2$ be a closed ball centered on $(0,0)$ of radius $0<\delta<1$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and contain only one zero in $A$, namely $(0,0)$. In other ...
- 63.9k
10
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1
answer
668
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On Pareto functions
The Pareto principle says that the top 20% of wealthy people people hold over 80% of the wealth. Suppose we had a non-negative function on $\mathbb R^n$ that satisfied this principle on every open ...
- 6,367
2
votes
4
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742
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Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?
Is the generalized hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;-x^2\bigr)$ for $a>-1$ and $x>0$ an elementary function?
How about the positivity, monotonicity, and convexity of the ...
- 1,091
1
vote
1
answer
75
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Lower bound of $\frac{f(x)}{x^{n+1}}$
Let $f:[0,a]\to \Bbb{R}_{\geq 0}$ be real analytic, $a<1$. Furthermore, $f(0) = 0$ and $f$ is strictly increasing on $[0,a]$. Let $n\in \Bbb{N}$ be the smallest positive integer such that $f^{(n)}(...
- 128k
2
votes
1
answer
200
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Subset in $[0,1]^k$ with positive density
Given a positive constant $0<\gamma<1$, does there exists integer $k_0>0$ such that for any integer $k\geq k_0$ the following holds?:
For any $A\subseteq\left[0,1\right]^k$ with the measure ...
CommunityBot
- 1
2
votes
0
answers
135
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Estimating an integral of the Green function in the plane
Suppose $\Omega$ is a bounded, simply connected domain, $z_{0}\in{\Omega}$ and for any $z\in{\Omega}$, $d_{z}:=\text{dist}(z,\partial{\Omega})$. I am interested in understanding the behavior of ...
- 63.9k
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0
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89
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Maximal function on mixed $L^{p}$
Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is
$$
\Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
- 31
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0
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36
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Sufficient condition for interpolation
If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying:
$Z_0=X$, $...
- 101
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1
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80
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Orthogonal space of polynomials
Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and
$$
\int_0^{+\infty} f(t) t^n ...
- 3,425
4
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1
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201
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How much can you improve a Hölder function by composing it with another?
Let $f: [0, 1] \to \mathbb R$ be a continuous function. Define the local Hölder exponent $H(f, x)$ of $f$ at $x \in [0, 1]$ by
$$H(f, x) := \sup\left\{0 \leq \alpha \leq 1\mid\lim_{\delta \to 0_+} \...
- 6,367
1
vote
1
answer
125
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Integrability of modified diagonalizable Jacobian
I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as
$$
J_f(x)=S(x)\Lambda(x) {S(x)}^{-1},
$$
where the ...
- 108k
2
votes
0
answers
188
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A sharp version of a Tauberian theorem
The following Tauberian theorem is true (see Theorem I.11.1 of ''Tauberian theory: A century of developments''). Let $ a_n $ a sequence of real numbers.
If $f(x) = \sum_{n=1}^\infty a_n x^n $ ...
2
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0
answers
207
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History of bump functions
When were the standard bump function examples such as $e^{-1/(1-x^2)}$ first understood, and what was the context or motivation at the time?
As an upper bound I would guess that they must have been ...
- 2,247
1
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1
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99
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Maximum column norm of random $A^{-1}B$
Suppose that $A$ is an $n$ by $n$ Gaussian matrix (each component i.i.d. normal distributed with mean 0 and variance 1). Let $b$ be a $n$-Gaussian vector. Then it could be easily proven that the ...
- 3,741
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126
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A question about associated operator on continuous functions space equiped with L2 norm
For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
- 21.5k
1
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1
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175
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Analyzable functions and accelero-summation
Is there a complete and rigorous, yet concise, definition of what an analyzable function is, along with the related notion of accelero-summation, both in the sense of Écalle? All of the definitions I ...
CommunityBot
- 1
4
votes
1
answer
182
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Extracting a subsequence Cesàro converging to the limsup of the Cesàro sums
Let $X_n$ be a sequence of uniformly bounded random variables — that is, there exists some $K > 0$ such that $|X_n| \leq K$ almost surely for all $n \in \mathbb N$.
Write $\bar X_N := \frac{1}{N} \...
- 12.9k
1
vote
1
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79
views
PDF of the difference of two Beta Prime distribution
I am struggling to find the PDF of the difference of two Beta Prime distribution.
Definition
A random variable is said to have a Beta Prime distribution $\text{B}'(\alpha, \beta)$ with $\alpha, \beta&...
- 128k
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2
answers
3k
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Gluing two diffeomorphisms together
A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have
$\psi(...
- 28k
0
votes
0
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60
views
The size of super level sets and the symmetry on a sphere
Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define
$$
S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}.
$$
Suppose ...
- 869
4
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1
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165
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Dual spaces of Banach-valued $L^{p}$-spaces
Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\...
- 9,007
3
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2
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516
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Several definitions of approximate continuity of real functions
I found the definition of approximate continuity stated as follows:
A function $f:\mathbb R \to \mathbb R$ is approximately continuous at $x_0$ iff there exists a set $A\in \mathcal L$ such that $x_0\...
- 1,309
2
votes
1
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139
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Domain of the infinitesimal generator of a composition $C_0$-semigroup
In the paper [1] the following $C_0$-group is presented,
$$
T(t)f(x) = f(e^{-t} x) , \quad x \in (0,\infty) \quad f \in E
$$
where $E$ is an ($L^1,L^\infty$)-interpolation space. In mi case, I'm just ...
- 6,035
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0
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54
views
Isoperimetric Inequalities in Annular Regions
Let $\Omega$ be an open set in $\mathbb{R}^2$ whose boundary is a rectifiable Jordan curve. Then an old result by Alfred Huber states that
$$
\left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(...
- 434
7
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1
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370
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Duality of $H^1$ and BMO
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
- 183
-1
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1
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204
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Cauchy reduction formula with measure (a variation)
The Cauchy reduction formula conveniently compresses $n$ integrations of a function $F(x)$ into a single integral. Here I am interested in reducing the following "curved-space" ...
- 141
3
votes
2
answers
248
views
Exceptional set for Marstrand's projection theorem
If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $...
- 6,206
3
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2
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434
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Closed form for $ \int_{0}^{1} \dotsi \int_{0}^{1} \frac{x_1^q + \dotsb + x_n^q}{x_1^p + \dotsb + x_n^p} \, \mathrm{d}x_1 \dotsm \mathrm{d}x_n $
I asked this question on MSE, but received no answer.
Recently, reading this problem, I found out that
$$ \lim_{n\to \infty} \int_{0}^{1} \dotsi \int_{0}^{1} \frac{x_1^q + \dotsb + x_n^q}{x_1^p + \...
- 128k
3
votes
1
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346
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Prove that $\lim\limits_{n\to\infty}\left(\sum\limits_{r=0}^{n-1}\sqrt{1-\frac{r^2}{n^2}}-\frac{\pi}{4}n\right)=\frac{1}{2}$
I came across the above question in a mathematical problem. It is not difficult to see that
$$
\lim\limits_{n\to\infty}\left(\frac{1}{n}\sum\limits_{r=0}^{n-1}\sqrt{1-\frac{r^2}{n^2}}\right)=\int\...
- 6,367
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3
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2k
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Every function on reals a sum of two surjective real functions?
From this question, and the answer thereof, we can see that every real valued function on reals is a sum of two injective functions. Is the same true if we replace injectivity by surjectivity.
For ...
- 503
1
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0
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78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
2
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75
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Regularity of solutions to an elliptic boundary value problem
Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
- 969
0
votes
1
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100
views
Projection on a countable union of linear subspace
For any natural number $n$, $V_n$ denotes a closed linear subspace of a $L_2(m)$ space, which is an Hilbert Space, where $m$ denotes a finite measure. Moreover $(V_n)$ is increasing, that is $V_n$ is ...
- 128k
0
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0
answers
22
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Approximation of Lipschitz functions by convex combinaison of Lipschitz functions depending on projections
Let $K\subset \mathbb R^2$ be compact. For any $c>0$, denote by ${\rm Lip}_c(K)$ the collection of Lipschitz functions $f:K\to\mathbb R$ whose Lipschitz constant is less than or equal to $c$. Set $...
- 1,334
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0
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66
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Random elliptical potential lemma
Elliptical Potential Lemma: Let $V_0 \in \mathbb{R}^{d \times d}$ be positive definite and $a_1,a_2,...,a_n \in \mathbb{R}^{d}$ be a sequence of vectors with $||a_t ||_2 \leq L < \infty$ for all $t ...
- 61
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8
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1k
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$n$-th derivative of $\exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right)$
Let $\lambda$ and $\mu$ be two positive real numbers and let denote $f$ the function defined as:
$$\forall x>0,~f(x):= \exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right).$$
I am struggling to find ...
- 17k
0
votes
2
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178
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"Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇"
This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that ...
- 1
4
votes
0
answers
158
views
Measurability of $L^{p}(L^{q})$ integrable functions
Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that
$
\int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty
$
In addition we ...
- 31
7
votes
2
answers
2k
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Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
- 11
6
votes
1
answer
528
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A functional equation
I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \...
- 3,599
3
votes
1
answer
459
views
Limit of an infinite series with quadratic arguments
I have encountered a limiting process on some infinite series. So, I would like to ask:
QUESTION. Assume $n$ is an even positive integer. Is this true?
$$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
- 5,624
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votes
0
answers
53
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Vectors of complex exponentials span $\mathbf{C}^N$
Let $Q = [0,1]\times [0,1]$. Let $1\leq k < \infty$ and $\{(x_l,\xi_l)\}_{l=1}^{k}\subseteq Q$ be such that $(x_i,\xi_i)\ne (x_j, \xi_j)$ for $i\ne j$. Additionally, for $1\leq l \leq k$, let $n_l\...
- 63.9k
122
votes
5
answers
27k
views
Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
- 1,016
11
votes
1
answer
676
views
Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
- 1,190
1
vote
0
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95
views
Distance between two convex sets
Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
- 125
7
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2
answers
324
views
For this continuous non differentiable function $f$ How to determine $\sup\{a\}$ s.t $\lim\limits_{h\to0}\frac{f(x+h)-f(x)}{h^\alpha}=0$ for all $x$?
I asked this question on MSE here.
Define $g(x)= |x|$ for $|x|\in [-1,1]$ , $g(x+2)=g(x)$
$$f(x)= \sum_{n \ge 1} \frac{3^n g\left(4^n x\right) }{4^n}$$
This function is a famous example of a ...
- 6,367
1
vote
0
answers
96
views
Sequential definitions of continuity and related classes
It is well-known that the usual 'epsilon-delta' definition of continuity is equivalent to the sequential definition (assuming countable choice). Less well-known is the sequential definition of ...
- 4,359