All Questions
5,629 questions
5
votes
1
answer
2k
views
Baire's simple limit theorem "almost everywhere"
The Baire's simple limit theorem states that if the functions $f_n : \mathbb{R} \to \mathbb{R}$ are continuous and converge everywhere to a function $f$ then $f$ has a dense set of continuity points. ...
- 115
0
votes
1
answer
95
views
Estimating pointwise multiplication conjugated by a Fourier multiplier
I asked this question first on MSE but there was no activity.
Let $m(D)$ be a Fourier multiplier and $f$ a known function. I'm trying to estimate the operator
$$Tu=m^{-1}(D)(f(x)m(D)u)$$
in say $H^s$....
- 247
11
votes
1
answer
1k
views
The Hölder inequality for fractional order Sobolev seminorm?
This question is post on MSE a week ago. I move it here to draw more attention.
Let $u\in C^\infty(\bar I)$ be given where $I=(0,1)$. Define
$$
t(\alpha):=\left(\int_I\int_I \frac{|u(x)-u(y)|^\alpha}{...
- 679
3
votes
0
answers
155
views
asymptotics of the largest real root
Suppose you have a family of polynomials
$$P_n(x)=\sum_{k=0}^n(-1)^ka_k^{(n)}x^k$$
for $n=0,1,2,\dots$.
Further assumptions:
(1) the coefficients $a_k^{(n)}$ are recursively related to the $a_j^{(n-1)...
- 43.2k
4
votes
2
answers
562
views
Reference request: concave/convex envelope
I'm seeking the references concerning on the regularity analysis of concave envelopes, i.e. given some measurable function $f:\mathbb R^d\to\mathbb R$ that is bounded from above ($d=1$ or $d\ge 1$), ...
- 1,835
4
votes
1
answer
1k
views
Product of two non-measurable sets
Let $A\subset\mathbb{R}^p$ and $B\subset\mathbb{R}^q$, it’s not difficult to show that $$m^*(A\times B)\leq m^*(A)\cdot m^*(B)$$, where $m^*()$stands for the outter measure in Lebesgue meaning.
If A ...
- 549
8
votes
1
answer
5k
views
Lax's proof of the change of variables theorem
I am teaching a course on Multivariable Calculus for Graduate students. I came across this nice article by Lax where a special case of the change of variables theorem is proved:
Theorem. Let $f:\...
- 1,169
1
vote
0
answers
128
views
determine when $e^{ikx}$ can be boundary value of a holomorphic function
Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.
My question is, for what curves $...
- 31
8
votes
2
answers
634
views
Existence of a uniformly continuous function $g$ on $\mathbb{R}$ where $f = g$ a.e.?
Suppose $f \in L^\infty(\mathbb{R})$, $f_h(x) = f(x + h)$, and$$\lim_{h \to 0} \|f_h - f\|_\infty = 0.$$Does there exist a uniformly continuous function $g$ on $\mathbb{R}$ such that $f = g$ almost ...
- 113
1
vote
0
answers
311
views
Estimating an integral involving Bessel functions
I would like to preface this question by saying that I have asked a series of questions on this topic on Math Stack Exchange, but have almost never received any fruitful responses, with the exception ...
- 161
3
votes
1
answer
193
views
Inequality of a concave function
Let $G:\mathbb R\to\mathbb R$ be a concave function, define $G_{\epsilon}: \mathbb R\to\mathbb R$ by
$$G_{\epsilon}(x)~~:=~~\max_{y\in [x-\epsilon, x+\epsilon]}G(y).$$
My question is the following: ...
- 1,835
1
vote
0
answers
53
views
Given a fixed convex domain $\Omega$ in 3D, for what value $c$ the function $f(c) := \int_{\partial \Omega} |x-c| d \sigma_x$ gets its minimum?
Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^3$, then consider the following minimization problem:
$$\inf_{c \in \overline{\Omega}} f(c), \quad f(c) := \int_{\partial \Omega} |x-c| ...
- 1,350
3
votes
1
answer
139
views
Bilipschitzian maps and densities
Let $ A \subseteq \mathbf{R}^{m} $ and suppose that $ \mathbf{R}^{m} \setminus A $ has $ m $ dimensional density equals $ 0 $ at a point $ a \in A $. Let $ B \subseteq \mathbf{R}^{m} $ and let $ f : A ...
- 541
1
vote
0
answers
123
views
Generalization of concave envelope
Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
- 1,835
6
votes
1
answer
728
views
Intuition behind the non-Borel Lusin example
Among the concrete examples of a non-borel subset of $\mathbb{R}$,
I know only the Lusin example.
This is the set $L$ of all irrational numbers whose
continued fraction representation $(a_0,a_1,\...
- 549
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k
13
votes
3
answers
820
views
Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?
This question is related to another one that I asked two days ago.
Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with
the following two properties?
The ...
- 1,396
2
votes
0
answers
228
views
Integrating an n-fold Cauchy product of a Fourier series
I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found here.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be ...
- 161
1
vote
0
answers
58
views
A question on Integral inequality
Let $0 < \epsilon < 1$. Consider $\{a_n\}_{n \geq 1} \in l_2$ and $L(t) = 1+\epsilon t$. Let $x$ be fixed such that $0 < x < L(t)$. Does there exist $\tau \geq 0$ such that the following ...
- 11
3
votes
1
answer
144
views
Operator norm of almost mathieu operator
The almost Mathieu operator has become famous since it is the central object of the ten martini problem.
In this paper here a bound on the operator norm is given. Although the bound is of course ...
- 31
3
votes
1
answer
156
views
How many steps do I have tto complete? Recursive sequence
Maybe it's a simple question... Fix a positive $N$. Let $a_{n}$ be the recursive sequence:
$$a_{1} = N$$
$$a_{n}=a_{n-1}-(a_{n-1})^{\frac{2}{3}}.$$
How many steps do I have to complete in order to ...
10
votes
2
answers
1k
views
Can the integration of integrable sections of a measurable function of two variables ever result in a non-measurable function?
I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct ...
- 1,396
1
vote
1
answer
131
views
Maximum of gradient of convex functions [closed]
The question comes from the page 472, Elliptic partial differential equations of second order/ David Gilbarg, Neil S. Trudinger. In one dimension it's obviously true, but it seems more involved in ...
- 31
1
vote
2
answers
371
views
Weak convergence in vector-valued Hilbert space
Let $V$ be a separable Hilbert space and define $X=L^2(0,T;V)$. Then $u_m\to u$ weakly in $X$ means
for every $v\in X'=L^2(0,T;V')$
$$
\int_0^T\langle v(t),u_m(t)\rangle\ dt\to\int_0^T\langle v(...
user14319
9
votes
3
answers
383
views
convergence of 2nd eigenvalue
Fix $0<h_1<h_2<h_3<1$ reals. All matrices below are $3\times3$ real.
Suppose the sequence of matrices $M(n)$ are symmetric positive definite and these converge (point-wise) to a symmetric ...
- 43.2k
0
votes
2
answers
144
views
Optimization function of two variables
Let $A, B, C, D \in \mathbb{R^*_+}$.
Is it possible to solve
$$
\max_{ \substack{0 \leq x\leq A \\ 0\leq y\leq B}} \frac{1+x+y}{(1+Cx)(1+Dy)}
$$
The KKT conditions give for an extrema $(x^*,y^*)$
...
user83947
0
votes
0
answers
42
views
What (analytical or numerical) method can I use to solve scalar optimal problem?
I got the following optimization problem in mind and I am looking for some (analytic or numerical) methods to solve it. Can anyone have any ideas? Here is problem
\begin{aligned}
& {\text{...
- 221
4
votes
0
answers
147
views
The asymptotic behavior of the ratio between the largest two of $n$ i.i.d. chi-square random variables
My question is about the asymptotic behavior of the ratio between the largest and second largest values of $n$ independent chi-square random variables.
Let $X_1, \ldots, X_n$ be $n$ independent and ...
- 1,127
21
votes
3
answers
3k
views
Approximate intermediate value theorem in pure constructive mathematics
The ordinary intermediate value theorem (IVT) is not provable in constructive mathematics. To show this, one can construct a Brouwerian "weak counterexample" and also promote it to a precise ...
- 66.8k
2
votes
1
answer
1k
views
From bounded variation to 1-Lipschitz function
Let $f\colon[0,1]\to \mathbb{R^2}$ be continuous such that $f(0)=f(1)$.
If want to find a 1-Lipschitz function $g : [0,a]\to f([0,1])$ such that $g(0)=g(a)$ and $g$ is surjective ($a>0$).
I had ...
- 21
0
votes
2
answers
139
views
On the existence of $ \lim_{x \to 0^{+}} \frac{\log(f(x))}{\log(x)} $ under some constraints
I am considering a smooth-enough real-valued function $ f: (0,1) \to (0,\infty) $ such that
$ f $ is decreasing,
$\lim_{x\rightarrow0^{+}}f(x)=\infty $,
$ x \mapsto x^{2} f'(x) $ is decreasing,
$\...
- 301
5
votes
1
answer
780
views
Do real analytic functions on $\mathbb{C}\mathbb{P}^n$ form a Noetherian ring?
Question: Is the ring of real-analytic functions on $\mathbb{C}\mathbb{P}^n$ (real valued)
a Noetherian ring?
References or counterexamples are welcome.
I know that the ring of germs of holomorphic ...
- 589
2
votes
0
answers
45
views
Maximizing the sum of a decreasing function over a separated set
Fix $d>0$. Let $f:[0,\infty)\to(0,\infty)$ be a decreasing function of $x$ for $x\geq d$. Let $S_d\subset\mathbb{R}^n$ represent a set of points containing the origin such that the (Euclidean) ...
- 21
1
vote
1
answer
186
views
A problem involving power series
We define an entire function on $\mathbb{C}^m$ by
$$
f(z_1,\cdots,z_m)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}t^{2n}(z_1^2+\cdots+z_m^2)^n,
$$
here $t$ is some (positive) real number. Of course, $f(x)=...
- 1,906
3
votes
0
answers
1k
views
Concentration of Sub-exponential random Vectors
I was wondering if there is a similar definition of multivariate sub-exponential distribution as the sub-Gaussian case.
Specifically, a random vector $X \in \mathbf{R}^d$ is sub-Gaussian if
\begin{...
- 1,127
1
vote
1
answer
310
views
inequality involving increasing functions
Let $a_k$ and $b_k$ be ascending positive numbers for $1\leq k \leq K+1$.
If it is known that
$$\frac{K\left(\exp\left(\frac{1}{K}\sum_{k=1}^K b_k\right)-1\right)}{\left(\sum_{k=1}^K \sqrt{a_k} \sqrt{\...
4
votes
0
answers
95
views
Approximating martingales given marginal distributions
Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e.
$$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$
and increasing in ...
- 1,835
2
votes
0
answers
63
views
Sensitivity of a function against its random arguments
Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
- 482
3
votes
0
answers
228
views
Sub-multiplicative function in expectation or pointwise? [closed]
Consider the function that satisfies
$$ \mathbb{E}[f(X)f(Y)]\leq \mathbb{E}[f(XY)],$$
where $X\in\mathbb{R}$ and $Y\in\mathbb{R}$ are Gaussian random variables with mean $0$ and variance $1$, and ...
11
votes
3
answers
618
views
smooth functional to detect whether a function has a zero
Does there exist a function $F : C^\infty(\mathbb{R}, [0, \infty)) \to \mathbb{R}$ with the following properties:
$F(f) = 0$ if and only if there exists an $x \in [0,1]$ such that $f(x) = 0$.
$F$ is ...
- 1,092
3
votes
1
answer
531
views
An argument in the proof of a compactness theorem
In the proof of a compactness theorem involving fractional derivatives in Temam's Navier-Stokes Equations, an argument as the following is made.
Suppose $X_0,X,X_1$ are Hilbert spaces such that
...
user14319
-1
votes
1
answer
197
views
Closed form for sum involving digamma? [closed]
Let $\Gamma(n)$ be Euler's Gamma function and $\psi_0$ = $\frac{\Gamma'(n)}{\Gamma(n)}$ be the Digamma function.
Is there a closed form for
$$\sum_{n=1}^{\infty} \frac{\psi_0(n)}{n^2}=?$$
I've done ...
2
votes
0
answers
385
views
(Quasi) convexity of separately convex homogeneous functions
Consider a function $f:\mathbb{R}^n_{\geq 0}\rightarrow \mathbb{R}$ that is separately convex, i.e. such that $\frac{d^2f}{dx_i^2}\geq 0$ for all $i\in \{1,\dots n\}$. Assume also that $f$ is ...
- 389
2
votes
1
answer
3k
views
A simple question about the Hardy-Littlewood maximal function
Let $f\in L^1(\mathbb{R}^n)$. It is well known that the Hardy-Littlewood maximal function $Mf\notin L^1(\mathbb{R}^n)$ (if $f \ne 0$ a.e.), though there is a weak-type (1,1) bound for this maximal ...
- 171
-1
votes
1
answer
69
views
Proof of $\lim_{i\to\infty}\lambda_i^{-1}\left|f(\hat{x}+\lambda_ix,u_i) - f(\hat{x},u_i) - D_xf(\hat{x},u_i)(\lambda_ix)\right| = 0$
I am trying to prove or disprove the next statement that seems necessary for the proof of Proposition 2.9 of this book.
Let $U\subset R^k$ be compact and $f:R^n\times U \to R^m$ be twice ...
- 183
2
votes
1
answer
116
views
Bounding a function with second moments
Let $f(x,y)$ be a non-negative function with $x,y \in \mathbb R^3$ that satisfies
$$
I_1(f) := \iint_{\mathbb R^3 \times \mathbb R^3 } f(x,y) \, dx \ dy < \infty
$$
and
$$
I_2(f) := \iint_{\...
- 183
0
votes
1
answer
172
views
Taking away the "almost sure" [closed]
Given an arbitrary sequence of random variables (or say measurable functions on a finite-measure space) $\xi_n$, one can show by a truncation and Borel-Cantelli argument that there always exists a ...
- 87
0
votes
1
answer
109
views
How to show $a\mapsto \frac{\gamma(a,x)}{\Gamma(a)}$ is decreasing on $\mathbb{R}_+^*$?
Let $a>0,x\geq 0$, the lower regularized incomplete gamma function is defined as : $$P(a,x)=\frac{\gamma(a,x)}{\Gamma(a)} = \int_0^x \frac{e^{-t}t^{a-1}}{\Gamma(a)}dt.$$
I have read in the paper ...
- 131
2
votes
1
answer
363
views
On a derivative involving the Riemann zeta function
Let $n$ be a positive real number. Can the equality
$$\dfrac{d^{n}}{ds^{n}}\Big[s^{n-1}\ln\Big(\pi^{-s/2}\Gamma\Big(1+\frac{s}{2}\Big)\Big)\Big]\Bigg|_{s=1} = - \dfrac{d^{n}}{ds^{n}}\Big[s^{n-1}\ln\...
- 47
2
votes
1
answer
127
views
Variation of trace of symmetric powers
Consider the space $\mathrm{SU}(2)^\natural$ of conjugacy classes in $\mathrm{SU}(2)$. It has a natural identification with the interval $[0,\pi]$ with Haar measure $\frac{2}{\pi} \sin^2\theta\, \...
- 5,759