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0 votes
1 answer
97 views

Numerically bounding a Exponential-Trigonometric Integral [closed]

I am having some trouble with this (undergrad) problem. The Twitter account I found this from was deleted so I unfortunately have not found an answer. I have tried decomposing into Riemann sum and ...
0 votes
1 answer
57 views

Lower bounding an alternating series with signs from a martingale difference sequence

Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that $$M_n := \sum_{i = 0}^n \epsilon_i$$ is a martingale. We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
3 votes
1 answer
227 views

"Essential values" of a function at a point?

Recall that the essential range $\operatorname{ess.im} f$ of a measurable function $f \in L^\infty(\mathbb{R})$ is a compact set. Denote by $f_k$ the restriction of $f$ to the interval $[-1/k,1/k]$, ...
2 votes
2 answers
154 views

Closure of $C([0,1]^2)$ via weak*-topology [closed]

Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$. The dual space of $C([0,1]^2)$, denoted by $C^*([...
3 votes
0 answers
219 views

Strictly contracting solutions to the Eikonal equation on Riemannian manifolds

Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$. Question: Does there exist, on every complete ...
4 votes
1 answer
297 views

Oscillation of monotone real-analytic function

Let $f:(a,\infty)\rightarrow \mathbb{R}$ be a real-analytic and strictly monotone function. I have been wondering how much this function can "oscillate". Namely, can we always find a ...
2 votes
1 answer
276 views

Estimating a sum over set partitions

Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$. I would like to estimate the following alternating sum. QUESTION. Is this true? ...
1 vote
1 answer
124 views

$d(x,y) = \min\{|x_1−y_1|+|x_2−y_2|, 1−|x_1−y_1|+|x_2−(1−y_2)|\}$ defines a metric on $[0,1)\times[0,1]$? [closed]

For $x,y \in [0,1)\times[0,1]$, let $d(x,y)$ be the minimum of $|x_1−y_1|+|x_2−y_2|$ and $1−|x_1−y_1|+|x_2−(1−y_2)|$. Prove or disprove that $d$ is a metric. I was unable to find a counterexample to ...
7 votes
0 answers
619 views

Lavrentiev Phenomenon

Does there exist a (onedimensional) integral functional of calculus of variations $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt
 $$ such that not only $$ \inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
1 vote
0 answers
60 views

Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations

The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying $$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
5 votes
2 answers
372 views

Weak Archimedean property instead of Archimedean property

We say that a sequence $(z_n)$ of real numbers is a modulated Cauchy sequence, whenever there exists a function $\alpha:\mathbb{N} \rightarrow \mathbb{N}$ such that: $$ |z_i-z_j| \le \frac{1}{k} \quad ...
3 votes
0 answers
100 views

How to compute the partial derivatives of this function?

For any probability measure $\mu$ on $\mathbb R^2$ and $\theta\in [0,2\pi]$, denote by $\mu_\theta$ its projection along $v:=(\cos\theta,\sin\theta)$. Namely, if $X$ is a random variable distributed ...
0 votes
1 answer
231 views

Questions on the compactness of $L_1([0,1]^2)$'s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
6 votes
1 answer
193 views

The most even partition of $\mathbb R$ into measure dense sets

Notation: $\mu$ denotes the Lebesgue measure. Let $\mathcal D$ be the set of Lebesgue measurable subsets of $\mathbb R$ such that itself and its complement have nonzero Lebesgue measure in every ...
3 votes
1 answer
343 views

Asymptotic behavior of a recursion

Let $x_n(0)=1$, $$ x_n(N+1) = \frac{1}{N+1}\sum_{k=0}^N \sum_{j=1}^n x_j(k)x_{n+1-j}(N-k) + \frac{10}{N+1} x_{n+1}(N) , \quad\quad N\ge 0 . $$ So the recursion is on $N$, and at each level, we compute ...
2 votes
1 answer
149 views

Proof that superlinearly convergent sequence converges faster than linearly convergent sequence

Given real sequences $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$, both converging to the same limit $A$ and such that $|a_n-A|\neq 0$ and $|b_n-A|\neq 0$ for every $n$ sufficiently large, we ...
3 votes
2 answers
118 views

Does the derivative of the antiderivative of a BV function $f$ agree with $f$ at all but countably many points of differentiability?

Let $f: (a, b) \to \mathbb R$ be a function of bounded variation, and write $$F(x) := \int_a^x f(t) \, dt$$ for the antiderivative. Is it true that at all but countably points of differentiability of $...
0 votes
1 answer
101 views

Limit sequence of regular function in $L_1$‘s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
23 votes
4 answers
2k views

Identity for an infinite product

Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes". QUESTION. Is this true? $$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
9 votes
1 answer
553 views

Does the sequence formed by Intersecting angle bisector in a pentagon converge?

I asked this question on MSE here. Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
15 votes
2 answers
1k views

Converse of mean value theorem

Note: This is an attempt to narrow down conditions under which the conjecture stated in this previous post is true. As stated, it is false as shown by the counterexample provided in the answers by the ...
5 votes
0 answers
156 views

What is the Hausdorff dimension of the set on which this exponential sum is bounded?

This is a direct follow up to For which rationals is this exponential sum bounded? Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$. What is the Hausdorff dimension of the ...
3 votes
1 answer
490 views

Space derivative of flow of ODE with monotone source

Consider the ODE $$ \begin{cases} \partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\ \Phi(0,x) = x, & x \in \mathbb R \end{cases} $$ where $f$ is function which is a non-...
1 vote
1 answer
179 views

For fixed $f \in L^2$ and $T>0$, choose $g$ so that $ \mathbb{E}^x[g(T-\tau)\chi_{X_\tau=1}]=-\mathbb{E}^x[f(X_T)\chi_{\tau \ge T}]$

Let $f \in L^2(0,1)$ and $T>0$ be fixed. How can I choose $g \in L^2(0,T)$ such that \begin{align*} 0\equiv \mathbb{E}^x\left[f\left(X_T\right) \chi_{\tau \geqslant T}+g(T-\tau) \chi_{X_\tau=1}\...
4 votes
1 answer
204 views

Stationary phase formula for a complex valued phase

I'd be interested in computing an asymptotic expansion when $h \rightarrow 0$, of an integral of the form $$ I_h = \int_{\mathbb{R}}{e^{\frac{i}{h}\varphi(x)}dx} $$ where $\varphi : \mathbb{R} \...
0 votes
0 answers
121 views

Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?

Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
12 votes
2 answers
866 views

Sets that project to zero measure on all lines except one

It is a (difficult) exercise to show that there exists a measurable set $E \subset [0,1]^2$ (necessarily with zero 2-dimensional Lebesgue measure) such that the projection on every line passing ...
5 votes
0 answers
285 views

How do you go about making ranges (for integer variables) independent?

Basic question: say you have a sum $$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$ where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
-1 votes
1 answer
167 views

Space of distributions on $[0,1]^2$: weakly compact or not?

Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$. Question 1: Does $\mathcal{X}$ ...
8 votes
1 answer
342 views

How large can the set of turbulent points be?

This question resisted attempts on MSE. Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold: $$\...
4 votes
1 answer
374 views

An open mapping theorem for homogeneous functions?

I am researching different generalizations of the familiar open mapping theorem from functional analysis. Every "proof" I attempt while simply assuming positive-homogeneity, even in the finite-dim ...
4 votes
1 answer
217 views

$2$ continuous, commuting functions doesn't always have a common fixed point

The question is as such: If two continuous mappings $f$ and $g$ of a closed interval into itself commute, that is, $f\circ g=g\circ f$, then they do not always have a common fixed point. -- Zorich ...
1 vote
1 answer
50 views

Increasing function of $\theta$ for the Ali-Mikhail-Haq Survival Copula

I have been trying to solve the following function is non-increasing (non-decreasing) with respect $\theta$ where $\theta \in (0,1)$ (resp. $\theta \in (-1,0)$) \begin{equation} f(\theta)= \frac{h(t,\...
9 votes
1 answer
492 views

Dispersion points of Lipschitz functions

For a function $f: \mathbb R^n \to \mathbb R^m$ with $m < n$, we say that $x \in \mathbb R^n$ is a dispersion point of $f$ if $$\liminf_{y \to x} \frac{|f(y) - f(x)|}{|y - x|} > 0.$$ Question: ...
21 votes
2 answers
2k views

Boundedness of sum of sin(sin(n))

Playing with desmos I have accidentally noticed that the sequence of partial sums $$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$ is bounded. However, I did not succeed in proving this ...
1 vote
1 answer
330 views

Does $\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$ converge?

I am trying to study the converge of the series $$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$ But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? ...
0 votes
1 answer
139 views

Proving negativeness of function involving $-\log t$

I have been trying to solve the following function is non-increasing with respect $\theta$ \begin{equation} h(t,\beta) = \frac{1-t-\frac{\beta(-\log t)^{\theta}}{\theta(-\log \beta)^{\theta -1}}}{1-\...
7 votes
1 answer
270 views

Sequential continuity and the Axiom of Choice

It is well-known that ZF cannot prove the following: "for a function $f$ from reals to reals and any real $x$, $f$ is continuous at $x$ if and only if $f$ is sequentially continuous at $x$."...
1 vote
2 answers
102 views

About the recursive inequality $w_p \geq (1-\frac {\pi}n)w_{p-2n} + 2\pi + o(1)$

Suppose we have a non-decreasing sequence of positive real numbers that tend to infinity: $0<w_1\leq w_2\leq w_3\leq...$ It is known that: For every $n$ and $p\geq 2n$, we have $w_p \geq (1-\frac {...
3 votes
1 answer
966 views

Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$): $\min_{x\in U}\|x-y\|$. I believe the minimizer $x_{U}^{*}$ is ...
2 votes
0 answers
99 views

Closed form for $\int_0^{+\infty} \ln^p(t) \frac{\sin^q(t)}{t^r}dt$

Do you know if there exists a closed form for the integral : $$I_{p,q,r} = \int_0^{+ \infty} \ln^p(t) \frac{ \sin^q (t)}{t^r} dt$$ where $p$, $q$, $r$ are natural integers such as this integral ...
0 votes
0 answers
73 views

An example of a groupoid that satisfy the following hypothesis

In the paper titled, 'Tannaka–Krein duality for compact groupoids I, Representation theory', the author proves the Peter Weyl theorem on compact groupoids. In the statement, he gives the hypothesis ...
0 votes
0 answers
60 views

Criteria for log-absolute-monotonicity

Consider a function $f: [0,1] \rightarrow \mathbb R$ defined by a power series $f(x) = a_0 + a_1 x + a_2 x^2 + \dots$, where all $a_i$ are positive. Is there are any criterion in terms of the ...
2 votes
1 answer
133 views

Points of differentiability of convex functions

Let $U$ be an open neighbourhood of $0 \in \mathbb{R}^2$ and $f\colon U \to \mathbb{R}$ a convex (and bounded) function. Denote by $D \subset U$ the set of points on which $f$ is totally ...
2 votes
0 answers
85 views

Higher cohomology groups for the trivial action of the reals on themselves

For a freely generated countable abelian group $A$ with the trivial action on itself ($a\cdot b = b$) the resulting cohomology groups are well-known and eventually vanish (see e.g. here). Coming from ...
-1 votes
1 answer
61 views

Asking for some references on correlations of joint optimization problems

Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
6 votes
1 answer
816 views

Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?

Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$. Let $\Omega$...
4 votes
1 answer
249 views

Does this functional admit an absolute minimizer?

This is a close relative of the following problem. Let $\Omega$ be an open, bounded subdomain of $\mathbb R^n$ with smooth boundary, and $f_i \in W^{1, \infty} (\Omega)$ a sequence of functions ...
7 votes
1 answer
346 views

Mean Cauchy sequences

Let $X$ be a complete metric space. Suppose a sequence of elements $x_n$ is Cauchy in mean, in the sense that $$\lim_{K \to \infty} \limsup_{N, M \to \infty} \frac{1}{NM} \sum_{i = K+1}^{K + N} \sum_{...
2 votes
1 answer
309 views

Reverse estimate on the Riesz potential $I_\alpha : L^{n/\alpha}\to \mathrm{BMO}$

$\newcommand\BMO{\mathrm{BMO}}$Consider the Riesz potential on $\mathbb{R}^n$ given by $$ I_\alpha f(x) = c_{n,\alpha} \int_{\mathbb{R}^n} \frac{f(y)}{\lvert x-y\rvert^{n-\alpha}} dy. $$ It is known ...