Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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36 views

Stochastic Fixpoint Approximations of Contractions

Context/Introduction Consider a contraction $f\colon\mathbb{R}^S\to\mathbb{R}^S$ with $f(X^*)=X^*$ where function evaluation at a certain position is only possible with some stochastic error. Where Y(...
5
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1answer
195 views

Is there any continuous ternary function which can not be represented by composition of continuous binary functions?

Let $f : X^3 \rightarrow X$. If $X$ is $\mathbb Z$, then there will be a couple of functions $g,h$ from $\mathbb Z^2$ to $\mathbb Z$ that satisfies $f(x,y,z) = g(h(x,y),z)$ since there is a bijection ...
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0answers
138 views

Largest weak(-like) topology with respect to which continuous functions are dense in the space of Borel functions

Let $X$ denote the space of bounded Borel functions $f\colon [0,1] \to \mathbb{R}$. Let $M$ denote the space of finite Borel measures on $[0,1]$. What is the largest family $F \subset M$ such that for ...
7
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1answer
211 views

Compactness of set of indicator functions

Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set $$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$ Is this set compact in $L^\infty(0,1)$ with respect ...
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1answer
162 views

About infinite products and Euler Gamma functions [closed]

I am interested in knowing how to calculate infinite products like (or reading any reference about it): $$\prod_{j=1}^{\infty}\left( 1-\left( \frac{x}{a+j\pi} \right) ^2 \right)$$ Inserting it into ...
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2answers
155 views

Integral formula involving Legendre polynomial

I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values. \begin{equation} \int_{-1}^{1}\sqrt{...
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65 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\...
3
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1answer
204 views

On convex functions which are non constant on every segment

I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the ...
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1answer
118 views

Trajectory leaving a set

Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...
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1answer
82 views

Uniformly Converging Metrization of Uniform Structure

This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question. Let $X$ be a set with a uniform structure ...
4
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1answer
284 views

Nearly eventually almost periodic functions

Call a function $f: [0, \infty) \to \mathbb R$ nearly eventually almost periodic with period $p > 0$ if for a.e. $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges. Suppose $f: ...
6
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2answers
270 views

Maximize $L^p$ norm over sphere

For $p \in \mathbb{R}$, consider the function $$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$ My goal is to maximize this function under the constraints that $$ \lambda_1^2 +...
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1answer
239 views

Prove that $x^{y^x}>y^{x^y}$ for $x>y>0.$ [closed]

Let $x>y>0$. Prove that $$x^{y^x}>y^{x^y}.$$ My attempts: Let $1>x>y>0$. In this case it's enough to prove that $$y^x<x^y$$ or $$x\ln\frac{1}{y}>y\ln\frac{1}{x},$$ which is ...
2
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1answer
100 views

How to compute volume of parametric regions?

I guess this is something pretty standard in calculus, but I was unable to google the answer. Assume I have unit hypercube $C_n = [0,1]^n$. I also have a function $f : \mathbb{R}^n \to \mathbb{R}^{n+...
6
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3answers
287 views

Computing the volume of a simplex-like object with constraints

For any $n \geq 2$, let $$D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] = \{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid \sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \},$$ where $r \...
4
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1answer
101 views

Construct a dense family in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ based on distance functions

Let $\Omega$ be a sufficiently regular domain, for example $\Omega=B(0,r)$, $r>0$, and $m(x)=\textrm{dist}(x,\partial \Omega)$ be the distance to boundary function. Suppose $\mathcal{F}$ is a ...
2
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1answer
214 views

An isoperimetric inequality for curve in the plane?

Let $f(x,y)=0$ be a (smooth) simple closed curve $C$ on the plane and $R$ the region bounded by $C$ (appropriately oriented). Assume the origin lies in the interior of $R$. QUESTION. Let $r=\sqrt{x^...
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0answers
85 views

Perturbation theory compact operator

Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$ $\Vert Kx-\lambda x \Vert \le \varepsilon.$ It is well-known ...
9
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2answers
347 views

Small uncountable cardinals related to $\sigma$-continuity

A function $f:X\to Y$ is defined to be $\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ ...
2
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1answer
165 views

Generalised raindrop function

Given a sequence of reals $(a_n)_{n > 0}$, let $f: [0, 1] \to R$ be the generalised raindrop function defined: $f(x) = a_q$ if $x$ is rational, with denominator $q$ in lowest form; $0$ otherwise. ...
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1answer
113 views

Reference request: Baire class 2 functions

There are many articles on Baire 1 functions, but not many on Baire 2 and above. Where can I find a nice comprehensive survey of them?
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1answer
351 views

Formal group law and Koenigs function conjecture?

Let $f(x,y)$ be a symmetric real function and a formal group law $$G(x + y) = f(G(x),G(y)). \tag{1}$$ Consider the equation $$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$ This equation has many ...
7
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1answer
512 views

Eventually almost periodic functions

Call a function $f: [0, \infty) \to \mathbb R$ eventually almost periodic with period $p > 0$ if for all $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges. Suppose $f: [0, \...
2
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1answer
60 views

Decaying of a certain ratio of binomial sums

Consider the two sequences $$a(n)=\sum_{k=1}^n\binom{n}k\sum_{j=1}^{k/2}\binom{k}{2j}\frac{(2j)!}{j!}$$ and $$b(n)=\sum_{k=0}^n\binom{n}k^2k!$$ QUESTION. Is this true? $$\frac{a(n)}{b(n)}\...
5
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1answer
148 views

Binomial Distributions and Inequality

Suppose $c>2$ is a real number, and $x$ solves equation $f(\frac{1}{2}+x,m,2m)=(2m+1)cx$ for some $m>0$ integer number, where $f$ is probability mass function for binomial distribution, with ...
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1answer
84 views

“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 ...
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1answer
351 views

Convergence of a certain sum

Suppose $ g_i: [0, 1] \to \Bbb R$, $i\in\Bbb N$, are $C^1$ functions and that there is some $c > 0$ such that for every $0 < \epsilon < c$, the functions $$ s(\epsilon)_i := \sum_{k=0}^i {\...
10
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1answer
275 views

Estimation of the Gromov–Wasserstein distance of spheres

Let $(X,d_X,\mu_X)$ and $(Y,d_Y,\mu_Y)$ be two metric measure spaces. A probability measure $\mu$ over $X\times Y$ is called a coupling if $(\pi_1)_\sharp \mu=\mu_X$ and $(\pi_2)_\sharp \mu=\mu_Y$. We ...
6
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1answer
166 views

Eigenvalue estimates for operator perturbations

I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading. What was behind ...
2
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1answer
161 views

Corollary of the Malgrange Preparation Theorem

(This question was previously posted on MSE and I decided to post it here too.) Let $f:\mathbb{R}\times \mathbb{R}^n \to \mathbb{R}$ be a smooth function, such that $$f(0,0)=0,\ \frac{\partial f}{\...
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1answer
61 views

On probabilistic extension for Bernstein polynomials

Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...
1
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1answer
119 views

Expansion of an integral

I have an integral of the form $$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular and positive function such that $I$ is finite ...
8
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4answers
269 views

Defining the value of a distribution at a point

Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
7
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0answers
384 views

A discontinuous construction

Suppose we have an uncountable family of functions $f_r: [0, 1] \to R$ indexed by $r \in [0, 1]$ such that for each $r$, there exists a unique $x$ in $[0, 1]$ such that $f_{r}$ is positive on $x$ and $...
8
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1answer
320 views

Simultaneous Riemann Rearrangement

Here all functions are $\mathbb R \to \mathbb R$. Fix $M$ a positive integer. For $i = 0, 1, ..., M,$ let $f_0 = Id$, and the other $f_i$ be continuous functions such that for all $0 \leq k < M$, $...
0
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1answer
131 views

A functional equation in real analysis

For what function $u:[0,1]\rightarrow R$ with bounded derivative, such that $\forall p\in[0,1]$, $\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k}u(\frac{k}{n})=u(p)$ ...
1
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0answers
73 views

Coboundary in the slow mixing systems

Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
0
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1answer
72 views

Inequality involving product-of-minus vs minus-of-product for positive integers

I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows: Consider positive integers $a_1$, $a_2$, $b_1$ and $b_2$ where $a_1>b_1$ and $a_2>...
12
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3answers
2k views

Does anyone recognize this inequality?

In some paper the authors make use of the following inequality without further explanation: Let $x\in\mathbb{R}^n$ with $x_1\le\cdots\le x_n$ and $\alpha\in[0,1]^n$ with $\sum_{i=1}^n \alpha_i=N\in\{1,...
4
votes
3answers
167 views

A Riccati type integral inequality

Let $x(t),t\in [1,\infty)$ be a nondecreasing positive function satisfying the following inequality: $$ x'(t) \le \int_t^{+\infty} x(s)\frac{k(s)}{s^2}\,ds, $$ for any $t \ge 1$, where $k(t),t\in [1,\...
2
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1answer
122 views

Must $q$ be analytic?

I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that $$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$ which also takes $\mathbb{R}^+ \to \...
2
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0answers
177 views

Shattering with sinusoids

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\...
1
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1answer
64 views

Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector

Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector. Question $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$ Observation This paper allows us to ...
3
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1answer
84 views

A problem with sequences with composition of $\log$s

If $(a_n)_{n \ge 1}$ is a non-negative sequence s.t., $$\sum\limits_{n = c_k}^\infty \frac{a_n}{\log^{(k)} n} < \infty, \, \forall k \ge 1 \overset{?}{\implies} \sum\limits_{n \ge 1} a_n < \...
1
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1answer
139 views

Why is this series summable?

Let $\delta, \epsilon \in \mathbb{R}$, $\delta >0$, $\epsilon >0$. Let $\{ a_k\}^\infty$,$\{ b_k\}^\infty$ be sequences of positive integers such that $\lim \sup_{k \rightarrow \infty} \frac{...
4
votes
1answer
161 views

Graphs that are not $\mathbb{R}^2$-realizable

We say that a finite, simple, undirected graph $G=(V,E)$ is $\mathbb{R}^2$-realizable if there is an injective map $\varphi:V\to \mathbb{R}^2$ such that for $v\neq w \in V$ we have $\{v,w\} \in E$ if ...
4
votes
2answers
173 views

Can one realize this as an ergodic process?

Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph. We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$ In other words: For ...
5
votes
1answer
126 views

Estimate of the difference quotients in terms of an $L^{1,\infty}$ function

Let $f \colon \mathbb R^d \to \mathbb R$ be a measurable function. Consider the following property: (P) there exist a negligible set $N \subset \mathbb R^d$ and function $T_f \in L^p(\mathbb R^d)$ ...
11
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0answers
142 views

Maximizing an integral w.r.t. a measure on the unit sphere

I would like to know if the answer to the following question is known. Let $d \ge 3$. What is the value of $$ \theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...
3
votes
0answers
51 views

system of Euler like ode's

I am interested in solving some linear elliptic system like $$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$ $$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...