Questions tagged [real-analysis]

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

Filter by
Sorted by
Tagged with
10 votes
1 answer
618 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 ...
No One's user avatar
  • 1,545
3 votes
1 answer
204 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 ...
Sascha's user avatar
  • 506
2 votes
1 answer
269 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}{\...
Matheus Manzatto's user avatar
-1 votes
1 answer
82 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\...
ZUN LI's user avatar
  • 101
1 vote
1 answer
2k 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 ...
John's user avatar
  • 31
9 votes
4 answers
806 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(...
B K's user avatar
  • 1,890
7 votes
0 answers
420 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 $...
James Baxter's user avatar
  • 2,029
8 votes
1 answer
435 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 $0 \leq k < M$, $...
James Baxter's user avatar
  • 2,029
0 votes
1 answer
168 views

A functional equation in real analysis

For what function $u:[0,1]\rightarrow R$ with bounded derivative, such that $\forall p\in[0,1]$, $\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k}u(\frac{k}{n})=u(p)$ ...
ZUN LI's user avatar
  • 101
1 vote
0 answers
86 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: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
jason's user avatar
  • 553
0 votes
1 answer
107 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>...
Piccadilly Dough's user avatar
11 votes
3 answers
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,...
Robert Rauch's user avatar
4 votes
3 answers
200 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,\...
Totoro's user avatar
  • 2,515
2 votes
1 answer
140 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 \...
Richard Diagram's user avatar
2 votes
0 answers
193 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\...
Rajesh D's user avatar
  • 704
2 votes
1 answer
440 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 ...
dohmatob's user avatar
  • 6,716
3 votes
1 answer
131 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 < \...
r9m's user avatar
  • 810
2 votes
1 answer
181 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{...
Angeliki Koutsoukou Argyraki's user avatar
4 votes
1 answer
173 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 ...
Dominic van der Zypen's user avatar
3 votes
2 answers
253 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 ...
user avatar
5 votes
1 answer
326 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 $T_f \in L^p(\mathbb R^d)$ ...
Romeo's user avatar
  • 960
10 votes
0 answers
164 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 \...
Romeo's user avatar
  • 960
3 votes
0 answers
55 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 ...
Math604's user avatar
  • 1,363
1 vote
0 answers
64 views

Regularity of superposition operator generated by function between Banach spaces

Let $E$, $F$ be Banach spaces, $D$ be open in $E$, and $K=[0,1]$. Given $\varphi\colon K\times D\to F$ I call $$ \varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot)) $$ the ...
Ramen's user avatar
  • 121
0 votes
1 answer
60 views

Empty interior lack of minima

Suppose that $U \subseteq \mathbb{R}^d$, and satsifies $U$ is dense in $\mathbb{R}^d$, U has empty interior, Then is it possible that $$ \inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x), $$ ...
ABIM's user avatar
  • 5,019
2 votes
1 answer
304 views

Logarithmic and polynomial functions with two roots

This is a question that I came across a few days ago,Although it is not particularly like a research problem, the following problem is that I study the zero distribution of a class of elementary ...
math110's user avatar
  • 4,230
0 votes
1 answer
89 views

Existence of a certain type of function

Trying to find functions with the given property: Given $M>0, K$ compact in $\mathbf{R^n}$,$f:U\rightarrow\mathbf{R}$ a $C^2$ function, where $U$ open in $\mathbf{R^n}$ and $K\subset U$such that $...
Partha's user avatar
  • 759
2 votes
2 answers
344 views

Ergodic theorem and products

If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that $$ \lim_{n \rightarrow \infty} \frac{f_n}{...
user avatar
0 votes
0 answers
61 views

Feller semigroups and fractional operators

Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user avatar
4 votes
1 answer
314 views

Proving two inequalities involving the gamma and digamma functions

I'm having trouble proving the following inequality: $$\forall p>1 \quad \forall m\geq 0 \quad \dfrac{m^2\Gamma(\dfrac{2m}{p})\Gamma(\dfrac{2m}{q})}{\Gamma(\dfrac{2m+2}{p})\Gamma(\dfrac{2m+2}{q})}\...
Yonatan Shelah's user avatar
10 votes
2 answers
982 views

Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?

This question is an old question from mathstackexchange. Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $ And let $ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $ It appears that we have $$\sup ...
mick's user avatar
  • 703
2 votes
0 answers
149 views

Can a local extremum of a function be an asymptotically stable equilibrium of corresponding gradient dynamics?

Let's first describe the setup: we consider a(say smooth enough) function $f: \mathbb{R}^d \to \mathbb{R}$ and write it as $(x,y) \to f(x,y)$, where $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
Markus Holzleitner's user avatar
17 votes
1 answer
1k views

Continuous functions of three variables as superpositions of two variable functions

Could we always locally represent a continuous function $F(x,y,z)$ in the form of $g\left(f(x,y),z\right)$ for suitable continuous functions $f$, $g$ of two variables? I am aware of Vladimir Arnold's ...
KhashF's user avatar
  • 2,588
27 votes
2 answers
1k views

Rademacher theorem

If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
Piotr Hajlasz's user avatar
1 vote
0 answers
96 views

How is the dominated convergence theorem applied in the proof of Lyapunov’s criterion?

Let $$\Gamma(f,g):=\frac12f'g'\;\;\;\text{for }f,g\in C^1(\mathbb R),$$ $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ with a continuously differentiable and positive density $\...
0xbadf00d's user avatar
  • 161
5 votes
0 answers
264 views

Is there any geometrical/homological intuition behind symmetrized gradient?

The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in ...
Romeo's user avatar
  • 960
63 votes
6 answers
12k views

Why isn't integral defined as the area under the graph of function?

In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...
user57888's user avatar
  • 1,229
1 vote
0 answers
229 views

Short question on functions of bounded variation

For a function $f: \mathbb R \to \mathbb R$ of locally bounded variation, when is $$\liminf_{e \to 0} V(f)[x, x+e]/e $$finite everywhere? Here $V(f)[a, b]$ denotes the total variation of the function ...
James Baxter's user avatar
  • 2,029
8 votes
2 answers
487 views

How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?

Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...
0xbadf00d's user avatar
  • 161
0 votes
1 answer
216 views

Cross entropy loss is not twice differentiable?

I was reading a recent theory paper in machine learning by Kenji Kawaguchi and Leslie Pack Kaelbling https://arxiv.org/pdf/1901.00279.pdf and the authors seem to suggest in section 2.2 that cross-...
ted's user avatar
  • 271
0 votes
0 answers
109 views

Harnack Inequality for uniformly elliptic PDE via constructing a singularity

I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
David's user avatar
  • 1
2 votes
1 answer
516 views

Density in fractional Sobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1−\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right), $$ $$ H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}. $$ Q: Is $C^\...
Guohuan Zhao's user avatar
6 votes
1 answer
1k views

Uniformly differentiable functions

Note: Here all functions are $\mathbb R \to \mathbb R$. $Id$ denotes the identity function. Let $g_i$ be a family of functions indexed by some (potentially uncountable) index set $I$. Given a ...
James Baxter's user avatar
  • 2,029
8 votes
1 answer
313 views

A strictly decreasing function between uncountable subsets of the reals

By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line) it is possible to prove the following ...
Taras Banakh's user avatar
  • 40.8k
1 vote
0 answers
206 views

Quantitative Lusin’s theorem

We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$. Let $f$ be measurable. For every $e$ in $...
James Baxter's user avatar
  • 2,029
6 votes
1 answer
347 views

Looking for infinite series resembling an exponential

I'm looking for some $f(x)$ that has the following property: $\sum_{x=1}^\infty f(kx) = r^k$ for some real $0 < r < 1$, and at least for strictly positive integer $k$. Does such an $f(x)$ ...
Mike Battaglia's user avatar
0 votes
1 answer
116 views

Integrable function [closed]

Suppose that $a, b, c_1$ and $c_2$ are real constant. Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e. $$\int_1^{\infty}\int_1^{\...
Xiaopai Song's user avatar
12 votes
2 answers
271 views

Show that $f(t)=\sum_{i=1}^n a_i e^{-(x_i-t)^2}-c$ has at most $2n$ zeros

Let \begin{align} f(t)=\sum_{i=1}^n a_i e^{-(x_i-t)^2}-c \end{align} where $x_1<x_2<...< x_n$ and $a_i>0$. For some positive constant $c$. Can we show that $f(t)$ has at most $2n$ ...
Boby's user avatar
  • 631
1 vote
0 answers
67 views

Continuous injection of metric ball into Euclidean ball

This is a follow-up to this post. Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by $$ \kappa_X(\epsilon)\triangleq\sup\left\{ k : \exists x_0,\dots,x_k \...
ABIM's user avatar
  • 5,019
2 votes
1 answer
293 views

Necessary and Sufficient conditions for integrable function [closed]

Suppose that $a, b$ and $c$ are constant. Is there the necessary and sufficient conditions of $a ,b, c$ for the following integration is integrable? i.e. $$\int_0^\infty \int_0^\infty \int_0^\infty ...
Xiaopai Song's user avatar

1
55 56
57
58 59
106