Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5,268
questions
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 ...
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 ...
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
$$f(0,0)=0,\ \frac{\partial f}{\...
-1
votes
1
answer
82
views
On probabilistic extension for Bernstein polynomials
Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...
1
vote
1
answer
2k
views
Expansion of an integral
I have an integral of the form
$$ 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 ...
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(...
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 $...
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$, $...
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)$
...
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$ ...
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>...
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,...
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,
$$
for any $t \ge 1$, where $k(t),t\in [1,\...
2
votes
1
answer
140
views
Must $q$ be analytic?
I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that
$$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$
which also takes $\mathbb{R}^+ \to \...
2
votes
0
answers
193
views
Shattering with sinusoids
Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\...
2
votes
1
answer
440
views
Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector
Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector.
Question
$\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$
Observation
This paper allows us to ...
3
votes
1
answer
131
views
A problem with sequences with composition of $\log$s
If $(a_n)_{n \ge 1}$ is a non-negative sequence s.t., $$\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 < \...
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{...
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 ...
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 ...
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 $T_f \in L^p(\mathbb R^d)$ ...
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
$$
\theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...
3
votes
0
answers
55
views
system of Euler like ode's
I am interested in solving some linear elliptic system like
$$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$
$$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
1
vote
0
answers
64
views
Regularity of superposition operator generated by function between Banach spaces
Let $E$, $F$ be Banach spaces, $D$ be open in $E$, and $K=[0,1]$. Given $\varphi\colon K\times D\to F$ I call
$$
\varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot))
$$
the ...
0
votes
1
answer
60
views
Empty interior lack of minima
Suppose that $U \subseteq \mathbb{R}^d$, and satsifies
$U$ is dense in $\mathbb{R}^d$,
U has empty interior,
Then is it possible that
$$
\inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x),
$$
...
2
votes
1
answer
304
views
Logarithmic and polynomial functions with two roots
This is a question that I came across a few days ago,Although it is not particularly like a research problem, the following problem is that I study the zero distribution of a class of elementary ...
0
votes
1
answer
89
views
Existence of a certain type of function
Trying to find functions with the given property:
Given $M>0, K$ compact in $\mathbf{R^n}$,$f:U\rightarrow\mathbf{R}$ a $C^2$ function, where $U$ open in $\mathbf{R^n}$ and $K\subset U$such that $...
2
votes
2
answers
344
views
Ergodic theorem and products
If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that
$$ \lim_{n \rightarrow \infty} \frac{f_n}{...
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 ...
4
votes
1
answer
314
views
Proving two inequalities involving the gamma and digamma functions
I'm having trouble proving the following inequality:
$$\forall p>1 \quad \forall m\geq 0 \quad \dfrac{m^2\Gamma(\dfrac{2m}{p})\Gamma(\dfrac{2m}{q})}{\Gamma(\dfrac{2m+2}{p})\Gamma(\dfrac{2m+2}{q})}\...
10
votes
2
answers
982
views
Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?
This question is an old question from mathstackexchange.
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that we have
$$\sup ...
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 $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
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 ...
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 ...
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 $\...
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 ...
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 ...
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 ...
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 $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...
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-...
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 ...
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^\...
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 ...
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 ...
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 $...
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 $f(x)$ ...
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.
$$\int_1^{\infty}\int_1^{\...
12
votes
2
answers
271
views
Show that $f(t)=\sum_{i=1}^n a_i e^{-(x_i-t)^2}-c$ has at most $2n$ zeros
Let
\begin{align}
f(t)=\sum_{i=1}^n a_i e^{-(x_i-t)^2}-c
\end{align}
where $x_1<x_2<...< x_n$ and $a_i>0$. For some positive constant $c$.
Can we show that $f(t)$ has at most $2n$ ...
1
vote
0
answers
67
views
Continuous injection of metric ball into Euclidean ball
This is a follow-up to this post.
Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by
$$
\kappa_X(\epsilon)\triangleq\sup\left\{
k : \exists x_0,\dots,x_k \...
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 ...