The real-analysis tag has no wiki summary.

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### Convex interaction energy

Does anybody know examples of absolutely continuous probability measures $\mu_0,\mu_1$ on $\mathbb{R}^n$ with finite 2nd moments such that
$$
\frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ...

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61 views

### Theorem with an example

i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?

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86 views

### The problem of Reimann zeta function [on hold]

$\zeta(2)=\sum_0^\infty 1/n^{2}<\pi^2/6=1.644934<2$
From the popular knowledge
$\zeta(2)\Gamma(2)=\int_0^\infty x/(e^x-1)dx$
but
$\int_0^\infty x/(e^x-1)dx=\int_0^\infty ...

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102 views

### The problem of Riemann zeta function [on hold]

$\zeta(2)=\sum_0^\infty 1/n^{2}<1.8$
From the popular knowledge
$\zeta(2)\Gamma(2)=\int_0^\infty x/(e^x-1)dx$
but
$\int_0^\infty x/(e^x-1)dx=\int_0^\infty xe^{-x}/(1-e^{-x})dx$
$=\int_0^\infty ...

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46 views

### The class of bounded uniformly continuous functions in viscosity solution theory for Hamilton-Jacobi equations

Dumb question: Usually in viscosity solution theory for Hamilton Jacobi equations (with convex, coercive Hamiltonians), solutions are said to be in the class $BUC(\mathbb{R}^n)$ or ...

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**1**answer

60 views

### Equivalent of Stirling-like numbers

let $b_{n,k}$ be the numbers defined formally by $$X^n=\sum_{k=0}^n b_{n,k}\binom{X}{k}$$ where $\binom{X}{n}=\frac{1}{n!}\prod_{k=0}^{n-1}(X-k)$.
I am looking for an equivalent of $b_{n,k}$ when $k$ ...

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176 views

### A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is motivated ...

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553 views

### For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to ...

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49 views

### Under condition of Zygmund is the following inequality true?

Let $f:R\rightarrow R$ be a continuous function and satisfies the following Zygmund condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, ...

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251 views

### Pros and cons of probability model for permutations

I am studying probability model of random permetuation
Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k
inversions ($inv(\pi)$). The analytic approach was considered by ...

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87 views

### Is it possible to list $\mathbb{Q}$ so that the result set to be a monotoic sequence? [migrated]

Let $\mathbb{Q}$ be the set of rational numbers. Is it possible, relabeling if needed, to list $\mathbb{Q}$ such that the result set to be a monotonic sequence? If not, why? If it is true, where is ...

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**1**answer

466 views

### Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$:
$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...

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**1**answer

76 views

### Young transform reference

The Young transform of nonnegative function $f(x)$, $x \in \mathbb R^n_+$ is defined to be
$$
(\mathscr Yf)(y) = \inf \left[ \left. \frac{x_1 y_1 + \ldots + x_n y_n}{f(x)} \; \right|\; x \colon ...

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106 views

### Regarding sub-additive sequences and Fekete's lemma

A non-negative sequence $\{a_n\}$ is sub-additive if $a_{m+n}\leq a_m + a_n.$ Fekete's lemma says that for any non-negative sub-additive sequence:
$$\lim_{n\to\infty} \frac{a_n}{n} = \inf_{n} ...

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### Can we affirm that this function is absolutely continuous?

Suppose you have a sequence of functions $g_{\tau_n}$ ($0<\tau_n \rightarrow0$) from $]0; +\infty[$ in $\mathbb{R}$ that converges uniformly to a function $g$ in every compact interval.
Suppose ...

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103 views

### Smooth convex extensibility of combination of two line segments

This is a refined version of my earlier question Convex extensibility of combination of two lines.
Is there a smooth function $f:[0,1]\times [0,1]\rightarrow\mathbb R$ such
that for all $x\in ...

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**2**answers

391 views

### Pathological behavior of Borel sets?

Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the ...

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39 views

### minimum distance between sets and relation with functions [on hold]

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem.
$$ \min ...

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**1**answer

91 views

### Countable vs. ultra-negligible sets [duplicate]

A subset $A\subset\mathbb{R}$ is negligible if for each $\epsilon>0$ there exists a sequence $(I_n)$ of intervals such that $A\subset\cup_n I_n$ and $\sum_n \vert I_n \vert \leq \epsilon$. Let us ...

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### Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...

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187 views

### Distance between two sets

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem.
$$ \min ...

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79 views

### Convergence of the Double Integral of a Polynomial Reciprocal

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:
(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;
(ii) $f$ is non-degenerate, in the sense that there isn't a ...

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122 views

### Level sets of a Weierstrass nowhere-differentiable function

Can anyone describe level sets of a Weierstrass nowhere-differentiable function? For example, let $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos( 4^n \pi x)$. For some $c \in (-2,2)$, what is known ...

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147 views

### Smallest positive zero of Weierstrass nowhere differentiable function

Consider the Weierstrass nowhere differentiable function $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos(4^n \pi x)$. It seems that the smallest positive zero of $f(x)$ occurs at $x=\frac{1}{5}$, but I ...

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### Feature relationship based class separability [closed]

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description.
I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...

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378 views

### Integral inequality for convex function

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true:
\begin{equation}
\frac{1}{b-a} \int_a^b ...

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676 views

### An Entropy Inequality (generalized)

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability ...

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### Show that uniform continuity implies stochastic equicontinuity

Let $\Theta$ be a metric space and assume it is compact. Let $W_t: \Omega \rightarrow \mathbb{R}^k$ be a random variable for $t\leq T$. Let $m(.,\theta): \mathbb{R}^k\rightarrow\mathbb{R}^s$. Let ...

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### reference needed for sobolev type estimates

I'm reading a paper and the authors applied the following sobolev type estimates
$$
||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2}
$$
for $\alpha>\frac{1}{4}$,
where $v$ ...

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129 views

### Question on Morse inequalities

I want to understand why: From K.C Chang's book "Infinite Dimensional
Morse Theory and Multiple Solution Problems":
if i have
then $(4.1)$ is formal : it means that
EDIT1: $(4.1)$ tel us that ...

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### Looking for a trigonometric function with known shape [migrated]

I am looking for a trigonometric function where I know the shape of
and also have some function values.
I almost have it but my formula is not precise.
If you plot 3/4-1/4*sin(3*t)-3/4*cos(6*t) you ...

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### The Largest Root of Associated Laguerre Polynomial

The Laguerre polynomial $L_n(x)$ is the solution to the Laguerre differential equation
\begin{equation*}
x\,y'' + (1 - x)\,y' + n\,y = 0.
\end{equation*}
The associated Laguerre polynomial ...

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54 views

### Multivariate monotonic function

Let $f(x_1, \dots, x_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ ...

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135 views

### Is there such a matrix in $SO(n)$?

Given two $n$ dimensional positive definite matrices $A', B'$, is there a matrix $O \in SO(n)$ such that $A=O A', B=O B'$ and
$$
\frac{A_{ij}}{\sqrt{A_{ii}A_{jj}}} = ...

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60 views

### k-th largest root in common interlacing polynomials

In their proof of the celebrated Kadison-Singer conjecture, Marcus, Spielman and Srivastava exploited so-called interlacing families which are originally defined for their work on Ramanujan graphs. ...

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109 views

### Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...

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### Equivalence of negative Sobolev norm of derivative to $L^2$-norm

Let $S:=(0,1)^2$ be the unit square in $\mathbb{R}^2$, and let $M:=\{u\in L^2(S)\mid \int_S u=0\}$ be the space of (real-valued) $L^2$-functions with mean value zero. On $M$ we can consider the ...

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98 views

### Limits of functions with converging zeros

What can one say about the derivatives of a smooth function of several variables that is a limit of smooth functions with converging zeros?
More precisely, suppose that $f_i: R^n \to R^m$ is a ...

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### Finding f such that f(f(x))=g(x) given g

Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...

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### Boundary gradient estimate

Assume $U$ is the unit disk and $\bar U$ its closure and let $u\in C^2(U)\cap C(\bar U)$ be a real function, with $u(z)=0$ for $z\in \partial U$. If $$|\Delta u|\le A|\nabla u|^2+g(z),$$ for some ...

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### Is a function with nowhere vanishing derivatives analytic?

My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic?
EDIT. According to ...

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### Geometric interpretation of Euler's identity for homogeneous functions [closed]

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is called homogeneous of degree $d \geq 0$ if $$f(\lambda x_1, \ldots, \lambda x_n ) = \lambda^d f(x_1, \ldots, x_n)$$ Differentiating both sides ...

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92 views

### Find sufficient and necessary conditions on $f$ in which the level curve $f(x,y)=0$ implies only one case $x=a$ for all real $y$ [closed]

Let $f:ℝ²→ℝ$ be an arbitrary harmonic function. A level curve in two dimensions is a curve on which the value of a function $f(x,y)$ is a constant. My question is: Find sufficient and necessary ...

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187 views

### dual space of a subspace of the space of bounded measures

Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...

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### Tubular neighbourhood which is nowhere piecewise linear

I recently asked this question.
I think, if the following were true, then I would solve my problem.
Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...

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157 views

### Beurling density and interpolation

Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density ...

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56 views

### Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$.
I would like to know how many extrema $p$ has on the standard simplex
...

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69 views

### Is the countable intersection of residual sets in [0,1] with Hausdorff dimension 1 of full Hausdorff dimension?

Let $E_k\subset [0,1]$ be residual subsets (i.e. containing dense $G_\delta $ set) with $E_{k+1}\subset E_k$ and $\dim_HE_k=1, \forall k.$ My question is : $\dim_H\bigcap_k E_k=1?$ Thanks.

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### Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum
$$\sum_{i=1}^n f(x_i) w_i$$
where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...

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514 views

### Which functions are Wiener-integrable?

I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them.
Background
The Wiener integral is an analytic tool to define certain ...