Related papers: Some remarks on the Cassinian metric
We present tight Bell inequalities expressed by probabilities for three four- and five-dimensional systems. The tight structure of Bell inequalities for three $d$-dimensional systems (qudits) is proposed. Some interesting Bell inequalities…
We first prove a weighted inequality of Moser-Trudinger type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than -1. Without symmetry assumption, it…
The Hilbert metric between two points $x,y$ in a bounded convex domain $G$ is defined as the logarithm of the cross-ratio of $x,y$ and the intersection points of the Euclidean line passing through the points $x,y$ and the boundary of the…
We comment on progress in measurements of the Casimir force and discuss what is the actual reliability of different experiments. In this connection a more rigorous approach to the usage of such concepts as accuracy, precision, and measure…
We investigate integer numbers which possess at the same time the properties to be triangulars and squares, that are, numbers $a$ for which do exist integers $m$ and $n$ such that $ a = n^2 = \frac{m \cdot (m+1)}{2} $. In particular, we are…
A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the…
We present a new proof of a recent $\epsilon$ regularity of G. Tian and J.Viaclovsky. Moreover, our idea also also works with a kind of $L^p, p<\dim M/2$ assumptions on the curvature.
This is a comprehensive set of notes on the ArXiV paper math.CA/0609815 by Dmitry Bilyk and the author. The focus of that paper is a new inequality for sums of hyperbolic Haar functions in three variables, extending a famous result of J…
We discuss some basic properties of the Sibony functions and pseudometrics.
This article describes some aspects of Cauchy integrals and related geometry of sets and measures in Euclidean spaces, etc.
We study a hyperbolic type metric $h_{G,c}$ introduced by Dovgoshey, Hariri, and Vuorinen. We find the best constant $c>0$, for which this function $h_{G,c}$ is a metric in specific choices of $G$. We give several sharp inequalities between…
In this paper, ideas of open ball, closed ball, compact set are introduced and some related basic properties are studied. Some topological properties and some other well known results of metric spaces including Cantor intersection theorem…
We show explicit forms for extremals of some fourth-order sharp trace inequalities on the unit balls recently proved by Ache-Chang. We also give a classification result of the bi-harmonic equation on $\mathbb{R}^4_+$ with some conformally…
A conflict has been reported between the baryon density inferred from deuterium observations and that found from recent CMB observations by BOOMERanG and MAXIMA. Despite the flurry of papers that attempt to resolve this conflict by adding…
We prove the existence of a $C^{1,1}$ conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to $-1$ plus terms of order $e^{-2r}$ where $r$ is the distance from any fixed compact set. This metric has…
In the paper, we provide an alternative and united proof of a double inequality for bounding the arithmetic-geometric mean.
Two simple proofs of the triangle inequality for the Jaccard distance in terms of nonnegative, monotone, submodular functions are given and discussed.
It is shown that there exist infinitely many triangular numbers (congruent to 3 mod 12) which cannot be the distance between two perfect numbers.
We study convexity and starlikeness of quasihyperbolic and distance ratio metric balls on Banach spaces. In particular, problems related to these metrics on convex domains, and on punctured Banach spaces, are considered.
The isodiametric inequality is derived from the isoperimetric inequality trough a variational principle, establishing that balls maximize the perimeter among convex sets with fixed diameter. This principle brings also quantitative…