Related papers: Alexandrov's embedding theorem
Here I give a description of Alexandrov 4-point comparison via quadratic forms and then propose a natural 5-point condition which might have some future. Consider this note as a letter from me --- do not take it seriously.
Alexandrov's estimate states that if $\Omega$ is a bounded open convex domain in ${\mathbb R}^n$ and $u:\bar \Omega\to {\mathbb R}$ is a convex solution of the Monge-Ampere equation $\det D^2 u = f$ that vanishes on $\partial \Omega$, then…
An extension of the Artin Braid Group with new operators that generate double and triple intersections is considered. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple…
Let $\Omega$ be a John domain, and let $\Gamma\subset \partial \Omega$ be an $h$-set. For some functions $h$ and some weight functions depending on distance from $\Gamma$, embedding theorems for a weighted Sobolev class is obtained.
The original proof of the Sharkovsky theorem is presented in full detail. The proof should be accessible to readers with basic Real Analysis background. Although nowadays there are several alternative proofs of this classical result, we…
This the first of a set of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by…
In 2017, Jones studied the unitary representations of Thompson's group $F$ and defined a method to construct knots and links from $F$. One of his results is that any knot or link can be obtained from an element of this group, which is…
In this note, the correction to the proof of one theorem in some our previous paper [arXiv:1302.0589] will be given.
The two of us have shared a fascination with James Victor Uspensky's 1937 textbook $Introduction \, to \, Mathematical \, Probability$ ever since our graduate student days: it contains many interesting results not found in other books on…
Suppose we wish to embed an (associative) $k$-algebra $A$ in a $k$-algebra $R$ generated in some specified way; e.g., by two elements, or by copies of given $k$-algebras $A_1,$ $A_2,$ $A_3.$ Several authors have obtained sufficient…
This is an overview article on the Kontsevich integral written for the Encyclopedia of Mathematical Physics, to be published by Elsevier.
This article is dedicated to the centenary of the birth of Aleksandr D. Alexandrov (1912-1999). His functional-analytical approach to the solving of the Minkowski problem is examined and applied to the extremal problems of isoperimetric…
We prove embedding theorems for fully anisotropic Besov spaces. More concrete, inequalities between modulus of continuity in different metrics and of Sobolev type are obtained. Our goal is to get sharp estimates for some anisotropic cases…
In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of…
Braidoids generalize the classical braids and form a counterpart theory to the theory of planar knotoids, just as the theory of braids does for the theory of knots. In this paper, we introduce basic notions of braidoids, a closure operation…
This is an English translation of four remarkable articles, originally written in Russian, by Sergei Ivanovich Adian (1931--2020), supplemented by the translation of two closely related articles by Andrei Andreevich Markov Jr (1903--1979).…
The paper is written for Kluwer's Encyclopaedia of Mathematics.
In this paper, we give a comparison version of Pythagorean Theorem to judge the lower or upper bound of the curvature of Alexandrov spaces (including Riemannian manifolds).
Alexander Andreevich Andrianov was a remarkable personality in Russian physics during the last decades. A member of th prestigious school of theoretical physics in Saint Petersburg, he made relevant contributions to a number of topics along…
These are notes for a mini-course of 3 lectures given at the St. Petersburg School in Probability and Statistical Physics (June 2012). My aim was to explain, on the example of a particular model, how ideas from the representation theory of…