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Victor Alexandrov

Publications and source records attributed to Victor Alexandrov.

At least 19 recordsLinked to original sources
arXiv2025

Steffen's flexible polyhedron is embedded. A proof via symbolic computations

A polyhedron is flexible if it can be continuously deformed preserving the shape and dimensions of every its face. In the late 1970's Klaus Steffen constructed a sphere-homeomorphic embedded flexible polyhedron with triangular faces and with 9 vertices only, which is well-known in the theory of flexible polyhedra. At about the same time, a hypothesis was formulated that the Steffen polyhedron has the least possible number of vertices among all embedded flexible polyhedra without boundary. A counterexample to this hypothesis was constructed by Matteo Gallet, Georg Grasegger, Jan Legersk{ý}, and Josef Schicho in 2024 only. Surprisingly, until now, no proof has been published in the mathematical literature that the Steffen polyhedron is embedded. Probably, this fact was considered obvious to everyone who made a cardboard model of this polyhedron. In this article, we prove this fact using computer symbolic calculations.

math.MG↗
arXiv2025

Additional first order equation for infinitesimal bendings of smooth surfaces in the isothermal coordinates

The article contributes to the theory of infinitesimal bendings of smooth surfaces in Euclidean 3-space. We derive a linear differential equation of the first order, which previously did not appear in the literature and which is satisfied by any Darboux rotation field of a smooth surface. We show that, for some surfaces, this additional equation is functionally independent of the three standard equations that the Darboux rotation field satisfies (and by which it is determined). As a consequence of this additional equation, we prove the maximum principle for the components of the Darboux rotation field for a class of disk-homeomorphic surfaces containing not only surfaces of positive Gaussian curvature.

math.DG↗
arXiv2022

Recognition of affine-equivalent polyhedra by their natural developments

The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two polyhedra are isometric or not by using their developments only. In this article, we study a similar problem about whether it is possible, using only the developments of two convex polyhedra of Euclidean 3-space, to understand that these polyhedra are (or are not) affine-equivalent.

math.MG↗
arXiv2020

Around Efimov's differential test for homeomorphism

In 1968, N.\,V.~Efimov proved the following remarkable theorem: \textit{Let $f:\mathbb{R}^2\to\mathbb{R}^2\in C^1$ be such that $\det f'(x)<0$ for all $x\in\mathbb{R}^2$ and let there exist a function $a(x)>0$ and constants $C_1\geqslant 0$, $C_2\geqslant 0$ such that the inequalities $|1/a(x)-1/a(y)|\leqslant C_1 |x-y|+C_2$ and $|\det f'(x)|\geqslant a(x)|\operatorname{curl}f(x)|+a^2(x)$ hold true for all $x, y\in\mathbb{R}^2$. Then $f(\mathbb{R}^2)$ is a convex domain and $f$ maps $\mathbb{R}^2$ onto $f(\mathbb{R}^2)$ homeomorphically.} Here $\operatorname{curl}f(x)$ stands for the curl of $f$ at $x\in\mathbb{R}^2$. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of global inverse functions, as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.

math.DG↗
arXiv2020

Kolmogorov's legacy: Algorithmic Theory of Informatics and Kolmogorov Programmable Technology

In this survey, we explore Andrei Nikolayevich Kolmogorov's seminal work in just one of his many facets: its influence Computer Science especially his viewpoint of what herein we call 'Algorithmic Theory of Informatics.' Can a computer file 'reduce' its 'size' if we add to it new symbols? Do equations of state like second Newton law in Physics exist in Computer Science? Can Leibniz' principle of identification by indistinguishability be formalized? In the computer, there are no coordinates, no distances, and no dimensions; most of traditional mathematical approaches do not work. The computer processes finite binary sequences i.e. the sequences of 0 and 1. A natural question arises: Should we continue today, as we have done for many years, to approach Computer Science problems by using classical mathematical apparatus such as 'mathematical modeling'? The first who drew attention to this question and gave insightful answers to it was Kolmogorov in 1960s. Kolmogorov's empirical postulate about existence of a program that translates 'a natural number into its binary record and the record into the number' formulated in 1958 represents a hint of Kolmogorov's approach to Computer Science. Following his ideas, we interpret Kolmogorov algorithm, Kolmogorov machine, and Kolmogorov complexity in the context of modern information technologies showing that they essentially represent fundamental elements of Algorithmic Theory of Informatics, Kolmogorov Programmable Technology, and new Komputer Mathematics i.e. Mathematics of computers.

cs.GL↗
arXiv2019

Necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex

We derive fundamentally new equations that are satisfied by first-order flexes of a flexible polyhedron. Moreover, we indicate two sources of such new equations. These sources are the Dehn invariants and rigidity matrix. The equations derived provide us with fundamentally new necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex.

math.MG↗
arXiv2019

The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in $\mathbb R^d$ does not always remain unaltered during the flex

Being motivated by the theory of flexible polyhedra, we study the Dirichlet and Neumann eigenvalues for the Laplace operator in special bounded domains of Euclidean $d$-space. The boundary of such a domain is an embedded simplicial complex which allows a continuous deformation (a flex), under which each simplex of the complex moves as a solid body and the change in the spatial shape of the domain is achieved through a change of the dihedral angles only. The main result of this article is that both the Dirichlet and Neumann spectra of the Laplace operator in such a domain do not necessarily remain unaltered during the flex of its boundary.

math.MG↗
arXiv2019

The global rigidity of a framework is not an affine-invariant property

It is well-known that the property of a bar-and-joint framework `to be infinitesimally rigid' is invariant under projective transformations of Eucliean $d$-space for every $d\geqslant 2$. It is less known that the property of a bar-and-joint framework `to be globally rigid' is not invariant even under affine transformations of the Euclidean plane. In this note, we prove of the latter statement for Euclidean $d$-space for every $d\geqslant 2$.

math.MG↗
arXiv2018

A sufficient condition for a polyhedron to be rigid

We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed continuously by changing its dihedral angles only. We prove that the polyhedron is not flexible if for each of its edges the following holds true: the length of this edge is not a linear combination with rational coefficients of the lengths of the remaining edges. We prove also that if a polyhedron is flexible, then some linear combinations of its dihedral angles remain constant during the flex. In this case, the coefficients of such a linear combination do not alter during the flex, are integers, and do not equal to zero simultaneously.

math.MG↗
arXiv2017

Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?

We choose some special unit vectors $\boldsymbol{n}_1,\dots,\boldsymbol{n}_5$ in $\mathbb{R}^3$ and denote by $\mathscr{L}\subset\mathbb{R}^5$ the set of all points $(L_1,\dots,L_5)\in\mathbb{R}^5$ with the following property: there exists a compact convex polytope $P\subset\mathbb{R}^3$ such that the vectors $\boldsymbol{n}_1,\dots,\boldsymbol{n}_5$ (and no other vector) are unit outward normals to the faces of $P$ and the perimeter of the face with the outward normal $\boldsymbol{n}_k$ is equal to $L_k$ for all $k=1,\dots,5$. Our main result reads that $\mathscr{L}$ is not a locally-analytic set, i.\,e., we prove that, for some point $(L_1,\dots,L_5)\in\mathscr{L}$, it is not possible to find a neighborhood $U\subset\mathbb{R}^5$ and an analytic set $A\subset\mathbb{R}^5$ such that $\mathscr{L}\cap U=A\cap U$. We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces.

math.MG↗
arXiv2015

On the number of solutions of a quadratic equation in a normed space

We study an equation $Qu=g$, where $Q$ is a continuous quadratic operator acting from one normed space to another normed space. Obviously, if $u$ is a solution of such equation then $-u$ is also a solution. We find conditions implying that there are no other solutions and apply them to the study of the Dirichlet boundary value problem for the partial differential equation $uΔu =g$.

math.FA↗
arXiv2015

Kondo Breakdown in Topological Kondo Insulators

Motivated by the observation of light surface states in SmB6, we examine the effects of surface Kondo breakdown in topological Kondo insulators. We present both numerical and analytic results which show that the decoupling of the localized moments at the surface disturbs the compensation between light and heavy electrons and dopes the Dirac cone. Dispersion of these uncompensated surface states are dominated by inter-site hopping, which leads to a much lighter quasiparticles. These surface states are also highly durable against the effects of surface magnetism and decreasing thickness of the sample.

cond-mat.str-el↗