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Vladimir G. Tkachev

Publications and source records attributed to Vladimir G. Tkachev.

At least 37 records · Page 2Linked to original sources
arXiv2018

Variety of idempotents in nonassociative algebras

In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least $n-1$ nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension $n$. We also discuss the exceptionality of the eigenvalue $λ=\frac12$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.

math.RA↗
arXiv2017

Global stability and persistence of complex foodwebs

We develop a novel approach to study the global behaviour of large foodwebs for ecosystems where several species share multiple resources. The model extends and generalize some previous works and takes into account self-limitation. Under certain conditions, we establish the global convergence and persistence of solutions.

math.DS↗
arXiv2017

Permanency of the age-structured population model on several temporally variable patches

We consider a system of nonlinear partial differential equations that describes an age-structured population inhabiting several temporally varying patches. We prove existence and uniqueness of solution and analyze its large-time behavior in cases when the environment is constant and when it changes periodically. A pivotal assumption is that individuals can disperse and that each patch can be reached from every other patch, directly or through several intermediary patches. We introduce the net reproductive operator and characteristic equations for time-independent and periodical models and prove that permanency is defined by the net reproductive rate for the whole system. If the net reproductive rate is less or equal to one, extinction on all patches is imminent. Otherwise, permanency on all patches is guaranteed. The proof is based on a new approach to analysis of large-time stability.

math.DS↗
arXiv2016

Persistence analysis of the age-structured population model on several patches

We consider a system of nonlinear partial differential equations that describes an age-structured population living in changing environment on $N$ patches. We prove existence and uniqueness of solution and analyze large time behavior of the system in time-independent case, for periodically changing and for irregularly varying environment. Under the assumption that every patch can be reached from every other patch, directly or through several intermediary patches, and that net reproductive operator has spectral radius larger than one, we prove that population is persistent on all patches. If the spectral radius is less or equal one, extinction on all patches is imminent.

math.DS↗
arXiv2016

New examples of higher-dimensional minimal hypersurfaces

The main results of the paper are Proposition 3 and 4 which provide an effective way to construct minimal hypersurfaces in a Euclidean space. We demonstrate our technique by several new examples. This note is English translation of an earlier draft version written (in Russian) in September 1999. The final version of the paper will be published somewhere else.

math.DG↗
arXiv2012

A note on isoparametric polynomials

We show that any homogeneous polynomial solution of |\nabla F(x)|^2=m^2|x|^(2m-2), m>1, is either a radially symmetric polynomial F(x)=\pm |x|^m (for even m's) or it is a composition of a Chebychev polynomial and a Cartan-Münzner polynomial.

math.AP↗
arXiv2011

On the exponential transform of lemniscates

It is known that the exponential transform of a quadrature domain is a rational function for which the denominator has a certain separable form. In the present paper we show that the exponential transform of lemniscate domains in general are not rational functions, of any form. Several examples are given to illustrate the general picture. The main tool used is that of polynomial and meromorphic resultants.

math.CV↗
arXiv2011

On the exponential transform of multi-sheeted algebraic domains

We introduce multi-sheeted versions of algebraic domains and quadrature domains, allowing them to be branched covering surfaces over the Riemann sphere. The two classes of domains turn out to be the same, and the main result states that the extended exponential transform of such a domain agrees, apart from some simple factors, with the extended elimination function for a generating pair of functions. In an example we discuss the algebraic curves associated to level curves of the Neumann oval, and determine which of these give rise to multi-sheeted algebraic domains.

math.CV↗
arXiv2010

Non-isoparametric solutions of the eikonal equation

In this paper, we prove that a quartic polynomial solution of the eikonal equation $|\nabla_x f|^2=16x^{6}$ in $\R{n}$ is either an isoparametric polynomial or congruent to a polynomial $f=(\sum_{i=1}^n x_i^2)^2-8(\sum_{i=1}^k x_i^2)(\sum_{i=k+1}^{n} x_i^2)$, $k=0,1,\...,[\frac{n}{2}]$.

math.DG↗
arXiv2010

On a classification of minimal cubic cones in R^n

We establish a classification of cubic minimal cones in case of the so-called radial eigencubics. Our principal result states that any radial eigencubic is either a member of the infinite family of eigencubics of Clifford type, or belongs to one of 18 exceptional families. We prove that at least 12 of the 18 families are non-empty and study their algebraic structure. We also establish that any radial eigencubic satisfies the trace identity $\det \mathrm{Hess}^3 (f)=αf$ for the Hessian matrix of $f$, where $α\in \R{}$. Another result of the paper is a correspondence between radial eigencubics and isoparametric hypersurfaces with four principal curvatures.

math.DG↗
arXiv2010

Minimal cubic cones via Clifford algebras

We construct two infinite families of algebraic minimal cones in $R^{n}$. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any $n\ge4$, $n\ne 16k+1$, there is at least one minimal cone in $R^{n}$ given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in $R^{m^2}$, $m\ge2$, defined by an irreducible homogeneous polynomial of degree $m$. These examples provide particular answers to the questions on algebraic minimal cones posed by Wu-Yi Hsiang in the 1960's.

math.DG↗
arXiv2010

A generalization of Cartan's theorem on isoparametric cubics

We give a generalization of the well-known result of E. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation $|\nabla f|^2=9|x|^4$ must be rotationally equivalent to either $x_n^3-3x_n(x_1^2+...+x_{n-1}^2)$, or to one of four exceptional Cartan cubic polynomials in dimensions $n=5,8,14,26$.

math.DG↗
arXiv2009

On the Jacobian of the harmonic moment map

In this paper we represent harmonic moments in the language of transfinite functions, that is projective limits of polynomials in infinitely many variables. We obtain also an explicit formula for the Jacobian of a generalized harmonic moment map.

math.CV↗