# Vladimir G. TkachevLinköping University | LiU · Department of Mathematics (MAI)

Vladimir G. Tkachev

D. Sci. (Habil.)1998 Sobolev Institute of Math.

## About

100

Publications

8,232

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451

Citations

Citations since 2017

Introduction

Some of my current interests:
• nonassociative algebras and applications
• minimal submanifolds and minimal cones
• resultants and the exponential transform
• population dynamics, epidemiological models
• Hele-Shaw problem, moment problem

Additional affiliations

July 2012 - present

January 2005 - December 2008

July 1987 - December 2008

## Publications

Publications (100)

We prove that the number s(n) of disjoint minimal graphs supported on domains in R^n is bounded by e(n+1)^2. In the two-dimensional case we show that s(2) is at most three (the conjectured number is two).

We introduce a notion of the resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness
in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and
give explicit formulas for the algebraic dependence between two meromorphic functions on a compact...

This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutio...

In this paper we will explain an interesting phenomenon which occurs in general nonasso-ciative algebras. More precisely, we establish that any finite-dimensional commutative nonassociative algebra over a field satisfying an identity always contains 1/2 in its Peirce spectrum. We also show that the corresponding 1/2-Peirce module satisfies the Jord...

In contrast to an infinite family of explicit examples of two-dimensional $p$-harmonic functions obtained by G.Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of $p$-harmonic and biharmonic functions. Remarkably, for some...

The purpose of this paper is to give a systematic study of two new classes of commutative nonassociative algebras, the so-called isospectral and medial algebras. An isospectral algebra $\mathbb{A}$ is a generic commutative nonassociative algebra whose idempotents have the same Peirce spectrum. A medial algebra is algebra with identity $(xy)(zw)=(xz...

We perform a bifurcation analysis on an SIR model involving two pathogens that influences each other. Partial cross-immunity is assumed and coinfection is thought to be less transmittable then each of the diseases alone. The susceptible class has density dependent growth with carrying capacity $K$. Our model generalizes the model developed in our p...

In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also hel...

In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987 , 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfec...

We consider an age-structured density-dependent population model on several tempo-
rally variable patches. There are two key assumptions on which we base model setup
and analysis. First, intraspecific competition is limited to competition between individ-
uals of the same age (pure intra-cohort competition) and it affects density-dependent
mortalit...

In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in the 1st part of the paper. We look for coexistence equilibrium points, their stability and dependence on the carrying capacity $K$. Two sets of parameter value are determined, each giving rise to different scenar...

In this paper we will explain an interesting phenomenon which occurs in general nonassociative algebras. More precisely, we establish that any finite-dimensional commutative nonassociative algebra over a field satisfying an identity always contains 12 in its Peirce spectrum. We also show that the corresponding 12-Peirce eigenspace satisfies the Jor...

In this paper we develop an SIR model for coinfection. We discuss how the underlying dynamics depends on the carrying capacity $K$: from a simple dynamics to a more complicated. This can help in understanding of appearance of more complicated dynamics, for example, chaos etc. The density dependent population growth is also considered. It is present...

Multiple viruses are widely studied because of their negative effect on the health of host as well as on whole population. The dynamics of coinfection is important in this case. We formulated a SIR model that describes the coinfection of the two viral strains in a single host population with an addition of limited growth of susceptible in terms of...

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 n² − 1 nontrivial obstructions (syzyg...

The population size has far-reaching effects on the fitness of a population that in turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behavior. In this paper, we develop a rigorous mathematical anal...

An SIR model with the coinfection of the two infectious agents in a single host population is considered. The model includes the environmental carry capacity in each class of population. A special case of this model is analyzed and several threshold conditions are obtained which describes the establishment of disease in the population. We prove tha...

An SIR model with the coinfection of the two infectious agents in a single host population is considered. The model includes the environmental carry capacity in each class of population. A special case of this model is analyzed and several threshold conditions are obtained which describes the establishment of disease in the population. We prove tha...

In this short survey we give a background and explain some recent developments in algebraic minimal cones and nonassociative algebras. A good deal of this paper is recollections of my collaboration with my teacher, PhD supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that motivated the present research. This paper is dedicat...

In this short survey we give a background and explain some recent developments in algebraic minimal cones and nonassociative algebras. A good deal of this paper is recollections of my collaboration with my teacher, PhD supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that motivated the present research. This paper is dedicat...

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.

Sildes of my talk on Seminario di Matematica, Scuola Normale Superiore, 14 March 2019

The population size has far-reaching effects on the fitness of the population, that, in its turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behaviour. In this paper, we develop a rigourous mathema...

In this note, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? The aim of the paper is twofold. Firstly, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on re...

Understanding and finding of general algebraic constant mean curvature surfaces in the Euclidean spaces is a hard open problem. The basic examples are the standard spheres and the round cylinders, all defined by a polynomial of degree 2. In this paper, we prove that there are no algebraic hypersurfaces of degree 3 in Rn\documentclass[12pt]{minimal}...

In this paper, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? The aim of the paper is twofold. Firstly, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on r...

We establish sharp inequalities for the Riesz potential and its gradient in $\mathbb{R}^{n}$ and indicate their usefulness for potential analysis, moment theory and other applications.

Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic O...

We characterize the combinatoric stratification of the variety of generic algebras

Multiple viruses are widely studied because of their negative effect on the health of host as well as on whole population. The dynamics of coinfection is important in this case. We formulated a SIR model that describes the coinfection of the two viral strains in a single host population with an addition of limited growth of susceptible in terms of...

We study the biodiversity problem for resource competition systems with extinctions and self-limitation effects. Our main result establishes estimates of biodiversity in terms of the fundamental parameters of the model. We also prove the global stability of solutions for systems with extinctions and large turnover rate. We show that when the extinc...

To mathematically show the existence and stability of large foodwebs, large and complex as foodwebs in nature, is still one of the key problems in theoretical ecology. A specific part of this theoretical issue is that many species can share just a few resources, yet the competitive exclusion principle asserts that such foodwebs should not exist. To...

Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic O...

If $V$ is a finite-dimensional unital commutative (maybe nonassociative) algebra carrying an associative positive definite bilinear form then there exist a nonzero idempotent $c\ne e$ ($e$ being the algebra unit) of the shortest possible length $|c|^2$. In particular, $|c|^2\le \frac12|e|^2$. We prove that the equality holds exactly when $V$ is a J...

In their paper of 1993, Meyer and Neutsch established the existence of a 48-dimensional associative subalgebra in the Griess algebra. By exhibiting an explicit counter example, the present paper shows a gap in the proof one of the key results in Meyer and Neutsch's paper, which states that an idempotent a in the Griess algebra is indecomposable if...

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 (syzygi...

It is pointed out that despite of the non-linearity of the underlying equations, there do exist rather general methods that allow to generate new minimal surfaces from known ones.

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 individ...

A scanned copy of the paper of 1999:
Sergienko, V. V.; Tkachev, V. G. Doubly periodic maximal surfaces with singularities, Siberian Adv. Math. 12 (2002), no. 1, 77–91.
MathReview MR1927296

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...

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 note written by the authors (in Russian) in September 1999. The final version of the paper will be publishe...

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 a...

English translation of a short note of the author published (in Russian) in the Proceedings of XXIId Conference "A Student and The Scientific Progress", Novosibirsk, 1984, p.~66--68

By using a nonassociative algebra argument, we prove that $u\equiv0$ is the
only cubic homogeneous polynomial solution to the $p$-Laplace equation
$\mathrm{div} |Du|^{p-2}Du(x)=0 $ in $\mathbb{R}^n$ for any $n\ge2$ and
$p\not\in\{0,2\}$.

We establish a natural correspondence between (the equivalence classes of)
cubic solutions of an eiconal type equation and (the isomorphy classes of)
cubic Jordan algebras.

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-M\"unzner polynomial.

We show how to construct a non-smooth solution to Hessian fully nonlinear second-order uniformly elliptic equation using the Cartan isoparametric cubic in 5 dimensions.

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...

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 eli...

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 a...

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 $f=(\sum_{i=1}^n x_i^2)^2-8(\sum_{i=1}^k x_i^2)(\sum_{i=k+1}^{n} x_i^2)$, $k=0,1,\...,[\frac{n}{2}]$. Comment: Typos corrected. References added. Section Concluding...

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 $n=5,8,14,26$.

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...

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.

A surface is called a tube if its level-sets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the so-called flow vector. We study how the geometry of the Gaussian image of a higher-dimensional minimal tube M is controlled by the angle alpha(M) between the axis and the...

We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in ℝN
. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, that a minimal surface of codi...

A surface M is called p-minimal if one of the coordinate functions is p-harmonic in the inner metric. We show that in the twodimensional case the Gaussian map of such surfaces is quasiconformal. In the case when the surface is a tube we study the geometrical structure of such surfaces. In particularly, we establish the second order differential ine...

In the present paper we study two-dimensional maximal surfaces with harmonic level-sets. As a corollary we obtain a new class of one-periodic maximal surfaces.

Our main result states that the function (1 − Eρ
is subharmonic, where 0 ≤ ρ ≤ 1 is a density function in ℝn, n ≥ 3, and \(E_p \left( x \right) = \exp \left( { - \tfrac{2}
{n}\rlap{--} \smallint \tfrac{{\rho \left( \zeta \right)d\zeta }}
{{\left| {\zeta - x} \right|^n }}} \right)\), is the exponential transform of ρ. This answers in affirmative the...

We obtain an explicit representation for quasiradial $\gamma$-harmonic functions, which shows that these functions have essentially algebraic nature. In particular, we give a complete description of all $\gamma$ which admit algebraic quasiradial solutions. Unlike the cases $\gamma=\infty$ and $\gamma=1$, only finitely many algebraic solutions is sh...

We establish the complex analogue of Ullemar's formula for polynomial domains. We show that the Jacobian of the complex moment mapping is equal to the self-resultant of the defining polynomial.

Let $Q(z,w)=-\prod_{k=1}^n [(z-a_k)(\bar{w}-\bar{a}_k)-R_k^2]$. M. Putinar and B. Gustafsson proved recently that the matrix $Q(a_i,a_j)$, $1\leq i,j\leq n$, is positive definite if disks $|z-a_i|<R_i$ form a disjoint collection. We extend this result on symmetric collections of discs with overlapping. More precisely, we show that in the case when...

We obtain an explicit representation for quasiradial $\gamma$-harmonic
functions, which shows that these functions have essentially algebraic nature.
In particular, we give a complete description of all $\gamma$ which admit
algebraic quasiradial solutions. Unlike the cases $\gamma=\infty$ and
$\gamma=1$, only finitely many algebraic solutions is sh...

We prove the complex analogue of Ullemar’s formula for the Jacobian of the complex moment mapping. This formula was previously established in the real case.

Our main result is an extension of the classical Cauchy inequality for the case of bounded densities. In particular, this implies subharmonicity of the function Mn(E), where Vn(x) is the critical Riesz potential in Rn (α=n) of a density 0≤ρ≤1 and Mn(t) is the profile function: the solution of y′(t)=1−yn/2(t), y(0)=0. We show thath this result is op...

The complex moment sequence m(P) is assigned to a univalent polynomial P by the Cauchy transform of the P(D), where D is the unit disk. We establish the representation of the Jacobian det dm(P) in terms of roots of the derivative P'. Combining this result with the special decomposition for the Hurwitz determinants, we prove a formula for the Jacobi...

Our main result states that if M is a properly immersed minimal hypersurface with compact boundary and the multiplicity of projection of M on a hyperplane does not exceed s ∈ Z then M has quadratic volume growth and Θn(M) ≤ sκn, where Θn(M) is the volume density at infinity and κn ∼ πn/8 as n → ∞. This implies that any entire n-dimensional minimal...

We study the boundary of the nonnegative trigonometric polynomials from the algebraic point of view. In particularly, we show that it is a subset of an irreducible algebraic hypersurface and established its explicit form in terms of resultants.

We study metric and analytic properties of generalized lemniscates E
t
(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E
t
(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E
t
(f)| is convex on any interval free of critical points of ln|f|. A...

We construct and study a family of double-periodic almost entire solutions of the maximal surface equation. The solutions are parameterized by a submanifold of $3\times 3$-matrices (the so-called generating matrices). We show that the constructed solutions are either space-like or of mixed type with the light-cone type isolated singularities.

The author defines a tube to be an immersed submanifold u:Mp→Rn+1 and the interval of existence τ(Mp) to be the interval of those t for which the intersection Σt of u(Mp) with the hyperplane xn+1=t ...

A surface M is said to be p-minimal if one of the coordinate functions is p-harmonic in the inner metric. We show that in the two dimensional case the Gaussian map of such surfaces is quasiconformal. In the case when the surface is a tube we study the
geometrical structure of such surfaces. In particular, we establish the second order differential...

We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for $\p$-harmonic functions without requiring the geodesic completeness requirement of a manifold. Moreover, an upper e...

A least upper bound for the inner radiusR of an opening in a complete minimal hypersurface contained in a parallel layer is given. Namely, if Δ is the width of this
layer, thenR≤Δ/(2c
p), wherec
p is an absolute constant depending only on the dimensionp of the minimal hypersurface.

We obtain various estimates of the life-time of two-dimensional minimal tubes in R^3 by potential theory methods.

We prove that any entire convex $C^2$-solution to a Hessian type equation
with a sublinear growth at infinity is an affine function.

Questions about the behavior of the mean curvature of surfaces given in the form of graph Xn+1 = f(x) over an arbitrary domain Ωin ℝn are considered. It is proved, for example, that if mean curvature H is a continuously monotonically increasing function of coordinates xn+1 in ℝn+1, then the following assertions are fulfilled: a) if Ω = ℝn, then H =...

A tubular surface is an immersion in R^n for which the section \Pi\cap u(M) by an arbitrary hyperplane \Pi orthogonal to a fixed vector e\in{\mathbf R}^n is a compact set.For tubular minimal surfaces in R^n we prove that (a) if dim M=2 and u(M) lies in a half-space, then u(M) also lies in some hyperplane; and (b) if dim M >2 , then a tubular minima...

By using capacity estimates, we establish a Bernstein-type result for entire graphs in R^2 whose mean curvature is a monotone function of the 3-d coordinate function. This class includes in particular (nonparametric) minimal hypersurfaces, hypersurfaces with constant mean curvature, capillary surfaces. Our main result generalizes a recent result of...

## Projects

Projects (5)

Since the famous Bernstein results (1915) on entire (i.e. defined in the whole R2) minimal graphs, the study of entire solutions attracts much attention. We areinterested in entire solutions to the minimal surface equation, the p-Laplace equation, and some further classes of PDEs (normally of elliptic type). This project has a natural connection to my other projects on minimal submanifolds and nonassociative structures.

We use function-theoretic techniques (including capacity, the extremal length, the fundamental frequency, the projective volume etc) to study two-dimensional and higher-dimensional minimal (zero mean curvature) submanifolds of Rn and their generalizations (constant mean curvature hypersurfaces, zero mean curvature sumbanifolds in Lorentz spaces etc).

The project cover some topics concerning harmonic moments of plane domains, their relationship to the exponential transform and to the meromorphic resultant.