Vladimir Gennad'Evich Tkachev
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
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Research Subjects
Vladimir Gennad'Evich Tkachev has a total of 27 co-authors. Here are the top co-authors by number of shared publications:
- Uno Wennergren
Department of Physics, Chemistry, and Biology, Linköping University, Linköping, Sweden
9 shared publications - Vladimir Kozlov
Department of Mathematics, Linköping University, Linköping, Sweden
9 shared publications - Björn Gustafsson
Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
8 shared publications - Jonathan Andersson
Department of Mathematics, Linköping University, Linköping, Sweden
4 shared publications Samia Ghersheen
Department of Mathematics Linköping University Linköping Sweden
3 shared publications- Samia Ghersheen
Department of Mathematics, Linköping University, Linköping, Sweden
3 shared publications
Latest Publications
- Open Access
Algebraic constructions of cubic minimal cones
Hsiang algebras are a class of nonassociative algebra defined in terms of a relation quartic in elements of the algebra. This class arises naturally in relation to the construction of real algebraic minimal cones. Additionally, Hsiang algebras were crucial in the construction of singular (trulsy viscosity) solutions of nonlinear uniformly elliptic partial differential equations in a series of papers by Nikolai Nadirashvili and Serge Vlăduţ. The classification of Hsiang algebras is a challenging problems, based on a Peirce decomposition like that used to study Jordan algebras, although more complicated. This paper introduces two new tools for studying Hsiang algebras: a distinguished class of algebras called \textit{quasicomposition} that generalize the Hurwitz algebras (the reals, complexes, quaternions, and octonions) and cross-product algebras and a tripling construction associating with a given algebra one of three times the dimension of the original algebra, that can be thought of as a kind of analogue of the Cayley-Dickson doubling process. One main result states that the triple is an exceptional Hsiang algebra if and only if the original algebra is quasicomposition. The quasicomposition and tripling notions are interesting in their own right and will be considered in a more general form elsewhere.
Open AccessAlgebraic constructions of cubic minimal cones
Hsiang algebras are a class of nonassociative algebra defined in terms of a relation quartic in elements of the algebra. This class arises naturally in relation to the construction of real algebraic minimal cones. Additionally, Hsiang algebras were crucial in the construction of singular (trulsy viscosity) solutions of nonlinear uniformly elliptic partial differential equations in a series of papers by Nikolai Nadirashvili and Serge Vlăduţ. The classification of Hsiang algebras is a challenging problems, based on a Peirce decomposition like that used to study Jordan algebras, although more complicated. This paper introduces two new tools for studying Hsiang algebras: a distinguished class of algebras called \textit{quasicomposition} that generalize the Hurwitz algebras (the reals, complexes, quaternions, and octonions) and cross-product algebras and a tripling construction associating with a given algebra one of three times the dimension of the original algebra, that can be thought of as a kind of analogue of the Cayley-Dickson doubling process. One main result states that the triple is an exceptional Hsiang algebra if and only if the original algebra is quasicomposition. The quasicomposition and tripling notions are interesting in their own right and will be considered in a more general form elsewhere.
- Open Access
- 9 September 2024
- Published by Centre pour la Communication Scientifique Directe (CCSD) in Communications in Mathematics
- Vol. ume 32 (20
Inner isotopes associated with automorphisms of commutative associative algebras
The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.
- 9 September 2024
- Published by Centre pour la Communication Scientifique Directe (CCSD) in Communications in Mathematics
- Vol. ume 32 (20
Open AccessInner isotopes associated with automorphisms of commutative associative algebras
The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.
- Open Access
Sharp pointwise gradient estimates for Riesz potentials with a bounded density
We establish sharp pointwise inequalities for the Riesz potential and its gradient in $\mathbb{R}^{n}$ and indicate their usefulness for potential analysis, moment theory and other applications.Open AccessSharp pointwise gradient estimates for Riesz potentials with a bounded density
We establish sharp pointwise inequalities for the Riesz potential and its gradient in $\mathbb{R}^{n}$ and indicate their usefulness for potential analysis, moment theory and other applications. - Open Access
- 4 December 2021
- Published by Springer Nature in Journal of Mathematical Biology
- Vol. 83 (6-7) , 1-62
Global stability of an age-structured population model on several temporally variable patches
We consider an age-structured density-dependent population model on several temporally variable patches. There are two key assumptions on which we base model setup and analysis. First, intraspecific competition is limited to competition between individuals of the same age (pure intra-cohort competition) and it affects density-dependent mortality. Second, dispersal between patches ensures that each patch can be reached from every other patch, directly or through several intermediary patches, within individual reproductive age. Using strong monotonicity we prove existence and uniqueness of solution and analyze its large-time behavior in cases of constant, periodically variable and irregularly variable environment. In analogy to the next generation operator, we introduce the net reproductive operator and the basic reproduction number $$R_0$$ for time-independent and periodical models and establish the permanence dichotomy: if $$R_0\le 1$$ , extinction on all patches is imminent, and if $$R_0>1$$ , permanence on all patches is guaranteed. We show that a solution for the general time-dependent problem can be bounded by above and below by solutions to the associated periodic problems. Using two-side estimates, we establish uniform boundedness and uniform persistence of a solution for the general time-dependent problem and describe its asymptotic behaviour.- 4 December 2021
- Published by Springer Nature in Journal of Mathematical Biology
- Vol. 83 (6-7) , 1-62
Open AccessGlobal stability of an age-structured population model on several temporally variable patches
We consider an age-structured density-dependent population model on several temporally variable patches. There are two key assumptions on which we base model setup and analysis. First, intraspecific competition is limited to competition between individuals of the same age (pure intra-cohort competition) and it affects density-dependent mortality. Second, dispersal between patches ensures that each patch can be reached from every other patch, directly or through several intermediary patches, within individual reproductive age. Using strong monotonicity we prove existence and uniqueness of solution and analyze its large-time behavior in cases of constant, periodically variable and irregularly variable environment. In analogy to the next generation operator, we introduce the net reproductive operator and the basic reproduction number $$R_0$$ for time-independent and periodical models and establish the permanence dichotomy: if $$R_0\le 1$$ , extinction on all patches is imminent, and if $$R_0>1$$ , permanence on all patches is guaranteed. We show that a solution for the general time-dependent problem can be bounded by above and below by solutions to the associated periodic problems. Using two-side estimates, we establish uniform boundedness and uniform persistence of a solution for the general time-dependent problem and describe its asymptotic behaviour. - Open Access
- 30 September 2021
- Published by Springer Nature in Analysis and Mathematical Physics
- Vol. 11 (4) , 1-40
Effect of density dependence on coinfection dynamics: part 2
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 coinfection dynamics.arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacityK. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized byK. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product ofKand the rate $${\bar{\gamma }}$$ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.- 30 September 2021
- Published by Springer Nature in Analysis and Mathematical Physics
- Vol. 11 (4) , 1-40
Open AccessEffect of density dependence on coinfection dynamics: part 2
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 coinfection dynamics.arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacityK. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized byK. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product ofKand the rate $${\bar{\gamma }}$$ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear. - Open Access
- 21 September 2021
- Published by Springer Nature in Analysis and Mathematical Physics
- Vol. 11 (4) , 1-25
Effect of density dependence on coinfection dynamics
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 help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number $$R_0\approx 1$$ . We show even more, that for the values $$R_0>1$$ there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).- 21 September 2021
- Published by Springer Nature in Analysis and Mathematical Physics
- Vol. 11 (4) , 1-25
Open AccessEffect of density dependence on coinfection dynamics
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 help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number $$R_0\approx 1$$ . We show even more, that for the values $$R_0>1$$ there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019). - Open Access
Effect of density dependence on coinfection dynamics
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 presented that pathogens can invade in population and their invasion depends on the carrying capacity $K$ which shows that the progression of disease in population depends on carrying capacity. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra type models.Open AccessEffect of density dependence on coinfection dynamics
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 presented that pathogens can invade in population and their invasion depends on the carrying capacity $K$ which shows that the progression of disease in population depends on carrying capacity. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra type models. - 27 March 2020
- Published by Elsevier in Differential Geometry and its Applications
- Vol. 70, 101629
New explicit solutions to the p-Laplace equation based on isoparametric foliations
- 27 March 2020
- Published by Elsevier in Differential Geometry and its Applications
- Vol. 70, 101629
New explicit solutions to the p-Laplace equation based on isoparametric foliations
- 4 February 2020
- Published by American Mathematical Society (AMS) in St. Petersburg Mathematical Journal
- Vol. 31 (2) , 223-240
Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs
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? Our aim in the paper is twofold. First, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on recent results on nonclassical solutions to fully nonlinear equations. Second, to define an appropriate algebraic formalism, which makes the analytic part of the construction of nonclassical solutions more transparent.
- 4 February 2020
- Published by American Mathematical Society (AMS) in St. Petersburg Mathematical Journal
- Vol. 31 (2) , 223-240
Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs
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? Our aim in the paper is twofold. First, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on recent results on nonclassical solutions to fully nonlinear equations. Second, to define an appropriate algebraic formalism, which makes the analytic part of the construction of nonclassical solutions more transparent.
- 17 August 2019
- Published by Springer Nature in Journal of Mathematical Sciences
- Vol. 242 (1) , 2-24
Density–Dependent Feedback in Age–Structured Populations
- 17 August 2019
- Published by Springer Nature in Journal of Mathematical Sciences
- Vol. 242 (1) , 2-24
Density–Dependent Feedback in Age–Structured Populations