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Tkachev, V. (2020). New explicit solutions to the p-Laplace equation based on isoparametric foliations. Differential geometry and its applications (Print), 70, Article ID 101629.
Open this publication in new window or tab >>New explicit solutions to the p-Laplace equation based on isoparametric foliations
2020 (English)In: Differential geometry and its applications (Print), ISSN 0926-2245, E-ISSN 1872-6984, Vol. 70, article id 101629Article in journal (Refereed) Published
##### Abstract [en]

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 distinguished values of p and the ambient dimension n this yields first examples of rational and algebraic p-harmonic functions. Moreover, we show that there are no p-harmonic polynomials of the isoparametric type. This supports a negative answer to a question proposed in 1980 by J. Lewis.

##### National Category
Geometry Mathematical Analysis Algebra and Logic
##### Identifiers
urn:nbn:se:liu:diva-164928 (URN)10.1016/j.difgeo.2020.101629 (DOI)
Available from: 2020-04-02 Created: 2020-04-02 Last updated: 2020-04-02
Perdomo, O. & Tkachev, V. G. (2019). Algebraic CMC hypersurfaces of order 3 in Euclidean spaces. Journal of Geometry, 110(1), Article ID 110:6.
Open this publication in new window or tab >>Algebraic CMC hypersurfaces of order 3 in Euclidean spaces
2019 (English)In: Journal of Geometry, ISSN 0047-2468, E-ISSN 1420-8997, Vol. 110, no 1, article id 110:6Article in journal (Refereed) Published
##### Abstract [en]

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 thereare no algebraic hypersurfaces of degree 3 in higherdimensional (n>2) Euclidean spaces, with nonzero constant mean curvature.

##### Place, publisher, year, edition, pages
Basel, Switzerland: Birkhaeuser Science, 2019
##### Keywords
Constant mean curvature; algebraic surfaces
Geometry
##### Identifiers
urn:nbn:se:liu:diva-153452 (URN)10.1007/s00022-018-0461-z (DOI)2-s2.0-85058816832 (Scopus ID)
Available from: 2018-12-18 Created: 2018-12-18 Last updated: 2020-03-27Bibliographically approved
Andersson, J., Kozlov, V., Radosavljevic, S., Tkachev, V. & Wennergren, U. (2019). Density-Dependent Feedback in Age-Structured Populations. Journal of Mathematical Sciences, 242(1), 2-24
Open this publication in new window or tab >>Density-Dependent Feedback in Age-Structured Populations
2019 (English)In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 242, no 1, p. 2-24Article in journal (Refereed) Published
##### Abstract [en]

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 mathematical analysis to study positive and negative effects of increased population density in the classical nonlinear age-structured population model introduced by Gurtin \& MacCamy in the late 1970s. One of our main results expresses the global stability of the system in terms of the newborn function only. We also derive the existence of a threshold population size implying the population extinction, which is well-known in population dynamics as an Allee effect.

##### Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2019
##### Keywords
Age-Structured Populations
##### National Category
Mathematics Other Biological Topics
##### Identifiers
urn:nbn:se:liu:diva-157057 (URN)
Available from: 2019-05-24 Created: 2019-05-24 Last updated: 2019-06-13Bibliographically approved
Ghersheen, S., Kozlov, V., Tkachev, V. & Wennergren, U. (2019). Dynamical behaviour of SIR model with coinfection: The case of finite carrying capacity. Mathematical methods in the applied sciences, 42(17), 5805-5826
Open this publication in new window or tab >>Dynamical behaviour of SIR model with coinfection: The case of finite carrying capacity
2019 (English)In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 42, no 17, p. 5805-5826Article in journal (Refereed) Published
##### Abstract [en]

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 are important in this case. We formulated an susceptible infected recovered (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 carrying capacity. The model describes five classes of a population: susceptible, infected by first virus, infected by second virus, infected by both viruses, and completely immune class. We proved that for any set of parameter values, there exists a globally stable equilibrium point. This guarantees that the disease always persists in the population with a deeper connection between the intensity of infection and carrying capacity of population. Increase in resources in terms of carrying capacity promotes the risk of infection, which may lead to destabilization of the population.

##### Place, publisher, year, edition, pages
John Wiley & Sons, 2019
##### Keywords
carrying capacity, coinfection, global stability, linear complementarity problem, SIR model
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:liu:diva-160293 (URN)10.1002/mma.5671 (DOI)000496512900014 ()2-s2.0-85066086382 (Scopus ID)
Available from: 2019-09-17 Created: 2019-09-17 Last updated: 2020-01-02Bibliographically approved
Kozlov, V., Tkachev, V., Vakulenko, S. & Wennergren, U. (2019). Global stability and persistence of complex foodwebs. Annali di Matematica Pura ed Applicata, 198(5), 1693-1709
Open this publication in new window or tab >>Global stability and persistence of complex foodwebs
2019 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 198, no 5, p. 1693-1709Article in journal (Refereed) Published
##### Abstract [en]

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 generalizes some previous works and takes into account self-limitation. Under certain explicit conditions, we establish the global convergence and persistence of solutions.

Springer, 2019
##### Keywords
Global stability, Persistence, Period-two-points, Non-increasing maps, Complex foodwebs, Self-limitation, Multiple resources
##### National Category
Mathematical Analysis Other Biological Topics
##### Identifiers
urn:nbn:se:liu:diva-155083 (URN)10.1007/s10231-019-00840-1 (DOI)000492034000012 ()
Available from: 2019-03-15 Created: 2019-03-15 Last updated: 2019-11-11Bibliographically approved
Ghersheen, S., Kozlov, V., Tkachev, V. & Wennergren, U. (2019). Mathematical analysis of complex SIR model with coinfection and density dependence. Paper presented at 2019/09/17. Computational and Mathematical Methods, 1(4)
Open this publication in new window or tab >>Mathematical analysis of complex SIR model with coinfection and density dependence
2019 (English)In: Computational and Mathematical Methods, ISSN 2577-7408, Vol. 1, no 4Article in journal (Refereed) Published
##### Abstract [en]

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 diseases in the population. We prove that, for small carrying capacity K, there exists a globally stable disease-free equilibrium point. Furthermore, we establish the continuity of the transition dynamics of the stable equilibrium point, that is, we prove that, (1) for small values of K, there exists a unique globally stable equilibrium point, and (b) it moves continuously as K is growing (while its face type may change). This indicates that the carrying capacity is the crucial parameter and an increase in resources in terms of carrying capacity promotes the risk of infection.

##### Place, publisher, year, edition, pages
Wiley-Blackwell Publishing Inc., 2019
##### Keywords
carrying capacity, coinfection, global stability, SIR model
##### National Category
Mathematics Immunology
##### Identifiers
urn:nbn:se:liu:diva-160284 (URN)10.1002/cmm4.1042 (DOI)
##### Conference
2019/09/17
Available from: 2019-09-17 Created: 2019-09-17 Last updated: 2019-10-16Bibliographically approved
Tkachev, V. (2019). On an extremal property of Jordan algebras of Clifford type. Communications in Algebra, 47(3), 1057-1066
Open this publication in new window or tab >>On an extremal property of Jordan algebras of Clifford type
2019 (English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 47, no 3, p. 1057-1066Article in journal (Refereed) Published
##### Keywords
Commutative nonassociative algebras; Associative bilinear form; Idempotents; Jordan algebras; Jordan algebras of Clifford type; Axial algebras
##### National Category
Algebra and Logic
##### Identifiers
urn:nbn:se:liu:diva-150010 (URN)10.1080/00927872.2018.1499924 (DOI)000464586900013 ()
Available from: 2018-08-07 Created: 2018-08-07 Last updated: 2019-06-04Bibliographically approved
Tkachev, V. (2019). Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs. St. Petersburg Mathematical Journal, 31(2), 255-278
Open this publication in new window or tab >>Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs
2019 (English)In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 31, no 2, p. 255-278Article in journal (Refereed) Published
##### Abstract [en]

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 recent results on nonclassical solutions to fully nonlinear equations. Secondly, to define an appropriate algebraic formalism which makes the analytic part of the construction of nonclassical solutions more transparent.

##### Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2019
##### Keywords
Ellitpic type PDEs, Nonassociative algebras, Idempotents
##### National Category
Mathematical Analysis Algebra and Logic
##### Identifiers
urn:nbn:se:liu:diva-154520 (URN)
Available from: 2019-02-20 Created: 2019-02-20 Last updated: 2019-03-25Bibliographically approved
Krasnov, Y. & Tkachev, V. (2019). Varieties of idempotents in nonassociative algebras. In: Bernstein, Swanhild (Ed.), Topics in Clifford Analysis: . Paper presented at Special Volume in Honor of Wolfgang Sprössig (pp. 80-112). Springer International Publishing
Open this publication in new window or tab >>Varieties of idempotents in nonassociative algebras
2019 (English)In: Topics in Clifford Analysis / [ed] Bernstein, Swanhild, Springer International Publishing , 2019, p. 80-112Conference paper, Published paper (Refereed)
##### Abstract [en]

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 $\lambda=\frac12$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras. solution.

##### Place, publisher, year, edition, pages
Springer International Publishing, 2019
##### Series
Trends in Mathematics, ISSN 2297-0215
##### National Category
Algebra and Logic
##### Identifiers
urn:nbn:se:liu:diva-160301 (URN)10.1007/978-3-030-23854-4 (DOI)978-3-030-23853-7 (ISBN)
##### Conference
Special Volume in Honor of Wolfgang Sprössig
Available from: 2019-09-17 Created: 2019-09-17 Last updated: 2019-12-18
Tkachev, V. (2019). V.M. Miklyukov: from Dimension 8 to Nonassociative Algebras. Mathematical Physics and Computer Simulation, 22(2), 33-50
Open this publication in new window or tab >>V.M. Miklyukov: from Dimension 8 to Nonassociative Algebras
2019 (English)In: Mathematical Physics and Computer Simulation, ISSN 2222-8896, Vol. 22, no 2, p. 33-50Article in journal (Refereed) Published
##### Abstract [en]

In this short survey we give a background and explain some recent developments in algebraic minimal cones and nonassociative algebras. A part 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.

##### Place, publisher, year, edition, pages
Volgograd, Russian Federation: Volgograd State University, 2019
##### Keywords
Entire solutions, Nonassociative algebras, Quasielliptic equations
##### National Category
Mathematical Analysis Algebra and Logic Geometry
##### Identifiers
urn:nbn:se:liu:diva-159675 (URN)10.15688/mpcm.jvolsu.2019.2.3 (DOI)
Available from: 2019-08-16 Created: 2019-08-16 Last updated: 2019-11-18Bibliographically approved
##### Identifiers
ORCID iD: orcid.org/0000-0002-8422-6140

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