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 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.

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.

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 previous papers by introducing the possibility for coinfected individuals to spread only one of the diseases when in contact with a susceptible. We perform a bifurcation analysis and prove the existence of a branch of stable equilibrium points parmeterized by $K$. The branch bifurcates for some $K$ resulting in changes in which compartments are present as well as the overall dynamics of the system. Depending on the parameters different transition scenarios occur.

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 𝔸 is a generic commutative nonassociative algebra whose idempotents have the same Peirce spectrum. A medial algebra is algebra with identity (xy)(zw)=(xz)(yw). We show that these two classes are essentially coincide. We also prove that any medial spectral algebra is isomorphic to a certain isotopic deformation of the commutative associative quotient algebra 𝐊[z]/(z^n-1).

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 Jordan type fusion laws. The present approach is based on an explicit representation of the Peirce polynomial for an arbitrary algebra identity. To work with fusion rules, we develop the concept of the Peirce symbol and show that it can be explicitly determined for a wide class of algebras. We also illustrate our approach by further applications to genetic algebras and algebra of minimal cones (the so-called Hsiang algebras).

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 [Formula: see text] for time-independent and periodical models and establish the permanence dichotomy: if [Formula: see text], extinction on all patches is imminent, and if [Formula: see text], 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.

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 R0≈1 . We show even more, that for the values R0>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).

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.

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.

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.

An analytical solution of a three-level model of symmetry breaking in excited A(L)-D-A(R) quadrupolar triads with an electron donor D and identical electron acceptors A(L) and A(R) is derived, in particular, an analytical expression for the dissymmetry parameter (difference in charges, in electron charge units, on the left and right arms of the molecule) is obtained. The model predicts the threshold dependence of the symmetry breaking degree on the parameters of the molecule and its interaction with the solvent. It is shown that for typical molecular parameters, symmetry breaking occurs as a charge transfer from one arm of the molecule to the other with nearly invariable donor charge. A considerable variation of the donor charge in the course of symmetry breaking is predicted for triads with small energy gap between the ground and first excited states. Analysis of the results shows that for a large parameter area, they are very similar to those obtained in a much simpler two-level model, which suggests that instead of a more realistic three-level model, we can use a two-level model to describe symmetry breaking in excited quadrupole molecules. The theory of symmetry breaking effect on the intramolecular vibrational spectra is developed. A comparison of the effect of solvent polarity on IR spectra changes due to an increase in the degree of symmetry breaking with the available experimental data shows that the model adequately describes this phenomenon.

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.

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 that for small carrying capacity $K$ there exist a globally stable disease free equilibrium point. Furthermore, we establish the continuity of the transition dynamics of the stable equilibrium point, i.e. 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 indicate that carrying capacity is the crucial parameter and increase in resources in terms of carrying capacity promotes the risk of infection.

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 analysis of the study of positive and negative effects of the increased population density in the classical nonlinear age-structured population model. One of the main results expresses the global stability of the system in terms of the newborn function only. We establish the existence of a threshold population size implying the population extinction, which is well known in the population dynamics as the Allee effect.

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

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.

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 \(\mathbb {R}^n\), \(n\ge 3\), with nonzero constant mean curvature.

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=1/2 which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.

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.

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 Jordan algebra of Clifford type.