a chain of evolution algebras abstract we introduce a notion of chain of evolution algebras the sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of chapman kolmogorov equation we give several examples time homogenous time non homogenous periodic etc of such chains for a periodic chain of evolution algebras we construct a continuum set of non isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set we obtain a criteria for an evolution algebra to be baric and give a concept of a property transition for several chains of evolution algebras we describe the behavior of the baric property depending on the time for a chain of evolution algebras given by the matrix of a two state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely with respect to lebesgue measure we also construct examples of the almost surely baric chains of evolution algebras we show that there are chains of evolution algebras such that if it has a unique resp infinitely many absolute nilpotent element at a fixed time then it has unique resp infinitely many absolute nilpotent element any time also there are chains of evolution algebras which have not such property for an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time but for other values of parameters there is a critical time t c such that the chain has only two idempotent elements if time t t c and it has four idempotent elements if time t t c

gaussian wave packet transform based numerical scheme for the semi classical schrodinger equation with random inputs abstract in this work we study the semi classical limit of the schrodinger equation with random inputs and show that the semi classical schrodinger equation produces o e oscillations in the random variable space with the gaussian wave packet transform the original schrodinger equation is mapped to an ordinary differential equation ode system for the wave packet parameters coupled with a partial differential equation pde for the quantity w in rescaled variables further we show that the w equation does not produce e dependent oscillations and thus it is more amenable for numerical simulations we propose multi level sampling strategy in implementing the gaussian wave packet transform where in the most costly part i e simulating the w equation it is sufficient to use e independent samples we also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm and hopefully shed light on possible future directions

nilpotent evolution algebras over arbitrary fields abstract the paper is devoted to the study of annihilator extensions of evolution algebras and suggests an approach to classify finite dimensional nilpotent evolution algebras subsequently nilpotent evolution algebras of dimension up to four are classified

classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field abstract evolution algebras were introduced into genetics to deal with the mechanism of inheritance of asexual organisms their distribution into isotopism classes is uniquely related with the mutation of alleles in non mendelian genetics this paper deals with such a distribution by means of computational algebraic geometry we focus in particular on the two dimensional case which is related to the asexual reproduction processes of diploid organisms specifically we determine the existence of four isotopism classes whatever the base field is and we characterize the corresponding isomorphism classes

fuzzy groups some group theoretic analogs abstract we introduce the concepts of 1 order of a fuzzy subgroup of a finite group 2 a fuzzy abelian group and 3 a fuzzy solvable group we obtain some results involving these notions which are analogs of results from group theory also we prove analogs of two basic theorems of group theory namely cayley s theorem and lagrange s theorem

an evolution algebra in population genetics abstract we consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type we study basic properties of the algebra this algebra is commutative and hence flexible not associative and not necessarily power associative in general we prove that being alternative is equivalent to being associative we find conditions to be an associative a fourth power associative or a nilpotent algebra we also prove that if the algebra is not alternative then to be power associative is equivalent to be jordan moreover it is not unital in a general case we describe the full set of idempotent elements and the full set of absolute nilpotent elements the set of all operators of left right multiplications is described under some conditions it is proved that the corresponding algebra is centroidal moreover the classification of 2 dimensional and some 3 dimensional algebras are obtained

lattices of fuzzy subgroupoids fuzzy submonoids and fuzzy subgroups we redefine the sup min product of fuzzy subsets and discuss the redefined sup min products of fuzzy subgroupoids fuzzy submonoids and fuzzy subgroups in addition we study lattice structures of the lattices of fuzzy subgroupoids fuzzy submonoids and fuzzy subgroups

the gaussian wave packet transform efficient computation of the semi classical limit of the schrodinger equation part 1 formulation and the one dimensional case in this paper an efficient and accurate method for simulating the propagation of a localized solution of the schrodinger equation with a smooth potential near the semiclassical limit is presented we are interested computing arbitrarily accurate solutions when the non dimensional planck s constant e is small but not negligible the method is based on a time dependent transformation of the independent variables closely related to gaussian wave packets a rescaled wave function w satisfies a new schrodinger equation with a time dependent potential which is a perturbation of the harmonic oscillator the perturbation being o e so that all stiffness in space and time is greatly reduced in fact for integration in a fixed time interval the number of modes required to fully resolve the problem decreases when e is decreased the original wave function may be reconstructed by fourier interpolation although expectation values of the observables can be computed directly from the function w itself if the initial condition is a gaussian wave packet very few modes are necessary to fully resolve the w variable so for short time very accurate solutions can be obtained at low computational cost initial conditions other than gaussians wave packets can also be used in this paper the gaussian wave packet transform is carefully outlined and applied to the schrodinger equation in one dimension detailed numerical tests show the efficiency and accuracy of the approach in the sequel of this paper this approach has been extended to the multidimensional case

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

commutative fuzzy sets and nilpotent fuzzy groups abstract the notions of a commutative fuzzy subset and a nilpotent fuzzy subgroup are introduced and some of their properties are investigated in particular it is shown that all nilpotent groups can be characterized in terms of their nilpotent fuzzy subgroups

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

evolution algebras of arbitrary dimension and their decompositions abstract we study evolution algebras of arbitrary dimension we analyze in deep the notions of evolution subalgebras ideals and non degeneracy and describe the ideals generated by one element and characterize the simple evolution algebras we also prove the existence and unicity of a direct sum decomposition into irreducible components for every non degenerate evolution algebra when the algebra is degenerate the uniqueness cannot be assured the graph associated to an evolution algebra relative to a natural basis will play a fundamental role to describe the structure of the algebra concretely a non degenerate evolution algebra is irreducible if and only if the graph is connected moreover when the evolution algebra is finite dimensional we give a process called the fragmentation process to decompose the algebra into irreducible components

on nilpotent index and dibaricity of evolution algebras abstract an evolution algebra corresponds to a quadratic matrix a of structural constants it is known the equivalence between nil right nilpotent evolution algebras and evolution algebras which are defined by upper triangular matrices a we establish a criterion for an n dimensional nilpotent evolution algebra to be with maximal nilpotent index 2 n 1 1 we give the classification of finite dimensional complex evolution algebras with maximal nilpotent index moreover for any s 1 n 1 we construct a wide class of n dimensional evolution algebras with nilpotent index 2 n s 1 we show that nilpotent evolution algebras are not dibaric and establish a criterion for two dimensional real evolution algebras to be dibaric

evolution algebra of a bisexual population abstract we introduce an evolution algebra identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra the basic properties of the algebra are studied we prove that this algebra is commutative and hence flexible not associative and not necessarily power associative we show that the evolution algebra of the bisexual population is not a baric algebra but a dibaric algebra and hence its square is baric moreover we show that the algebra is a banach algebra the set of all derivations of the evolution algebra is described we find necessary conditions for a state of the population to be a fixed point or a zero point of the evolution operator which corresponds to the evolution algebra we also establish upper estimate of the limit points set for trajectories of the evolution operator using the necessary conditions we give a detailed analysis of a special case of the evolution algebra of bisexual population in which type 1 of females and males have preference for such a special case we describe the full set of idempotent elements and the full set of absolute nilpotent elements

some characterizations of fuzzy subgroup via fuzzy p subsets and fuzzy p subgroups abstract the notion of a p of a fuzzy subgroup a is introduced using the notion we characterize fuzzy subgroups and show the every commutative fuzzy subgroup characterized as the intersection of its all minimal fuzzy p subgroups

algebraic computation of genetic patterns related to three dimensional evolution algebras the mitosis process of an eukaryotic cell can be represented by the structure constants of an evolution algebra any isotopism of the latter corresponds to a mutation of genotypes of the former this paper uses computational algebraic geometry to determine the distribution of three dimensional evolution algebras over any field into isotopism classes and hence to describe the spectrum of genetic patterns of three distinct genotypes during a mitosis process their distribution into isomorphism classes is also determined in case of dealing with algebras having a one dimensional annihilator

constrained evolution algebras and dynamical systems of a bisexual population abstract consider a bisexual population such that the set of females can be partitioned into finitely many different types indexed by 1 2 n and similarly that the male types are indexed by 1 2 recently an evolution algebra of bisexual population was introduced by identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra in this paper we study constrained evolution algebra of bisexual population in which type 1 of females and males have preference for such algebras sets of idempotent and absolute nilpotent elements are known we consider two particular cases of this algebra giving more constraints on the structural constants of the algebra by the first our constraint we obtain an n dimensional algebra with a matrix of structural constants containing only 0 and 1 in the second case we consider n 2 but with general constraints in both cases we study dynamical systems generated by the quadratic evolution operators of corresponding constrained algebras we find all fixed points limit points and some 2 periodic points of the dynamical systems moreover we study several properties of the constrained algebras connecting them to the dynamical systems we give some biological interpretation of our results

redefined sup min product of fuzzy subsets and join of fuzzy subgroups abstract we redefine the sup min product of two fuzzy subsets give an explicit formula for the join of two fuzzy subgroups of a group and show that the lattice of all normal fuzzy subgroups of a group is a sublattice of the lattice of all fuzzy subgroups of the group

nilpotent fuzzy groups abstract the commutator of two fuzzy subsets of a group is defined and its properties are studied using this idea the descending central chain of a fuzzy subgroup is introduced a definition of a nilpotent fuzzy group is proposed and different properties of nilpotent fuzzy groups are discussed