a new method of constructing symmetry adapted linear combinations for a finite group ii correspondence theorem and basis operators abstract a correspondence theorem is introduced to describe the relation between an elementary basis function and a symmetry adapted linear combination belonging to the same irreducible representation of a point group by means of basis operators the general expressions are given for the symmetry adapted linear combinations the coupling coefficients the lattice harmonics and the hybrid atomic orbitals illustrative examples are discussed to demonstrate the effectiveness of the formalism

persistence and extinction in stochastic delay logistic equation by incorporating ornstein uhlenbeck process abstract the persistence and extinction pe are interesting topics in mathematics this research analyzed pe of stochastic logistic equations sle by incorporating the ornstein uhlenbeck process slop and stochastic delay logistic equation sdle by incorporating the ornstein uhlenbeck process sldop firstly we proved that slop and sldop have positive solutions likewise for stochastic permanence sp weak persistence in the mean wpm non persistence in the mean npm and extinction the sufficient conditions are established for slop and sldop subsequently for numerical simulation we used 4 stage stochastic runge kutta srk4 to illustrate the effectiveness of the results

some properties of fuzzy groups on definit des sous groupes flous normaux des sous groupes flous normaux de niveau et leurs homomorphismes et on donne certaines de leurs proprietes

fuzzy subgroups having the property ii abstract in 3 the notion of the property of a fuzzy subgroup of a group which is a more fuzzified notion of the notion of a generalized characteristic fuzzy subgroup of a finite group is introduced in this paper we show that a generalized characteristic fuzzy subgroup of any group has always the property

some characterizations of fuzzy subgroups abstract we show that if a fuzzy subgroup can be written as the intersection of all its minimal fuzzy p subgroups or the intersection of all its minimal fuzzy p subgroups then some properties of the fuzzy subgroup characterize the properties of all its minimal fuzzy p subgroups or all its minimal fuzzy p subgroups and investigate interrelationships among some properties of fuzzy subgroups

fuzzy normal subgroups and fuzzy cosets abstract we introduce here the notion of 1 i a fuzzy normal subgroup and 2 ii a fuzzy coset we prove that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g further we show that the level subgroups of a fuzzy normal subgroup are all normal we obtain several results which are analogs of some basic theorems of group theory in particular a fuzzy version of lagrange s theorem is established

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

on generalized characteristic fuzzy subgroups of a finite group abstract in 1991 asaad 2 suggested the following two questions i what can be said about the structure of a fuzzy subgroup a of a finite group g when information is known about the structure of g ii what can be said about the structure of group g when information is known about the structure of a fuzzy subgroup a of g the purpose of this paper is to continue the investigation of the above questions

persistence and extinction in general non autonomous logistic model with delays and stochastic perturbation this is a continuation of the paper liu and wang 2012 14 firstly we consider a general non autonomous logistic model with delays and stochastic perturbation then sufficient conditions for extinction are established as well as nonpersistence in the mean weak persistence and stochastic permanence the threshold between weak persistence and extinction is obtained finally numerical simulations are introduced to support the theoretical analysis results

fuzzy subgroups having the property abstract we introduce the notion of the property for a fuzzy group and characterize all finite cyclic groups in terms of this notion this property concerns the transition of the order of an element of a group from the classical to the fuzzy setting

global stability and stochastic permanence of a non autonomous logistic equation with random perturbation abstract this paper discusses a randomized non autonomous logistic equation d n t n t a t b t n t d t t d b t where b t is a 1 dimensional standard brownian motion in d q jiang n z shi a note on non autonomous logistic equation with random perturbation j math anal appl 303 2005 164 172 the authors show that e 1 n t has a unique positive t periodic solution e 1 n p t provided a t b t and t are continuous t periodic functions a t 0 b t 0 and 0 t a s 2 s d s 0 we show that this equation is stochastically permanent and the solution n p t is globally attractive provided a t b t and t are continuous t periodic functions a t 0 b t 0 and min t 0 t a t max t 0 t 2 t by the way the similar results of a generalized non autonomous logistic equation with random perturbation are yielded

a new method of constructing the symmetry coordinates of molecular vibrations based on the correspondence theorem a general method of constructing the symmetry coordinates of molecular vibrations is presented via the correspondence theorem which describes the parallelism between a salc symmetry adapted linear combination and an elementary basis function belonging to the same irreducible representation of the symmetry group of the molecule

a new method of constructing the symmetry adapted linear combinations based on the correspondence theorem and induced representations the theory of induced representations is incorporated into the method of constructing symmetry adapted linear combinations for a symmetry group based on the correspondence theorem developed previously the theory is effective for a highly symmetric molecule with a set of equivalent radicals each of which again consists of a number of equivalent atoms an illustrative example is given to construct the internal symmetry coordinates of molecular vibrations of c ch3 4 with the frozen configuration belonging to the point group td

a unified theory of the point groups iii classification and basis functions of improper point groups this paper introduces a new system of classification of improper point groups which is most effective for describing their general irreducible representations the complete set of the general angular momentum eigenfunctions is classified according to the general irreducible representations of the seven sets of the improper point groups corresponding to cn c and dn d for an aribtrary n

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

analysis of a stochastic logistic model with diffusion stochastic logistic model with diffusion and levy jumps is proposed critical value between stability and extinction for species in each patch is obtained critical value has a close relationship with levy jumps numerical simulations are introduced to validate the results taking both white noise and levy jump noise into account a stochastic logistic model with diffusion is proposed and considered under some simple assumptions the almost complete parameters analysis of the model is carried out in each case it is shown that the population in each patch is either stable in time average or extinct depending on the parameters of the model especially depending on the intensity of the levy jump noise some simulation figures are introduced to validate the theoretical results

a new method of constructing symmetry adapted linear combinations for a finite group i salc s of equivalent invariant functions abstract a theory of representation for a finite group g has been presented based on the basis functions defined on a set of equivalent points s n with respect to the symmetry operations of g the general expressions are given for the symmetry adapted linear combinations of equivalent invariant functions and the bonding hybrid atomic orbitals

level subgroups and union of fuzzy subgroups abstract in this paper we investigate the conditions under which a given fuzzy subgroup of a given group can or can not be realized as a union of two proper fuzzy subgroups moreover we formulate the notion of fuzzy subgroup generated by a given fuzzy subset by level subgroups we also give characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups also we discuss the relationship between level subgroups of a given fuzzy subgroup and the level subgroups of its homomorphic image

a characterization of fuzzy subgroups of some abelian groups abstract we introduce the notion of fuzzy p subgroups and characterize fuzzy subgroups of torsion groups and cyclic groups by their minimal fuzzy p subgroups and minimal fuzzy p subgroups

tl subgroups having the property abstract we introduce the notion of a tl subgroup having the property that is a generalization of the notion of a generalized characteristic tl subgroup as abou zaid s notion was generalized in kim 1994 5 and investigate interrelationship between the notion of a tl subgroup having the property and the notion of a generalized characteristic tl subgroup

fuzzy orders relative to fuzzy subgroups abstract in this paper we introduce the notion of fuzzy orders of the elements of a group and show that most of the basic properties of orders of elements in the group theory are valid in the theory of fuzzy groups where ordinary orders of elements are replaced by fuzzy orders of elements reports of further development of the subject will follow in subsequent essays

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

orders of fuzzy subgroups and fuzzy p subgroups abstract in this paper we introduce the notion of orders of fuzzy subgroups and the notion of fuzzy p subgroups by using the notion of fuzzy orders of elements and investigate some properties of fuzzy subgroups

generated and cyclic fuzzy groups abstract subgroups and normal subgroups of a fuzzy group are defined and their criteria are established the fuzzy order of an element of a group is defined and its relationship with the order of the element is examined the construction of the smallest fuzzy group containing a given arbitrary fuzzy set is systematized a cyclic fuzzy group that is a restriction of a fuzzy group or that is generated by a fuzzy point is brought to discussion the usual interactions between a cyclic and an abelian fuzzy group are pointed out the possibility that an abelian fuzzy group might be isomorphic to a direct product of some of its cyclic subgroups is touched upon

stochastic population dynamics under regime switching ii this is a continuation of our paper q luo x mao stochastic population dynamics under regime switching j math anal appl 334 2007 69 84 on stochastic population dynamics under regime switching in this paper we still take both white and color environmental noise into account we show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population in some special but important situations we precisely describe the limit of the average in time of the population

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

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

classification of configurations of systems of identical particles on finite sets all possible configurations of a system of n identical particles are classified for the case when the configuration space q of a single particle reduces to a finite set the classification is achieved by means of a combinatorial analogue of the duality of weyl two groups the pauli group permuting identical particles and the symmetric group on q act on the nth cartesian power of q these actions mutually commute and their orbit structure is classified by appropriate epikernels moreover each stratum of each group is invariant under the dual action which yields a coarsening of permutation representations acting on appropriate sets of orbits it is shown that such a coarsening recovers the geometric symmetry for the case when particles are indistinguishable

persistence and extinction in stochastic non autonomous logistic systems this paper studies two widely used stochastic non autonomous logistic models for the first system sufficient conditions for extinction non persistence in the mean weak persistence and stochastic permanence are established the critical number between weak persistence and extinction is obtained for the second system sufficient criteria for extinction non persistence in the mean weak persistence in the mean strong persistence in the mean and stochastic permanence are established the critical number between weak persistence in the mean and extinction is obtained it should be pointed out that this research is systematical and complete in fact the behaviors of the two models in every coefficient cases are cleared up by the results obtained in this paper

a remark on stochastic logistic model with diffusion stochastic logistic system with diffusion is proposed and studied almost sufficient and necessary conditions for stability in time average and extinction are established some numerical examples are introduced to support the main results

the lattice of fuzzy left right ideals of a ring is modular we use the technique of nested set to study the structure of the lattice of all fuzzy left right ideals of a given ring we give a way to depict the supremum of fuzzy left right ideals by nested set and prove that the lattice of all fuzzy left right ideals of a given ring is modular

a unified theory of the point groups ii the general projective representations and their application to space groups the general expressions of all inequivalent irreducible projective representations of crystallographic and noncrystallographic point groups except for the icosahedral group are given in terms of the vector representations of the proper double point groups its application to the representations of the space group of wave vector is discussed construction of the basis sets is discussed

a unified theory of the point groups and their general irreducible representations point groups and their general irreducible vector and projective representations are characterized by the subgroup conditions for su 2 in a unified scheme these conditions are given by simple polynomial equations imposed on the matrix elements of the 2 2 unitary matrices of su 2 the general irreducible representations for the point groups d dn with arbitrary integer n 2 o and t are given by four simple and effective tables

groups and fuzzy subgroups abstract the purpose of this paper is to deal with the following two problems 1 what can be said about the structure of a fuzzy subgroup a of a group g when information is known about the structure of g 2 what can be said about the structure of a group g when information is known about the structure of a fuzzy subgroup a of g