deformed shape invariance and exactly solvable hamiltonians with position dependent effective mass known shape invariant potentials for the constant mass schrodinger equation are taken as effective potentials in a position dependent effective mass pdem one the corresponding shape invariance condition turns out to be deformed its solvability imposes the form of both the deformed superpotential and the pdem a lot of new exactly solvable potentials associated with a pdem background are generated in this way a novel and important condition restricting the existence of bound states whenever the pdem vanishes at an end point of the interval is identified in some cases the bound state spectrum results from a smooth deformation of that of the conventional shape invariant potential used in the construction in others one observes a generation or suppression of bound states depending on the mass parameter values the corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials

soft groups and normalistic soft groups soft set theory proposed by molodtsov has been regarded as an effective mathematical tool to deal with uncertainties in this paper first we correct some of the problematic cases in a previous paper by aktas and email protected man h aktas n email protected man soft sets and soft groups inf sci 177 2007 2726 2735 moreover we introduce the concepts of normalistic soft group and normalistic soft group homomorphism study their several related properties and investigate some structures that are preserved under normalistic soft group homomorphisms

application of fuzzy soft set in decision making problems based on grey theory there are many uncertain problems in practical production and life which need decisions made with soft sets and fuzzy soft sets however the basis of evaluation of the decision method is single and simple the same decision problem can obtain different results from using a different evaluation basis in this paper in order to obtain the right result we discuss fuzzy soft set decision problems a new algorithm based on grey relational analysis is presented the evaluation bases of the new algorithm are multiple there is more information in a decision result based on multiple evaluation bases which is more easily accepted and logical to one s thinking for the two cases examined the results show that the new algorithm is efficient for solving decision problems

transitive closures and orderings on soft sets in this paper an attempt is made to extend some standard results in set theory on the basis of soft set relations antisymmetric relation and transitive closure of a soft set relation are introduced and an analogue of warshall s algorithm is proposed for calculating the transitive closure of a soft set relation ordering on a soft set is defined and some set theoretical results based on this are proved

equivalence of unstable anharmonic oscillators and double wells we give a direct and general proof of the equality of the energy levels of a pair of different quantum problems the double well and the unstable anharmonic oscillator defined by complex translation the resonances of the two problems are also equal and are the analytic continuations of the anharmonic oscillator energy levels

applications of soft sets in ideal theory of bck bci algebras the notion of soft ideals and idealistic soft bck bci algebras is introduced and several examples are given relations between soft bck bci algebras and idealistic soft bck bci algebras are provided the intersection union and operation and or operation of soft ideals and idealistic soft bck bci algebras are established

soft sets and soft groups molodtsov introduced the concept of soft set theory which can be used as a generic mathematical tool for dealing with uncertainty in this paper we introduce the basic properties of soft sets and compare soft sets to the related concepts of fuzzy sets and rough sets we then give a definition of soft groups and derive their basic properties using molodtsov s definition of the soft sets

fuzzy filters of mtl algebras characterizations of fuzzy filters in mtl algebras are given the fuzzy filter generated by a fuzzy set is considered the notion of boolean fuzzy filters and mv fuzzy filters are introduced and related properties are investigated a condition for a fuzzy filter to be boolean is provided a characterization of a boolean fuzzy filter is given a congruence relation on a mtl algebra induced by a fuzzy filter is established and we show that the set of all congruence relations induced by a fuzzy filter is a completely distributive lattice

soft p ideals of soft bci algebras molodtsov d molodtsov soft set theory first results comput math appl 37 1999 19 31 introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches jun y b jun soft bck bci algebras comput math appl 56 2008 1408 1413 applied first the notion of soft sets by molodtsov to the theory of bck bci algebras in this paper we introduce the notion of soft p ideals and p idealistic soft bci algebras and then investigate their basic properties using soft sets we give characterizations of fuzzy p ideals in bci algebras we provide relations between fuzzy p ideals and p idealistic soft bci algebras

notes on interval valued intuitionistic fuzzy soft sets and their properties comput math appl 60 2010 906 918 in this paper we point out two definitions definition 18 and definition 19 in a previous paper by jiang et al y jiang y tang q chen h liu j tang interval valued intuitionistic fuzzy soft sets and their properties comput math appl 60 2010 906 918 are not appropriate accordingly the proofs of some relevant theorems are also not right we redefine the two definitions and reprove these theorems by using the new definitions

on soft equality molodtsov introduced the concept of soft sets which can be seen as a new mathematical tool for dealing with uncertainty in this paper we deal with the algebraic structure of soft sets the lattice structures of soft sets are constructed the concept of soft equality is introduced and some related properties are derived it is proved that soft equality is a congruence relation with respect to some operations and the soft quotient algebra is established

dirac equation with position dependent effective mass and solvable potentials in the schrodinger equation we study the one dimensional non hermitian imaginary potential with a real energy spectrum in the framework of the position dependent effective mass dirac equation the dirac equation is mapped into the exactly solvable schrodinger like equation endowed with position dependent effective mass that we present a new procedure to solve it the point canonical transformation in non relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function the imaginary potential can be obtained the spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials we also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials

a general help inequality connected with symmetric operators in this paper a general help hardy everitt littlewood polya inequality is considered which is connected with a symmetric operator in a hilbert space and abstract boundary mappings a criterion for the validity of such an inequality in terms of the abstract titchmarsh weyl function is proved and applied to sturm liouville operators difference operators a hamiltonian system and a block operator matrix

non self adjoint difference operators and their spectrum initially this paper is a discrete analogue of the work of brown et al 1999 proc r soc a 455 12351257 on second order differential equations with complex coefficients that is we investiga

orthogonal polynomials and extensions of copson s inequality this is a survey of recent results on a class of series inequalities involving second order difference operators which includes a well known inequality of copson s a connection has been established between these inequalities and the properties of the hellinger nevanlinna m function for an associated recurrence relation mx aw x a e c the validity of the inequality the value of the best constant and the nontrivial equalising sequences if they exist are determined in terms of m the function m is shown to have an integral representation in terms of a measure with respect to which polynomial solutions of the recurrence relation are orthogonal and this is used to examine a number of examples of the inequality the best constants in some cases have only been evaluated numerically

fuzzy subinclines ideals of incline algebras the fuzzification of subinclines ideals in inclines are considered and some related properties are investigated we also state the product of fuzzy subinclines ideals and the projections of fuzzy subinclines ideals we discuss fuzzy relations fuzzy characteristic subinclines ideals

algebraic properties of the dirac oscillator an algebraic representation independent analysis is presented for the dirac oscillator in an angular momentum basis the analysis is based on shift operators for energy and angular momentum and it is similar to that for a non relativistic isotropic harmonic oscillator the shift operators generate all the eigenkets of the dirac oscillator from a vacuum ket the shift operations yield energy eigenvalues and certain matrix elements the relationship to the factorization method is discussed

vague soft hemirings abstract xu et al introduced the concept of vague soft sets which is an extension to the soft set and the vague set in this paper we apply the concept of vague soft sets to hemiring theory the notion of q vague soft left h ideals of a hemiring is introduced and some related properties are investigated

the normal parameter reduction of soft sets and its algorithm this paper is concerned with the reduction of soft sets and fuzzy soft sets firstly the problems of suboptimal choice and added parameter set of soft sets are analyzed then we introduce the definition of normal parameter reduction in soft sets to overcome these problems in addition a heuristic algorithm of normal parameter reduction is presented two new definitions parameter important degree and decision partition are proposed for analyzing the algorithm of normal parameter reduction furthermore the normal parameter reduction is also investigated in fuzzy soft sets

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

soft matrix theory and its decision making in this work we define soft matrices and their operations which are more functional to make theoretical studies in the soft set theory we then define products of soft matrices and their properties we finally construct a soft max min decision making method which can be successfully applied to the problems that contain uncertainties

kernels and closures of soft set relations and soft set relation mappings in this paper the notions of anti reflexive kernel symmetric kernel reflexive closure and symmetric closure of a soft set relation are first introduced respectively then their accurate calculation formulae and some properties are obtained finally soft set relation mappings and inverse soft set relation mappings are proposed and some related properties are discussed

asymptotic solvability of an imaginary cubic oscillator with spikes for complex potentials v x ix 3 2 ix 2 2 ix 1 2 which are symmetric we show that in 1 strong coupling regime the low lying bound states almost coincide with harmonic oscillators whenever the spectrum remains real this means at all critical 1

exclusive disjunctive soft sets soft sets theory initiated by molodtsov is an emerging tool to deal with uncertain problems and has been studied by scholars in both theory and practice this paper proposes the notion of exclusive disjunctive soft sets and studies some of its operations such as restricted relaxed and operations dependency between exclusive disjunctive soft sets and bijective soft sets exclusive disjunctive soft decision systems reduction of exclusive disjunctive soft decision systems core of exclusive disjunctive soft decision systems decision rules in exclusive disjunctive decision soft sets moreover this study gives an application of exclusive disjunctive soft sets which shows that it can be applied to attribute reduction of incomplete information system

on weakly cut stable maps a mapping f p q between posets p and q is called weakly cut stable if f can be naturally extended to a weakly complete lattice homomorphism i e preserving non empty meets and joins f n p n q between the corresponding dedekind macneille completions these mappings are studied and described by the 1st order language

on fuzzy h ideals in hemirings the fuzzy setting of a left h ideal in a hemiring is constructed and basic properties are investigated using a collection of left h ideals of a hemiring s fuzzy left h ideals of s are established the notion of a finite valued fuzzy left h ideal is introduced and its characterization is given fuzzy relations on a hemiring s are discussed

fuzzy strong implicative hyper bck ideals of hyper bck algebras the fuzzy setting of strong implicative hyper bck ideals in hyper bck algebras is considered some of their properties are investigated relations among fuzzy strong hyper bck ideals fuzzy implicative hyper bck ideals and fuzzy strong implicative hyper bck ideals are given a characterization of a fuzzy strong implicative hyper bck ideal is provided the hyper homomorphic preimage of a fuzzy strong implicative hyper bck ideal is discussed

soft bck bci algebras molodtsov d molodtsov soft set theory first results comput math appl 37 1999 19 31 introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches in this paper we apply the notion of soft sets by molodtsov to the theory of bck bci algebras the notion of soft bck bci algebras and soft subalgebras are introduced and their basic properties are derived

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

solvable potentials associated with su 1 1 algebras a systematic study we consider a specific differential realization of the su 1 1 algebra and use it to explore such algebraic structures associated with shape invariant potentials our approach combines elements of various methods of solving the schrodinger equation such as supersymmetric quantum mechanics or the factorization method algebraic techniques and special function theory in fact it amounts to reformulating transformations mapping the schrodinger equation into the differential equation of orthogonal polynomials in group theoretical terms our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings the procedure presented here implicitly contains a similar treatment of the compact su 2 algebra as well possible generalizations of this approach involving different realizations of the su 1 1 algebra other algebraic structures and larger classes of potentials are also outlined

some kinds of c ccq interval valued fuzzy ideals of bci algebras in this paper we first introduce the notions of positive implicative implicative and commutative interval valued fuzzy ideals of bci algebras which are generalizations of positive implicative implicative and commutative fuzzy ideals respectively and investigate some of their related properties the concept of quasi coincidence of an interval valued fuzzy point within an interval valued fuzzy set is introduced in fact the concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we further introduce the notions of positive implicative implicative and commutative q interval valued fuzzy ideals of bci algebras and investigate some of their related properties some characterization theorems of these generalized interval valued fuzzy ideals are derived the relationship among these generalized interval valued fuzzy ideals of bci algebras is also considered

a combined forecasting approach based on fuzzy soft sets forecasting the export and import volume in international trade is the prerequisite of a government s policy making and guidance for a healthier international trade development however an individual forecast may not always perform satisfactorily while combination of forecasts may result in a better forecast than component forecasts we believe the component forecasts employed in combined forecasts are a description of the actual time series which is fuzzy this paper attempts to use forecasting accuracy as the criterion of fuzzy membership function and proposes a combined forecasting approach based on fuzzy soft sets this paper also examines the method with data of international trade from 1993 to 2006 in the chongqing municipality of china and compares it with a combined forecasting approach based on rough sets and each individual forecast the experimental results show that the combined approach provided in this paper improves the forecasting performance of each individual forecast and is free from a rough sets approach s restrictions as well it is a promising forecasting approach and a new application of soft sets theory

soft semirings molodtsov introduced the concept of soft sets which can be seen as a new mathematical tool for dealing with uncertainty in this paper we initiate the study of soft semirings by using the soft set theory the notions of soft semirings soft subsemirings soft ideals idealistic soft semirings and soft semiring homomorphisms are introduced and several related properties are investigated

position dependent effective mass dirac equations with pt symmetric and non pt symmetric potentials we present a new procedure to construct the one dimensional non hermitian imaginary potential with a real energy spectrum in the context of the position dependent effective mass dirac equation with the vector coupling scheme in 1 1 dimensions in the first example we consider a case for which the mass distribution combines linear and inversely linear forms the dirac problem with a pt symmetric potential is mapped into the exactly solvable schrodinger like equation problem with the isotonic oscillator by using the local scaling of the wavefunction in the second example we take a mass distribution with smooth step shape the dirac problem with a non pt symmetric imaginary potential is mapped into the exactly solvable schrodinger like equation problem with the rosen morse potential the real relativistic energy levels and corresponding wavefunctions for the bound states are obtained in terms of the supersymmetric quantum mechanics approach and the function analysis method

on some new operations in soft set theory molodtsov introduced the theory of soft sets which can be seen as a new mathematical approach to vagueness in this paper we first point out that several assertions proposition 2 3 iv vi proposition 2 4 and proposition 2 6 iii iv in a previous paper by maji et al p k maji r biswas a r roy soft set theory comput math appl 45 2003 555 562 are not true in general by counterexamples furthermore based on the analysis of several operations on soft sets introduced in the same paper we give some new notions such as the restricted intersection the restricted union the restricted difference and the extended intersection of two soft sets moreover we improve the notion of complement of a soft set and prove that certain de morgan s laws hold in soft set theory with respect to these new definitions

algebraic structures of soft sets associated with new operations recently new operations have been defined for soft sets in this paper we study some important properties associated with these new operations a collection of all soft sets with respect to new operations give rise to four idempotent monoids then with the help of these monoids we can study semiring hemiring structures of soft sets some of these semirings hemirings are actually lattices finally we show that soft sets with a fixed set of parameters are mv algebras and bck algebras

monte carlo approach to dendritic growth the connection between diffusion limited aggregation and the equations of dendritic growth is critically examined a different type of monte carlo simulation is proposed and used to construct two dimensional dendrite like patterns the wavelength occurring for short times is in good agreement with the linear stability analysis the time dependence of the characteristic wavelengths is also determined

soft bl algebras based on fuzzy sets by means of soft sets and q soft sets some characterizations of implicative positive implicative and fantastic filteristic soft bl algebras are investigated finally we prove that a soft set is an implicative filteristic soft bl algebra if and only if it is both a positive implicative filteristic soft bl algebra and a fantastic filteristic soft bl algebra

discrete inequalities orthogonal polynomials and the spectral theory of difference operators in a recent paper the first two authors studied a class of series inequalities associated with a three term recurrence relation which includes a well known inequality of copson s it was shown that the validity of the inequality and the value of the best constant are determined in terms of the so called hellinger nevanlinna m function the theory is the discrete analogue of that established by everitt for a class of integro differential inequalities in this paper the properties of the m function are investigated and connections with the theory of orthogonal polynomials and the hamburger moment problem are explored the results are applied to give examples of the series inequalities associated with the classical orthogonal polynomials

soft hyperstructure in this paper we study soft hypergroupoids firstly we introduce the notion of soft hypergroupoids and some examples are given also we show that soft hypergroupoids are closely related to l subhypergroupoid secondly using the notion of soft hypergroupoid some new properties of soft hypergroupoids are obtained lastly we investigate some properties of soft subhypergroupoids

soft polygroups soft set theory introduced by molodtsov has been considered as an effective mathematical tool for modeling uncertainties in this paper we initiate the study of algebraic hyperstructures of soft sets the concepts of soft polygroups normal soft polygroups soft subpolygroups and normal soft subpolygroups are introduced and some related properties are investigated furthermore we propose the homomorphism and isomorphism of soft polygroups and establish three isomorphism theorems for soft polygroups

some kinds of fuzzy ideals of bci algebras the concepts of c c q d fuzzy positive implicative implicative and commutative ideals and c c q d fuzzy positive implicative implicative and commutative ideals in bci algebras are introduced some new characterizations are investigated in particular we prove that a fuzzy set m of a bci algebra x is an c c q d fuzzy implicative ideal of x if and only if it is both an c c q d fuzzy positive implicative ideal and an c c q d fuzzy commutative ideal finally we give some characterizations of three particular classes of bci algebras by these generalized fuzzy ideals

redefined fuzzy implicative filters using the belongs to relation and quasi coincidence with relation q between fuzzy points and fuzzy sets the concept of q j fuzzy implicative filters where q j are any two of q email protected q email protected q with q email protected q is introduced and related properties are discussed relations between email protected q email protected q fuzzy implicative filters and email protected q fuzzy implicative filters are investigated and conditions for an email protected q fuzzy implicative filter to be an fuzzy implicative filter are provided characterizations of email protected q fuzzy implicative filters are given and conditions for a fuzzy set to be a q email protected q fuzzy implicative filter are provided

application of level soft sets in decision making based on interval valued fuzzy soft sets molodtsov s soft set theory was originally proposed as a general mathematical tool for dealing with uncertainty research on fuzzy soft set based decision making has received much attention in recent years this paper aims to give deeper insights into decision making involving interval valued fuzzy soft sets a hybrid model combining soft sets with interval valued fuzzy sets the concept called reduct fuzzy soft sets of interval valued fuzzy soft sets is introduced using reduct fuzzy soft sets and level soft sets flexible schemes for decision making based on weighted interval valued fuzzy soft sets are proposed and some illustrative examples are employed to show that the proposals presented here are not only more reasonable but more efficient in practical applications

soft subsets and soft product operations molodtsov s soft set theory provides a general mathematical framework for dealing with uncertainty it is known that soft subsets and soft equal relations are of vital importance in soft set theory this paper aims to give a systematic study on several types of soft subsets and various soft equal relations induced by them we give some equivalent characterizations of different soft subsets and endeavor to ascertain the interrelations among these notions illustrated by a number of concrete examples we also consider ontology based soft sets and show that soft l subsets generalize both soft m subsets and ontology based soft subsets moreover by means of soft l subsets and some related notions we give a theoretical study concerning soft product operations such as products and products we consummate some incomplete results concerning soft product operations existing in the literature and investigate the algebraic properties of soft product operations in detail finally we consider free soft algebras associated with soft product operations it is shown that soft l equal relations are congruence relations over free soft algebras and the corresponding quotient structures form commutative semigroups

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

soft ideals and soft filters of soft ordered semigroups it is shown that soft ordered semigroups over an ordered semigroup are actually soft ordered subsemigroups of s a soft idealistic semigroups over an ordered semigroup are soft ideals of soft ordered semigroup s a the concept of soft filters of a soft ordered semigroup is introduced it is shown that restricted complement of a soft filter is a soft prime ideal of s a

soft rough fuzzy sets and soft fuzzy rough sets fuzzy set theory soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related feng et al introduced the notions of rough soft set soft rough set and soft rough fuzzy set by combining fuzzy set rough set and soft set all together this paper is devoted to the further discussion of the combinations of fuzzy set rough set and soft set a new soft rough set model is proposed and its properties are derived furthermore fuzzy soft set is employed to granulate the universe of discourse and a more general model called soft fuzzy rough set is established the lower and upper approximation operators are presented and their related properties are surveyed

fuzzy positive implicative and fuzzy associative filters of lattice implication algebras we fuzzify the concept of positive implicative filters and associative filters in lattice implication algebras we prove that i every fuzzy positive implicative filter is a fuzzy implicative filter and ii every fuzzy associative filter is a fuzzy filter we provide equivalent conditions for both fuzzy positive implicative filter and fuzzy associative filter

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

on operations of soft sets soft set theory proposed by molodtsov has been regarded as an effective mathematical tool to deal with uncertainties in this paper first we prove that certain de morgan s law hold in soft set theory with respect to different operations on soft sets then we discuss the basic properties of operations on soft sets such as intersection extended intersection restricted union and restricted difference moreover we illustrate their interconnections between each other also we define the notion of restricted symmetric difference of soft sets and investigate its properties the main purpose of this paper is to extend the theoretical aspect of operations on soft sets

soft set theory applied to ideals in d algebras the notions of transitive soft d algebras soft edge d algebras soft d algebras soft d ideals soft d ideals soft d ideals and d idealistic d idealistic or d idealistic soft d algebras are introduced also their related properties are surveyed

the bijective soft set with its operations soft set theory is a newly emerging tool to deal with uncertain problems and has been studied by scholars in theory and practice this paper proposes the concept of bijective soft set and some of its operations such as the restricted and and the relaxed and operations on a bijective soft set dependency between two bijective soft sets bijective soft decision system significance of bijective soft set with respect to bijective soft decision system reduction of bijective soft set with respect to bijective soft decision system and decision rules in bijective soft decision system with these notions and operations an application of bijective soft set in decision making problems is also discussed

generalized fuzzy interior ideals in semigroups using the idea of quasi coincidence of a fuzzy point with a fuzzy set the concept of an a b fuzzy interior ideal which is a generalization of a fuzzy interior ideal in a semigroup is introduced and related properties are investigated

on an extension of copson s inequality for infinite series in 1979 copson proved the following analogue of the hardy littlewood inequality if is a sequence of real numbers such that are convergent where a n a n 1 a n and 2 a n a n then is convergent and the constant 4 being best possible equality occurs if and only if a n 0 for all n in this paper we give a result that extends copson s result to inequalities of the form where mx n p n l x n l q n x n n 0 1 the validity of such an inequality and the best possible value of the constant k are determined in terms of the analogue of the titchmarsh weyl m function for the difference equation mx n w n x n n 0 1

pruning method for optimal solutions of intm intn decision making scheme abstract this paper aims to find a faster method for optimal solutions of feng et al s int m int n decision making scheme we first give theoretical characterizations of optimal decision sets then we develop a pruning method which filters out those objects that cannot be elements of any optimal decision sets in the beginning experimental results have shown that our method has higher efficiency in computing the optimal solutions of this scheme particularly when we are processing soft sets with a great quantity of data

a personal history of the m coefficient this is a short personal history of the m coefficient for sturm liouville differential and difference equations now associated historically with the names of weyl hellinger nevanlinna and titchmarsh

soft set relations and functions the traditional soft set is a mapping from a parameter to the crisp subset of universe molodtsov introduced the theory of soft sets as a generalized tool for modeling complex systems involving uncertain or not clearly defined objects in this paper the concepts of soft set relations are introduced as a sub soft set of the cartesian product of the soft sets and many related concepts such as equivalent soft set relation partition composition function etc are discussed

pseudo d algebras we introduce the notion of a pseudo d algebra as a generalization of the idea of a d algebra the class of pseudo d algebras can be included in the class of coupled d algebras we show that the class of pseudo d algebras behaves well with respect to standard mappings such as direct products and standard mappings such as homomorphisms indeed we are able to prove a fundamental theorem of homomorphisms and we are able to identify kernel objects as d ideals in this theory we also discuss several large classes of examples of pseudo d algebras which occur in different settings and we identify properties sufficient for or equivalent to membership in important subclasses of the class of pseudo d algebras in these settings