on soft mappings in this paper an idea of soft mappings is given and some of their properties are studied images and inverse images of crisp sets and soft sets under soft mappings are also studied an application of soft mapping in medical diagnosis has been shown

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

l fuzzy soft sets based on complete boolean lattices in this paper the concept of l fuzzy soft sets based on complete boolean lattices which can be seen as a generalization of fuzzy soft sets is introduced the topological and lattice structure of l fuzzy soft sets are obtained some operations on l fuzzy soft sets are discussed and new types of l fuzzy soft sets such as full keeping intersection and keeping union l fuzzy soft sets are defined and supported by some illustrative examples a pair of l fuzzy soft rough approximations is proposed and their properties are given based on l fuzzy soft rough approximations the concept of l fuzzy soft rough sets is introduced and their structures are studied the fact that every finite l fuzzy topological space is an l fuzzy soft approximation space is revealed in addition we obtain two one to one correspondence relationships associated with l fuzzy soft sets which expounds the broad application prospect of l fuzzy soft sets

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

soft nexuses in this paper we define soft nexuses over a nexus n after that we define the notions cyclic soft nexus prime and maximal soft subnexuses and we obtain some results about the above definitions finally we define the integer number induced by soft nexus a a and denoted by a a and show that for any n there exists soft nexus a a such that a a n

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

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

collectivity and geometry iv sp 6 sp 2 o 3 basis states for open shells using generating function methods branching rules for sp 6 sp 2 o 3 are derived the branching rules suggest an integrity basis or set of elementary permissible diagrams in terms of which the subgroup basis states are defined they correspond to vibrational or bohr mottelson type states in the nuclear symplectic model

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

internal labelling problem an algorithmic procedure combining the decomposition of casimir operators induced by the embedding of a subalgebra into a semisimple lie algebra with the properties of commutators of subgroup scalars an analytical algorithm for the computation of missing label operators with the commutativity requirement is proposed two new criteria for subgroups scalars to commute are given the algorithm is completed with a recursive method to construct orthonormal bases of states as examples to illustrate the procedure four labelling problems are explicitly studied

a comparative study of fuzzy sets and rough sets this paper reviews and compares theories of fuzzy sets and rough sets two approaches for the formulation of fuzzy sets are reviewed one is based on many valued logic and the other is based on modal logic two views of rough sets are presented set oriented view and operator oriented view rough sets under set oriented view are closely related to fuzzy sets which leads to non truth functional fuzzy set operators both of them may be considered as deviations of classical set algebra in contrast rough sets under operator oriented view are different from fuzzy sets and may be regarded as an extension of classical set algebra

ordinary representations and modular forms this paper establishes criteria for a representation gal q q gl2 qp to be modular in the sense that there is a newform f such that f fontaine and mazur have conjectured that is modular provided it is unramified outside of a finite set of primes irreducible has odd determinant and the restriction to a decomposition group dp at p satisfies certain conditions the following conjecture is a special case of fm95 ref 1 conjecture 3c

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

soft substructures of rings fields and modules soft set theory proposed by molodtsov has been regarded as an effective mathematical tool to deal with uncertainties in this paper we introduce and study soft subrings and soft ideals of a ring by using molodtsov s definition of the soft sets moreover we introduce soft subfields of a field and soft submodule of a left r module some related properties about soft substructures of rings fields and modules are investigated and illustrated by many examples

on finite and continuous little groups of representations of semi simple lie groups the multiplicity of the identity representation occurring in the reduction of representations of semi simple lie groups to their finite or continuous subgroups is given for many group subgroup pairs

a note on soft topological spaces shabir and naz 2011 12 introduced and studied the notions of soft topological spaces soft interior soft closure and soft separation axioms but we found that some results are incorrect see their remark 3 23 so the purpose of this note is first to point out some errors in remark 4 and example 9 of shabir and naz 2011 12 and second to investigate properties of soft separation axioms defined in shabir and naz 2011 12 in particular we investigate the soft regular spaces and some properties of them we show that if a soft topological space x t e is soft t 1 and soft regular i e a soft t 3 space then x e is soft closed for each email protected x their theorem 3 21

generalised fuzzy soft sets in this paper we define generalised fuzzy soft sets and study some of their properties application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

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

on the modularity of ℚ curves a q curve is an elliptic curve over a number field k which is geometrically isogenous to each of its galois conjugates ribet 16 asked whether every q curve is modular and showed that a positive answer would follow from serre s conjecture on mod p galois representations we answer ribet s question in the affirmative subject to certain local conditions at 3 msc classification 11g18 14g35 14h52

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

rough sets and fuzzy sets some remarks on interrelations abstract this short paper refers to two remarks given in 3 and disputes them any rough set can be expressed by means of a membership function u 0 0 5 1 we show that this representation can be extended to union and intersection

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

topological structure of fuzzy soft sets in this paper the concept of fuzzy soft topology is introduced and some of its structural properties such as neighborhood of a fuzzy soft set interior fuzzy soft set fuzzy soft basis fuzzy soft subspace topology are studied

on serre s conjecture for 2 dimensional mod p representations of gal q q we prove the existence in many cases of minimally ramified p adic lifts of 2 dimensional continuous odd absolutely irreducible mod p representations of the absolute galois group of q it is predicted by serre s conjecture that such representations arise from newforms of optimal level and weight

automorphic lifts of prescribed types we prove a variety of results on the existence of automorphic galois representations lifting a residual automorphic galois representation we prove a result on the structure of deformation rings of local galois representations and deduce from this and the method of khare and wintenberger a result on the existence of modular lifts of specified type for galois representations corresponding to hilbert modular forms of parallel weight 2 we discuss some conjectures on the weights of n dimensional mod p galois representations finally we use recent work of taylor to prove level raising and lowering results for n dimensional automorphic galois representations

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

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

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

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

combination of interval valued fuzzy set and soft set the soft set theory proposed by molodtsov can be used as a general mathematical tool for dealing with uncertainty by combining the interval valued fuzzy set and soft set models the purpose of this paper is to introduce the concept of the interval valued fuzzy soft set the complement and and or operations are defined on the interval valued fuzzy soft sets the demorgan s associative and distribution laws of the interval valued fuzzy soft sets are then proved finally a decision problem is analyzed by the interval valued fuzzy soft set some numerical examples are employed to substantiate the conceptual arguments

fuzzy soft semigroups and fuzzy soft ideals in this paper the notions of fuzzy soft semigroup and fuzzy soft left right ideal are given and the a level set union and intersection of them were studied for each a 0 1 finally fuzzy soft image and fuzzy soft inverse image of fuzzy soft semigroup ideal are discussed

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

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

modularity of certain potentially barsotti tate galois representations where is the reduction of p these results were subject to hypotheses on the local behavior of p at x i e the restriction of p to a decomposition group at x and to irreducibility hypotheses on 1p in this paper we build on the methods of 461 451 and 121 and relax the hypotheses on local behavior in particular we treat certain adic representations which are not semistable at x but potentially semistable we do this using results of 61 generalizing a theorem of ramakrishna 321 see fontaine mazur 22 131 for a slightly different point of view the results in 61 show that certain potentially barsotti tate deformation problems are smooth allowing us to define certain universal deformations for with the necessary galoistheoretic properties to apply wiles method to carry out the proof that these deformations are indeed realized in the cohomology of modular curves i e that the universal deformation rings are hecke algebras we need to identify the corresponding cohomology groups and prove they have the modular theoretic properties needed to apply wiles method as in 151 and 121 the identification is made by matching local behavior of automorphic representations and galois representations via the local langlands correspondence together with fontaine s theory at the prime we work directly with cohomology of modular curves instead of

soft sets and soft rings molodtsov 1999 introduced the concept of soft sets in 1 then maji et al 2003 defined some operations on soft sets in 2 aktas and ca man 2007 defined the notion of soft groups in 3 finally soft semirings are defined by feng et al 2008 in 5 in this paper we have introduced initial concepts of soft rings

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

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

modularity lifting beyond the taylor wiles method we prove new modularity lifting theorems for p adic galois representations in situations where the methods of wiles and taylor wiles do not apply previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology in practice this imposes several restrictions one must be in a shimura variety setting and the automorphic forms must be of regular weight at infinity in this paper we essentially show how to remove these restrictions our most general result is a modularity lifting theorem which on the automorphic side applies to automorphic forms on the group mathrm gl n over a general number field it is contingent on a conjecture which in particular predicts the existence of galois representations associated to torsion classes in the cohomology of the associated locally symmetric space we show that if this conjecture holds then our main theorem implies the following if e is an elliptic curve over an arbitrary number field then e is potentially automorphic and satisfies the sato tate conjecture in addition we also prove some unconditional results for example in the setting of mathrm gl 2 over mathbf q we identify certain minimal global deformation rings with the hecke algebras acting on spaces of p adic katz modular forms of weight 1 such algebras may well contain p torsion moreover we also completely solve the problem for p odd of determining the multiplicity of an irreducible modular representation overline rho in the jacobian j 1 n where n is the minimal level such that overline rho arises in weight two

further remarks on the relation between rough and fuzzy sets abstract based on results of pawlak and wygralak some remarks on the relation between rough sets and fuzzy sets are given an improved definition of rough set is also proposed

commutativity of missing label operators in terms of berezin brackets we obtain a criterion on the commutativity of polynomials in the enveloping algebra of a lie algebra in terms of an involution condition with respect to the berezin bracket as an application it is shown that the commutativity requirement of missing label operators for reduction chains in the missing label problem can be solved analytically

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

modularity of fibres in rigid local systems k as is explained for example in w1 and w2 fontaine and mazur fm conjectured that this is always the case signi cantprogress on this conjecture was achieved w3 by proving particular instancesof the following lifting conjecture conjecture 1 1 suppose that l is odd and that the residual represen tation attached to is modular then itself is modular conjecture 1 1 is proved in w3 and tw when k q and the restriction of to the decomposition groups at the primes above l are semistable in the senseof ddt 2 4 this is enough using the primes l 3 and 5 to establish theshimura taniyama conjecture for semistable elliptic curves thanks to a keyresult of langlands and tunnell progressively stronger cases of conjecture 1 1were subsequently proved by di cdt fu and sw1 in sw1 skinnerand wiles obtain quite general results in the context where k is any totallyreal eld the principal assumption being that is ordinary at the primesabove l in this note we consider galois representations which occur in rigid fam ilies and establish their modularity under conjecture 1 1 this implies themodularity over suitable real abelian extensions of the galois representationsoccurring in the cohomology of the curvesy

sp 6 states in an su 3 u 1 basis we find all missing label operators and also a complete set of analytic nonorthonormal basis states for the group subgroup sp 6 su 3 u 1 both for the compact version of sp 6 and for the noncompact sp 6 r relevant to the symplectic nuclear collective model

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

collectivity and geometry i general approach in the last decade an extensive literature appeared in which the microscopic collective behavior of nuclei was associated with definite irreducible representations irreps of either the o n or sp 6 groups where n a 1 and a is the number of nucleons it became clear that the two approaches are equivalent as problems with 3n degrees of freedom are characterized by a definite irrep of the group sp 6n and for its subgroup sp 6 o n the irrep of o n determines that of sp 6 and vice versa thus one can consider that collective effects appear when one introduces the constraint that in the many body system the states are restricted to a definite irrep of o n and thus also of sp 6 and the hamiltonians are in the enveloping algebra of sp 6 rather than in that of sp 6n once sp 6 becomes the paramount group of collective motions the problem is to determine the matrix elements of the generators of sp 6 in a basis characterized by irreps of its subgroups what subgroups to choose rowe and rosensteel

g2 van der waerden invariant the g2 van der waerden invariant is given it solves the external labeling problem connected with direct products of irreducible representations of g2

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

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

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

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

on a conjecture of conrad diamond and taylor we prove a conjecture of conrad diamond and taylor on the size of certain deformation rings parametrizing potentially barsotti tate ga lois representations to achieve this we extend results of breuil and m ezard classifying galois lattices in semistable representations in terms of strongly divisible modules to the potentially crystalline case in hodge tate weights 0 1 we then use these strongly divisible modules to compute the desired deformation rings as a corollary we obtain new results on the modularity of potentially barsotti tate representations

letter to the editor comment on a fuzzy soft set theoretic approach to decision making problems the algorithm for identification of an object in a previous paper of a r roy et al a r roy p k maji a fuzzy soft set theoretic approach to decision making problems j comput appl math 203 2007 412 418 is incorrect using the algorithm the right choice cannot be obtained in general the problem is illustrated by a counter example

the taylor wiles construction and multiplicity one wiles proof 17 of the modularity of semistable elliptic curves over q relies on a construction of taylor and wiles 16 showing that certain hecke algebras are complete intersections these hecke algebras are defined by considering the action of hecke operators on spaces of modular forms of minimal level or equivalently on homology groups or jacobians of modular curves taylor and wiles proceed roughly speaking by patching algebras arising from forms of different levels one of the deep results used in their construction was the fact that the homology of the modular curve becomes a free module of rank two over the hecke algebra upon localization at certain maximal ideals this result known as a multiplicity one result is a generalization of a theorem of mazur 11 its proof relied on the q expansion principle of deligne rapoport and katz and the comparison of mod betti and de rham cohomologies see sect 2 1 of 17 multiplicity one was thought to be a crucial ingredient of the taylor wiles construction as well as other parts of wiles proof the purpose of this paper is to explain how to alter the arguments of 16 and 17 so that multiplicity one results are a byproduct rather than an ingredient the key conceptual change underlying this improvement is the following rather than prove that after localization the hecke algebra can be identified with the universal deformation ring of a mod galois representation we prove that the homology of the modular curve is a free module over this deformation ring to carry this out we change the taylor wiles construction1 by 1 patching the modules as well as the algebras and 2 applying the auslander buchsbaum

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

moduli of finite flat group schemes and modularity we prove that under some mild conditions a two dimensional p adic galois representation which is residually modular and potentially barsotti tate at p is modular this provides a more conceptual way of establishing the shimura taniyama weil conjecture especially for elliptic curves which acquire good reduction over a wildly ramified extension of 3 the main ingredient is a new technique for analyzing flat deformation rings it involves resolving them by spaces which parametrize finite flat group scheme models of galois representations introduction