the novikov conjecture for algebraic k theory of the group algebra over the ring of schatten class operators abstract in this paper we prove the algebraic k theory novikov conjecture for group algebras over the ring of schatten class operators the main technical tool in the proof is an explicit construction of the connes chern character

on the negative k theory of schemes in finite characteristic abstract we show that if x is a d dimensional scheme of finite type over an infinite perfect field k of characteristic p 0 then k i x 0 and x is k i regular for i d 2 whenever the resolution of singularities holds over k this proves the k dimension conjecture of weibel c weibel k theory and analytic isomorphisms invent math 61 1980 177 197 2 9 except for d 1 i d 2 in all characteristics assuming the resolution of singularities

on the general linear group and hochschild homology our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring before stating the result we establish some notation and conventions rings are associative and with unit if a is a ring then gl a uk 0glk a is its infinite general linear group an a bimodule is an abelian group b which is both a left a module and a right a module and satisfies alb a2 al ba2 for ai ea be b for example b a if b is an a bimodule then m b uk omk b is the infinite additive group of matrices with entries in b conjugation defines an action the adjoint action of gl a on m b note that an a q bimodule is just an a bimodule which is also a rational vector space if b is an a q bimodule then hj a c q b denotes the hochschild homology of a q with coefficients in b our main result it appears in slightly more detailed form as theorem v 3 is

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

banach strong novikov conjecture for polynomially contractible groups abstract we prove the banach strong novikov conjecture for groups having polynomially bounded higher order combinatorial functions this includes all automatic groups

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

a general decomposition theorem for the k server problem the first general decomposition theorem for the k server problem is presented whereas all previous theorems are for the case of a finite metric with k 1 points the theorem given here allows an arbitrary number of points in the underlying metric space this theorem implies o polylog k competitive randomized algorithms for certain metric spaces consisting of a polylogarithmic number of widely separated subspaces and takes a first step toward a general o polylog k competitive algorithm the only other cases for which polylogarithmic competitive randomized algorithms are known are the uniform metric space and the weighted cache metric space with two weights

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

a polylogarithmic competitive algorithm for the k server problem we give the first polylogarithmic competitive randomized online algorithm for the k server problem on an arbitrary finite metric space in particular our algorithm achieves a competitive ratio of o log3 n log2 k for any metric space on n points our algorithm improves upon the deterministic 2k 1 competitive algorithm of koutsoupias and papadimitriou koutsoupias and papadimitriou 1995 for a wide range of n

nil k theory maps to cyclic homology algebraic k theory breaks into two pieces nil k theory and karoubi villamayor k theory karoubi has constructed chern classes from the latter groups into cyclic homology we construct maps from nil k theory to cyclic homology which are compatible with karoubi s maps but with a degree shift several recent results show that in characteristic zero our map is often an isomorphism ever since a connes discovered cyclic homology in 1982 it has been widely held that cyclic homology is closely related to algebraic k theory supporting this belief karoubi defined chern classes limep kn a limhcn 2p a in k1 and goodwillie defined a related map ey kn a hcn a in g other maps between k theory and variants of cyclic homology have been constructed by loday unpublished burghelea bur 1 ogle ogle 1 and others of course all of these maps are related to the dennis trace map to hochschild homology d kn a hhn a which r k dennis discovered in 1975 d in this paper we prove three results the first is that karoubi s map factors through goodwillie s theorem a for every associative algebra there is a commutative diagram kn a

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

bi relative algebraic k theory and topological cyclic homology it is well known that algebraic k theory preserves products of rings however in general algebraic k theory does not preserve fiber products of rings and one defines bi relative algebraic k theory so as to measure the deviation it was proved recently by cortinas 6 that rationally bi relative algebraic k theory and bi relative negative cyclic homology agree in this paper we show that with finite coefficients bi relative algebraic k theory and bi relative topological cyclic homology agree as an application we show that for a possibly singular curve over a perfect field k of positive characteristic p the cyclotomic trace map induces an isomorphism of the p adic algebraic k groups and the p adic topological cyclic homology groups in non negative degrees as a further application we show that the difference between the p adic k groups of the integral group ring of a finite group and the p adic k groups of a maximal z order in the rational group algebra can be expressed entirely in terms of topological cyclic homology let f be a functor that to a unital associative ring a associates a symmetric spectrum f a if i a is a two sided ideal the relative term f a i is defined to be the homotopy fiber of the map f a f a i induced from the canonical projection if further f a b is a ring homomorphism such that f i f i is an isomorphism onto a two sided ideal of b there is an induced map f a i f b f i and the bi relative term f a b i is defined to be the homotopy fiber of this map then there is a distinguished triangle in the stable homotopy category

comparing k theories for complex varieties the semi topological k theory of a complex variety was defined in a recent paper by the authors with the expectation that it would prove to be a theory lying part way between the algebraic k theory of the variety and the topological k theory of the associated analytic space and thus would share properties with each of these other theories in this paper we realize these expectations by proving among other results that 1 the algebraic k theory with finite coefficients and the semi topological k theory with finite coefficients coincide on all projective complex varieties 2 semi topological k theory and topological k theory agree on certain types of generalized flag varieties and 3 assuming a result asserted by cohen and lima filho the semi topological k theory of any smooth projective variety becomes isomorphic to the topological k theory of the underlying analytic space once the bott element is inverted to illustrate the utility of our results we observe that a new proof of the quillen lichtenbaum conjecture for smooth complete curves is obtained as a corollary

the obstruction to excision in k theory and in cyclic homology let f a b be a ring homomorphism of not necessarily unital rings and i triangleleft a an ideal which is mapped by f isomorphically to an ideal of b the obstruction to excision in k theory is the failure of the map between relative k groups k a i k b f i to be an isomorphism it is measured by the birelative groups k a b i similarly the groups hn a b i measure the obstruction to excision in negative cyclic homology we show that the rational jones goodwillie chern character induces an isomorphism ch k a b i otimes mathbb q overset sim to hn a otimes mathbb q b otimes mathbb q i otimes mathbb q

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

algebraic k theory of stable c algebras abstract let a be a unital c algebra and let l denote the calkin algebra the bounded operators on a separable hilbert space modulo the compact operators k we prove the following conjecture of m karoubi the algebraic and topological k theory groups of the tensor product c algebra a l are equal the algebra a l may be regarded as a suspension of the more elementary c algebra a k thus karoubi s conjecture asserts roughly speaking that the algebraic and topological k theories of stable c algebras agree

central stability for the homology of congruence subgroups and the second homology of torelli groups abstract we prove a representation stability result for the second homology groups of torelli subgroups of mapping class groups and automorphism groups of free groups this strengthens the results of boldsen hauge dollerup and day putman we also prove a new representation stability result for the homology of certain congruence subgroups partially improving upon the work of putman sam these results follow from a general theorem on syzygies of certain modules with finite polynomial degree

cyclic homology cdh cohomology and negative k theory we prove a blow up formula for cyclic homology which we use to show that infinitesimal k theory satisfies cdh descent combining that result with some computations of the cdh cohomology of the sheaf of regular functions we verify a conjecture of weibel predicting the vanishing of algebraic k theory of a scheme in degrees less than minus the dimension of the scheme for schemes essentially of finite type over a field of characteristic zero

algebraic geometry of topological spaces i we use techniques from both real and complex algebraic geometry to study k theoretic and related invariants of the algebra c x of continuous complex valued functions on a compact hausdorff topological space x for example we prove a parameterized version of a theorem by joseph gubeladze we show that if m is a countable abelian cancellative torsion free semi normal monoid and x is contractible then every finitely generated projective module over c x m is free the particular case m mathbb n 0 n gives a parameterized version of the celebrated theorem proved independently by daniel quillen and andrei suslin that finitely generated projective modules over a polynomial ring over a field are free the conjecture of jonathan rosenberg which predicts the homotopy invariance of the negative algebraic k theory of c x follows from the particular case m mathbb z n we also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on c algebras and for a homology theory of commutative algebras to vanish on c algebras these criteria have numerous applications for example the vanishing criterion applied to nil k theory implies that commutative c algebras are k regular as another application we show that the familiar formulas of hochschild kostant rosenberg and loday quillen for the algebraic hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic hochschild and cyclic homology of c x applications to the conjectures of be linson soule and farrell jones are also given

on metric ramsey type phenomena the main question studied in this article may be viewed as a nonlinear analogue of dvoretzky s theorem in banach space theory or as part of ramsey theory in combinatorics given a finite metric space on n points we seek its subspace of largest cardinality which can be embedded with a given distortion in hilbert space we provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion our main theorem states that for any 0 every n point metric space contains a subset of size at least n1 which is embeddable in hilbert space with o log 1 distortion the bound on the distortion is tight up to the log 1 factor we further include a comprehensive study of various other aspects of this problem

on the k server conjecture we prove that the work function algorithm for the k server problem has a competitive ratio at most 2 k 1 manasse et al 1988 conjectured that the competitive ratio for the k server problem is exactly k it is trivially at least k previously the best known upper bound was exponential in k our proof involves three crucial ingredients a quasiconvexity property of work functions a duality lemma that uses quasiconvexity to characterize the configuration that achieve maximum increase of the work function and a potential function that exploits the duality lemma

the nilpotence conjecture in k theory of toric varieties it is shown that all nontrivial elements in higher k groups of toric varieties are annihilated by iterations of the natural frobenius type endomorphisms this is a higher analog of the triviality of vector bundles on affine toric varieties

bivariant algebraic k theory we show how methods from k theory of operator algebras can be applied in a completely algebraic setting to define a bivariant m1 stable homotopy invariant excisive k theory of algebras over a fixed unital ground ring h a b 7 kk a b which is universal in the sense that it maps uniquely to any other such theory it turns out kk is related to c weibel s homotopy algebraic k theory kh we prove that if h is commutative and a is central as an h bimodule then kk h a kh a we show further that some calculations from operator algebra kk theory such as the exact sequence of pimsner voiculescu carry over to algebraic kk

p adic deformation of algebraic cycle classes we study the p adic deformation properties of algebraic cycle classes modulo rational equivalence we show that the crystalline chern character of a vector bundle on the closed fibre lies in a certain part of the hodge filtration if and only if rationally the class of the vector bundle lifts to a formal pro class in k theory on the p adic scheme

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

a decomposition theorem for task systems and bounds for randomized server problems a lower bound of omega sqrt log k log log k is proved for the competitive ratio of randomized algorithms for the k server problem against an oblivious adversary the bound holds for arbitrary metric spaces having at least k 1 points and provides a new lower bound for the metrical task system problem as well this improves the previous best lower bound of omega log log k for arbitrary metric spaces h j karloff y rabani and y ravid siam j comput 23 1994 pp 293 312 and more closely approaches the conjectured lower bound of omega log k for the server problem on k 1 equally spaced points on a line which corresponds to a natural motion planning problem a lower bound of omega frac log k log log k is obtained the results are deduced from a general decomposition theorem for a simpler version of both the k server and the metrical task system problems called the pursuit evasion game it is shown that if a metric space cal m can be decomposed into two spaces cal m l and cal m r such that the distance between them is sufficiently large compared to their diameter then the competitive ratio for this game on cal m can be expressed nearly exactly in terms of the ratios on each of the two subspaces this yields a divide and conquer approach to bounding the competitive ratio of a space

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

regulators on additive higher chow groups as an attempt to understand motives over k x x m we define the cubical additive higher chow groups with modulus for all dimensions extending the works of s bloch h esnault and k r ulling on 0 dimensional cycles we give an explicit construction of regulator maps on the groups of 1 cycles with an aid of the residue theory of a parshin and v lomadze

excision in algebraic k theory introduction 1 excision and the homology of affine groups 2 volodin s spaces 3 excision for rings with the triple factorization property 4 lie analogue of volodin s construction 5 the homology of nilpotent groups and nilpotent lie algebras 6 a proof of the main theorems 7 the tensor product of h unital algebras 8 the matrix ring of a small category 9 the acyclicity of triangular complexes 10 a proof of karoubi s conjecture

higher topological cyclic homology and the segal conjecture for tori abstract we investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory higher topological cyclic homology is constructed from the fixed points of a version of topological hochschild homology based on the n dimensional torus and we propose it as a computationally tractable cousin of n fold iterated algebraic k theory the fixed points of toral topological hochschild homology are related to one another by restriction and frobenius operators we introduce two additional families of operators on fixed points the verschiebung indexed on self isogenies of the n torus and the differentials indexed on n vectors we give a detailed analysis of the relations among the restriction frobenius verschiebung and differentials producing a higher analog of the structure hesselholt and madsen described for 1 dimensional topological cyclic homology we calculate two important pieces of higher topological cyclic homology namely topological restriction homology and topological frobenius homology for the sphere spectrum the latter computation allows us to establish the segal conjecture for the torus which is to say to completely compute the cohomotopy type of the classifying space of the torus

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

set valued information systems set valued information systems are generalized models of single valued information systems incomplete information systems can be viewed as disjunctively interpreted set valued information systems since some objects in set valued information systems may have more than one value for an attribute so we define tolerance relation and use the maximal tolerance classes to classify the universe of discourse in order to derive optimal decision rules from set valued decision information systems we propose the concept of relative reduct of maximal tolerance classes and define a kind of discernibility function to compute the relative reduct by boolean reasoning techniques finally we define three kinds of relative reducts for set valued information systems and used them to evaluate the significance of attributes

pro unitality and pro excision in algebraic k theory and cyclic homology the purpose of this paper is to study pro excision in algebraic k theory and cyclic homology after suslin wodzicki cuntz quillen cortinas and geisser hesselholt as well as continuity properties of andre quillen and hochschild homology a key tool is first to establish the equivalence of various pro tor vanishing conditions which appear in the literature this allows us to prove that all ideals of commutative noetherian rings are pro unital in a suitable sense we show moreover that such pro unital ideals satisfy pro excision in derived hochschild and cyclic homology it follows hence and from the suslin wodzicki criterion that ideals of commutative noetherian rings satisfy pro excision in derived hochschild and cyclic homology and in algebraic k theory in addition our techniques yield a strong form of the pro hochschild kostant rosenberg theorem an extension to general base rings of the cuntz quillen excision theorem in periodic cyclic homology a generalisation of the f igin tsygan theorem a short proof of pro excision in topological hochschild and cyclic homology and new artin rees and continuity statements in andre quillen and hochschild homology msc 19d55 primary 16e40 13d03 secondary

additive polylogarithms and their functional equations let k varepsilon 2 k varepsilon varepsilon 2 the single valued real analytic n polylogarithm mathcal l n mathbb c to mathbb r is fundamental in the study of weight n motivic cohomology over a field k of characteristic 0 in this paper we extend the construction in unver algebra number theory 3 1 34 2009 to define additive n polylogarithms li n k varepsilon 2 to k and prove that they satisfy functional equations analogous to those of mathcal l n under a mild hypothesis we show that these functions descend to an analog of the nth bloch group b n k varepsilon 2 defined by goncharov adv math 114 197 318 1995 we hope that these functions will be useful in the study of weight n motivic cohomology over k e 2

better algorithms for unfair metrical task systems and applications unfair metrical task systems are a generalization of online metrical task systems in this paper we introduce new techniques to combine algorithms for unfair metrical task systems and apply these techniques to obtain improved randomized online algorithms for metrical task systems on arbitrary metric spaces

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

excision in cyclic homology and in rational algebraic k theory for a precise definition see 1 below by replacing everywhere k by k q one obtains the corresponding notion in rational algebraic k theory the above definition has an obvious counterpart for cyclic homology and algebras over a fixed field in the case of a general commutative ground ring k some restrictions on the class of allowable extensions seem inevitable due to the well known limitations of cyclic homology considered as a homology functor for algebras not flat over a ground ring an extension of k algebras will be called pure if it is pure as an extension of k modules in the sense of p m cohn 6 cf also appendix a 3 below the class of pure extensions contains e g i extensions which admit a k module splitting ii extensions a e r s with s flat over k in fact one among the several possible characterizations of purity says that an extension is pure if and only if the underlying extension of k modules is an inductive limit of split extensions see theorem a 4 of appendix a below everywhere in this paper the word excision used in the context of cyclic homology will mean excision with respect to the class of pure extensions the second purpose of the present paper is to give a complete characterization of the class of algebras possessing the excision property in cyclic homology before stating the corresponding result we need the following definition let us

an optimal on line algorithm for k servers on trees the k server problem is investigated when the metric space is a tree for this case an on line k competitive algorithm for k servers is presented the competitiveness ratio k is optimal the algorithm is memoryless in the sense that it does not use any information from the past

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

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

bass nk groups and cdh fibrant hochschild homology the k theory of a polynomial ring r t contains the k theory of r as a summand for r commutative and containing q we describe k r t k r in terms of hochschild homology and the cohomology of kahler differentials for the cdh topology we use this to address bass question whether kn r kn r t implies kn r kn r t1 t 2 the answer to this question is affirmative when r is essentially of finite type over the complex numbers but negative in general

algebraic k theory and descent for blow ups we prove that algebraic k theory satisfies pro descent for abstract blow up squares of noetherian schemes as an application we derive weibel s conjecture on the vanishing of negative k groups

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

the steinberg group of a monoid ring nilpotence and algorithms for a regular ring r and an affine monoid m the homotheties of m act nilpotently on the milnor unstable groups of r m this strengthens the k2 part of the main result of j gubeladze the nilpotence conjecture in k theory of toric varieties invent math 160 2005 173 216 in two ways the coefficient field of characteristic 0 is extended to any regular ring and the stable k2 group is substituted by the unstable ones the proof is based on a polyhedral combinatorial technique computations in steinberg groups and a substantially corrected version of an old result on elementary matrices by mushkudiani z mushkudiani k2 groups of monoid algebras over regular rings proc a razmadze math inst 113 1995 120 137 a similar stronger nilpotence result for k1 and algorithmic consequences for factorization of high frobenius powers of invertible matrices are also derived

k theory of endomorphism rings and of rings of invariants abstract this paper gives the following description ofk0of the endomorphism ring of a finitely generated projective module t heorem let t be a ring and p a finitely generated projective t module let i be the trace ideal of p then k0 end pt is isomorphic to a subgroup of k0 t i if further the natural map k1 t k1 t i is surjective then k0 end pt is isomorphic to the subgroup of k0 t generated by the direct summands of pn for n n as a corollary we can determinek0of the ring of invariants for many free linear actions in particular the following result is proved t heorem let v be a fixed point free linear representation of a finite group g over a field k of characteristic zero and let s v be the symmetric algebra of v let k be any finite dimensional k vector space then k0 s v g ks k s v g ks k similar results are given for suitable noncommutative versions ofs v

excision in algebraic k theory and karoubi s conjecture abstract we prove that the property of excision in algebraic k theory is for a q algebra a equivalent to the h unitality of the latter our excision theorem in particular implies karoubi s conjecture on the equality of algebraic and topological k theory groups of stable c algebras it also allows us to identify the algebraic k theory of the symbol map in the theory of pseudodifferential operators

sur la simplicite essentielle du groupe des inversibles et du groupe unitaire dans une c algebre simple let a be a simple approximately finite dimensional c algebra with unit let gl1 a be the group of invertible elements and let u1 a be that of unitaries in a in a previous work the abelianized groups gl1 a dgl1 a and u1 a du1 a have been described with grothendieck s group k0 a viewed as a dimension group for a here it is shown that the groups dgl1 a and du1 a are simple up to their centres the case of a stable c algebra and that of a c algebra with unit which is simple infinite are also studied in the last two cases moreover it is shown that there exists a natural homomorphism from milnor s group k2mil a onto k0 a

the k theory of toric varieties recent advances in computational techniques for k theory allow us to describe the k theory of toric varieties in terms of the k theory of fields and simple cohomological data

infinitesimal cohomology and the chern character to negative cyclic homology there is a chern character from k theory to negative cyclic homology we show that it preserves the decomposition coming from adams operations at least in characteristic zero

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

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

k regularity cdh fibrant hochschild homology and a conjecture of vorst it is a well known fact that algebraic if theory is homotopy invariant as a functor on regular schemes if x is a regular scheme then the natural map kn x kn x x 1 is an isomorphism for all n ez this is false in general for nonregular schemes and rings to express this failure bass introduced the terminology that for any contravari ant functor v defined on schemes a scheme x is called v regular if the pullback maps v x v x x ar are isomorphisms for all r 0 if x spec i we also say that r is p regular thus regular schemes are ifn regular for every n in contrast it was observed as long ago as in 2 that a nonreduced affine scheme can never be k regular in particular if a is an artinian ring that is a o dimensional noetherian ring then a is regular that is reduced if and only if a is ifi regular in 17 vorst conjectured that for an affine scheme x of finite type over a field f and of dimension d regularity and k i regularity are equivalent vorst proved this conjecture for d 1 by proving that regularity implies normality in this paper we prove vorst s conjecture in all dimensions provided the char acteristic of the ground field f is zero in fact we prove a stronger statement we say that x is regular in codimension n in x note that for all n z if a ring r is ifn regular then it is kn i regular this is proved in 17 for n 1 and in 6 4 4 for n kh x where k x is the algebraic if theory spectrum of x and kh x is the homotopy if theory of x defined in 19 we write fk r for fk spec r theorem 0 1 let r be a commutative ring which is essentially of finite type over a field f of characteristic 0 then the following hold a if tk r is n connected then r is regular in codimension khi x is an isomorphism for i n and a surjection for i n 1 so that fk x

commutator structure of operator ideals commutator structure of operator ideals ken dykema 1 tadeusz figiel gary weiss and mariusz wodzicki department of mathematics texas a m university college station tx 77843 3368 usa b instytut matematyczny polskiej akademii nauk 81 825 sopot poland department of mathematical sciences university of cincinnati cincinnati oh 45221 0025 usa department of mathematics university of california berkeley ca 94720 3840 usa

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

comparison between algebraic and topological k theory of locally convex algebras abstract this paper is concerned with the algebraic k theory of locally convex c algebras stabilized by operator ideals and its comparison with topological k theory we show that if l is locally convex and j a frechet operator ideal then all the different variants of topological k theory agree on the completed projective tensor product l j and that the obstruction for the comparison map k l j k top l j to be an isomorphism is absolute algebraic cyclic homology we prove the existence of an exact sequence theorem 6 2 1 we show that cyclic homology vanishes in the case when j is the ideal of compact operators and l is a frechet algebra whose topology is generated by a countable family of sub multiplicative seminorms and admits an approximate right or left unit which is totally bounded with respect to that family theorem 8 3 3 this proves the generalized version of karoubi s conjecture due to mariusz wodzicki and announced in his paper m wodzicki algebraic k theory and functional analysis in first european congress of mathematics vol ii paris 1992 in progr math vol 120 birkhauser basel 1994 pp 485 496 we also consider stabilization with respect to a wider class of operator ideals called sub harmonic every frechet ideal is sub harmonic but not conversely for example the schatten ideal l p is sub harmonic for all p 0 but is frechet only if p 1 we prove a variant of the exact sequence above which essentially says that if a is a c algebra and j is sub harmonic then the obstruction for the periodicity of k a c j is again cyclic homology theorem 7 1 1 this generalizes to all algebras a result of wodzicki for h unital algebras announced in m wodzicki algebraic k theory and functional analysis in first european congress of mathematics vol ii paris 1992 in progr math vol 120 birkhauser basel 1994 pp 485 496 the main technical tools we use are the diffeotopy invariance theorem of cuntz and the second author which we generalize in theorem 6 1 6 and the excision theorem for infinitesimal k theory due to the first author