algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

the additivity of traces in triangulated categories abstract we explain a fundamental additivity theorem for euler characteristics and generalized trace maps in triangulated categories the proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation the refinement consists of several new axioms relating products and distinguished triangles the axioms hold in the examples and shed light on generalized homology and cohomology theories

rings modules and algebras in infinite loop space theory we give a new construction of the algebraic k theory of small permutative categories that preserves multiplicative structure and therefore allows us to give a unified treatment of rings modules and algebras in both the input and output this requires us to define multiplicative structure on the category of small permutative categories the framework we use is the concept of multicategory elsewhere also called colored operad a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction our method ends up in the hovey shipley smith category of symmetric spectra with an intermediate stop at a category of functors out of a particular wreath product

symmetric powers in abstract homotopy categories abstract we study symmetric powers in the homotopy categories of abstract closed symmetric monoidal model categories in both unstable and stable settings as an outcome we prove that symmetric powers preserve the nisnevich and etale homotopy type in the unstable and stable motivic homotopy theories of schemes over a base more precisely if f is a weak equivalence of motivic spaces or a weak equivalence between positively cofibrant motivic spectra with respect to the nisnevich or etale topology then all symmetric powers sym n f are weak equivalences too this gives left derived symmetric powers in the corresponding motivic homotopy categories of schemes over a base which aggregate into a categorical structures on these categories

hz algebra spectra are differential graded algebras we show that the homotopy theory of differential graded algebras coincides with the homotopy theory of h algebra spectra namely we construct quillen equivalences between the quillen model categories of unbounded differential graded algebras and h algebra spectra we also construct quillen equivalences between the differential graded modules and module spectra over these algebras we use these equivalences in turn to produce algebraic models for rational stable model categories we show that basically any rational stable model category is quillen equivalent to modules over a differential graded algebra with many objects

the nsluc property and klee envelope a notion called norm subdifferential local uniform convexity nsluc is introduced and studied it is shown that the property holds for all subsets of a banach space whenever the norm is either locally uniformly convex or k fully convex the property is also valid for all subsets of the banach space if the norm is kadec klee and its dual norm is g teaux differentiable off zero the nsluc concept allows us to obtain new properties of the klee envelope for example a connection between attainment sets of the klee envelope of a function and its convex hull it is also proved that the klee envelope with unit power plus an appropriate distance function is equal to some constant on an open convex subset as large as we need as a consequence of obtained results the subdifferential properties of the klee envelope can be inherited from subdifferential properties of the opposite of the distance function to the complement of the bounded convex open set moreover the problem of singleton property of sets with unique farthest point is reduced to the problem of convexity of chebyshev sets

on the formalization of fuzzy random variables abstract in this paper we develop a discussion on the mathematical formalization of the concept of fuzzy random variable this discussion is mainly focused on finding an adequate notion of measurability to be coherent with the notions on the space these random elements take values

fuzzy random programming with equilibrium chance constraints to model fuzzy random decision systems this paper first defines three kinds of equilibrium chances via fuzzy integrals in the sense of sugeno then a new class of fuzzy random programming problems is presented based on equilibrium chances also some convex theorems about fuzzy random linear programming problems are proved the results provide us methods to convert primal fuzzy random programming problems to their equivalent stochastic convex programming ones so that both the primal problems and their equivalent problems have the same optimal solutions and the techniques developed for stochastic convex programming can apply after that a solution approach which integrates simulations neural network and genetic algorithm is suggested to solve general fuzzy random programming problems at the end of this paper three numerical examples are provided since the equivalent stochastic programming problems of the three examples are very complex and nonconvex the techniques of stochastic programming cannot apply in this paper we solve them by the proposed hybrid intelligent algorithm the results show that the algorithm is feasible and effectiveness

an approach for solving a fuzzy multiobjective programming problem abstract in this paper we present a new approach based on the nearest interval approximation operator for dealing with a multiobjective programming problem with fuzzy valued objective functions by the way we have established a karush kuhn tucker k k t kind of pareto optimality conditions for the resulting interval multiobjective program to this end we made use of gh differentiability of involved interval valued functions two algorithms play a pivotal role in the proposed method the first one returns a nearest interval approximation to a given fuzzy number the other one makes use of k k t conditions to deliver a pareto optimal solution of the above mentioned resulting interval program

uniqueness of the multiplicative cyclotomic trace abstract making use of the theory of noncommutative motives we characterize the topological dennis trace map as the unique multiplicative natural transformation from algebraic k theory to topological hochschild homology thh and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology tc moreover we prove that the space of all multiplicative structures on algebraic k theory is contractible we also show that the algebraic k theory functor from small stable categories to spectra is lax symmetric monoidal which in particular implies that e n ring spectra give rise to e n 1 ring algebraic k theory spectra along the way we develop a multiplicative morita theory establishing a symmetric monoidal equivalence between the category of small idempotent complete stable categories and the morita localization of the category of spectral categories

solving fuzzy stochastic linear programming problems using superiority and inferiority measures in this paper the author presents a model to measure the superiority and inferiority of fuzzy numbers fuzzy stochastic variables then the new measures are used to convert the fuzzy stochastic linear program into the corresponding deterministic linear program numerical examples are provided to illustrate the effectiveness of the proposed method

a multiplicative comparison of segal and waldhausen k theory in this paper we establish a multiplicative equivalence between two multiplicative algebraic k theory constructions elmendorf and mandell s version of segal s k theory and blumberg and mandell s version of waldhausen s s bullet construction this equivalence implies that the ring spectra algebra spectra and module spectra constructed via these two classical algebraic k theory functors are equivalent as ring algebra or module spectra respectively it also allows for comparisons of spectrally enriched categories constructed via these definitions of k theory as both the elmendorf mandell and blumberg mandell multiplicative versions of k theory encode their multiplicativity in the language of multicategories our main theorem is that there is multinatural transformation relating these two symmetric multifunctors that lifts the classical functor from segal s to waldhausen s construction along the way we provide a slight generalization of the elmendorf mandell construction to symmetric monoidal categories

localization sequences for logarithmic topological hochschild homology we study the logarithmic topological hochschild homology of ring spectra with logarithmic structures and establish localization sequences for this theory our results apply for example to connective covers of periodic ring spectra like real and complex topological k theory mathematics subject classification 14f10 19d55 55p43

a uniqueness theorem for stable homotopy theory roughly speaking the stable homotopy category of algebraic topology is obtained from the homotopy category of topological spaces by inverting the suspension functor yielding a linear approximation to the homotopy category of spaces the isomorphism classes of objects in the stable homotopy category represent the generalized cohomology theories defined by the eilenberg steenrod axioms es45 without the dimension axiom which distinguishes ordinary from generalized cohomology theories the first construction of the full stable homotopy category was given by boardman b69 nowadays manymodels for the stable homotopy category are known most of which have the additional structure of a closed model category in the sense of quillen q67 in m83 h r margolis introduced a short list of axioms and conjectured that they characterize the stable homotopy category up to an equivalence of categories in theorem 3 2 we prove that these axioms do uniquely specify the stable homotopy category whenever there is some underlying quillen model category we also prove amore structured version theuniqueness theorembelow which states that the model category of spectra itself is uniquely determined by certain equivalent conditions up to so called quillen equivalence of

solving linear programming problems under fuzziness and randomness environment using attainment values in this paper the author presents a model to measure attainment values of fuzzy numbers fuzzy stochastic variables these new measures are then used to convert the fuzzy linear programming problem or the fuzzy stochastic linear programming problem into the corresponding deterministic linear programming problem numerical comparisons are provided to illustrate the effectiveness of the proposed method

fuzzy stochastic linear programming survey and future research directions in many real life problems one has to base decision on information which is both fuzzily imprecise and probabilistically uncertain although consistency indexes providing a union nexus between possibilistic and probabilistic representation of uncertainty exist there are no reliable transformations between them this calls for new paradigms for incorporating the two kinds of uncertainty into mathematical models fuzzy stochastic linear programming is an attempt to fulfill this need it deals with modelling and problem solving issues related to situations where randomness and fuzziness co occur in a linear programming framework in this paper we provide a survey of the essential elements methods and algorithms for this class of linear programming problems along with promising research directions being a survey the paper includes many references to both give due credit to results in the field and to help readers obtain more detailed information on issues of interest

fuzzy and semi infinite mathematical programming abstract in this paper linkups between imprecision and infinity in an optimization framework are highlighted inspired by these connections the new potential opened up by semi infinite optimization is gainfully exploited to push forward an approach for solving a fuzzy mathematical program issues regarding the viability of the approach as well as the satisfying character of solutions yielded by this way are also addressed furthermore we present a cutting plane method for solving the resulting semi infinite program we end up by some concluding remarks and indicate directions for further developments

coassembly and the k theory of finite groups abstract we study the k theory and swan theory of the group ring r g when g is a finite group and r is any ring or ring spectrum in this setting the well known assembly map for k r g has a companion called the coassembly map we prove that their composite is the equivariant norm of k r this gives a splitting of both assembly and coassembly after k n localization a new map between whitehead torsion and tate cohomology and a partial computation of k theory of representations in the category of spectra

the smash product for derived categories in stable homotopy theory abstract an e 1 or a ring spectrum r has a derived category of modules d r an e 2 structure on r endows d r with a monoidal product r an e 3 structure on r endows r with a braiding if the e 3 structure extends to an e 4 structure then the braided monoidal product r is symmetric monoidal

robust granular optimization a structured approach for optimization under integrated uncertainty solving optimization problems under hybrid uncertainty bears a heavy computational burden in this study we propose a unified structured optimization approach termed robust granular optimization rgo to tackle the optimization problems under hybrid manifold uncertainties in a computationally tractable manner essentially the rgo can be regarded as a complementary fusion of granular computing and robust optimization techniques the paradigm of rgo consists of three core phases 1 uncertainty identification 2 information granulation in which basic granular units bgus are formed and 3 robust optimization realized over the bgus following the proposed paradigm we develop two classes of rgo models for general single stage and two stage optimization problems with separable and higher order hybrid uncertainties respectively it is shown that both types rgo models can be equivalently transformed into linear programs or mixed integer linear programs that can be handled efficiently by off the shelf solvers furthermore a target based tradeoff model is developed to enhance the flexibility of the rgo models in balancing the granularity level or robustness level and the solution conservativeness the tradeoff model can also be efficiently solved by a binary search algorithm finally sufficient computational studies are presented and comparisons with the existing approaches show that the rgo models can bring much higher computational efficiency and scalability without losing much optimality and the rgo solutions exhibit a stronger resistance to the uncertainty

homology and cohomology of e ring spectra every homology or cohomology theory on a category of e ring spectra is topological andre quillen homology or cohomology with appropriate coefficients analogous results hold more generally for categories of algebras over operads

fuzzy optimization an appraisal abstract this paper takes a general look at core ideas that make up the burgeoning body of fuzzy mathematical programming emphasizing the methodological view although fuzzy mathematical programming has enjoyed a rapidly increasing acceptance within the scientific community some technical hurdles exist to hinder a unanimity reasons for this as well as possible ways for improvement are also discussed

time value of delays in unreliable production systems with mixed uncertainties of fuzziness and randomness abstract this paper proposes a parametric programming approach to address the notion of the time value of delays in the presence of mixed random and fuzzy uncertainties that result from unreliable systems to consider different types of delay time values the system states are appropriately and carefully identified and defined and a cost based fuzzy decision model that incorporates several unreliability factors is constructed then the proposed model is transformed into a pair of nonlinear programs parameterized by the possibility level to identify the lower and upper bounds on the minimal total cost per unit time at and thus construct the membership function to provide analytical expressions a special case with analytical results is also presented in contrast to existing studies the results derived from the proposed solution procedure conserve the fuzziness of the input information representing a significant difference from the crisp results obtained using approaches based on probability theory the results indicate that the proposed approach can provide more precise information to managers and improve decision making in practical system design

conditional value at risk in stochastic programs with mixed integer recourse in classical two stage stochastic programming the expected value of the total costs is minimized recently mean risk models studied in mathematical finance for several decades have attracted attention in stochastic programming we consider conditional value at risk as risk measure in the framework of two stage stochastic integer programming the paper addresses structure stability and algorithms for this class of models in particular we study continuity properties of the objective function both with respect to the first stage decisions and the integrating probability measure further we present an explicit mixed integer linear programming formulation of the problem when the probability distribution is discrete and finite finally a solution algorithm based on lagrangean relaxation of nonanticipativity is proposed

abstract representation theory of dynkin quivers of type a abstract we study the representation theory of dynkin quivers of type a in abstract stable homotopy theories including those associated to fields rings schemes differential graded algebras and ring spectra reflection functors partial coxeter functors and serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules certain explicitly constructed spectral bimodules in fact these universal tilting modules are spectral refinements of classical tilting complexes as a consequence we obtain split epimorphisms from the spectral picard groupoid to derived picard groupoids over arbitrary fields these results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators as further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field this includes yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral serre duality result finally using abstract representation theory of linearly oriented a n quivers we construct canonical higher triangulations in stable derivators and hence a posteriori in stable model categories and stable categories

transfer maps and the cyclotomic trace we analyze the equivariant restriction or transfer maps in topological hochschild homology associated to inclusions of group rings of the form r h r g where r is a symmetric ring spectrum g is a discrete group and h g is a subgroup of finite index this leads to a complete description of the associated restriction or transfer maps in topological cyclic homology

a class of fuzzy random optimization expected value models fuzzy random variable is a measurable mapping from a probability space to a collection of fuzzy variables this concept may be regard as an extension of both random variable and fuzzy variable in this paper the linearity of a scalar value expected value operator of fuzzy random variable is discussed and a fuzzy random simulation approach is suggested to evaluate the expected value of a fuzzy random variable in addition three types of fuzzy random expected value models are presented to model fuzzy random decision systems moreover a hybrid intelligent algorithm which incorporates simulation neural network and genetic algorithm is designed in order to solve general fuzzy random expected value models at the end of this paper the effectiveness of this algorithm is showed via three numerical examples

operadic multiplications in equivariant spectra norms and transfers we study homotopy coherent commutative multiplicative structures on equivariant spaces and spectra we define n operads equivariant generalizations of e operads algebras in equivariant spectra over an n operad model homotopically commutative equivariant ring spectra that only admit certain collections of hill hopkins ravenel norms determined by the operad analogously algebras in equivariant spaces over an n operad provide explicit constructions of certain transfers this characterization yields a conceptual explanation of the structure of equivariant infinite loop spaces to explain the relationship between norms transfers and n operads we discuss the general features of these operads linking their properties to families of finite sets with group actions and analyzing their behavior under norms and geometric fixed points a surprising consequence of our study is that in stark contract to the classical setting equivariantly the little disks and linear isometries operads for a general incomplete universe u need not determine the same algebras our work is motivated by the need to provide a framework to describe the flavors of commutativity seen in recent work of the second author and hopkins on localization of equivariant commutative ring spectra

fuzzy random chance constrained programming by fuzzy random programming we mean the optimization theory dealing with fuzzy random decision problems this paper presents a new concept of chance of fuzzy random events and constructs a general framework of fuzzy random chance constrained programming we also design a spectrum of fuzzy random simulations for computing uncertain functions arising in the area of fuzzy random programming to speed up the process of handling uncertain functions we train a neural network to approximate uncertain functions based on the training data generated by fuzzy random simulation finally we integrate the fuzzy random simulation neural network and genetic algorithm to produce a more powerful and effective hybrid intelligent algorithm for solving fuzzy random programming models and illustrate its effectiveness by some numerical examples

interactive multiobjective fuzzy random linear programming maximization of possibility and probability this paper considers multiobjective linear programming problems with fuzzy random variables coefficients a new decision making model is proposed to maximize both possibility and probability which is based on possibilistic programming and stochastic programming an interactive algorithm is constructed to obtain a satisficing solution satisfying at least weak pareto optimality

optimization problems for curved mechanical structures we study optimal design problems for three dimensional curved rods and for shells under minimal regularity assumptions for the geometry the results that we establish concern the existence of optimal shapes and the sensitivity analysis we also compute some numerical examples for the optimization of curved rods the models used have been investigated in our previous works a ignat j sprekels and d tiba math methods appl sci 25 2002 pp 835 854 j sprekels and d tiba adv math sci appl 12 2002 pp 175 190 a complete study of kirchhoff love arches and of related minimization questions has been performed in a ignat j sprekels and d tiba siam j control optim 40 2001 pp 1107 1133

on fuzzy random multiobjective quadratic programming in this paper a multiobjective quadratic programming problem having fuzzy random coefficients matrix in the objective and constraints and the decision vector are fuzzy pseudorandom variables is considered first we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series optimal solutions of relative scalar fuzzy quadratic programming some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients having decision vectors as fuzzy variables at the end numerical examples are illustrated in the support of the obtained results

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

on the nonexistence of elements of kervaire invariant one we show that the kervaire invariant one elements j 2j 1 2s exist only for j 6 by browder s theorem this means that smooth framed manifolds of kervaire invariant one exist only in dimensions 2 6 14 30 62 and possibly 126 except for dimension 126 this resolves a longstanding problem in algebraic topology

radon nikody m theorem and conditional expectation of fuzzy valued measures and variables abstract the notions of fuzzy valued measure and conditional expectation of a fuzzy valued variable are defined the main result of the paper is a proof of the existence of the radon nikod m derivative of the fuzzy valued measure a convergence theorem for conditional expectations is proved as well

derived galois deformation rings abstract we define a derived version of mazur s galois deformation ring it is a pro simplicial ring r classifying deformations of a fixed galois representation to simplicial coefficient rings its zeroth homotopy group 0 r recovers mazur s deformation ring we give evidence that these rings r occur in the wild for suitable galois representations the langlands program predicts that 0 r should act on the homology of an arithmetic group we explain how the taylor wiles method can be used to upgrade such an action to a graded action of r on the homology

a stochastic possibilistic programming model for bank hedging decision problems abstract for a practical bank hedging decision optimization problem interest rates and price of futures contract may involve both fuzziness and randomness for subjective nature of satisfaction maximum desired values of loan demand deposit supply and ratio of desired loan to deposit are often fuzzy in this study we consider and solve a stochastic possibilistic programming model of bank hedging decision problems with the above characters we first use the expected value to obtain an auxiliary possibilistic linear programming problem which is further resolved by use of level cut an crisp auxiliary bi objective linear programming model is then proposed and solved by our augmented maximum approach for illustration purpose a numerical bank hedging decision problem is solved

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

a model structure for coloured operads in symmetric spectra we describe a model structure for coloured operads with values in the category of symmetric spectra with the positive model structure in which fibrations and weak equivalences are defined at the level of the underlying collections this allows us to treat r module spectra where r is a cofibrant ring spectrum as algebras over a cofibrant spectrum valued operad with r as its first term using this model structure we give sufficient conditions for homotopical localizations in the category of symmetric spectra to preserve module structures

the bousfield kuhn functor and topological andre quillen cohomology we construct a natural transformation from the bousfield kuhn functor evaluated on a space to the topological andre quillen cohomology of the k n local spanier whitehead dual of the space and show that the map is an equivalence in the case where the space is a sphere this results in a method for computing unstable v n periodic homotopy groups of spheres from their morava e cohomology as modules over the dyer lashof algebra of morava e theory we relate the resulting algebraic computations to the algebraic geometry of isogenies between lubin tate formal groups

optimisation under hybrid uncertainty abstract fuzzy stochastic optimisation is emerging as a subfield of mathematical programming the disciplinary matrix of which consists of analysis of mathematical programs under fuzziness and randomness along with methods for solving them the primum movens of this paper is to describe a unifying methodological approach that is suitable for finding a satisfying solution of a mathematical program in the presence of fuzzy data and random variables properties of fuzzy random variables frvs serve as the backdrop to this approach which also lends itself better to handling mathematical programs with fuzzy random coefficients for the paper to be somewhat self contained the notion of frv is briefly discussed and a synopsis of fuzzy stochastic optimisation provided a systematically solved example aimed at illustrating the proposed approach is also included

an algebraic model for free rational g spectra for connected compact lie groups g we show that the category of free rational g spectra for a connected compact lie group g is quillen equivalent to the category of torsion differential graded modules over the polynomial ring h bg the ingredients are the enriched morita equivalences of schwede and shipley topology 42 1 103 153 2003 the functors of shipley am j math 129 351 379 2007 making rational spectra algebraic koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring

derived representation theory of lie algebras and stable homotopy categorification of slk abstract we set up foundations of representation theory over s the sphere spectrum which is the initial ring of stable homotopy theory in particular we treat s lie algebras and their representations characters g l n s verma modules and their duals harish chandra pairs and zuckermann functors as an application we construct a khovanov s l k stable homotopy type with a large prime hypothesis which is a new link invariant using a stable homotopy analogue of the method of j sussan

linear programming under randomness and fuzziness this paper is concerned with the use of fuzzy sets theory well suited for nuanced human behavior to extend the applicability of linear programming in a way to let both randomness and fuzziness uncertainties be taken into account a fuzzy stochastic linear programming problem is considered and ways to handle it suggested the basic idea behind our approaches is to translate the original problem in deterministic terms via tools provided by fuzzy sets theory to get a classical one some conditions for the resulting problems to be solved without extra computational effort are stressed and illustrative examples given

s modules and symmetric spectra we study a symmetric monoidal adjoint functor pair between the category of s modules and the category of symmetric spectra the functors induce equivalences between the respective homotopy categories of spectra module spectra and ring spectra then at approximately the same time two categories of spectra with nice smash products were discov ered elmendorf kriz mandell and may constructed the category of s modules ekmm and je smith introduced symmetric spectra hss both categories are quillen model categories and have associated notions of ring and module spectra however these two categories arise in completely dierent ways and even though the homotopy categories are equivalent it is not a priori clear if both frameworks give rise to the same homotopy theory of rings and modules both categories have their merits described in detail in the introductions of ekmm and hss and it is desirable to be able to translate results obtained in one category into conclusions valid in the other the present paper describes an easy mechanism which facilitates such comparisons below we dene a lax symmetric monoidal functor ms sp from the category of s modules to the category of symmetric spectra the functor preserves homotopy groups and has a strong symmetric monoidal left adjoint we show that the two functors induce inverse equivalences of the homotopy categories of spectra ring spectra commutative ring spectra and module spectra main theorem the functor from the category ms of s modules to the category sp of symmetric spectra passes to a symmetric monoidal equivalence of homotopy categories

equivariant properties of symmetric products sp n x x n n we denote by sp n fsp n s m gm 0 the spectrum whose terms are the n th symmetric products of spheres a celebrated theorem of dold and thom 7 asserts that sp 1 s m is an eilenberg mac lane space of type z m for m 1 so sp 1 is an eilenberg mac lane spectrum for the group z the resulting ltration

a unified approach to fuzzy random variables the concept of fuzzy random variable was introduced as an analogous notion to random variables in order to extend statistical analysis to situations when the outcomes of some random experiment are fuzzy sets but in contrary to the classical statistical methods no unique definition has been established yet in this paper a set theoretical concept of fuzzy random variable will be presented this notion provides a useful framework to compare different concepts of fuzzy random variables using methods of general topology and drawing on results from topological measure theory and the theory of analytic spaces as the main result it will be shown that the introduced concept of fuzzy random variable is a unification of the already known ones

value at risk methodology under soft conditions approach fuzzy stochastic approach the paper describes methodology of dealing with financial modelling under uncertainty with risk and vagueness aspects an approach to modelling risks by the value at risk methodology under imprecise and soft conditions is solved it is supposed that the input data and problem conditions is difficult to determine as real numbers or as some precise distribution function thus vagueness is modelled through the fuzzy numbers of the linear t number type the combination of risk and vagueness is solved by fuzzy stochastic methodology illustrative example is introduced

interactive fuzzy linear programming abstract in this study an interactive fuzzy linear programming approach is investigated to improve the flexibility and robustness of linear programming lp technique this paper provides a practical modelling method which is a symmetric integration of zimmermann s werners s vergegay s and chanas s fuzzy lp approaches and additionally provides a decision support system for solving a specific domain of practical lp problems the interactive concept provides a learning process about the system whereby the decision maker can learn to recognize good solutions relative importance of factors in the system and then design a high productivity system instead of optimizing a given system this interactive fuzzy lp system provides integration oriented adaptation and learning features by considering all the possibilities of a specific domain of lp problems which are integrated in logical order using an if then rule

universal toda brackets of ring spectra we construct and examine the universal toda bracket of a highly structured ring spectrum r this invariant of r is a cohomology class in the mac lane cohomology of the graded ring of homotopy groups of r which carries information about r and the category of r module spectra it determines for example all triple toda brackets of r and the first obstruction to realizing a module over the homotopy groups of r by an r module spectrum for periodic ring spectra we study the corresponding theory of higher universal toda brackets the real and complex k theory spectra serve as our main examples

on some optimisation models in a fuzzy stochastic environment this paper is on fuzzy stochastic optimisation an area that is quickly coming to the forefront of mathematical programming under uncertainty an even stronger motivating factor for the growing interest in this area can be found in the ubiquitous nature of decision problems involving hybrid imprecision more precisely we consider a range of situations in which random factors and fuzzy information co occur in an optimisation setting related hybrid optimisation models are discussed and converted into deterministic terms through appropriate tools like probabilistic set uncertain probability and fuzzy random variable making good use of uncertainty principles we also discuss ways to deal with the resulting problems numerical examples carried out using class optimisation software demonstrate the efficiency of the proposed approaches we shall end this article by pointing out some of the challenges that currently occupy researchers in this emerging field

on fuzzy stochastic optimization abstract this paper extends the applicability of optimization models to situations where both fuzzy and random data are in the state of affairs the major mathematical tool used to this end is the concept of fuzzy random variables which provides a synergistic way for dealing with the two kinds of imprecision the extension principle is canonically extended to fuzzy random variables and exploited to derive a way for handling robust fuzzy stochastic mathematical programming problems finally we provide some possible avenues for further fruitful developments in this field

monoidal uniqueness of stable homotopy theory we show that the monoidal product on the stable homotopy category of spectra is essentially unique this strengthens work of this author with schwede on the uniqueness of models of the stable homotopy theory of spectra also the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra s modules w spaces orthogonal spectra simplicial functors with symmetric spectra as an application we show that with an added assumption about underlying model structures margolis axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence

the stable homotopy category has a unique model at the prime 2 the stable homotopy category has been extensively studied by algebraic topologists for a long time for many applications it is convenient or even necessary to work with point set level models of spectra as opposed to working up to homotopy and the outcome of a calculation can depend on the choice of model in recent years many new models for the stable homotopy category have been constructed it is especially useful to have the structure of a closed model category in the sense of quillen qui and many examples of spectra categories t into this context bf rob87 ekmm hss lyd mmss moreover all known examples capture the same homotopy theory in technical terms one speaks of quillen equivalent model categories hov def 1 3 12 hence not only the homotopy categories but also higher order information such as toda brackets homotopy colimits and homotopy types of function spaces coincide in two quillen equivalent model categories the answer to every homotopy theoretic question comes out the same in a model category one can pass to the associated homotopy category by formally inverting the class of weak equivalences however passage to the homotopy category loses information and in general the homotopy theory can not be recovered from the homotopy category see 2 1 and 2 2 for two examples in this paper we show that in contrast to the general case the stable homotopy category completely determines the stable homotopy theory 2 locally we prove a uniqueness theorem which says that there is essentially only one model category structure underlying the stable homotopy category of 2 local spectra the stable homotopy category has no exotic models at the prime 2

localization of algebras over coloured operads we give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories our approach encompasses a number of previous results about preservation of structures under localizations such as loop spaces or infinite loop spaces and provides new results of the same kind for instance under suitable assumptions homotopical localizations preserve ring spectra in the strict sense not only up to homotopy modules over ring spectra and algebras over commutative ring spectra as well as maps between ring spectra module maps and algebra maps it is principally the treatment of module spectra and their maps that led us to the use of coloured operads also called enriched multicategories in this context

virtual vector bundles and graded thom spectra we introduce a convenient framework for constructing and analyzing orthogonal thom spectra arising from virtual vector bundles this framework enables us to set up a theory of orientations and graded thom isomorphisms with good multiplicative properties the theory is applied to the analysis of logarithmic structures on commutative ring spectra

crystals and derived local moduli for ordinary k3 surfaces abstract it is shown that the k3 spectra which refine the local rings of the moduli stack of ordinary p primitively polarized k3 surfaces in characteristic p allow for an e structure which is unique up to equivalence this uses the e obstruction theory of goerss and hopkins and the description of the deformation theory of such k3 surfaces in terms of their hodge f crystals due to deligne and illusie furthermore all automorphisms of such k3 surfaces can be realized by e maps which are unique up to homotopy and this can be rigidified to an action if the automorphism group is tame

the transfer is functorial abstract we prove that the becker gottlieb transfer is functorial up to homotopy for all fibrations with finitely dominated fibers this resolves a lingering foundational question about the transfer which was originally defined in the late 1970s in order to simplify the proof of the adams conjecture our approach differs from previous attempts in that we closely emulate the geometric argument in the case of a smooth fiber bundle this leads to a multiplicative description of the transfer different from the standard presentation as the trace of a diagonal map

diagram spaces and symmetric spectra abstract we present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra the main motivating examples are the i spaces which are diagrams indexed by finite sets and injections and j spaces which are diagrams indexed by quillen s localization construction 1 on the category of finite sets and bijections we show that the category of i spaces provides a convenient model for the homotopy category of spaces in which every e space can be rectified to a strictly commutative monoid similarly the commutative monoids in the category of j spaces model graded e spaces using the theory of j spaces we introduce the graded units of a symmetric ring spectrum the graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures

picard groups grothendieck rings and burnside rings of categories abstract we discuss the picard group the grothendieck ring and the burnside ring of a symmetric monoidal category and we consider examples from algebra homological algebra topology and algebraic geometry

homotopy theory of spectral categories abstract we construct a quillen model structure on the category of spectral categories where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories

tilting theory via stable homotopy theory we show that certain tilting results for quivers are formal consequences of stability and as such are part of a formal calculus available in any abstract stable homotopy theory thus these results are for example valid over arbitrary ground rings for quasi coherent modules on schemes in the differential graded context in stable homotopy theory and also in the equivariant motivic or parametrized variant thereof in further work we will continue developing this calculus and obtain additional abstract tilting results here we also deduce an additional characterization of stability based on goodwillie s strongly co cartesian n cubes as applications we construct abstract auslander reiten translations and abstract serre functors for the trivalent source and verify the relative fractionally calabi yau property this is used to offer a new perspective on may s axioms for monoidal triangulated categories

derived l adic zeta functions abstract let k 0 v k be the grothendieck group of k varieties campbell and zakharevich have constructed a higher algebraic k theory spectrum k v k such that 0 k v k k 0 v k in this paper we construct non trivial classes in the higher homotopy groups of k v k when k is finite or a subfield of c to do this we give a recipe for lifting motivic measures k 0 v k k 0 e to maps of spectra k v k k e we consider two special cases the classical local zeta function thought of as a homomorphism k 0 v f q k 0 end q l and the compactly supported euler characteristic thought of as a homomorphism k 0 v c k 0 q we use lifts of these motivic measures to prove that the grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map s k v k is nontrivial in higher dimensions when k is finite or a subfield of c and moreover that when k is finite this map is not surjective on higher homotopy groups