exotic elliptic algebras of dimension 4 abstract let e be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3 let be a translation automorphism of e that is not of order 2 in a previous paper we studied an algebra a a e that depends on this data a e s e m 2 k where s e is the 4 dimensional sklyanin algebra associated to e m 2 k is the ring of 2 2 matrices over k and is z 2 z 2 acting in a particular way as automorphisms of s and m 2 k the action of on s is compatible with the translation action of the 2 torsion subgroup e 2 on e following the ideas and results in papers of artin tate van den bergh smith stafford and levasseur smith this paper examines the line modules point modules and fat point modules over a and their incidence relations the right context for the results is non commutative algebraic geometry we view a as a homogeneous coordinate ring of a non commutative analogue of p 3 that we denote by proj n c a point modules and fat point modules determine points in proj n c a line modules determine lines in proj n c a line modules for a are in bijection with certain lines in p a 1 p 3 and therefore correspond to the closed points of a certain subscheme l of the grassmannian g 1 3 shelton vancliff call l the line scheme for a we show that l is the union of 7 reduced and irreducible components 3 quartic elliptic space curves and 4 plane conics in the ambient plucker p 5 and that deg l 20 the union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in p a 1 thus the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for a

hochschild cohomology and atiyah classes abstract in this paper we prove that on a smooth algebraic variety the hkr morphism twisted by the square root of the todd genus gives an isomorphism between the sheaf of poly vector fields and the sheaf of poly differential operators both considered as derived gerstenhaber algebras in particular we obtain an isomorphism between hochschild cohomology and the cohomology of poly vector fields which is compatible with the lie bracket and the cupproduct the latter compatibility is an unpublished result by kontsevich our proof is set in the framework of lie algebroids and so applies without modification in much more general settings as well

resolution of a system of the max product fuzzy relation equations using lou factorization in this paper the lu factorization is extended to the fuzzy square matrix with respect to the max product composition operator called l u factorization equivalently we will find two fuzzy lower and upper triangular matrices l and u for a fuzzy square matrix a such that a l u where is the max product composition an algorithm is presented to find the matrices l and u furthermore some necessary and sufficient conditions are proposed for the existence and uniqueness of the l u factorization for a given fuzzy square matrix a an algorithm is also proposed to find the solution set of a square system of fuzzy relation equations fre using the l u factorization the algorithm finds the solution set without finding its minimal solutions and maximum solution it is shown that the two algorithms have a polynomial time complexity as o n 3 since the determination of the minimal solutions is an np hard problem the algorithm can be very important from the practical point of view

the inverse problem for distributive lattices abstract the inverse problem givena in vl n b in vl m andb in vl m find all a in vl n such that a a b b where l is distributive lattice denotes max min composition vl n denotes the set of all vector with n member on l and x a1 a2 an b1 b2 bn ai bj n m is investigated the necessary and sufficient condition for the existence of a solution is shown and an analytical solution is given

fuzzy linear systems abstract fuzzy linear systems of equations and inequalities over a bounded chain are studied both from theoretical and computational points of view a unified approach is presented for solving such systems completed with polynomial time algorithms the main results are concerned with establishing the consistency of the system computing all kinds of solutions or marking the contradictory equations respectively inequalities if the system is inconsistent applications in fuzzy linear programming multivalued logic fuzzy matrix equations or inequalities fuzzy relation equations or inclusions fuzzy automata and medicine are presented

fuzzy relational equations and the inverse problem abstract the inverse problem concerned with fuzzy relations is investigated the conditions for the existence of a solution are shown and an analytical solution is given a method for the improvement of the solution is proposed

formality theorem for gerbes abstract the main result of the present paper is an analogue of kontsevich formality theorem in the context of the deformation theory of gerbes we construct an l deformation of the schouten algebra of multi vectors which controls the deformation theory of a gerbe

hochschild cohomology of abelian categories and ringed spaces this paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors we show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of hochschild cohomology for abelian categories we then show that this hochschild cohomology coincides with the one defined by gerstenhaber schack and swan in the case of module categories over diagrams and schemes and also with the hochschild cohomology for exact categories introduced recently by keller in addition we show in complete generality that hochschild cohomology satisfies a mayer vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf

the hochschild cohomology ring of a global quotient orbifold abstract we study the cup product on the hochschild cohomology of the stack quotient x g of a smooth quasi projective variety x by a finite group g more specifically we construct a g equivariant sheaf of graded algebras on x whose g invariant global sections recover the associated graded algebra of the hochschild cohomology of x g under a natural filtration this sheaf is an algebra over the polyvector fields t x p o l y on x and is generated as a t x p o l y algebra by the sum of the determinants det n x g of the normal bundles of the fixed loci in x we employ our understanding of hochschild cohomology to conclude that the analog of kontsevich s formality theorem for the cup product does not hold for deligne mumford stacks in general we discuss in the case of a symplectic group action on a symplectic variety x relationships with orbifold cohomology and ruan s cohomological conjectures in describing the hochschild cohomology in the symplectic situation we employ compatible trivializations of the determinants det n x g which requires for the cup product a nontrivial normalization missing in previous literature

the cohomology of presheaves of algebras i presheaves over a partially ordered set to each presheaf over a poset of associative algebras a we associate an algebra a we define a full exact embedding of the category of presheaf a bimodules in that of a bimodules we show that this embedding preserves neither enough relative injectives nor enough relative projectives but nonetheless preserves relative yoneda cohomology the cohomology isomorphism links the deformations of manifolds algebraic presheaves and algebras it also implies that the cohomology of any triangulable space is isomorphic to the hochschild cohomology of an associative algebra the latter isomorphism preserves all known cohomology operations we conclude the paper by exhibiting for each associative algebra and triangulable space a product which is again an associative algebra 0 introduction presheaves are rather familiar objects and arise in many situations some of which are discussed below our primary purpose in this paper is to prove a theorem which reduces questions about the cohomology of presheaves of rings to questions about the classical cohomology of rings specifically fix a commutative ring k and a partially ordered set poset jy let a be a presheaf of associative k algebras over j hereinafter a diagram then a has naturally associated with it an abelian category of bimodules and a relative yoneda cohomology bifunctor ext a all the foregoing formalism is introduced in 1 we define for each such diagram a a single k algebra a and a full exact embedding m m of the category of a bimodules in the category of a bimodules 2 the embedding induces a natural transformation of yoneda cohomology bifunctors w exta a ext l 2 our central result is the special cohomology comparison theorem cct namely for an arbitrary diagram a over an arbitrary poset w exta exta h is an isomorphism the proof occupies 3 4 the scheme of the proof will be given shortly first we describe some of the history and utility of the cct in gs1 we proved a primitive version of the cct applicable to arbitrary diagrams defined over arbitrary finite posets and to a narrow class of diagrams defined over a limited class of infinite posets the proofs given there for the finite received by the editors march 6 1987 1980 mathematics subject classification 1985 revision primary 16a58 16a61 16a62 18g10 18g25 secondary 18e25 32g05 55n25 55u10

on drinfeld s dg quotient abstract the possibility of constructing quotients of differential graded dg categories is essential in non commutative algebraic geometry the first construction of dg quotients appeared in keller s work keller 1994 21 and it was recently followed by drinfeld s elegant approach drinfeld 2004 9 although drinfeld s dg quotient admits a very simple construction it didn t seem to be intrinsically defined in this article we complete this aspect of drinfeld s work by providing three different characterizations of drinfeld s dg quotient in terms of simple universal properties

fuzzy complex analysis ii integration abstract this paper continues the author s research in fuzzy complex analysis where we now define and study the basic properties of a fuzzy contour integral for fuzzy mappings which map a rectifiable curve in the complex plane into fuzzy complex numbers

the first obstructions to enhancing a triangulated category in this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement we show that these obstructions do not vanish in an example of triangulated category without models we also obtain cohomological characterizations of pre triangulated dg a infinity and spectral categories

the cauchy problem for complex fuzzy differential equations abstract we discuss the existence of a solution to the cauchy problem for fuzzy differential equations that accommodates the notion of fuzzy sets defined by complex valued membership functions we first propose definitions of complex fuzzy sets and discuss entailed results which parallel those of regular fuzzy numbers we then give two existence results relevant to the cauchy problem for fuzzy differential equations in the case of bounded integral operators these results require either holder continuous or lipschitz continuous response functions

moduli of non commutative polarized schemes we construct using geometric invariant theory a quasi projective deligne mumford stack of stable graded algebras we also construct a derived enhancement which classifies twisted bundles of stable graded a infty algebras the tangent complex of the derived scheme is given by graded hochschild cohomology which we relate to ordinary hochschild cohomology we obtain a version of hilbert stability for non commutative projective schemes

cohomology of exact categories and non additive sheaves abstract we use non additive sheaves to introduce an absolute notion of hochschild cohomology for exact categories as ext s in a suitable bisheaf category we compare our approach to various definitions present in the literature

resolution of the cartesian product of fuzzy sets abstract the problem given a in v n b in v m and b in v m find all a in v n such that a a b b where denotes max min composition x denotes cartesian product and v n denotes the set of all fuzzy vectors with n members is investigated the condition for the existence of a solution is shown and an analytical solution is given

a software routine to solve the generalized inverse problem of fuzzy systems abstract a software routine to produce the solution of the generalized inverse problem of fuzzy relational equations is presented given the fuzzy matrices r and b defined on u v and w v respectively the routine produces all fuzzy matrices a defined on the product space w u which satisfy a r b where denotes maxmin composition

the complete resolution of cartesian products of fuzzy sets abstract a complete solution is given to the cartesian problem given the fuzzy sets a u b v and b v find all fuzzy sets a u such that a a b b where denotes the max min composition this work is an extension of the work by pappis fuzzy sets and systems 26 1988 387 391 and kim et al fuzzy sets and systems 41 1991 293 297

resolution of cartesian products of fuzzy sets abstract the problem given the fuzzy subsets a u b v and b v find all a u such that a a b b where denotes max min composition and x denotes cartesian product is investigated the condition for the existence of a solution is shown and an analytical solution is given

ancfis a neurofuzzy architecture employing complex fuzzy sets complex fuzzy sets cfss are an extension of type 1 fuzzy sets in which the membership of an object to the set is a value from the unit disc of the complex plane although there has been considerable progress made in determining the properties of cfss and complex fuzzy logic there has yet to be any practical application of this concept we present the adaptive neurocomplex fuzzy inferential system ancfis which is the first neurofuzzy system architecture to implement complex fuzzy rules and in particular the signature property of rule interference we have applied this neurofuzzy system to the domain of time series forecasting which is an important machine learning problem we find that ancfis performs well in one synthetic and five real world forecasting problems and is also very parsimonious experimental comparisons show that ancfis is comparable with existing approaches on our five datasets this work demonstrates the utility of complex fuzzy logic on real world problems

on global deformation quantization in the algebraic case abstract we give a proof of yekutieli s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems instead we use an argument inspired by algebraic de rham cohomology

deformation theory of abelian categories in this paper we develop the basic infinitesimal deformation theory of abelian categories this theory yields a natural generalization of the well known deformation theory of algebras developed by gerstenhaber as part of our deformation theory we define a notion of flatness for abelian categories we show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems

fuzzy complex numbers abstract this paper first defines fuzzy complex numbers and shows closure under the basic arithmetic operations it is shown that arithmetic of fuzzy complex numbers may be performed in terms of cuts next two special types of fuzzy complex numbers based on the forms z x iy and z re i for regular complex numbers are investigated introducing a metric on the space of fuzzy complex numbers then allows us to discuss continuity and differentiability of fuzzy complex functions

the gerstenhaber bracket as a schouten bracket for polynomial rings extended by finite groups we apply new techniques to compute gerstenhaber brackets on the hochschild cohomology of a skew group algebra formed from a polynomial ring and a finite group in characteristic 0 we show that the gerstenhaber brackets can always be expressed in terms of schouten brackets on polyvector fields we obtain as consequences some conditions under which brackets are always 0 and show that the hochschild cohomology is a graded gerstenhaber algebra under the codimension grading strengthening known results

a computer algorithm for the solution of the inverse problem of fuzzy systems abstract a computer algorithm based on a method for dealing with the so called inverse problem is presented the algorithm produces the solution semi lattice formed by the fuzzy subsets a u satisfying the fuzzy relational equation a r b where denotes maxmin composition given a fuzzy relation r u v and a fuzzy subset b v

complex fuzzy logic a novel framework for logical reasoning termed complex fuzzy logic is presented in this paper complex fuzzy logic is a generalization of traditional fuzzy logic based on complex fuzzy sets in complex fuzzy logic inference rules are constructed and fired in a manner that closely parallels traditional fuzzy logic the novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets characterized by complex valued membership functions the range of these membership functions is extended from the traditional fuzzy range of 0 1 to the unit circle in the complex plane thus providing a method for describing membership in a set in terms of a complex number several mathematical properties of complex fuzzy sets which serve as a basis for the derivation of complex fuzzy logic are reviewed in this paper these properties include basic set theoretic operations on complex fuzzy sets namely complex fuzzy union and intersection complex fuzzy relations and their composition and a novel form of set aggregation vector aggregation complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic while benefiting from the properties of complex numbers and complex fuzzy sets the introduction of complex valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties and considerable potential for further research and application

fuzzy complex numbers representations operations and its analysis abstract as a key theoretical basis of fuzzy complex analysis a new representation of fuzzy complex numbers is proposed in this paper which unifies the exponential form trigonometric form and algebraic form of fuzzy complex numbers and some arithmetic operations are investigated the concepts of strong sum and strong difference are defined in order to simplify the addition and subtraction operations of complex fuzzy numbers and establish the calculus theory of fuzzy complex valued functions the metric between two fuzzy complex numbers the derivative differentiability and analyticity of fuzzy complex valued functions are defined and characterized the necessary and sufficient conditions of the analyticity of fuzzy complex valued functions are investigated furthermore the integral of fuzzy complex valued functions and cauchy integral theorem are given and discussed

an example of a non fourier mukai functor between derived categories of coherent sheaves orlov s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a fourier mukai functor in this paper we show that this result is false without the fully faithfulness hypothesis we also show that our functor does not lift to the homotopy category of spectral categories if the ground field is mathbb q

comment on pythagorean and complex fuzzy set operations dick et al introduced a relation leq on pi i numbers and proved that pi i leq is a bounded distributive complete lattice in the present paper we point out that the relation leq is not a partial order and that pi i leq is not a lattice in addition we study some partial orders on pi i numbers

scalar extensions of derived categories and non fourier mukai functors abstract orlov s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a fourier mukai functor this result has been extended by lunts and orlov to include functors from perfect complexes to quasi coherent complexes in this paper we show that the latter extension is false without the full faithfulness hypothesis our results are based on the properties of scalar extensions of derived categories whose investigation was started by pawel sosna and the first author

fuzzy complex analysis i differentiation abstract we consider functions mapping real numbers into the set of generalized complex fuzzy numbers c we define two derivatives of such functions which are generalizations of the derivative of real fuzzy mappings presented by dubois and prade 4 5 sufficient conditions on the function are given so that derivative will be a function mapping real numbers into c if the functional value is always a rectangular complex fuzzy number made up to lr real fuzzy numbers then we show that the derivative is easily computed from the derivatives of the lr fuzzy numbers other properties of the derivatives are presented

ellipsoids and ellipsoid shaped fuzzy sets as natural multi variate generalization of intervals and fuzzy numbers how to elicit them from users and how to use them in data processing in this paper we show that ellipsoids are natural multi variate generalization of intervals and ellipsoid shaped fuzzy sets are a natural generalization of fuzzy numbers we explain how to elicit them from users and how to use them in data processing

dg quotients of dg categories abstract keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with verdier s notion of quotient of a triangulated category modulo a triangulated subcategory this work is an attempt to further develop his theory

a method for multiple periodic factor prediction problems using complex fuzzy sets multiple periodic factor prediction mpfp problems exist widely in multisensor data fusion applications development of an effective prediction method should integrate information for multiple periodically changing factors because the uncertainty and periodicity coexist in the information used the prediction method should be able to handle them simultaneously in this study complex fuzzy sets are used to represent the information with uncertainty and periodicity a product sum aggregation operator psao is developed for a set of complex fuzzy sets which is used to integrate information with uncertainty and periodicity and a psao based prediction psaop method is then proposed to generate a solution of mpfp problems this study illustrates the details of the psaop method through two real applications in annual sunspot number prediction and bushfire danger rating prediction experiments indicate that the proposed psaop method effectively handles the uncertainty and periodicity in the information of multiple periodic factors simultaneously and can generate accurate predictions for mpfp problems

on deformations of triangulated models abstract this paper is the first part of a project aimed at understanding deformations of triangulated categories and more precisely their dg and a models and applying the resulting theory to the models occurring in the homological mirror symmetry setup in this first paper we focus on models of derived and related categories based upon the classical construction of twisted objects over a dg or a algebra for a hochschild 2 cocycle on such a model we describe a corresponding curvature compensating deformation which can be entirely understood within the framework of twisted objects we unravel the construction in the specific cases of derived a and abelian categories homotopy categories and categories of graded free qdg modules we identify a purity condition on our models which ensures that the structure of the model is preserved under deformation this condition is typically fulfilled for homotopy categories but not for unbounded derived categories

algebroid prestacks and deformations of ringed spaces for a ringed space x o we show that the deformations of the abelian category mod o of sheaves of o modules lowen and van den bergh 2006 are obtained from algebroid prestacks as introduced by kontsevich in case x is a quasi compact separated scheme the same is true for qch o the category of quasi coherent sheaves on x it follows in particular that there is a deformation equivalence between mod o and qch o

deformations of algebroid stacks abstract in this paper we consider deformations of an algebroid stack on an etale groupoid we construct a differential graded lie algebra dgla which controls this deformation theory in the case when the algebroid is a twisted form of functions we show that this dgla is quasiisomorphic to the twist of the dgla of hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe