quantum locally compact metric spaces abstract we introduce the notion of a quantum locally compact metric space which is the noncommutative analogue of a locally compact metric space and generalize to the non unital setting the notion of quantum metric spaces introduced by rieffel we then provide several examples of such structures including the moyal plane compact quantum metric spaces and locally compact metric spaces this paper provides an answer to the question raised in the literature about the proper notion of a quantum metric space in the non unital setup and offers important insights into noncommutative geometry for non compact quantum spaces

a local global principle for regular operators in hilbert c modules abstract hilbert c modules are the analogues of hilbert spaces where a c algebra plays the role of the scalar field with the advent of kasparov s celebrated kk theory they became a standard tool in the theory of operator algebras while the elementary properties of hilbert c modules can be derived basically in parallel to hilbert space theory the lack of an analogue of the projection theorem soon leads to serious obstructions and difficulties in particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check in this paper we present a new criterion for regularity in terms of the hilbert space localizations of an unbounded operator we discuss several examples which show that the criterion can easily be checked and that it leads to nontrivial regularity results

a fuzzy gould type integral in this paper we introduce and study a set valued gould type integral of a real function with respect to a fuzzy multimeasure taking values in the family of all nonempty closed subsets of a banach space x some classical properties of the integral are obtained only using the hypothesis of monotonicity for the set valued fuzzy measure

pointwise lipschitz functions on metric spaces abstract for a metric space x we study the space d x of bounded functions on x whose pointwise lipschitz constant is uniformly bounded d x is compared with the space lip x of bounded lipschitz functions on x in terms of different properties regarding the geometry of x we also obtain a banach stone theorem in this context in the case of a metric measure space we also compare d x with the newtonian sobolev space n 1 x in particular if x supports a doubling measure and satisfies a local poincare inequality we obtain that d x n 1 x

moyal planes are spectral triples axioms for nonunital spectral triples extending those introduced in the unital case by connes are proposed as a guide and for the sake of their importance in noncommutative quantum field theory the spaces r2n endowed with moyal products are intensively investigated some physical applications such as the construction of noncommutative wick monomials and the computation of the connes lott functional action are given for these noncommutative hyperplanes

nonunital spectral triples and metric completeness in unbounded kk theory abstract we consider the general properties of bounded approximate units in non self adjoint operator algebras such algebras arise naturally from the differential structure of spectral triples and unbounded kasparov modules our results allow us to present a unified approach to characterising completeness of spectral metric spaces existence of connections on modules self adjointness and regularity of induced operators on tensor product c modules and the lifting of kasparov products to the unbounded category in particular we prove novel existence results for quasicentral approximate units in non self adjoint operator algebras allowing us to strengthen kasparov s technical theorem and extend it to this realm finally we show that given any two composable kk classes we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the kasparov product

set valued choquet integrals revisited this paper performs a further investigation on the set valued choquet integral given by jang and kwon fuzzy sets and systems 112 2000 233 some incorrect properties are revised it is pointed out that the definition for convergence of sequence of sets or set valued functions is unreasonable by using the usual kuratowski s convergence some new convergence theorems for the set valued choquet integral are shown meanwhile some other properties are obtained

sums of regular self adjoint operators in hilbert c modules abstract we introduce a notion of weak anticommutativity for a pair s t of self adjoint regular operators in a hilbert c module e we prove that the sum s t of such pairs is self adjoint and regular on the intersection of their domains a similar result then holds for the sum s 2 t 2 of the squares we show that our definition is closely related to the connes skandalis positivity criterion in kk theory as such we weaken a sufficient condition of kucerovsky for representing the kasparov product our proofs indicate that our conditions are close to optimal

on the representation of choquet integrals of set valued functions and null sets in this paper we consider the definition of set valued choquet integrals we obtain some properties and the representation of set valued choquet integrals and also we discuss some fundamental relations between null sets and set valued choquet integrals

fuzzy measures defined by fuzzy integral and their absolute continuity abstract given a measurable space x f a fuzzy measure on x f and a nonnegative functionfonxthat is measurable with respect to f we can define a new set function on x f by the fuzzy integral it is known that is a lower semicontinuous fuzzy measure on x f and moreover if is finite then is a finite fuzzy measure as well in this paper we generalize in several different ways the concept of absolute continuity of set functions as defined in classical measure theory in addition we investigate the relationship among these generalizations by using the structural characteristics of set functions such as null additivity and autocontinuity and determine which types of absolute continuity of fuzzy measures are possessed by the fuzzy measure or the lower semicontinuous fuzzy measure obtained by the fuzzy integral

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

a gould type integral of fuzzy functions abstract in this paper a new type of integral of a fuzzy valued function relative to a non negative set function is proposed as a new useful tool in modeling uncertainties various properties of this new type of integral are established concerning the behavior with respect to the fuzzy function or to the set function several examples linking our integral with birkhoff and lebesgue integrals are provided also problems related to atomicity are addressed

noncommutative geometry for symmetric non self adjoint operators abstract we introduce the notion of a pre spectral triple which is a generalisation of a spectral triple a h d where d is no longer required to be self adjoint but closed and symmetric despite having weaker assumptions pre spectral triples allow us to introduce noncompact noncommutative geometry with boundary in particular we derive the hochschild character theorem in this setting we give a detailed study of dirac operators with dirichlet boundary conditions on domains in r d d 2

pseudo atoms and darboux property for set multifunctions in this paper we introduce and study the concepts of pseudo atom and darboux property for set multifunctions we establish relationships between the darboux property and non atomicity in the multivalued case we also prove that the range of a multimeasure having the darboux property is a convex set

a lusin type theorem for regular monotone uniformly autocontinuous set multifunctions in this paper we continue our study concerning monotone uniformly autocontinuous regular set multifunctions as an application a lusin type theorem under a suitable type of measurability is obtained and several applications for finitely purely atomic multisubmeasures are proved

on a technical theorem of kasparov abstract a short proof of the technical theorem underlying g g kasparov s kk theory is presented the theorem is concerned with the separation of two orthogonal subalgebras of an outer multiplier algebra m d d by an element in a relative commutant the proof given here is a simple application of the notion of quasicentral approximate unit

the general pettis sugeno integral of vector multifunctions relative to a vector fuzzy multimeasure abstract in this paper we define and study the general pettis sugeno integral of a vector multifunction relative to a vector fuzzy multimeasure both taking values in a locally convex space x ordered by a closed convex pointed cone x with nonempty interior several classic properties of this integral and some estimation and comparative results with the pettis lebesgue integral are established

corrigendum to a local global principle for regular operators in hilbert c modules j funct anal 262 10 2012 4540 4569 abstract we clarify that one of the main results from a local global principle for regular operators in hilbert c modules is already contained in an earlier paper by francois pierrot moreover in the paper by pierrot this result is proved in a stronger version that was only stated as a conjecture in our local global principle paper we also present a short proof based on pierrot s argument of the core issue in both of the two papers namely a result on the separation of a point from a closed submodule

preconditioned iterative methods for the nine point approximation to the convection diffusion equation iterative methods preconditioned by incomplete factorizations and sparse approximate inverses are considered for solving linear systems arising from fourth order finite difference schemes for convection diffusion problems simple recurrences for implementing the ilu 0 factorization of the nine point scheme are derived different sparsity patterns are considered for computing approximate inverses for the coefficient matrix and the quality of the preconditioner is studied in terms of plots of the field of values of the preconditioned matrices in terms of algebraic properties of the preconditioned matrices our experimental results show that incomplete factorizations give a preconditioner of better quality than approximate inverses comparison of the convergence rates of gmres applied to the preconditioned linear systems is done with respect to the field of values ritz and harmonic ritz values of the preconditioned matrices numerical results show that the gmres residual norm decreases rapidly when the difference between the ritz and harmonic ritz values becomes small we also describe the results of experiments when some well known krylov subspace methods are used to solve the linear system arising from the compact fourth order discretizations

spectral flow and the unbounded kasparov product abstract we present a fairly general construction of unbounded representatives for the interior kasparov product as a main tool we develop a theory of c 1 connections on operator modules we do not require any smoothness assumptions our unitality assumptions are minimal furthermore we use work of kucerovsky and our recent local global principle for regular operators in hilbert c modules as an application we show that the spectral flow theorem and more generally the index theory of dirac schrodinger operators can be nicely explained in terms of the interior kasparov product

algebra extensions and nonsingularity this paper is concerned with a notion of nonsingularity for noncommutative algebras which arises naturally in connection with cyclic homology let us consider associative unital algebras over the complex numbers we call an algebra a quasi free when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism a r i where i is a nilpotent ideal in r can be lifted to a homomorphism a r if we restrict to the category of finitely generated commutative algebras then this lifting property characterizes smooth algebras the ones corresponding to nonsingular affine varieties in this way quasi free algebras appear as noncommutative analogues of smooth algebras stretching the analogy we can even regard quasi free algebras as analogues of manifolds one of the aims of this paper is to develop the analogy further by showing that quasi free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds to give an example let us consider the analogue of an embedding an extension a r i where a and r are quasi free algebras playing the role of the submanifold and ambient manifold respectively in the manifold situation i i2 is the module of linear functions on the nor2 mal bundle and the symmetric algebra sa iii is the algebra of polynomial functions now in passing from commutative to noncommutative algebras the symmetric algebra of a module is replaced by the tensor algebra of a bimodule

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

fuzzy integrals of fuzzy valued functions abstract this paper based on the fuzzy measures and integrals given by sugeno first defines fuzzy integrals of set valued functions and then the fuzzy integral of fuzzy valued functions is obtained and discussed next it defines the fuzzy integrals of fuzzy mappings then pays much effort to the fuzzy integral of interval valued functions the properties and convergence theorems are obtained all these are generalizations of earlier results

on the regularities of fuzzy set multifunctions with applications in variation extensions and fuzzy set valued integrability problems in this paper we continue our studies 6 7 concerning regularity in this sense we obtain extension results by preserving regularity and we establish basic properties and continuity properties for the variation of fuzzy set multifunctions this enables us to get relationships among different types of regularity and also to point out regularity properties of the fuzzy set valued integral 20

fuzzy integrals of set valued mappings and fuzzy mappings abstract the paper based on sugeno 1974 first defines the concept of fuzzy integrals of set valued mappings by the similar way of aumann 1965 then it is discussed some results similar to aumann integral are obtained these include convexity closedness fatou s lemmas and the lebesgue convergence theorem etc at last the fuzzy integral of fuzzy mappings is defined the extended results corresponding with fuzzy integrals of set valued mappings are showed

a serre swan theorem for bundles of bounded geometry abstract the serre swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions this theorem is here generalized to manifolds of bounded geometry in this context it states that the category of hilbert bundles of bounded geometry is equivalent to the category of operator modules over the operator algebra of continuously differentiable functions which vanish at infinity operator modules are generalizations of hilbert c modules where the category of c algebras has been replaced by a more flexible category of involutive algebras of bounded operators the operator algebras operator modules play an important role in the study of the unbounded kasparov product

a set valued egoroff type theorem in this paper an egoroff type theorem for a fuzzy multimeasure defined on a s algebra of subsets of an abstract space and taking values in the family of all nonvoid closed subsets of a real normed space is established some necessary and or sufficient conditions under which egoroff s theorem is valid are also presented

unbounded bivariant k theory and correspondences in noncommutative geometry by adapting the algebraic notion of universal connection to the setting of unbounded kk cycles we show that the kasparov product of such cycles can be defined directly by an algebraic formula in order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable c module the theory of operator spaces provides the required tools finally the above mentioned kk cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples

on set valued fuzzy measures so far there have been many discussions on sugeno s fuzzy measures and fuzzy integrals but most of them are concentrated on single valued functions motivated by aumann s set valued integral j math anal appl 12 1965 1 we have introduced the fuzzy integral of set valued functions fuzzy sets syst 56 1993 237 fuzzy sets syst 76 1995 365 fuzzy sets syst 78 1996 341 ph d dissertation harbin institute of technology 1998 it is a well behaved extension of fuzzy integrals of single valued functions it is well known that arstein s set valued measure trans am math soc 165 1972 103 is an important branch in set valued analysis set valued analysis birkhauser berlin 1990 or theory of correspondences theory of correspondences wiley new york 1984 compared to it the present paper will try to establish the basic idea of set valued fuzzy measures which are monotone set valued set functions it is also a natural generalization of single valued fuzzy measures as well as an extension of set valued measures in the case of one dimension these works include the concept of set valued fuzzy measures set valued fuzzy measures defined by set valued fuzzy integrals and set valued pseudo additive measures it can be viewed as a continuation of previous work fuzzy sets syst 56 1993 237 fuzzy sets syst 76 1995 365 fuzzy sets syst 78 1996 341

non atomicity and the darboux property for fuzzy and non fuzzy borel baire multivalued set functions in this paper we establish results concerning non atomicity and the darboux property for multisubmeasures fuzzy multivalued set functions defined on the baire respectively borel d ring b 0 respectively b of a locally compact hausdorff space

on set valued fuzzy integrals abstract in the paper we extend fuzzy integrals of point valued functions which were defined by ralescu and adams to set valued functions some properties similar to auman integrals are shown and these include convexity monotonicity fatou s lemma lebesgue convergence theorem etc

generalized fuzzy integrals of set valued functions in this paper based on wu s generalized fuzzy integrals of point valued function c wu et al fuzzy sets and systems 70 1995 75 a theory of generalized fuzzy integrals of fuzzy valued functions will be investigated it includes definitions properties and various kinds of convergence theorems these are extensions of results in c wu et al fuzzy sets and systems 70 1995 75 d zhang and z wang fuzzy sets and systems 54 1993 63

iterative and parallel performance of high order compact systems representative transport pde problems in one two and three dimensions are approximated by a class of high order compact hoc difference schemes and their iterative and parallel performance are studied the eigenvalues and condition numbers of the hoc schemes are analyzed and the performance of standard krylov space methods is compared for hoc central differencing and standard first order upwinding schemes finally cpu times mflop rates and speedup curves are presented for fixed problem size cases and scaled problem size cases

poincare duality for cuntz pimsner algebras abstract we present a new approach to poincare duality for cuntz pimsner algebras we provide sufficient conditions under which poincare self duality for the coefficient algebra of a hilbert bimodule lifts to poincare self duality for the associated cuntz pimsner algebra with these conditions in hand we can constructively produce fundamental classes in k theory for a wide range of examples we can also produce k homology fundamental classes for the important examples of cuntz krieger algebras following kaminker putnam and crossed products of manifolds by isometries and their non commutative analogues

set valued lusin type theorem for null null additive set multifunctions in a previous paper we presented a set valued egoroff type theorem for monotone set multifunctions in this paper a new version of this set valued egoroff type theorem is obtained it enables a set valued lusin type theorem for null null additive monotone continuous set multifunctions to be established several applications of this lusin type theorem are given pointing out some relationships among m measurability a measurability and totally measurability of real functions we also compare our new lusin type theorem with lusin s theorem previously obtained by the second author

field of values analysis of preconditioned iterative methods for nonsymmetric elliptic problems the convergence rate of krylov subspace methods for the solution of nonsymmetric systems of linear equations such as gmres or fom is studied bounds on the convergence rate are presented which are based on the smallest real part of the field of values of the coefficient matrix and of its inverse estimates for these quantities are available during the iteration from the underlying arnoldi process it is shown how these bounds can be used to study the convergence properties in particular the dependence on the mesh size and on the size of the skew symmetric part for preconditioners for finite element discretizations of nonsymmetric elliptic boundary value problems this is illustrated for the hierarchical basis and multilevel preconditioners which constitute popular preconditioning strategies for such problems

autocontinuity of set valued fuzzy measures and applications the main object of this paper is to further the study of set valued fuzzy measures introduced by gavrilut in 2009 some asymptotic structural characteristics of set valued fuzzy measures such as null additivity and autocontinuity are investigated in terms of applications convergence theorems for sequences of measurable functions are established

non atomicity for fuzzy and non fuzzy multivalued set functions in this paper we extend the concept of set valued fuzzy measures introduced and studied by guo and zhang 2004 on set valued fuzzy measures information sciences 160 2004 13 25 to the class of multivalued set functions taking values in the family of non empty closed subsets of a real normed space also we introduce the notions of atom and non atomicity of a multivalued set function and present some of their properties we also establish an extension result by preserving non atomicity for a fuzzy multisubmeasure multimeasure

strong integral of multifunctions relative to a fuzzy measure abstract in this paper we define and study an integral for measurable multifunctions taking values in the family of all nonvoid subsets of 0 with respect to a fuzzy measure unlike various ways of defining such an integral e g aumann way using different selections dunford way riemann way we use the sugeno method directly applied to multifunctions the originality of our definition is that the value of the integral is a real scalar and not usually a set some classical integral properties a convergence theorem and relationships with other integrals are presented