harmonic analysis on nilpotent groups and singular integrals ii singular kernels supported on submanifolds on etudie des operateurs de convolution singuliers sur des groupes de lie nilpotents generaux et plus specifiquement des operateurs dont les noyaux ont pour supports des varietes de faible dimension et qui peuvent aussi contenir des facteurs polynomiaux exponentiels oscillatoires

completely homogeneous fuzzy closure spaces and lattice of fuzzy closure operators abstract we determine completely homogeneous fuzzy closure spaces and prove that the set l x of all fuzzy closure operators on a fixed set x forms a complete lattice some other properties of the lattice including the complementation problem are discussed

on the local time of random processes in random scenery random walks in random scenery are processes defined by z n sum k 1 n xi x 1 x k where basically x k k ge 1 and xi y y in mathbb z are two independent sequences of i i d random variables we assume here that x 1 is zz valued centered and with finite moments of all orders we also assume that xi 0 is zz valued centered and square integrable in this case h kesten and f spitzer proved that n 3 4 z nt t ge 0 converges in distribution as n to infty toward some self similar process delta t t ge 0 called brownian motion in random scenery in a previous paper we established that mathbb p z n 0 behaves asymptotically like a constant times n 3 4 as n to infty we extend here this local limit theorem we give a precise asymptotic result for the probability for z to return to zero simultaneously at several times as a byproduct of our computations we show that delta admits a bi continuous version of its local time process which is locally holder continuous of order 1 4 delta and 1 6 delta respectively in the time and space variables for any delta 0 in particular this gives a new proof of the fact previously obtained by khoshnevisan that the level sets of delta have hausdorff dimension a s equal to 1 4 we also get the convergence of every moment of the normalized local time of z toward its continuous counterpart

fuzzy closure spaces abstract fuzzy topological spaces do not constitute a natural boundary for the validity of theorems but many results can be extended to what are called fuzzy closure spaces or fcs s for short the notions of a subspace a sum and a product are extended to fcs s the hereditary additivity and productivity behaviour of compactness in fcs s is investigated and some weak forms of compactness and fuzzy continuous functions in fcs s are introduced the interaction between fuzzy proximity spaces and fcs s is investigated a necessary background is included for completeness

fuzzy nets and their application fuzzy net theory is enriched by the introduction of the fuzzy net of fuzzy sets the limsup liminf and limit of a fuzzy net of fuzzy sets are defined and their various properties are discussed alternative characterizations based on the notion of a fuzzy net are provided for several fuzzy topological concepts including open and closed fuzzy sets fuzzy continuity maps with closed fuzzy graph open fuzzy mapping etc thus the net theoretic approach is shown to be a promising tool in fuzzy topology

lp estimates for singular integrals and maximal operators associated with flat curves on the heisenberg group the maximal function along a curve t gamma t t gamma t on the heisenberg group is discussed the l sp p boundedness of this operator is shown under the doubling condition of gamma sp prime for convex gamma in mathbb r sp this condition also applies to the singular integrals when gamma is extended as an even or odd function the proof is based on angular littlewood paley decompositions in the heisenberg group

on t 0 and t 1 fuzzy closure spaces abstract in this paper we introduce subspace of a fuzzy closure space sum of a family of pairwise disjoint fuzzy closure spaces and product of a family of fuzzy closure spaces for the fuzzy closure spaces defined in srivastava et al 1994 we also introduce the notion of a t1 fuzzy closure space we have studied here t0 introduced earlier in srivastava et al 1994 and t1 fuzzy closure spaces in detail several results have been proved which establish the appropriateness of the definitions in particular we observe that t0 and t1 satisfy the hereditary productive and projective properties and in addition both are good extensions of the corresponding concepts in a closure space

generalization of l closure spaces abstract this paper introduces the concepts of l closure spaces and the convergence in l closure spaces in the variable basis setting of such closure spaces continuous generalized order homomorphisms are characterized by the convergence theory of nets filters and ideals furthermore in the fixed basis setting of such closure spaces some constructions on l closure spaces such as the subspace of an l cs the sum l cs the product of l cs and the induced l cs of a crisp closure space are investigated we obtain an important result that the category l closure is a topological category over set w r t the forgetful functor from l closure to set we obtain another important result that there is an adjunction between the category of crisp closure spaces and the full subcategory of those l closure spaces which are induced by some crisp closure space

separation axioms of fuzzy initial convergence spaces abstract we define a fuzzy t 0 t 1 hausdorff regular and symmetric convergence space and we investigate some properties of fuzzy initial convergence structure with respect to separation axioms

topological modification of a fuzzy closure space abstract fuzzy topological spaces do not constitute a natural boundary for the validity of theorems and many results can be extended to what are called fuzzy closure spaces fuzzy closure spaces have been introduced in 6 as a generalization of closure spaces 1 in the present paper we introduce the concept of topological modification of a fuzzy closure operator we prove that the topological modification of a fuzzy closure operator u on x is the finest topological fuzzy closure operator on x coarser than u the interaction between the concepts of a subspace and the topological modification of a fuzzy closure operator is also examined

pseudo closure operators in fuzzy topological spaces abstract the concept of a pseudo closure operator s of an fts x has been introduced in 6 in this paper we give an axiomatic characterization of pseudo closure operators also we give a necessary and sufficient condition on an fts x so that is the unique fuzzy topology which induces s at the end of the paper we give a characterization of fts s of structure a 6 in terms of pseudo closure operators

fuzzy cores and their extensions abstract we introduce core theory simultaneously for general topology and for fuzzy topology a fuzzy core on a set x is an operator from a subfamily of i x into i x subject to certain conditions we identify all the extension of a fuzzy core fuzzy cech closure operators and to fuzzy kuratowski operators the main purpose of this theory is to characterize as we do here some properties of functions as being equivalent to their continuity with respect to certain fuzzy topologies on their domains and codomains we nominate three existing theories as special instances of core theory one is from general topology and is strongly related to applications the theory of the scott topology on a partially set the other two theories are from fuzzy topology klein s theory of closure operators and the mashhour et al theory of pseudo closure operators

large deviations for local times and intersection local times of fractional brownian motions and riemann liouville processes in this paper we prove exact forms of large deviations for local times and intersection local times of fractional brownian motions and riemann liouville processes we also show that a fractional brownian motion and the related riemann liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times as a consequence of our large deviation estimates we derive laws of iterated logarithm for the corresponding local times the key points of our methods 1 logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional brownian motion 2 logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of riemann liouville processes 3 comparison of local and intersection local times based on embedding of a part of a fractional brownian motion into the reproducing kernel hilbert space of the riemann liouville process

relative remote neighborhood family and the characterizations of ultra fuzzy compactness abstract in this paper the concept of relative remote neighborhood family in l fuzzy topological spaces is introduced by which some characterizations of ultra fuzzy compactness are obtained using this concept we have proved alexander subbase lemma and tnxohob product theorem about ultra fuzzy compactness furthermore the ultra fuzzy paracompactness defined by strong locally finite relative remote neighborhood family is proved to be an l good extension

fuzzy points and local properties of fuzzy topological spaces abstract we prove that three theorems of c k wong on local properties of fuzzy topology are wrong thereby we discuss his notions of fuzzy point c 1 separability and local compactness

on the equivalence of four definitions of compactness of lf subsets in this paper the equivalence of four definitions of fuzzy compactness strong fuzzy compactness n compactness and ultra fuzzy compactness of l fuzzy subsets is discussed systematically a series of good results is obtained

interior and closure operators on texture spaces ii dikranjan giuli closure operators and hutton algebras in this work we discuss interior and closure operators on textures in the sense of dikranjan giuli first we define the category dficl of interior closure spaces and bicontinuous difunctions and show that it is topological over dftex whose objects are textures and morphisms are difunctions the category l closure of l closure spaces and zadeh type powerset operators and the counterparts of the lowen functors have been presented by wu neng zhou in a fixed basis setting we consider the closure operators on a hutton algebra l and in a natural way we define the category hcl of hutton closure spaces taking the morphisms of the opposite category of hutalg the category of hutton algebras fuzzy lattices and the mappings preserving arbitrary meets joins and involution in this case the categories l closure and h the category of hutton spaces and the morphisms in the sense of definition 2 1 can be considered as a subcategory and a full subcategory of hcl respectively using the fact that dficl and hcl o p are equivalent categories we guarantee the existence of products and sums in hcl finally we show that the generalized lowen functor can be also given in a textural framework for 0 1

pseudo closure operators in fuzzy neighbourhood base spaces abstract fuzzy neighbourhood base spaces and dual fuzzy neighbourhood spaces are two special types of fuzzy topological spaces they were introduced by morsi as useful tools for the theory explanation and application of lowen s fuzzy neighbourhood systems the notion of the pseudoclosure operator of a fuzzy topological space was introduced by mashhour et al as a tool in establishing a necessary and sufficient condition for the productivity of fuzzy kuratowski operators pseudo closure operators carry partial information about the fuzzy closure operators of their respective spaces we establish relationships between the pseudo closure operators of fuzzy neighbourhood base spaces their fuzzy closure operators and their level topologies we provide counterexamples to disprove other nonvalid relationships we also study some categorical aspects of the operation of taking the pseudo closure operators of fuzzy topological spaces

l fuzzy interior systems the aim of this work is to introduce the concept of l fuzzy interior systems and l fuzzy interior operators we start by establishing a connection between l fuzzy interior systems and l fuzzy interior operators it is shown that an l fuzzy interior system is precisely the fuzzy system in opposition to the crisp system and an l fuzzy interior operator is a suitable interior operator that has a close relation to an l fuzzy interior system it is also shown that there is a galois correspondence between the category of l fuzzy interior system spaces and that of l fuzzy interior spaces

some important applications of nets of l fuzzy sets in this paper some compactness adherence point and accumulation point in l fuzzy topological space are characterized by means of the nets of l fuzzy subsets

hilbert transforms and maximal functions along variable flat curves we study certain hilbert transforms and maximal functions along variable flat curves in the plane we obtain their l 2 r 2 boundedness by considering the oscillatory singular integrals which arise from an application of a partial fourier transform

a local limit theorem for random walks in random scenery and on randomly oriented lattices random walks in random scenery are processes defined by z n sum k 1 n xi x 1 x k where x k k ge 1 and xi y y in mathbb z are two independent sequences of i i d random variables we assume here that their distributions belong to the normal domain of attraction of stable laws with index alpha in 0 2 and beta in 0 2 respectively these processes were first studied by h kesten and f spitzer who proved the convergence in distribution when alpha neq 1 and as n to infty of n delta z n for some suitable delta 0 depending on alpha and beta here we are interested in the convergence as n to infty of n delta mathbb p z n lfloor n delta x rfloor when x in rr is fixed we also consider the case of random walks on randomly oriented lattices for which we obtain similar results

computing the lattice of all fixpoints of a fuzzy closure operator we present a fast bottom up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees along with the partial order on the set of all fixpoints fuzzy closure operators appear in several areas of fuzzy logic and its applications including formal concept analysis fca that we use as a reference area of application in this paper several problems in fca such as computing all formal concepts from data with graded attributes or computing non redundant bases of all attribute dependencies can be reduced to the problem of computing fixpoints of particular fuzzy closure operators associated with the input data the development of a general algorithm that is applicable in particular to these problems is the ultimate purpose of this paper we present the algorithm its theoretical foundations and experimental evaluation

l2 boundedness of hilbert transforms along variable curves abstract for the l 2 boundedness of the hilbert transforms along variable curves h f x 1 x 2 p v f x 1 t x 2 x 1 t d t t where c 2 r 1 odd or even 0 0 0 convex on 0 if 1 a nagel j vance s wainger and d weinberg got a necessary and sufficient condition on if is a polynomial j m bennett got a sufficient condition on in this paper we shall first give a counter example to show that under the condition of nagel vance wainger weinberg on the l 2 boundedness of h may fail even if c r 1 on the other hand we improve bennett s result by relaxing the condition on and simplifying the proof

l fuzzy basic proximity spaces abstract we use the concept of l quasi coincident in fuzzy set theory to define l fuzzy basic proximity which is a generalization of l fuzzy proximity as defined in 10 we study the following problems characterization of an l fuzzy basic proximity l fuzzy basic proximity neighbourhoods the l fuzzy closure induced by an l fuzzy basic proximity and the l fuzzy basic proximity compatible with an l fuzzy closure operator

regularity axioms in fuzzy topological spaces and fr i proximities abstract the purpose of this paper is to study the regularity axioms for the proximity notion in fuzzy topology we redefine the notion of cech proximity on i x which has been extended to fuzzy sets in 11 and we study the families m x c and m 1 x also we show that every fr i space can be derived from an r i proximity on i x and vice versa

on the character of convergence to brownian local time ii in this paper we consider the sequences of stochastic processes which converge weakly asn to brownian local time these processes are generated by a recurrent random walk with finite variance the main result is the following it is possible to redefine a random walk in such a way that for a wide class of processes the normalized differences between them and brownian local time converge in distribution to some stochastic process we also prove that such differences with probability one have the logarithmic upper bound it is so called strong invariance principles for local times