height pairings on shimura curves and p adic uniformization in a recent paper 16 of one of us it was shown that there is a close connection between the value of the height pairing of certain arithmetic 0 cycles on shimura curves and the values at the center of their symmetry of the derivatives of certain metaplectic eisenstein series of genus 2 on the one hand the height pairing can be written as a sum of local height pairings for example if the 0 cycles have disjoint support on the generic fiber then their height pairing is a sum of an archimedean contribution and a contribution from each of the finitely many finite primes p for which the cycles meet in the fiber at p on the other hand it turns out that the non singular part of the fourier expansion of the central derivative of the metaplectic eisenstein series also has a decomposition into a sum of contributions indexed by the places of q then one would like to compare the height pairing and the fourier coefficients by proving an identity of local contributions place by place in loc cit the identity for the archimedean place was proved and it was shown that the identity at a non archimedean place of good reduction is a consequence of results of gross and keating 10 for the algebraicgeometric side and of kitaoka 12 for the analytic side it then remains to consider the finite primes p where the shimura curve has bad reduction these are of three sorts i the primes p at which the quaternion algebra defining the shimura curve is split but which divide the level ii the primes p at which the quaternion algebra remains division but no level structure is imposed and iii the primes p at which the quaternion algebra remains division and where a higher level structure is imposed at p so far little is known concerning cases i and iii

a mass formula for unimodular lattices with no roots we derive a mass formula for n dimensional unimodular lattices having any prescribed root system we use katsurada s formula for the fourier coefficients of siegel eisenstein series to compute these masses for all root systems of even unimodular 32 dimensional lattices and odd unimodular lattices of dimension n 30 in particular we find the mass of even unimodular 32 dimensional lattices with no roots and the mass of odd unimodular lattices with no roots in dimension n 30 verifying bacher and venkov s enumerations in dimensions 27 and 28 we also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by the minkowski siegel mass constants

maass relations in higher genus for an arbitrary even genus 2n we show that the subspace of siegel cusp forms of degree 2n generated by ikeda lifts of elliptic cusp forms can be characterized by certain linear relations among fourier coefficients this generalizes the classical maass relations in degree two to higher degrees

a new design of binary mds array codes with asymptotically weak optimal repair binary maximum distance separable mds array codes are a special class of erasure codes for distributed storage that not only provides fault tolerance with minimum storage redundancy but also achieves low computational complexity they are constructed by encoding k information columns into r parity columns in which each element in a column is a bit such that any k out of the k r columns suffice to recover all information bits in addition to providing fault tolerance it is critical to improve repair performance in practical applications specifically if a single column fails then our goal is to minimize the repair bandwidth by downloading the least amount of bits from d healthy columns where k leq d leq k r 1 if one column of an mds code is failed it is known that we need to download at least 1 d k 1 fraction of the data stored in each of the d healthy columns if this lower bound is achieved for the repair of the failure column from accessing arbitrary d healthy columns we say that the mds code has optimal repair however if such lower bound is only achieved by d specific healthy columns then we say the mds code has weak optimal repair in this paper we propose two explicit constructions of binary mds array codes with more parity columns i e r geq 3 that achieve asymptotically weak optimal repair where k 1 leq d leq k lfloor r 1 2 rfloor and asymptotic means that the repair bandwidth achieves the minimum value asymptotically in d codes in the first construction have odd number of parity columns and asymptotically weak optimal repair for anyone information failure while codes in the second construction have even number of parity columns and asymptotically weak optimal repair for any one column failure

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

point set lattice theoretic topology this essay attempts to survey in a coherent way certain aspects of point set lattice theoretic or poslat topology by which we mean fuzzy topology grounded in notions of sets functions powersets and powerset operators the latter two being lattice theoretic in nature and examined from a lattice theoretic point of view using methods of category theory connections with other related issues and developments are frequently given

a new fuzzy compactness defined by fuzzy nets compactness is one of the most important notions in topology and is thoroughly investigated in general topology but it seems that the research about this notion in the area of fuzzy topology is not so effective as in general topology in fact it would seem that there was not even a good definition of compactness in this area until recently indeed soon after the famous article written by zadeh 8 was published chang 1 introduced the notion of fuzzy topological spaces and at the same time a certain fuzzy compactness was given from then on a series of different notions of fuzzy compactness have been discussed 2 3 71 and in 5 they were compared in detail but it seems unsatisfactory that all these fuzzy compactness ideas are defined for the whole fuzzy topological space but not for an arbitrary fuzzy subset therefore they are far from the crisp compactness for example it is impossible to discuss the hereditary property with respect to closed subsets of fuzzy compactness in 1977 pu and liu introduced the notion of fuzzy nets and a new concept of so called q neighborhood was given which could reflect the features of neighborhood structure in fuzzy topological spaces and by this new neighborhood structure the moore smith convergence theory was established spendidly 6 it furnishes us a basic tool to investigate the fuzzy compactness in a more general way to investigate the compactness of arbitrary fuzzy subsets in this paper a new fuzzy compactness defined by fuzzy nets we shall call it n compactness is given but we introduce the notion of r neighborhood see section 2 instead of qneighborhood to establish the moore smith convergence theory for it is perhaps more convennient to use n compactness possesses the following properties 1 n compactness is defined for arbitrary fuzzy subsets 2 for n compactness the hereditary property with respect to closed subsets holds 3 n compactness is a good extension 5 4 the continuous images of ncompact sets are n compact 5 for n compactness the tychonoff product theorem holds 6 if an n compact fuzzy topological space is t2 6 then it will satisfy a stronger separation axiom which will be called t4 in this paper 1 0022 247x 83 3 00

on newforms for kohnen plus spaces in this article we investigate the plus space of level n where 4 1n is a square free not necessarily odd integer this is a generalization of kohnen s work we define a hecke isomorphism wp k of mk 1 2 4m onto m k 1 2 8m for any odd positive integer m the methods of the proof of the newform theory are this isomorphism waldspurger s theorem and the dimension identity

special cycles on unitary shimura varieties i unramified local theory the supersingular locus in the fiber at p of a shimura variety attached to a unitary similitude group gu 1 n 1 over is uniformized by a formal scheme mathcal n in the case when p is an inert prime we define special cycles mathcal z bold x in mathcal n associated to collections bold x of m special homomorphisms with fundamental matrix t hermm ok when m n and t is nonsingular we show that the cycle mathcal z bold x is either empty or is a union of components of the ekedahl oort stratification and we give a necessary and sufficient condition in terms of t for mathcal z bold x to be irreducible when mathcal z bold x is zero dimensional in which case it reduces to a single point we determine the length of the corresponding local ring by using a variant of the theory of quasi canonical liftings we show that this length coincides with the derivative of a representation density for hermitian forms

a new approach to fuzzy topological spaces and various fuzzy topological notions defined by nets n fuzzy spaces introduced by the authors are further investigated various fuzzy topological motions such as fuzzy local compactness functions with closed fuzzy graph fuzzy subcontinuity inverse fuzzy subcontinuity etc are studied in the light of n i these notions are found to be more coherent in n i than in the chang s category

fourth order q difference equation for the first associated of the q classical orthogonal polynomials abstract we derive the fourth order q difference equation satisfied by the first associated of the q classical orthogonal polynomials the coefficients of this equation are given in terms of the polynomials and which appear in the q pearson difference equation dq defining the weight of the q classical orthogonal polynomials inside the q hahn tableau

on the lifting of elliptic cusp forms to siegel cusp forms of degree 2n i 1 cf 8 in this paper we will prove that their conjecture is true moreover we obtain a simple fourier coefficient formula for f z ibukiyama 23 also made a similar conjecture which was formulated in terms of the koecher maass series which we will not consider here let us describe our results throughout this paper we fix positive integers k and n such that k n mod 2 and put e 1 k 1 n for any positive rational number n we denote the absolute value of the discriminant of q vn q by dn the primitive dirichlet character corresponding to q v n q is denoted by xn let fn be the positive rational number such that n nf2nlet b e m2n q be a positive definite half integral symmetric matrix put db det 2b db d db fb f db and xb x db note that edb 0 1 mod 4 and that fb is an integer for each prime p let p qp cx be the unique additive character such that bp x exp 27ir ltx for x e z p 1 recall that the siegel series for b is defined by

hypergeometric type q difference equations rodrigues type representation for the second kind solution a rodrigues type representation for the second kind solution of a second order q difference equation of hypergeometric type is given this representation contains some q extensions of integrals related with relevant special functions for these integrals a general recurrence relation which only involves the coefficients of the q difference equation is also presented

an explicit formula for siegel series combining induction formulas for local densities with a functional equation for the siegel series we give an explicit formula for the siegel series by this formula we also give an explicit formula for the fourier coefficients of the siegel eisenstein series

fuzzy closed and fuzzy perfect mappings abstract it is shown that a fuzzy continuous map f is fuzzy perfect iff f i z is fuzzy closed for every fuzzy topological space z

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

fourth order differential equations for numerator polynomials the author gives explicitly the fourth order differential equation satisfied by the numerator polynomials associated polynomials of the classical orthogonal polynomials the coefficients of the differential equation are at most a quadratic combination of the polynomials sigma and tau and their derivatives defined via the relation sigma rho te this differential equation is therefore valid for the jacobi laguerre hermite and bessel associated polynomials

explicit formulas for the twisted koecher maas series of the duke imamoglu ikeda lift and their applications we give an explicit formula for the twisted koecher maas series of the duke imamoglu ikeda lift as an application we prove a certain algebraicity result for the values of twisted rankin selberg series at integers of half integral weight modular forms which was not treated by shimura j math soc japan 33 649 672 1981

minimal recurrence relations for connection coefficients between classical orthogonal polynomials discrete case abstract we present a simple approach in order to compute recursively the connection coefficients between two families of classical discrete orthogonal polynomials charlier meixner kravchuk hahn i e the coefficients cm n in the expression p n x n m 0 c m n q m x where pn x and qm x belong to the aforementioned class of polynomials this is scv2 done by adapting a general and systematic algorithm recently developed by the authors to the discrete classical situation moreover extensions of this method allow to give new addition formulae and to estimate cm n asymptotics in limit relations between some families

fourier coefficients of siegel eisenstein series of odd genus abstract we give a new expression of the t th fourier coefficients of the siegel eisenstein series of odd genus unlike even genus case only modified divisor sums and no generalized class numbers enter into our formula

on the local density formula and the gross keating invariant with an appendix the local density of a binary quadratic form by t ikeda and h katsurada ikeda and katsurada have developed the theory of the gross keating invariant of a quadratic form in their recent papers ikeda and katsurada am j math 140 1521 1565 2018 explicit formula of the siegel series of a quadratic form over a non archimedean local field 2017 arxiv 1602 06617 in particular they prove that the local factors of the fourier coefficients of the siegel eisenstein series are completely determined by the gross keating invariants with extra datums called the extended gk datums in ikeda and katsurada explicit formula of the siegel series of a quadratic form over a non archimedean local field 2017 arxiv 1602 06617 on the other hand such a local factor is a special case of the local density for a pair of two quadratic forms thus we propose a general question if the local density can be expressed in terms of a certain series of extended gk datums in this paper we prove that the answer to this question is affirmative for the local density of a single quadratic form defined over an unramified finite extension of mathbb z 2 and over a finite extension of mathbb z p with p odd in the appendix ikeda and katsurada compute the local density formula of a single binary quadratic form defined over any finite extension of mathbb z 2

connectedness in fuzzy topological spaces abstract we introduce four notions of connectedness for an arbitrary fuzzy set and discuss all the implications that exist between them using these notions various equivalence relations are defined on the set of fuzzy points components and quasi components are formed as the union of certain equivalence classes a fuzzy space is shown to be a dispoint union of its components in all the four cases and non overlapping union of its quasi components

fuzzy subcontinuity inverse fuzzy subcontinuity and a new category of fuzzy topological spaces the notions of subcontinuity and inverse subcontinuity are extended to the fuzzy setting adopting fuzzy net theoretic approach various properties of fuzzy subcontinuous and inversely fuzzy subcontinuous mappings are studied a category of fuzzy topological spaces is constructed canonical example is constructed by using fuzzy translation of fuzzy numbers introduced by rodabaugh to show that fuzzy subcontinuity does not imply kfuzzy continuity in the process n compactness is characterized and fuzzy unit interval is shown to be n compact also a direct proof is provided showing that fuzzy addition of fuzzy numbers is fuzzy continuous

various fuzzy topological notions and fuzzy almost continuity abstract we extend the notion of almost continuity in the sense of husain 1966 to the fuzzy setting we obtain a pointwise characterization of the notion by dual point and fuzzy net also we formulate conditions under which the two notions of fuzzy almost continuity the one defined in this paper and the one defined by ajmal and azad 1990 turn out to be equivalent the main idea behind this work is to show that using the localized approach developed in earlier papers the study of fuzzy almost continuity can be carried out as successfully as that of its counterpart in general topology in the process we exhibit the compatibility of various existing fuzzy topological notions such as fuzzy regularity c i connectedness i 3 4 c i local connectedness i 3 4 fuzzy t 1 axiom fuzzy t 2 axiom etc

a new approach to fuzzy topological spaces and fuzzy perfect mappings abstract a net theoretic approach is provided for defining a fuzzy topological space fuzzy topological spaces so defined form a category the main features and advantages of this category which is denoted by n here are discussed the notion of a perfect mapping is introduced and studied in the fuzzy setting it is found that in n a fts is n compact iff any mapping x x where x is any singleton is fuzzy perfect a fuzzy continuous mapping x y from an n compact fts x to a hausdorff fts y is fuzzy perfect in n however these results do not hold in c the chang s category in general it is found that the divergences in the study of fuzzy perfect mappings disappear if the study is carried out in n instead of c

bivariate second order linear partial differential equations and orthogonal polynomial solutions in this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second order linear partial differential equations which are admissible potentially self adjoint and of hypergeometric type general formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation using vector matrix notation moreover rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given as illustration these results are applied to a two parameter monic appell polynomials finally the non monic case is briefly discussed 2011 elsevier inc

category n and functions with closed fuzzy graph abstract a net theoretic approach for defining a fuzzy topological space is provided the fuzzy topological spaces so obtained form a category which we denote by n in this paper it is found that the limitations exhibited by the chang category particularly with respect to projections closure of a fuzzy set compactness etc no longer exist in n the notion of closed fuzzy graph is investigated the divergences shown by this notion are immediately put to rest if the study is carried out in n instead of c the chang category

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

connectedness and local connectedness in fuzzy topological spaces and heyting algebra valued sets abstract in this paper we introduce four notions of local connectedness in fuzzy topological spaces the question of these notions to be good extensions is discussed and their preservation under fuzzy quotients fuzzy sums and fuzzy products is investigated moreover we introduce the notion of quasi local connectedness which is a peculiarity of the fuzzy setting

an explicit arithmetic formula for the fourier coefficients of siegel eisenstein series of degree two and square free odd levels we give an explicit arithmetic formula for the fourier coefficients of siegel eisenstein series of degree two and square free odd levels

a categorical accommodation of various notions of fuzzy topology different definitions of fuzzy topology have been stated and developed in the literature but have not been satisfactorily related to each other and to ordinary topology to remedy this a new fuzzy topological category fuzz is defined it is a significant generalization of previous definitions it is the simplest setting yet constructed into which ordinary topology and these previous definitions may be placed by identifying each with a different subcategory of fuzz it therefore allows us to conveniently analyze the relative merits of previous definitions give coherence to known results and indicate appropriate directions for future development and it generates and may be viewed as both including and included in categories of fuzzy topological spaces which exhibit higher order fuzziness

extensions of some results of p humbert on bezout s identity for classical orthogonal polynomials in this paper the bezout s identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type differential equations relation with the starting family as well as recurrence relations and explicit representations are given for the bezout s pair extensions to classical orthogonal polynomials of a discrete variable and their q analogues are also presented applications of these results for the representation of the second kind functions are given