associative monotonic operations in fuzzy set theory abstract the properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets on fuzzy numbers and on fuzzy probabilistic sets

distributivity equation in the class of 2 uninorms this paper is mainly devoted to solving the functional equation of distributivity between aggregation operators with 2 neutral element our investigations are motivated by the couple of distributive logical connectives and their generalizations used in fuzzy set theory e g triangular norms conorms uninorms nullnorms and implications one of the recent generalizations covering both uninorms and nullnorms are 2 uninorms which form a class of commutative associative and increasing operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements in this work the distributivity of two binary operators from the class of 2 uninorms is considered in particular all possible solutions of the distributivity equation for the three defined subclasses of these operators depending on the position of its zero and neutral elements are characterized

organizing the arithmetic of elliptic curves abstract suppose that e is an elliptic curve defined over a number field k p is a rational prime and k is the maximal z p power extension of k in previous work b mazur k rubin elliptic curves and class field theory in ta tsien li ed proceedings of the international congress of mathematicians icm 2002 vol ii higher education press beijing 2002 pp 185 195 b mazur k rubin pairings in the arithmetic of elliptic curves in j cremona et al eds modular curves and abelian varieties progress in mathematics vol 224 2004 pp 151 163 we discussed the possibility that much of the arithmetic of e over k i e the mordell weil groups and their p adic height pairings the shafarevich tate groups and their cassels pairings over all finite extensions of k in k can be described efficiently in terms of a single skew hermitian matrix with entries drawn from the iwasawa algebra of k k in this paper using work of nekovar j nekovar selmer complexes preprint available at http www math jussieu fr nekovar pu we show that under not too stringent conditions such an organizing matrix does in fact exist we also work out an assortment of numerical instances in which we can describe the organizing matrix explicitly

distributivity equations in the class of semi t operators recently the distributivity equations have been discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms in this study we present the solutions to distributivity between semi t operators previous results related to the distributivity between nullnorms and between semi t operators and semi nullnorms can be obtained as simple corollaries

a formula for the derivative of the p adic l function of the symmetric square of a finite slope modular form let f be a modular form of weight k and nebentypus psi by generalizing a construction of dabrowski and delbourgo we construct a p adic l function interpolating the special values of the l function l s rm sym 2 f otimes xi where xi is a dirichlet character when s k 1 and xi psi 1 this p adic l function vanishes due to the presence of a so called trivial zero we give a formula for the derivative at s k 1 of this p adic l function when the form f is steinberg at p if the weight of f is even the conductor is even and squarefree and the nebentypus is trivial this formula implies a conjecture of benois

fixed point property of direct sums abstract for a uniformly convex space z we show that z direct sums x 1 x n z of banach spaces x 1 x n with r a x i 1 a for some a 0 1 have the fixed point property for nonexpansive mappings as a direct consequence the result holds for all direct sums with being strictly convex the same result is extended to all direct sums x y of spaces x and y with property m whenever 1 the permanence of properties that are sufficient for the fixed point property are obtained for z direct sums and then for direct sums such properties include the properties r x 2 wnus c nj a x 2 ukk and nuc

the distributive equations for idempotent uninorms and nullnorms the distributive equations only involving idempotent uninorms and nullnorms are investigated the following two cases are proved i as for a nullnorm with an idempotent uninorm having distributivity the necessary and sufficient conditions are given ii as for an idempotent uninorm with a nullnorm having distributivity the necessary and sufficient conditions are given also

diagonal cycles and euler systems ii the birch and swinnerton dyer conjecture for hasse weil artin l functions this article establishes new cases of the birch and swinnerton dyer conjecture in analytic rank 0 for elliptic curves over mathbb q viewed over the fields cut out by certain self dual artin representations of dimension at most 4 when the associated l function vanishes to even order ge 2 at its central point two canonical classes in the corresponding selmer group are constructed and shown to be linearly independent assuming the non vanishing of a garrett hida p adic l function at a point lying outside its range of classical interpolation the key tool for both results is the study of certain p adic families of global galois cohomology classes arising from gross kudla schoen diagonal cycles in a tower of triple products of modular curves

uniform convexity of ψ direct sums of banach spaces abstract let x and y be banach spaces and a continuous convex function on the unit interval 0 1 satisfying certain conditions let x y be the direct sum of x and y equipped with the associated norm with we show that x y is uniformly convex if and only if x y are uniformly convex and is strictly convex as a corollary we obtain that the l p q direct sum x p q y 1 q p not p q 1 nor is uniformly convex if and only if x y are where l p q is the lorentz sequence space these results extend the well known fact for the l p sum x p y 1 some other examples are also presented

a note on geometrical properties of banach spaces using ψ direct sums abstract let x be a banach space and a continuous convex function on 0 1 satisfying certain conditions let x x be the direct sum of x in this note we characterize the strict convexity uniform convexity and uniformly non squareness of banach spaces using direct sums which extends the well known characterization of these spaces

variation of iwasawa invariants in hida families let gq gl2 k be an absolutely irreducible modular galois representation over a finite field k of characteristic p assume further that is p ordinary and p distinguished in the sense that the restriction of to a decomposition group at p is reducible and non scalar the hida family h of is the set of all p ordinary p stabilized newforms f with mod p galois representation isomorphic to if is unramified at p then one must also fix an unramified line in and require that the ordinary line of f reduces to this fixed line these newforms are a dense set of points in a certain p adic analytic space of overconvergent eigenforms consisting of an intersecting system of branches i e irreducible components t a indexed by the minimal primes a of a certain hecke algebra to each modular form f h one may associate the iwasawa invariants an f an f alg f and alg f the analytic resp algebraic invariants are the number of zeroes of the p adic l function resp of the characteristic power series of the dual of the selmer group of f while the invariants are the exponents of the powers of p dividing the same objects in this paper we prove the following results on the behavior of these iwasawa invariants as f varies over h

absolute norms on cn abstract in this paper we study the absolute normalized norms on c n we show that the set of all absolute normalized norms on c n is in one to one correspondence with the set n of all continuous convex functions on n with some suitable conditions where n s1 sn 1 r n 1 s1 s2 sn 1 1 si 0 1 i n 1 as some applications we show that an absolute norm on c n is strictly convex if and only if the corresponding function on n is strictly convex further we consider the von neumann jordan constant of c n with an absolute normalized norm

the distributivity property of increasing binary operations abstract this paper is mainly devoted to solving the functional equations of distributivity and conditional distributivity of increasing binary operations with the unit our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory in particular some assumptions namely associativity and commutativity and results about conditional distributivity of uninorms and triangular norms and triangular conorms were simplified moreover some other properties e g componentwise convexity concavity for special operations from the class of quasi copulas is considered

a note on left and right distributivity equations for semi t operators and uninorms abstract dryga et al showed that lemma 3 3 i in 5 is false by a counterexample see example 2 in 4 for details the goal of this short note is to indicate that example 2 in 4 is wrong and lemma 3 3 i in 5 is valid

module de congruences pour gl 2 d un corps imaginaire quadratique et theorie d iwasawa d un corps cm biquadratique

computing the degree of the modular parametrization of a modular elliptic curve the weil taniyama conjecture states that every elliptic curve e q of conductor n can be parametrized by modular functions for the congruence subgroup t0 n of the modular group r f5l 2 z equivalently there is a nonconstant map ip from the modular curve xq n to e we present here a method of computing the degree of such a map for arbitrary n our method which works for all subgroups of finite index in i and not just r0 ar is derived from a method of zagier published in 1985 by using those ideas together with techniques which have recently been used by the author to compute large tables of modular elliptic curves we are able to derive an explicit and general formula which is simpler to implement than zagier s we discuss the results obtained including a table of degrees for all the modular elliptic curves of conductors up to 200

the modularity condition for semi t operators and semi uninorms abstract the aim of this paper is mainly to solve the functional equations given by the modularity condition several years ago the modularity equations for t norms t conorms uninorms and t operators which are commutative and associative have been studied our investigations are motivated by modularity condition for generalizations of these operators by removing associativity or commutativity in this work the following main conclusions are proved 1 a continuous t norm with respect to a continuous semicopula is modular if and only if they are equal the case for a semicopula with respect to a strict t norm is also the same a semicopula with respect to a co semicopula is modular if and only if the semicopula is min and the co semicopula is max the modularity condition does not hold for a co semicopula with respect to a semicopula 2 necessary and sufficient conditions are given for a semi t operator with respect to a semi uninorm a pseudo uninorm with respect to a semi t operator to satisfy the modularity condition equation new solutions to the modularity condition equations of the case 1 are characterized

congruences between modular forms cyclic isogenies of modular elliptic curves and integrality of p adic l functions let f be a congruence subgroup of type ni n2 and of level n we study congruences between weight 2 normalized newforms f and eisenstein series e on f modulo a prime p above a rational prime p assume that p t 6n e is a common eigenfunction for all hecke operators and f is ordinary at p we show that the abelian variety associated to f and the cuspidal subgroup associated to e intersect non trivially in their p torsion points let a be a modular elliptic curve over q with good ordinary reduction at p we apply the above result to show that an isogeny of degree divisible by p from the optimal curve a1 in the q isogeny class of elliptic curves containing a to a extends to an etale morphism of neron models over 7p if p 7 we use this to show that p adic distributions associated to the p adic l functions of a are 7p valued

complex strict convexity of absolute norms on cn and direct sums of banach spaces abstract we show that an absolute normalized norm on c 2 is complex strictly convex if and only if the associated convex function on 0 1 satisfies t max 1 t t for all 0 t 1 using this result we characterize when the direct sum x y of the complex banach spaces x and y is complex strictly convex and complex uniformly convex a characterization of complex strict convexity of c n for n 2 equipped with an absolute normalized norm is also given

newman polynomials with prescribed vanishing and integer sets with distinct subset sums we study the problem of determining the minimal degree d m of a polynomial that has all coefficients in 0 1 and a zero of multiplicity m at 1 we show that a greedy solution is optimal precisely when m 5 mirroring a result of boyd on polynomials with 1 coefficients we then examine polynomials of the form e e xe 1 where e is a set of m positive odd integers with distinct subset sums and we develop algorithms to determine the minimal degree of such a polynomial we determine that d m satisfies inequalities of the form 2m c1m d m 103 96 2m c2 last we consider the related problem of finding a set of m positive integers with distinct subset sums and minimal largest element and show that the conway guy sequence yields the optimal solution for m 9 extending some computations of lunnon

restricted distributivity for aggregation operators with absorbing element the problem of restricted distributivity i e a form of relaxed distributivity on the restricted domain plays an important role in many different fields such as utility theory and integration theory this paper considers the following two cases of restricted distributivity i a continuous nullnorm with respect to a continuous t conorm and ii a continuous nullnorm with respect to a uninorm of the form u m i n or u m a x with continuous underlying t norm and t conorm

on distributivity equations for uninorms over semi t operators recently dryga generalized nullnorms and t operators and introduced semi t operators by eliminating commutativity from the axioms of t operators distributivity equations were investigated in families of certain operations e g triangular norms conorms uninorms and nullnorms or called t operators in this paper we give out the solutions of distributivity equations for uninorms over semi t operators previous results about distributivity equations for uninorms over nullnorms can be obtained as corollaries

twists of hilbert modular forms the theory of newforms for hilbert modular forms is summarized including a statement of a strong multiplicity one theorem and a characterization of newforms as eigenfunctions for a certain involution whose dirichlet series has a prescribed euler product the general question of twisting hilbert modular newforms by arbitrary fecke characters is considered and the exact level of a character twist of a hilbert modular form is determined conditions under which the twist of a newform is a newform are given applications include a strengthening in the elliptic modular case of a theorem of atkin and li s regarding the characterization of imprimitive newforms as well as its generalization to the hilbert modular case and a decomposition theorem for certain spaces of newforms as the direct sum of twists of spaces of newforms of lower level

points de petite hauteur sur les courbes modulaires x0 n let n be a square free integer prime to 6 let the imbeding of x 0 n in its jacobian relative to the point we show that the set is finite and that is infinite this explicit form of the bogomolov conjecture is obtained by an estimation of the self intersection of the dualizing sheaf in the sense of arakelov theory of modular curves this result is obtained by estimating several quantities attached to the arakelov metric on x 0 n starting with petersson s trace formula

distributivity and conditional distributivity for uni nullnorms abstract the distributivity and conditional distributivity of a uninorm over a continuous t conorm is an open problem posed by klement in the linz seminar 2000 closing session after this the study of distributivity and conditional distributivity between two aggregation operators become specially interesting in this work we continue investigating the same topic for uni nullnorms then the full characterization of all pairs f g satisfying distributivity and conditional distributivity is given where f is a uni nullnorm g is a t norm or a t conorm or a uninorm from u min and u max

factorization of p adic rankin l series we prove that the p adic l series of the tensor square of a p ordinary modular form factors as the product of the symmetric square p adic l series of the form and a kubota leopoldt p adic l series this establishes a generalization of a conjecture of citro greenberg s exceptional zero conjecture for the adjoint follows as a corollary of our theorem our method of proof follows that of gross who proved a factorization result for the katz p adic l series associated to the restriction of a dirichlet character whereas gross s method is based on comparing circular units with elliptic units our method is based on comparing these same circular units with a new family of units called beilinson flach units that we construct the beilinson flach units are constructed using bloch s intersection theory of higher chow groups applied to products of modular curves we relate these units to special values of classical and p adic l functions using work of beilinson as generalized by lei loeffler zerbes in the archimedean case and bertolini darmon rotger as generalized by kings loeffler zerbes in the p adic case central to our method are two compatibility theorems regarding bloch s intersection pairing and the classical and p adic beilinson regulators defined on higher chow groups

distributivity and conditional distributivity for s uninorms abstract the distributivity condition plays an essential role in the framework of fuzzy connectives this paper studies both regular and conditional distributivity equations for s uninorms and gives the characterizations of distributivity equations between some binary aggregation operators and s uninorms in u min moreover we describe t norms t uninorms and semi t operators that are conditional distributive over s uninorms

some implications of the restricted distributivity of aggregation operators with absorbing elements for utility theory the issue of restricted distributivity conditional distributivity i e distributivity on the relaxed domain is crucial for many different areas such as utility theory and integration theory this paper considers restricted distributivity of a continuous nullnorm with respect to a continuous t conorm and some applications of the obtained results in utility theory

some remarks on the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication we establish several results towards the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields namely i the existence of an appropriate p adic l function building on works of hida and perrin riou ii the basic structure theory of the dual selmer group following works of coates hachimori venjakob et al and iii the implications of dihedral or anticyclotomic main conjectures with basechange the result of i is deduced from the construction of hida and perrin riou which in particular is seen to give a bounded distribution the result of ii allows us to deduce a corank formula for the p primary part of the tate shafarevich group of an elliptic curve in the z p 2 extension of an imaginary quadratic field finally iii allows us to deduce a criterion for one divisibility of the two variable main conjecture in terms of specializations to cyclotomic characters following a suggestion of greenberg as well as a refinement via basechange c 2011 elsevier inc all rights reserved

the distributivity equation for uninorms revisited abstract the distributivity equation has been widely studied involving different classes of aggregation functions from t norms and t conorms to uninorms nullnorms and generalizations of them it is important in the framework of logical connectives because of its applications in fuzzy logic and approximate reasoning as well as in image processing since uninorms have been used in these topics the study of the distributivity between two uninorms becomes specially interesting in a recent paper by the same authors the already known solutions were compiled and completed when the first uninorm is in any of the most studied classes of uninorms and the second uninorm is anyone in this paper we want to achieve this study by focusing on the reverse direction that is for the cases when the second uninorm lies in any of the most studied classes of uninorms and the first one is any uninorm we show along the paper that this new point of view leads to many new solutions

left and right distributivity equations for semi t operators and uninorms abstract recently the distributivity equation for different kind of operators has become a focus of research as the crucial tool in the numerous applications as utility and optimization theory or integration theory the main purpose of this paper is to study the distributivity equations between two special classes of aggregation operators namely semi t operators and uninorms without the commutativity assumption of semi t operators it is necessary to consider the left and the right distributivity conditions separately in both cases the obtained solutions are significantly different

a spectral proof of class number one we continue our previous work on the subject of re proving the heegner baker stark theorem giving another effective resolution of this conjecture of gauss namely there are exactly 9 imaginary quadratic fields mathbf q sqrt q with class number 1 specifically the list is q in 3 4 7 8 11 19 43 67 163 we again follow an ansatz of goldfeld considering a modular form l function of analytic rank 2 and reducing the situation to showing sufficient cancellation of the dirichlet series coefficients when restricted to the principal form previously we then chose a specific rank 2 elliptic curve with complex multiplication by mathbf q sqrt 1 and deduced the desired cancellation from an equi distribution result of hooley s for roots of a quadratic polynomial to varying moduli herein we instead use spectral techniques filling in details of work of templier and tsimerman to complete the proof relying on the duke iwaniec bound for fourier coefficients of half integral weight maass forms unlike the previous method we do not require complex multiplication and any analytic rank 2 modular form whose level has an even number of prime factors suffices

iwasawa theory for the symmetric square of a modular form we construct a compatible family of global cohomology classes an euler system for the symmetric square of a modular form and apply this to bounding selmer groups of the symmetric square galois representation and its twists

von neumann jordan constant of absolute normalized norms on c2 abstract we determine and estimate the von neumann jordan constant of absolute normalized norms on c 2 by means of their corresponding continuous convex functions on 0 1 this provides many new interesting examples including those of non lp type as well as some previous ones it is also shown that all such norms are uniformly non square except l1 and l norms

on the conditional distributivity of nullnorms over uninorms conditional distributivity of nullnorm over uninorm with continuous operators is discussed the structure of uninorms with idempotent operators is discussed the problem of conditional distributivity for different two aggregation operators plays an important role in many different fields such as utility theory and integration theory this paper deals with the conditional distributivity of a continuous nullnorm with respect to a uninorm with the continuous underlying t norm and t conorm

the main conjecture of iwasawa theory for totally real fields let p be an odd prime let mathcal g be a compact p adic lie group with a quotient isomorphic to p we give an explicit description of k 1 of the iwasawa algebra of mathcal g in terms of iwasawa algebras of abelian subquotients of mathcal g we also prove a result about k 1 of a certain canonical localisation of the iwasawa algebra of mathcal g which occurs in the formulation of the main conjectures of noncommutative iwasawa theory these results predict new congruences between special values of artin l functions which we then prove using the q expansion principle of deligne ribet as a consequence we prove the noncommutative main conjecture for totally real fields assuming a suitable version of iwasawa s conjecture about vanishing of the cyclotomic invariant in particular we get an unconditional result for totally real pro p p adic lie extension of abelian extensions of

the distributivity equations for semi t operators over uninorms recently the distributivity equation was discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms or called t operators in this paper we describe the solutions of distributivity for semi t operators over uninorms previous results about distributivity for nullnorms over uninorms can be obtained as a corollary

distributivity and conditional distributivity of a uninorm with continuous underlying operators over a continuous t conorm this paper deals with the distributivity and conditional distributivity of uninorms with continuous underlying operators over continuous triangular conorms in particular the involved triangular conorm is either maximum operator or an ordinal sum with only one summand in which the corresponding triangular conorm is strict from the obtained results it is deduced that distributivity and conditional distributivity are equivalent moreover we obtain the full characterization of some cases of this class of uninorms of which either the underlying triangular norm or triangular conorm is strict

two variable p adic l functions attached to eigenfamilies of positive slope consider a classical cusp eigenform f n 1 a n f q n of weight k 2 for 0 n with a dirichlet character mod n and let l f s n 1 n a n f n s denote the l function of f twisted with an arbitrary dirichlet character for a prime number p 5 consider a family of cusp eigenforms f k of weight k k f k n 1 a n f k q n containing f f k such that the fourier coefficients a n f k are given by certain p adic analytic functions k a n f k the purpose of this paper is to construct a two variable p adic l function attached to coleman s family f k of cusp eigenforms of a fixed positive slope v p p 0 where p p k is an eigenvalue which depends on k of the atkin operator u u p our p adic l function interpolates the special values l f k s at points s k with s 1 2 k 1 we give a construction using the rankin selberg method and the theory of p adic integration on a profinite group y with values in an affinoid k algebra a where k is a fixed finite extension of q p our p adic l functions are p adic mellin transforms of certain a valued measures in their turn such measures come from eisenstein distributions with values in certain banach a modules m m n a of families of overconvergent forms over a

the distributivity condition for uninorms and t operators in this work we study the functional equation given by the distributivity condition f x g y z g f x y f x z for all x y z 0 1 where the unknown functions f g are uninorms and or t operators we characterize all the solutions in the four possible cases i when f g are t operators ii when f is a t operator and g a uninorm iii when f is a uninorm and g a t operator and finally iv when f g are uninorms

on the conditional distributivity of continuous semi t operators over uninorms abstract the issue of conditional distributivity or how it is also called restricted distributivity which is a form of relaxed distributivity on the restricted domain is crucial for many different areas such as utility theory and integration theory the focus of this paper is on this specific form of distributivity for a continuous semi t operator with respect to a continuous t conorm and for a continuous semi t operator with respect to a uninorm of the form u min or u max with continuous underlying t norm and t conorm

distributivity and conditional distributivity for t uninorms abstract aggregation operators play an important role in many theoretical and practical aspects of applied mathematics recently the focus is on operators with an annihilator therefore the topic of this paper is distributivity both conditional and regular for certain classes of aggregation operators with this property the characterization of all pairs f g of aggregation operators that are satisfying distributivity law on both whole and restricted domain where f is a t uninorm in u max and g is a t conorm or a uninorm from u min or u max is given

on some solutions of the distributivity equation abstract the aim of this paper is to present some interesting results on the distributivity equation for which several pairs of binary operations on 0 1 are considered moreover different methods to solve distributivity equations are used in this work

the p adic gross zagier formula for elliptic curves at supersingular primes let p be a prime number and let e be an elliptic curve defined over of conductor n let k be an imaginary quadratic field with discriminant prime to pn such that all prime factors of n split in k b perrin riou established the p adic gross zagier formula that relates the first derivative of the p adic l function of e over k to the p adic height of the heegner point for k when e has good ordinary reduction at p in this article we prove the p adic gross zagier formula of e for the cyclotomic p extension at good supersingular prime p our result has an application for the full birch and swinnerton dyer conjecture suppose that the analytic rank of e over is 1 and assume that the iwasawa main conjecture is true for all good primes and the p adic height pairing is not identically equal to zero for all good ordinary primes then our result implies the full birch and swinnerton dyer conjecture up to bad primes in particular if e has complex multiplication and of analytic rank 1 the full birch and swinnerton dyer conjecture is true up to a power of bad primes and 2

distributivity and conditional distributivity of a uninorm and a continuous t conorm the open problem recalled by klement in the linz2000 closing session related to distributivity and conditional distributivity of a uninorm and a continuous t conorm is solved for the most usual known classes of uninorms from the obtained results it is deduced that distributivity and conditional distributivity are equivalent for these cases it is remarkable that solutions appear involving not only strict t conorms but also ordinal sums of the maximum with a strict t conorm conversely the distributivity of a t conorm over a uninorm is also studied leading only to already known solutions moreover the dual case of distributivity and conditional distributivity involving uninorms and continuous t norms is also solved proving again the equivalence of both kinds of distributivities

on the hida deformations of fine selmer groups abstract in this paper we study the fine selmer group of p adic galois representations and their deformations we show that for an infinite family of elliptic cuspforms if the invariant of the dual fine selmer group is zero for one member of the family then the same holds for all the other members further the invariants are equal for all but finitely many members in the family

canonical periods and congruence formulae the purpose of this article is to show how congruences between the fourier coefficients of hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated l functions this study was initiated by b mazur in his fundamental work on the eisenstein ideal see maz77 and maz79 where it was made clear that congruences for analytic l values were closely related to the integral structure of certain hecke rings and cohomology groups the results of maz79 also showed that congruences were useful in the study of nonvanishing of l functions this idea was then further developed by stevens ste82 and rubin wiles rw82 the work of rubin and wiles in particular used congruences to study the behavior of elliptic curves in towers of cyclotomic fields a key ingredient here was a theorem of washington which states roughly that almost l values in certain families are nonzero modulo p this theme has recently been taken up again in the work of ono skinner osa osb james jam and kohnen koh97 while the earlier history was primarily concerned with cyclotomic twists the current emphasis is on families of twists by quadratic characters here one wants quantitative estimates for the number of quadratic twists of a given modular form which have nonvanishing l function at s 1 we continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three this generalizes a beautiful example due to kevin james and provides new evidence for a conjecture of goldfeld gol79 it should however be pointed out that even the study of quadratic twists may be traced back to mazur the reader is urged to look at pages 212 213 of maz79 and especially at the footnote at the bottom of page 213 the theorems of davenport heilbronn dh71 and washington was78 which are crucial in this paper are both mentioned in mazur s article we want to begin by discussing the congruences that lie at the heart of this article thus let f anq n be an elliptic modular cuspform of level m and weight k 2 assume that f is a simultaneous eigenform for all the hecke operators and that a1 f 1 the l function associated to f is defined by the dirichlet series l s f ann s which converges for the real part of s sufficiently large and has analytic continuation to s c a fundamental theorem of shimura shi76 states that l s f enjoys the following algebraicity property

distributivity and conditional distributivity of semi uninorms over continuous t conorms and t norms the distributivity and conditional distributivity of a uninorm and a continuous t conorm present an open problem recalled by klement in the linz2000 closing session in 2006 ruiz and torrens 14 solved this problem for the most usual known classes of uninorms recently rak 13 solved the same problem for several classes of semi uninorms i e a special class of conjunctive semi uninorms a special class of disjunctive semi uninorms and the class of idempotent semi uninorms in this work we continue to investigate the same topic as the above by focusing on two other classes of semi uninorms i e representable semi uninorms and continuous semi uninorms the obtained results are different from the case of representable uninorms and also different from the cases of other known classes of semi uninorms moreover the dual case of distributivity and conditional distributivity of the above two classes of semi uninorms over continuous t norms is also investigated and similar results are obtained

on the distributivity property for uninorms abstract distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives for this reason the distributivity property has been extensively studied for several families of operations like triangular norms and conorms some kinds of uninorms and nullnorms also called t operators and even for some generalizations of them in this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms leading to many new solutions

on certain zeta functions attached to two hilbert modular forms i the case of hecke characters to make our exposition smooth let us first introduce some notational conventions for an algebraic number field k of finite degree we denote by jk the set of all embeddings of k into c and by ik the free z module generated by the elements of jk we then put rik ik0 r and cik ik0 c if p eapau e ik with a e jk and po e z we put xp fl xa pfor 0 x e k this is meaningful for p e cik if xa are all real and positive throughout the paper we denote by d the complex upper half plane and by f a totally real algebraic number field of degree n now the zeta function to be studied in this paper when suitably specialized has the form

the sign of the functional equation of the l function of an orthogonal motive we prove that the sign of the functional equation of the l function of an orthogonal motive of even weight is always positive if we admit the conjecture of the product formula for the epsilon factor of deligne d3 5 2 more precisely we define the sign independently of any conjecture by using the product formula as the definition and by working with one chosen good prime f and we prove its positivity the positivity has been established for the artin motives fr q d2 for the symmetric square of a modular elliptic curve c sc and in positive characteristic by serre fr q as in the proof in d2 we consider the second stiefel whitney class in zbr k of an orthogonal representation of the absolute galois group g of an algebraic number field k the positivity of the sign is then a consequence of the reciprocity law of the square residue symbol fixing a good prime f we work with an f adic representation and compare the local epsilon factor and the local stiefel whitney class for a prime vxr it is a theorem of deligne d2 for a prime vie we compute the stiefel whitney class assuming the representation is cristalline theorem 2 using a generalization of a theorem of fr6hlich fr to f adic representation of hodge tate type theorem 1 by combining them with an elementary computation for the hodge structures at r iot we conclude that the sign is positive theorem 3 the crucial step in the proof of theorem 2 given in section 1 is the determination by fontaine and lafaille fo l of the action of the tame inertia on the semisimplification of reduction modulo p of a cristalline representation the theory of orthogonal representations of galois group including a proof of fr6hlich s theorem is reviewed in a preliminary section 0

s adic l functions attached to the symmetric square of a newform in order to apply the ideas of iwasawa theory to the symmetric square of a newform we need to be able to define non archimedean analogues of its complex l series the interpolated p adic l function is closely connected via a main conjecture with certain selmer groups over the cyclotomic bbb z p extension of bbb q in the p ordinary case these functions are well understood in this article we extend the interpolation to an arbitrary set s of good primes not necessarily satisfying ordinarity conditions the corresponding s adic functions can be characterised in terms of certain admissibility criteria we also allow interpolation at particular primes dividing the level of the newform one interesting application is to the symmetric square of a modular elliptic curve e defined over bbb q our constructions yield p adic l functions at all primes of stable or semi stable reduction if p is ordinary or multiplicative the corresponding analytic function is bounded if p is supersingular our function behaves like log 2 1 t 1991 mathematics subject classification 11f67 11f66 11f33 11f30

structure of n uninorms in this paper we study binary operators on 0 1 which are associative monotone non decreasing in both variables and commutative amc with neutral element in this work we generalize the concept of neutral element and this generalization gives rise to a new class of amc binary operators on 0 1 called n uninorms n uninorms are denoted as u n where n comes from the generalization of the neutral element we study the structure of n uninorms the structure resembles an ordinal sum structure made up of n uninorms we characterize some special cases of them based on some continuity considerations and show that t norms t conorms uninorms and nullnorms t operators are special cases of n uninorms we also show that given n there are n 1 classes of operators in u n and each of them has many subclasses we also study the frank equation involving n uninorms and show that we need to consider only n uninorms for the study finally we show that the total number of subclasses of operators in u n follows the famous series called catalan numbers

distributivity between uninorms and nullnorms recently the distributivity equation was investigated in families of certain operations e g triangular norms conorms uninorms and nullnorms generally the assumption of associativity is not necessary in consideration of distributivity equation moreover if we omit commutativity assumption consideration of the left and right distributivity conditions is reasonable previous results about distributivity between uninorms and nullnorms can be obtained as simple corollaries

distributivity between semi t operators and mayor s aggregation operators the problem of distributivity was first posed over forty years ago and it has been investigated for families of certain operations such as t norms t conorms uninorms and nullnorms in this paper we investigate this topic further by focusing on mayor s aggregation operators and semi t operators which generalize the t operators by omitting commutativity the results that we obtain are complete and extend previous results concerning distributivity between mayor s aggregation operators and semi nullnorms

on convexity properties of ψ direct sums of banach spaces the direct sum of banach spaces is strictly convex respectively uniformly convex locally uniformly convex uniformly convex in every direction if each of the banach spaces are strictly convex respectively uniformly convex locally uniformly convex uniformly convex in every direction and the corresponding norm is strictly convex

distributivity between semi t operators and semi nullnorms recently the distributivity equation was discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms in this paper we describe the solutions of distributivity between semi t operators and semi nullnorms previous results about distributivity between nullnorms can be obtained as simple corollaries