closed subsets in duals of commutative table algebras a lower bound on the number of elements outside a closed subset of a c basis of a commutative c algebra with dual a table algebra is derived as is an equivalent condition for when the lower bound is met as corollaries lower bounds are obtained on the number of primitive idempotent matrices of rank greater than 1 in the adjacency algebra of a commutative imprimitive association scheme and for a given normal subgroup n of a finite group g on the number of irreducible characters of g whose kernels do not contain n and on the number of conjugacy classes of g not contained in n also found are equivalent conditions for when these lower bounds are met

permutation modules over table algebras and applications to association schemes abstract inspired by the notion of an action of a finite hypergroup on a finite set we introduce the more general concept of a permutation module over a table algebra it is easy to see that each permutation module over a table algebra is a direct sum of transitive permutation modules among the transitive permutation modules the w maximal and the maximal permutation modules are the most interesting ones we give various characterizations of these modules and we shall see that a standard table algebra admits a maximal transitive permutation module if and only if it arises from a finite association scheme we also show that the regular module of a standard table algebra is isomorphic to a direct summand of each w maximal transitive permutation module as a consequence one obtains 1 m for any irreducible character of the bose mesner algebra of a finite association scheme and its multiplicity m

on wreath products of c algebras table algebras and association schemes abstract in this paper we will first present characterizations of wreath products of c algebras and show that the dual of the wreath product of c algebras is isomorphic to the wreath product of the duals of those c algebras in the reverse order then we prove the krull schmidt type theorems for the wreath products of table algebras as well as for the wreath products of association schemes

reality based algebras with a two dimensional representation abstract it is shown that a two dimensional irreducible representation of a reality based algebra with a standard degree map is determined by the other irreducible representations of the algebra and the action of its given anti automorphism on the distinguished basis this generalizes known results for algebras of dimensions 5 6 7 and 8 explicit conclusions equivalent to those for dimensions 5 and 6 are recovered here by specializing the procedure of proof of the main theorem

noncommutative standard table algebras with at most one nontrivial multiplicity abstract standard table algebras that have an irreducible character of degree two and with all other irreducible characters of multiplicity one are classified to exact isomorphism

some symmetric identities on higher order q euler polynomials and multivariate fermionic p adic q integral on z p in this paper we derive some identities of symmetry related to higher order q euler polynomials by using the multivariate fermionic p adic q integral on z p furthermore some of these identities are also related to the q analogue of the alternating power sums and the multiplication theorem

the generalized terwilliger algebra and its finite dimensional modules when d 2 terwilliger j algebraic combin 1 1992 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations are often called the triple product relations they determine much of the structure of t though not all of it in general to illuminate the role these relations play the current author introduced j algebra233 2000 213 252 a generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c the dimensions of c and c are equal let d 1 denote this common dimension intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s triple product relations t is infinite dimensional and noncommutative in general in this paper we study t and its finite dimensional modules when d 2 and t has no extra vanishing intersection numbers or dual intersection numbers in this case we show t is c algebra isomorphic to m3 c a where m3 c denotes the c algebra consisting of all 3 by 3 matrices with entries in c and a denotes the associative c algebra with identity generated by the symbols e and f subject to the relations e2 e and f2 f we find a basis for a and we determine the center of a we classify the finite dimensional indecomposable a modules up to isomorphism there are four such a modules in every odd dimension and in every even dimension these modules are parameterized by a single complex number we also classify the finite dimensional irreducible a modules up to isomorphism using our results concerning a we find a basis for t we describe the center of t and we classify both the finite dimensional indecomposable and the finite dimensional irreducible t modules up to isomorphism

twisted h q bernoulli numbers and polynomials related to twisted h q zeta function and l function abstract in this paper by using q volkenborn integral we construct new generating functions of the new twisted h q bernoulli polynomials and numbers by applying the mellin transformation to these generating functions we obtain integral representations of the new twisted h q zeta function and twisted h q l function which interpolate the twisted h q bernoulli numbers and generalized twisted h q bernoulli numbers at non positive integers respectively furthermore relation between twisted h q zeta function and twisted h q l function are proved some new relations related to twisted h q bernoulli polynomials and numbers are given

a generalization of sylow s theorems on finite groups to association schemes let x g denote an association scheme or briefly a scheme in the sense of 5 and let p denote a prime number an element g in g will be called p valenced if ng is a power of p a subset of g will be called p valenced if each of its elements is p valenced a p valenced subset f of g will be called a p subset if nf is a power of p let us assume x g to be finite then by definition ng is finite we shall denote by sylp g the set of all closed p subsets h of g such that p does not divide ng h it is the purpose of the present note to prove the following theorem

on structures of quasi thin table algebras and applications to association schemes abstract in this paper we develop a structure theory for a class of table algebras i e the so called quasi thin table algebras they are direct generalizations of quasi thin association schemes although this class of table algebras is rather close to finite groups its structural properties are quite different from those of finite groups the thin radical and thin residue give rise to two group structures for every table algebra they play the central role in our discussions applications to association schemes are also discussed and some new results about quasi thin association schemes are obtained in particular we will give necessary and sufficient conditions under which a commutative quasi thin association scheme has multiplicities 1 or 2 we will need these results in the study of the characterization and classification of double quasi thin association schemes in 22

on morphisms of association schemes abstract in this paper we present a necessary and sufficient condition under which a combinatorial morphism between association schemes induces an algebra homomorphism between their scheme rings we will also discuss scheme ring homomorphisms of naturally valenced association schemes

varietal terwilliger algebras arising from wreath products of rank two association schemes abstract as abstractions of terwilliger algebras generalized terwilliger algebras and varietal terwilliger algebras have been studied in 5 6 17 in general it is difficult to determine the structure of a varietal terwilliger algebra as proposed in the concluding remarks in 4 in this paper we study the structures of those varietal terwilliger algebras arising from the wreath products of rank two association schemes in particular we give the explicit formulas for their central primitive idempotents applications to terwilliger algebras of association schemes are also discussed and an affirmative answer to the terwilliger conjecture mention in 4 is given

nonexistence of solutions for prescribed mean curvature equations on a ball abstract we prove two nonexistence results of radial solutions to the prescribed mean curvature type problem on a ball div d u 1 d u 2 f u x b r r n u 0 x b r where is a positive parameter f is a continuous function with f 0 0 under suitable assumptions on f we show that the problem with superlinear f has no nontrivial positive solutions for small while the problem with sublinear f has no nontrivial positive solutions for large the former covers many well known nonexistence results by finn serrin narukawa and suzuki ishimura pan and xing to the best of the authors knowledge the latter is the first nonexistence result involving sublinear mean curvature type equations in higher dimensions in particular the sublinear cases contain some important logistic type nonlinearities these nonexistence results differ greatly from those of semilinear problems

morphisms of naturally valenced association schemes and quotient schemes abstract let s and s be naturally valenced association schemes on sets x and x respectively and let be a combinatorial morphism from x s to x s in xu 2009 x2 a necessary and sufficient condition was given for to induce an algebra homomorphism from the scheme ring c s to the scheme ring c s the present paper provides new techniques with which this result can be proved without assuming ker to be finite to do this we will first need to prove that for any normal closed subset t of s whether t is finite or infinite the quotient s t is a naturally valenced association scheme on the set x t we will also need to discuss scheme ring homomorphisms of naturally valenced association schemes and prove some isomorphism theorems without assuming the kernels of the scheme ring homomorphisms to be finite as a direct consequence for a naturally valenced commutative association scheme s on a set x and any closed subset t of s the quotient s t is a naturally valenced commutative association scheme on the set x t the approach in this paper is different from xu 2009 x2 and quasi algebraic morphisms of naturally valenced association schemes are also studied

irreducible characters of wreath products in reality based algebras and applications to association schemes abstract wreath products in reality based algebras are generalizations of wreath products of table algebras and generalized camina frobenius pairs of c algebras in this paper we present characterizations of the wreath product in a reality based algebra by its irreducible characters and by the size of the zero submatrix of its character table applications to finite groups table algebras and association schemes are also discussed in particular we will show that the wreath product of one class association schemes is characterized by the zeros in its first eigenmatrix

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

a generalization of the terwilliger algebra abstract p m terwilliger 1992 j algebraic combin 1 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations determine much of the structure of t thought not all of it in general to illuminate the role these relations play we consider a certain generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c we define t by generators and relations intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s relations t is infinite dimensional and noncommutative in general we construct an irreducible t module which we call the primary module the dimension of this module is the same as that of c and c we find two bases of the primary module one diagonalizes c and the other diagonalizes c we compute the action of the generators of t on these bases we show t is a direct sum of two sided ideals t 0 and t 1 with t 0 isomorphic to a full matrix algebra we show that the irreducible module associated with t 0 is isomorphic to the primary module we compute the central primitive idempotent of t associated with t 0 in terms of the generators of t

nilpotent fusion categories abstract in this paper we extend categorically the notion of a finite nilpotent group to fusion categories to this end we first analyze the trivial component of the universal grading of a fusion category c and then introduce the upper central series of c for fusion categories with commutative grothendieck rings e g braided fusion categories we also introduce the lower central series we study arithmetic and structural properties of nilpotent fusion categories and apply our theory to modular categories and to semisimple hopf algebras in particular we show that in the modular case the two central series are centralizers of each other in the sense of m muger

decomposition of products in nilpotent table algebras abstract two theorems on the decomposition of products in nilpotent commutative table algebras are established this generalizes several results of adan bante on the decomposition of products of irreducible characters and of conjugacy classes of finite p groups and extends these results to fusion rule algebras and association schemes

fusion algebras with negative structure constants abstract we introduce fusion algebras with not necessarily positive structure constants and without identity element we prove that they are semisimple when tensored with c and that their characters satisfy orthogonality relations then we define the proper notion of subrings and factor rings for such algebras for certain algebras r we prove the existence of a ring r with nonnegative structure constants such that r is a factor ring of r we give some examples of interesting factor rings of the representation ring of the quantum double of a finite group then we investigate the algebras associated to hadamard matrices for an n n matrix the corresponding algebra is a factor ring of a subalgebra of z z 2 z n 2

on thin residues and basis digraphs of nilpotent table algebras and applications to nilpotent groups abstract in this paper we study characterizations of nilpotent table algebras in terms of thin residues thin residue matrices thin residue digraphs and basis digraphs we will also discuss applications to nilpotent groups in particular we give a conceptual explanation of the results in takegahara and yoshida 2008 ty

some structure theory of table algebras and applications to association schemes abstract in this paper we prove some basic structure theorems for arbitrary table algebras in particular we prove a krull schmidt theorem for arbitrary table algebras and give applications to association schemes

nilpotency property in table algebras abstract table algebras form an important class of c algebras the dual of a table algebra may not be a table algebra but just a c algebra it is not known under what conditions the dual of a table algebra is also a table algebra in this paper we prove that if a table algebra has nilpotency property then its dual also is a table algebra finally we conclude that if a b is a nilpotent table algebra then its dual also is a nilpotent table algebra

quotient structures homomorphisms and partial wreath products in reality based algebras abstract homomorphisms and quotient structures of reality based algebras are presented and described related central idempotents and quotient sets are characterized and the extent to which a homomorphism preserves the given anti automorphisms of the algebras is determined the notion of a partial wreath product an algebraic abstraction of the wedge product of association schemes is developed prototypical results previously obtained for various special cases including table algebras and commutative c algebras are thus uniformly generalized

the recognition problem for table algebras and reality based algebras given a finite dimensional noncommutative semisimple algebra a over c with involution we show that a always has a basis b for which a b is a reality based algebra for algebras that have a one dimensional representation we show that there always exists an rba basis for which is a positive degree map we characterize all rba bases of the 5 dimensional noncommutative semisimple algebra for which the algebra has a positive degree map and give examples of rba bases of c mn c for which the rba has a positive degree map for all n 2

fusion rings with few degrees abstract the main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p a well known theorem of isaacs and passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary numerically compatible nc fusion rings are defined as a potential generalization of the grothendieck rings of fusion categories and the nc fusion rings of order p p 2 p 3 pq and p q 2 where p q are primes are classified in terms of certain standard table algebras as another application of the main theorem

existence of nonnegative solutions for quasilinear elliptic equations with indefinite critical nonlinearities abstract we study the quasilinear elliptic equation div u u a x f u b x g u in with dirichlet boundary condition u 0 on where is a bounded domain in r n a x b x are sign changing continuous functions and g u has critical growth at infinity with respect to the principal part a nonnegative nontrivial solution is given under appropriate growth conditions on f u g u at 0 and

identities of symmetry for q bernoulli polynomials in this paper we derive eight basic identities of symmetry in three variables related to q bernoulli polynomials and the q analogue of power sums these and most of their corollaries are new since there have been results only concerning identities of symmetry in two variables these abundant symmetries shed new light even on the existing identities so as to yield some further interesting ones the derivations of the identities are based on the p adic integral expression of the generating function for the q bernoulli polynomials and the quotient of integrals that can be expressed as the exponential generating function for the q analogue of power sums

identities of symmetry for bernoulli polynomials arising from quotients of volkenborn integrals invariant under s3 in this paper we derive eight basic identities of symmetry in three variables related to bernoulli polynomials and power sums these and most of their corollaries are new since there have been results only about identities of symmetry in two variables these abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones the derivations of identities are based on the p adic integral expression of the generating function for the bernoulli polynomials and the quotient of integrals that can be expressed as the exponential generating function for the power sums

on semisimple varietal terwilliger algebras whose non primary ideals are 1 dimensional abstract as an abstraction of the terwilliger algebra of a commutative association scheme the notion of a varietal terwilliger algebra was introduced in 11 in this paper we first prove that there exists a varietal terwilliger algebra defined by any given standard c algebra and a generalized terwilliger algebra 6 is also a varietal terwilliger algebra then we present a necessary and sufficient condition under which a quotient algebra of a varietal terwilliger algebra is also a varietal terwilliger algebra we will also give a characterization of a semisimple varietal terwilliger algebra whose non primary ideals are all 1 dimensional and prove that a semisimple regular varietal terwilliger algebra with 1 dimensional non primary ideals is characterized by the structure of its base c subalgebra as direct consequences we obtain some results in 18 for finite association schemes and a conceptual explanation of the results in 9

fusion algebras for imprimitive complex reflection groups abstract we prove that the fourier matrices for the imprimitive complex reflection groups introduced by malle in gunter malle unipotente grade imprimitiver komplexer spiegelungsgruppen j algebra 177 3 1995 768 826 define fusion algebras with not necessarily positive but integer structure constants hence they define z algebras as a result we obtain that all known fourier matrices belonging to spetses define algebras with integer structure constants

on quasi thin association schemes with odd number of points abstract let x r be an association scheme in the sense of p h zieschang 1996 an algebraic approach to association schemes lecture notes in mathematics vol 1628 springer new york berlin where x is a finite set and r is a partition of x x we say that x r is quasi thin if each element of r has a valency of at most two in this paper we focus on quasi thin association schemes with an odd number of points and obtain that x r has a regular automorphism group when n o r is square free

partial differential equations involving subcritical critical and supercritical nonlinearities abstract we establish the existence of at least one nonnegative solution for the problem p div a u u f u in u 0 on where a and f satisfy conditions near zero here the novelty is that we do not need restrictions on the nonlinearities at infinity therefore we can consider subcritical critical and supercritical cases

on nakayama automorphisms of double frobenius algebras abstract double frobenius algebras or df algebras were recently introduced by the author the concept generalizes finite dimensional hopf algebras adjacency algebras of non commutative association schemes and c algebras or character algebras this paper studies basic properties of various nakayama automorphisms in a df algebra as an application the theorem of d e radford stating that the antipode s of a finite dimensional hopf algebra is of finite order is generalized to df algebras and so is his formula for s 4 n in terms of certain group like elements