maximal exponents of polyhedral cones iii article history received 31 may 2009 available online 14 november 2009 submitted by j bastero

some bismash products that are not group algebras ii abstract we show that an infinite family of hopf algebras that arise as a subfamily of the so called bismash products constructed from factorisations of symmetric groups do not have the structure as algebras of group algebras

on the existence of nodal domains for elliptic differential operators we prove that the selfadjoint elliptic differential equation 1 has rectangular nodal domains if the quadratic form of the equation takes on negative values the existence of nodal domains is closely connected with the position of the smallest point of the spectrum of the corresponding selfadjoint operator friedrichs extension if the smallest point of a second order selfadjoint differential operator with dirichlet boundary conditions is an eigenvalue then this eigenvalue is strictly increasing when the possibly unbounded domain where the coefficients of the differential operator are denned is shrinking theorem 4

on a conjecture about the exponent set of primitive matrices m lewin and y vitek conjecture 7 that every integer n 2 2n 2 2 1 is an exponent of some n n primitive matrix in this paper we prove three results related to lewin and vitek s conjecture 1 every integer n2 2n 2 4 1 is an exponent of some n n primitive matrix 2 the conjecture is true when n is sufficiently large 3 we give a counterexample to show that the conjecture is not true in the case when n 11

bounds on the exponent of primitivity which depend on the spectrum and the minimal polynomial abstract suppose a is an n n nonnegative primitive matrix whose minimal polynomial has degree m we conjecture that the well known bound on the exponent of primitivity n 1 2 1 due to wielandt can be replaced by m 1 2 1 the only case for which we cannot prove the conjecture is when m 5 the number of distinct eigenvalues of a is m 1 or m and the directed graph of a has no circuits of length shorter than m 1 but at least one of its vertices lies on a circuit of length not shorter than m we show that m m 1 is always a bound on the exponent this being an improvement on wielandt s bound when m n for the case in which a is also symmetric the bound which we obtain is 2 m 1 to obtain our results we prove a lemma which shows that for a general nonnegative matrix the number of its distinct eigenvalues is an upper bound on the length of the shortest circuit in its directed graph

on the exponent of a primitive digraph abstract we characterize the equality case of the upper bound d n s n 2 for the exponent of a primitive digraph in the case s 2 where n is the number of the vertices of the digraph d and s is the length of the shortest elementary circuit of d we also answer a question about the equality case when s 1 the exponent of an n n primitive nonnegative matrix a is the same as the exponent of the associated digraph d a of a so every theorem in this paper can be translated into a theorem about nonnegative matrices

clifford correspondence for finite dimensional hopf algebras abstract let b h be a crossed product algebra over an algebraically closed field with h a finite dimensional hopf algebra we give an explicit equivalence between the category of finite dimensional b h modules whose restriction to b is a direct sum of copies of a stable irreducible b module and the category of modules for a twisted product of h with the field this describes all finite dimensional irreducible b h modules containing a stable irreducible b submodule and thus generalizes the classical stable clifford correspondence for groups in case h is cocommutative we extend this correspondence to the nonstable case

hopf algebra extensions arising from semi direct products of groups hopf algebra extensions arising from a matched pair of groups were intro duced by g i kac kac he established an exact sequence the kac sequence that connects group cohomology and hopf algebra extensions this was re cently considered and generalized by other authors for instance a masuoka in mal ma2

the algebra of discrete torsion abstract we analyze the algebraic structures of g frobenius algebras which are the algebras associated to global group quotient objects a k a global orbifolds here g is any finite group first we show that these algebras are modules over the drinfel d double of the group ring k g and are moreover k g module algebras and k g comodule algebras we furthermore consider g frobenius algebras up to projective equivalence and define universal shifts of the multiplication and the g action preserving the projective equivalence class we show that these shifts are parameterized by z 2 g k and by h 2 g k when considering of g frobenius algebras up to isomorphism we go on to show that these shifts can be realized by the forming of tensor products with twisted group rings thus providing a group action of z 2 g k on g frobenius algebras acting transitively on the classes g frobenius related by universal shifts the multiplication is changed according to the cocycle while the g action transforms with a different cocycle derived from the cocycle defining the multiplication the values of this second cocycle also appear as a factor in front of the trace which is considered in the trace axiom of g frobenius algebras this allows us to identify the effect of our action by z 2 g k as so called discrete torsion effectively unifying all known approaches to discrete torsion for global orbifolds in one algebraic theory the new group action of discrete torsion is essentially derived from the multiplicative structure of g frobenius algebras this yields an algebraic realization of discrete torsion defined via the perturbation of the multiplication resulting from a tensor product with a twisted group ring additionally we show that this algebraic formulation of discrete torsion allows for a treatment of g frobenius algebras analogous to the theory of projective representations of groups group extensions and twisted group ring modules lastly we identify another set of discrete universal transformations among g frobenius algebras pertaining to their super structure and classified by hom g z 2 z which are essential for the application to mirror symmetry for singularities with symmetries

oscillation criteria for second order nonlinear delay dynamic equations on time scales by employing the generalized riccati transformation we establish some new oscillation criteria for second order delay dynamic equations on time scales the obtained results essentially improve the well know oscillation results for half linear dynamic equations such as kamenev type and philos type oscillation criteria some interesting examples are given to illustrate the versatility of our results

asymptotic behavior of solutions of second order nonlinear delay differential equations with impulses this paper is concerned with second order nonlinear delay differential equations with impulses of the form r t x t p t x t f t x email protected 0 t t k x t k i k x t k x t k j k x t k t t k k 1 2 t t 0 by impulsive differential inequality and riccati transformation sufficient conditions of asymptotic behavior of all solutions of second order nonlinear delay differential equations with impulses are obtained some examples are also inserted to illustrate the impulsive effect

the distance between zeros of an oscillatory solution to a half linear differential equation consider the oscillatory equation u t 1u t q t u t 1u t 0 where q t a r is locally integrable for some a 0 we prove some results on the distance between consecutive zeros of a solution of we apply also the results to the following equations r t u t 1u t q t u t 1u t 0 and i 1ndi du n 2diu c x u n 2u 0 a where i r c 0 0 and ar t 1 ii di xi d d1 dn a x rn x a is an exterior domain and c c a 0 iii 0 n 1 and n 2

on semisimple hopf algebras of dimension pq2 let p and q be distinct prime numbers we prove a result on the existence of nontrivial group like elements in a certain class of semisimple hopf algebras of dimension pqr we conclude the classification of semisimple hopf algebras a of dimension pq2 over an algebraically closed field k of characteristic zero such that both a and a are of frobenius type we also complete the classification of semisimple hopf algebras of dimension pq2 100

frobenius schur indicators for subgroups and the drinfel d double of weyl groups if g is any finite group and k is a field there is a natural construction of a hopf algebra over k associated to g the drinfel d double d g we prove that if g is any finite real reflection group with drinfel d double d g over an algebraically closed field k of characteristic not 2 then every simple d g module has frobenius schur indicator 1 this generalizes the classical results for modules over the group itself we also prove some new results about weyl groups in particular we prove that any abelian subgroup is inverted by some involution also if e is any elementary abelian 2 subgroup of the weyl group w then all representations of cw e are defined over q

modular representations and indicators for bismash products abstract we introduce brauer characters for representations of the bismash products of groups in characteristic p 0 p 2 and study their properties analogous to the classical case of finite groups we then use our results to extend to bismash products a theorem of thompson on lifting frobenius schur indicators from characteristic p to characteristic 0

oscillation criteria for pde with p laplacian via the riccati technique abstract the conditions on the function c x r n r are derived which ensure that the pde div u p 2 u c x u p 2u 0 p 1 is oscillatory i e belongs to the closure of the set of zeros of every solution defined on the domain x r n x 1 the main tool is the riccati technique combined with suitable a priori bounds

sturmian comparison theorem for half linear second order differential equations let be defined by s s p 2 s with p 1 a fixed number we extend sturm comparison theorem of the linear differential equation to the nonlinear differential equation by using the wirtinger inequality a lyapunov inequality and some oscillation criteria of e are also given

oscillation criteria for second order nonlinear neutral variable delay dynamic equations abstract this paper is concerned with oscillation of second order nonlinear neutral dynamic equations on time scales with a variable delay by using the generalized riccati technique and integral averaging technique new oscillation criteria are obtained for all solutions of the equation some results extend known results for difference equations when the time scale is the set z of positive integers and for differential equations when the time scale is r several examples are given to illustrate the results of the paper

hopf bimodules coquasibialgebras and an exact sequence of kac abstract based on the ideas of tannaka krein reconstruction we present a categorical construction that assigns to any cleft hopf algebra inclusion k h a coquasibialgebra having k as a hopf subalgebra as a special case the construction gives an intrinsic connection between the bismash product k q and the double cross product q k constructed from the same combinatorial data a cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a sweedler three cocycle thus our construction assigns to every bicrossproduct or hopf algebra extension of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a sweedler three cocycle based on this observation we use the construction to prove generalizations of kac s exact sequence for the group of hopf algebra extensions of a group algebra by a dual group algebra

interval criteria for oscillation of second order linear ordinary differential equations abstract new oscillation criteria are established for the equation py qy 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem and sharper than some previous results and can be applied to extreme cases such as t 0 q t dt

a bound on the exponent of primitivity in terms of diameter abstract an n n nonnegative matrix a is called primitive if for some positive integer k every entry in ak is positive ak 0 the exponent of primitivity of a is defined to be a min k z ak 0 where z denotes the set of positive integers two conjectures due to hartwig neumann and lin in 1984 or so are that a m 1 2 1 and a d2 1 where m is the degree of the minimal polynomial of a and d is the diameter of the directed graph of a it is well known that the latter is stronger than the former because d m 1 in a recent paper we have proved a m 1 2 1 in this paper we prove the conjecture a d2 1 on that basis

eigenvalues below the essential spectra of singular elliptic operators on developpe une nouvelle technique pour determiner si le nombre de valeurs propres au dessous du spectre essentiel d un operateur elliptique singulier est fini on donne une methode pour etablir des bornes inferieures au plus petit point du spectre essentiel en fonction du comportement des coefficients et du poids pres des singularites

maximal exponents of polyhedral cones i abstract let k be a proper i e closed pointed full convex cone in r n an n n matrix a is said to be k primitive if there exists a positive integer k such that a k k 0 int k the least such k is referred to as the exponent of a and is denoted by a for a polyhedral cone k the maximum value of a taken over all k primitive matrices a is called the exponent of k and is denoted by k it is proved that if k is an n dimensional polyhedral cone with m extreme rays then for any k primitive matrix a a m a 1 m 1 1 where m a denotes the degree of the minimal polynomial of a and the equality holds only if the digraph e p a k associated with a as a cone preserving map is equal to the unique up to isomorphism usual digraph associated with an m m primitive matrix whose exponent attains wielandt s classical sharp bound as a consequence for any n dimensional polyhedral cone k with m extreme rays k n 1 m 1 1 our work answers in the affirmative a conjecture posed by steve kirkland about an upper bound of k for a polyhedral cone k with a given number of extreme rays

on the algebra structure of some bismash products we study several families of semisimple hopf algebras arising as bismash products which are constructed from finite groups with a certain specified factorization first we associate a bismash product hq of dimension q q 1 q 1 to each of the finite groups pgl2 q and show that these hq do not have the structure as algebras of group algebras except when q 2 3 as a corollary all hopf algebras constructed from them by a comultiplication twist also have this property and are thus non trivial we also show that bismash products constructed from frobenius groups do have the structure as algebras of group algebras

classification of semisimple hopf algebras of dimension 16 abstract in this paper we completely classify nontrivial semisimple hopf algebras of dimension 16 we also compute all the possible structures of the grothendieck ring of semisimple non commutative hopf algebras of dimension 16 moreover we prove that non commutative semisimple hopf algebras of dimension pn p prime cannot have a cyclic group of grouplikes

oscillation theorems for nonlinear differential equations of second order abstract in this paper we present some new criteria for the oscillation of the differential equation a t x t x t p t x t q t f x t 0 the obtained results extend improve and correlate a number of existing criteria

integral averages and oscillation criteria for half linear partial differential equation the technique of weighted integral averages known in the oscillation theory of ordinary differential equations is extended to the half linear partial differential equation e div email protected email protected p email protected u c x u p 2u 0 p 1 this technique is used to obtain new oscillation criteria for e on the unbounded domains

the exponent set of symmetric primitive 0 1 matrices with zero trace abstract we prove that the exponent set of symmetric primitive 0 1 matrices with zero trace the adjacency matrices of the simple graphs is 2 3 2 n 4 s where s is the set of all odd numbers in n 2 n 1 2 n 5 we also obtain a characterization of the symmetric primitive matrices with zero trace whose exponents attain the upper bound 2 n 4

oscillation of the half linear pde in general exterior domains the variational approach abstract in the paper a variational technique is used to show how to derive new oscillation criteria for the half linear partial differential equation via the oscillation criteria for the ordinary half linear differential equation in contrast to the results usually found in the literature general types of unbounded domains can be considered

a diameter bound on the exponent of a primitive directed graph abstract a directed graph g is primitive if there exists a positive integer k such that for every pair of vertices u v g there is a walk from u to v of length k the least such k is called the exponent of g we define g k to be the directed graph having the same vertex set as g and arcs u v if and only if there is a walk in g of length k from vertex u to vertex v a well known upper bound for the exponent of a primitive directed graph g of order n is n 1 2 1 due to h wielandt in 1950 our main result is the following refinement of the wielandt bound if g is a primitive directed graph with diameter d then the exponent of g is at most d 2 1 we construct the primitive graphs for which equality is attained and we generalize the bound to the class of irreducible matrices in the course of proving the main result we find the following which is interesting in its own right if g is a primitive directed graph with diameter d then the diameter of g k is at most d for all positive integers k

self dual modules of semisimple hopf algebras abstract we prove that over an algebraically closed field of characteristic zero a semisimple hopf algebra that has a nontrivial self dual simple module must have even dimension this generalizes a classical result of w burnside as an application we show under the same assumptions that a semisimple hopf algebra that has a simple module of even dimension must itself have even dimension

cohomology of abelian matched pairs and the kac sequence the purpose of this paper is to introduce a cohomology theory for abelian matched pairs of hopf algebras and to explore its relationship to sweedler cohomology to singer cohomology and to extension theory an exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of hopf algebras generalizing those of kac and masuoka for matched pairs of finite groups and finite dimensional lie algebras the morphisms in the low degree part of this sequence are given explicitly enabling concrete computations

proof of a conjecture about the exponent of primitive matrices abstract an n n nonnegative matrix a is called primitive if for some positive integer k every entry in the matrix ak is positive ak 0 the exponent of primitivity of a is defined to be a min k z ak 0 where z denotes the set of positive integers the upper bound on a due to wielandt is a n 1 2 1 and a better bound for a due to hartwig and neumann is a m m 1 where m is the degree of the minimal polynomial of a also hartwig and neumann conjecture that a m 1 2 1 which had been suggested in 1984 in this paper we prove this conjecture

on lewin and vitek s conjecture about the exponent set of primitive matrices m lewin and y vitek conjecture that every integer w 1 i n2 2n 2 1 is the exponent of some n x n primitive matrix in this paper we prove that this conjecture is true except for n 11 the problem of determining the exponent set e is completely solved

oscillation criteria for half linear dynamic equations on time scales this paper is concerned with oscillation of the second order half linear delay dynamic equation r t x p t x t 0 on a time scale t where 1 is the quotient of odd positive integers p t and t t are positive rd continuous functions on t r t is positive and delta differentiable t t and limt t we establish some new sufficient conditions which ensure that every solution oscillates or converges to zero our results in the special cases when t r and t n involve and improve some oscillation results for second order differential and difference equations and when t hn t q n 0 and t n 2 our oscillation results are essentially new some examples illustrating the importance of our results are also included

on the connection between schrodinger and dirichlet forms relations between schrodinger forms associated with schrodinger operators in l2 dnx rn open n 1 and the corresponding dirichlet forms are investigated various concrete examples are presented

oscillation and asymptotics for second order half linear differential equations this paper discusses a class of second order half linear differential equations by using the generalized riccati technique and the averaging technique new oscillation criteria are obtained for all solutions of the equations to be oscillatory asymptotic behavior for forced equations is also discussed

central invariants and frobenius schur indicators for semisimple quasi hopf algebras abstract in this paper we obtain a canonical central element h for each semi simple quasi hopf algebra h over any field k and prove that h is invariant under gauge transformations we show that if k is algebraically closed of characteristic zero then for any irreducible representation of h which affords the character h takes only the values 0 1 or 1 moreover if h is a hopf algebra or a twisted quantum double of a finite group then h is the corresponding frobenius schur indicator we also prove an analog of a theorem of larson radford for split semi simple quasi hopf algebras over any field k using this result we establish the relationship between the antipode s the values of h and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero

post classification version of jordan s theorem on finite linear groups abstract using classification of finite simple groups i show that a finite subgroup g of gln c where c the complex numbers contains a commutative normal subgroup m of index at most n 1 nalogn b moreover if g is primitive and does not contain normal subgroups that are direct products of large alternating groups then the factor n 1 can be dropped i further show that similar statements hold also in characteristics p 2 if one takes m to be an extension of a group of lie type of characteristic p by a solvable group that has a normal p subgroup with commutative p quotient these results improve the celebrated theorems of jordan and of brauer and feit

extensions of hopf algebras and lie bialgebras let f g be finite dimensional lie algebras over a field of characteristic zero regard f and g the dual lie coalgebra of g as lie bialgebras with zero cobracket and zero bracket respectively suppose that a matched pair f g of lie bialgebras is given which has structure maps then it induces a matched pair u f ug of hopf algebras where u f is the universal envelope of f and ug is the hopf dual of ug we show that the group opext u f ug of cleft hopf algebra extensions associated with u f ug is naturally isomorphic to the group opext f g of lie bialgebra extensions associated with f g an exact sequence involving either of these groups is obtained which is a variation of the exact sequence due to g i kac if g g g there follows a bijection between the set ext u f ug of all cleft hopf algebra extensions of u f by ug and the set ext f g of all lie bialgebra extensions of f by g

a problem on the exponent of primitive digraphs abstract a digraph d a is called primitive if and only if a the 0 1 connection matrix of d a is primitive the exponent of primitivity of d a is defined to be d a min k z a k 0 where z denotes the set of positive integers in a recent paper we have proved the conjecture d a m 1 2 1 due to robert e hartwig and michael neumann where m is the degree of the minimal polynomial of a in this paper we characterize the equality case of the upper bound d a m 1 2 1

a bound on the exponent of a primitive matrix using boolean rank we present a bound on the exponent exp a of an n x n primitive matrix a in terms of its boolean rank b b a namely exp a b 1j2 2 further we show that for each 2 q b q n 1 there is an n x n primitive matrix a with b a b such that exp a b 1 2 2 and we explicitly describe all such matrices the new bound is compared with a well known bound of dulmage and mendelsohn and with a conjectured bound of hartwig and neumann several open problems are posed

oscillation and asymptotics for nonlinear second order differential equations this paper discusses a class of second order nonlinear differential equations by using the generalized riccati technique and the averaging technique new oscillation criteria are obtained for all solutions of the equation to be oscillatory asymptotic behavior for forced equations is also discussed

oscillation criteria for second order nonlinear perturbed differential equations abstract some criteria are obtained which guarantee that all solutions of the second order nonlinear perturbed differential equations r t y t p t y t q t y t h t y t y t are oscillatory

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

semisimple commutative algebras with positive bases abstract algebras that serve as models for concurrent studying of certain aspects of both the algebra of ordinary characters and the center of the group algebra of a finite group have been considered by various authors in this article we offer another such model the main differences between our model and the known ones are 1 our model includes brauer characters and principal indecomposable characters as special cases 2 our emphasis is on the eigenvalues of the regular representation of the algebra elements an approach that gives results on values of characters ordinary central brauer principal indecomposable as well as on their products

oscillation of damped pde with p laplacian in unbounded domains abstract in this paper new oscillation theorems for the damped partial differential equation pde with p laplacian div a x u p 2 u b x u p 2 u c x u p 2 u 0 are established in unbounded domains our theorems as two special cases when a i or a i b x 0 improve and complement some existing results in the literature to illustrate our main results we give some corollaries and examples

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

conjugacy criteria for half linear ode in the theory of pde with generalized p laplacian and mixed powers abstract in this paper we use conjugacy criteria for a half linear ordinary differential equation to prove some nonexistence results for a partial differential equation with generalized p laplacian and mixed derivatives div a x y p 2 y b x y p 2 y c x y p 2 y i 1 m c i x y p i 2 y e x the results can be used also to study equations with strongly non radial potential c x since the usual dependence on the mean value of c x over a sphere can be replaced by a more general term

finite quantum groups and quantum permutation groups abstract we give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups we also give examples of finite quantum groups which are not quantum permutation groups one such example occurs as a split abelian extension associated to the exact factorization s 4 z 4 s 3 and has dimension 24 we show that in fact this is the smallest possible dimension that a non quantum permutation group can have

cohomology and coquasi bialgebra extensions associated to a matched pair of bialgebras abstract by using braid diagrams we explicitly reconstruct the cohomology associated to a matched pair of cocommutative bialgebras in order to give a method of constructing coquasi bialgebras which generalize bialgebras and classifying them up to monoidal equivalence of their comodule categories an alternative homological proof is given for schauenburg s generalized kac sequence involving the abelian group opext h k of bialgebra extensions we define an abelian group opext h k of coquasi bialgebra extensions associated to a singer pair h k of bialgebras and prove a variant of schauenburg s sequence which involves the group it is also proved that there is a natural isomorphism opext h 1 k opext h 2 k that preserves monoidal equivalence classes if h 1 k and h 2 k arise from such matched pairs that are shifts of each other

integral averaging techniques for the oscillation of second order nonlinear differential equations abstract in this paper we enlist some known and establish some new oscillation criteria for the second order nonlinear differential equations of the form if a t x t t su p t t q t f x t 0 where the coefficients p t and q t are not assumed to be nonnegative for all large values of t these criteria are obtained by using integral averaging techniques and can be applied to some special cases where other classical oscillation results are not applicable a systematic study is attempted to extend improve and correlate a number of existing results

calculations of some groups of hopf algebra extensions w let us recall the theory of group extensions with abelian kernels mc x chap iv let g be a group and m a g module the equivalence classes of group extensions m a e a g giving rise to the g module action on m form a group with respect to the baer sum furthermore this group is 2 isomorphic to the cohomology group h g m the theory is extended to the theory of hopf algebra extensions which w x are abelian in some sense h s we restrict our interests to hopf algebra extensions over a field k which are of the form

oscillation criteria of nehari type for sturm liouville operators and elliptic differential operators of second order and the lower spectrum on etablit des criteres suffisants d oscillation de type nehari pour l equation differentielle u q t u t 0 0 t avec et sans restriction de signe sur q t on etend ces resultats aux equations de sturm liouville et aux equations elliptiques d ordre 2

bounds for finite primitive complex linear groups abstract in 1878 jordan showed that a finite complex linear group must possess a normal abelian subgroup whose index is bounded by a function of the degree n alone in this paper we study primitive groups when n 12 the optimal bound is n 1 achieved by the symmetric group of degree n 1 we obtain the optimal bounds in smaller degree also our proof uses known lower bounds for the degrees of the faithful representations of each quasisimple group for which the classification of finite simple groups is required in a subsequent paper m j collins on jordan s theorem for complex linear groups j group theory 10 2007 411 423 we will show that n 1 is the optimal bound in general for jordan s theorem when n 71

primitive normal matrices and covering numbers of finite groups abstract a primitive matrix is a square matrix m with nonnegative real entries such that the entries of ms are all positive for some positive integer s the smallest such s is called the primitivity index of m primitive matrices of normal type namely mmt and mtm have the same zero entries occur naturally in studying the so called conjugacy class covering number and character covering number of a finite group we show that if m is a primitive n n matrix of normal type with minimal polynomial of degree m then the primitivity index of m is at most n 2 1 m 1 this bound is then applied to improve known bounds for the various covering numbers of finite groups

oscillatory behavior of solutions of certain second order nonlinear differential equations abstract in this paper some simple inequalities are used to offer sufficient conditions for the oscillation of all solutions of the differential equation formula where 0 p q with p q odd integers or p even and q odd integers several examples which dwell upon the importance of our results are also included

finite dimensional cosemisimple hopf algebras in characteristic 0 are semisimple finite dimensional cosemisimple hopf algebras are a natural generalization of group algebras much of the representation theory of finite groups can be extended to such hopf algebras however there are still many unanswered basic questions for such hopf algebras one such question is a conjecture mentioned by kaplansky i the square of the antipode of a finite dimensional cosemisimple hopf algebra is the identity in this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 in the course of proving this we investigate two elements of the hopf algebra which are roughly analogous to the sum of the group elements in a group algebra we show that kaplansky s conjecture is equivalent to these two elements being equal for a finite dimensional hopf algebra a with antipode s over a field let l p q pq for p q e a denote the left module action of a on itself in this paper we study two elements of a defined by the trace function the element i defined by p a tr l p s for all pe a and the element x defined by p x tr l p for all p e a we show that 2 is a left integral and use it to prove theorem 3 3 the main result of this paper a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 is also semisimple hence its antipode has order 1 2 or 4 we extend the first corollary to 3 proposition 91 in theorems 4 3 and 4 4 by finding necessary and sufficient conditions for s to be a non zero left integral in which case x a and also study how x relates to the structure of a in particular when dim a 1 0 the right ideal xa may be thought of as a measure of the extent to which s i the elements ji and k are two examples of elements of a associated with an endomorphism of a in general for f end a we define a e a by p a tr l p cf for all pea thus i as2 and x a in section 2 267 0021 8693 88 33 00

s4 symmetry of 6j symbols and frobenius schur indicators in rigid monoidal c categories we show that a left rigid monoidal c category with irreducible monoidal unit is also a sovereign and spherical category defining a frobenius schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects as a main result we prove s4 invariance of 6j symbols in such a category

on a subfactor generalization of wallʼs conjecture abstract in this paper we discuss a conjecture on intermediate subfactors which is a generalization of wall s conjecture from the theory of finite groups we explore special cases of this conjecture and present supporting evidence in particular we prove special cases of this conjecture related to some finite dimensional kac algebras of izumi kosaki type which include relative version of wall s conjecture for solvable groups