on a conjecture concerning integral real roots of certain cubic polynomials abstract in this paper we determine all the rational pairs x n such that f n x 3 x 3 7 10 n x 2 2 6 n 2 11 n 8 x 4 n 3 6 n 2 10 n 24 0 it follows that for each positive integer n 5 there is no integer solution x for the polynomial this confirms a conjecture of li et al concerning the uniqueness of a class of optimal graphs in the study of an extremal graph theory problem

graded morita theory for infinite groups abstract having shown that not all graded rings are graded equivalent via classical morita theory to a skew group ring we extend classical morita theory which is based on rings with identity to a generalized graded morita theory for rings with local units this enables us to give necessary and sufficient conditions for graded morita equivalence between two rings graded by a group we show that the strongly graded property is a graded morita invariant and we show that a graded ring is graded morita equivalent to a skew group ring namely r g g if and only if it is strongly graded these results extend the classical cohen montgomery duality theory to rings graded by infinite groups the fundamental tools include rings with local units and the results of boisen

weyl groups of fine gradings on matrix algebras octonions and the albert algebra given a grading a g gag on a nonassociative algebra a by an abelian group g we have two subgroups of aut a the automorphisms that stabilize each component ag as a subspace and the automorphisms that permute the components by the weyl group of we mean the quotient of the latter subgroup by the former in the case of a cartan decomposition of a semisimple complex lie algebra this is the automorphism group of the root system i e the so called extended weyl group a grading is called fine if it cannot be refined we compute the weyl groups of all fine gradings on matrix algebras octonions and the albert algebra over an algebraically closed field of characteristic different from 2 in the case of the albert algebra

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

the brauer splitting theorem and projective representations of finite groups over rings in the theory of representations of finite groups the projective representations arise naturally when one studies the relations between the representations of the group and representations of certain subgroups the role played by the group rings in representation theory is taken by the twisted groups rings when one considers projective representations for representations over algebraically closed fields the theory of schur multipliers provides a very satisfactory tool that may be used to lift projective representations of a finite group to usual representations of a finite central extension of that group furthermore there exists a satisfactory theory of projective characters block theory for projective representations etc for which we may refer to karpilovsky s recent book 3 since for finite groups the twisted group rings defined over commutative rings are azumaya algebras when a mild condition holds it is natural to ask for an equivalent of the brauer splitting theorem for group rings in this paper we derive this splitting theorem for twisted group rings over commutative rings without nontrivial idempotents this is obtained by introducing a suitable extension of the schur multiplier theorem in such a way that this result may be applied to a single representation and over arbitrary fields or rings of the forementioned type along the way we get involved in some cocycle manipulations that will lead to the determination of the center of a twisted group ring in terms of ray classes the results derived in this paper should be of some use in the study of properties and

group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

graded identities of matrix algebras and the universal graded algebra in the last decade group gradings and graded identities of finite dimensional central simple algebras have been an active area of research we refer the reader to bahturin et al 6 and 7 there are two basic kinds of group grading elementary and fine it was proved by bahturin and zaicev 7 that any group grading of mn c is given by a certain composition of an elementary grading and a fine grading in this paper we are concerned with fine gradings on mn c and their corresponding graded identities let r be a simple algebra finite dimensional over its center k and g a finite group we say that r is fine graded by g if r g grg is a grading and dimk rg 1 thus any component is either 0 or isomorphic to k as a k vector space it is easy to show that supp r the subset of elements of g for which rg is not 0 is a subgroup of g moreover

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

an elementary proof of the blow up for semilinear wave equation in high space dimensions abstract this paper concerns the blow up of solutions to u tt u u p in high dimensions for n 4 and 1 p p 0 n where p 0 n is a critical exponent we proved that the solutions blow up in finite time by estimating the solutions near the wave front using elementary inequalities

finite time blow up for critical wave equations in high dimensions we prove that solutions to the critical wave equation 1 1 with dimension n 4 can not be global if the initial values are positive somewhere and nonnegative this completes the solution to the famous strauss conjecture about semilinear wave equations of the form u t2u u p 0 the rest of the cases the lower dimensional case n 3 and the sub or super critical cases were settled many years earlier by the work of several authors

on the existence and nonexistence of global solutions for certain semilinear exterior problems with nontrivial robin boundary conditions abstract we consider three types of semilinear equations elliptic parabolic and hyperbolic posed in the n dimensional exterior domain r n d where n 2 and d is the closed unit ball in r n a nontrivial robin boundary condition is imposed on the boundary of d using a test function approach with judicious choices of the test functions we show that the considered problems share a common critical behavior we discuss separately the cases n 2 and n 3 moreover in the case n 3 the dependence of the critical exponent on initial data is discussed to the best our knowledge the study of the critical behavior in an exterior domain with a nontrivial robin boundary condition has never been studied in the literature

global existence for the dissipative wave equations with space time dependent potential abstract this research paper aims at studying the long time behavior of solutions for wave equations with focusing nonlinearity we consider p c r n p n 2 n 2 and separable potential a t x x t with x 1 x and t 1 t satisfying conditions 3 4 which are given below the main result is that if the exponent p lies between p c r 1 4 1 2 n 1 2 p n 2 n 2 the solutions are global for sufficiently small data also the precise decay estimates for the energy l 2 and l p 1 norms of solutions in the region of global solutions have been found

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

graded artin algebras a graded artin algebra is a graded ring which neglecting the grading is an artin algebra although this definition has the merit of brevity it does not expose some of the basic properties of graded artin algebras for instance that there is a bound on absolute values of degrees of homogeneous elements indeed a different definition is given in section 1 and the discrepancy is rectified by theory 1 4 we admit that our chief interest in graded artin algebras lies in applications to the representation theory of artin algebras see 5 1 we feel nonetheless that graded artin algebras are interesting objects of study in their own right and some of our techniques especially in sections 5 and 6 are equally applicable to arbitrary graded rings thus this paper unlike its sequel 51 is designed to be read by a general ring theoretical audience it is in addition self contained from the standpoint of knowledge concerning graded rings required the major concern of the paper is the existence of gradings on given modules over a graded artin algebra we use the term gradable when a gradation exists our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable it follows immediately see section 3 that simple modules and projective modules over a graded artin algebra are gradable also inasmuch as we do not restrict ourselves to positively graded rings endomorphism rings of graded finitely generated modules over graded artin algebras are graded artin algebras in a similar vein duality and horn

adams inequalities for riesz subcritical potentials abstract we derive adams inequalities for potentials on general measure spaces extending and improving previous results in fontana and morpurgo 2011 the integral operators involved which we call riesz subcritical have kernels whose decreasing rearrangements are not worse than that of the riesz kernel on r n where the kernel is large but they behave better where the kernel is small the new element is a critical integrability condition on the kernel at infinity typical examples of such kernels are fundamental solutions of nonhomogeneous differential or pseudo differential operators another example is the riesz kernel itself restricted to suitable measurable sets which we name riesz subcritical domains such domains are characterized in terms of their growth at infinity as a consequence of the general results we obtain several new sharp adams and moser trudinger inequalities on r n on the hyperbolic space on riesz subcritical domains and on domains where the poincare inequality holds

hopf galois extensions and an exact sequence for h picard groups let h be a hopf algebra and a an h galois extension we investigate h morita autoequivalences of a introduce the concept of h picard group and we establish an exact sequence linking the h picard group of a and the picard group of acoh

a shift in the strauss exponent for semilinear wave equations with a not effective damping abstract in this note we study the global existence of small data solutions to the cauchy problem for the semilinear wave equation with a not effective scale invariant damping term namely v t t v 2 1 t v t v p v 0 x v 0 x v t 0 x v 1 x where p 1 n 2 we prove blow up in finite time in the subcritical range p 1 p 2 n and existence theorems for p p 2 n n 2 3 in this way we find the critical exponent for small data solutions to this problem our results lead to the conjecture p 2 n p 0 n 2 for n 2 where p 0 n is the strauss exponent for the classical semilinear wave equation with power nonlinearity

projective representations and relative semisimplicity abstract let r be a local commutative ring and let p be a prime not invertible in r let g be a finite group of order divisible by p it is well known that the group ring rg admits nonprojective lattices e g r itself with the trivial action for any element h2 g r one can form the twisted group ring r g the twisting problem asks whether there exists a class s t the corresponding twisted group ring admits only projective lattices for fields of characteristic p the answer is in e aljadeff and d j s robinson j pure appl algebra94 1994 1 15 here we answer this question for rings of the form z ps s 2 the main tools are the classification of modular representation of the klein 4 group over z 2 and a chouinard like theorem e aljadeff and y ginosar j algebra179 1996 599 606 for twisted group rings

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

reduction techniques for strongly graded rings and finite representation type abstract we present reduction techniques for studying the category of lattices over strongly graded orders in particular we apply these techniques in order to reduce the problem of classifying those strongly graded orders with finite representation type to the case where the coefficient ring is a maximal order in a division ring

group graded rings and duality we give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend known results on skew group rings to corresponding results for large classes of group graded rings finally we modify the construction slightly to handle infinite groups introduction in the first section we give an alternate construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend results on skew group rings and their modules to corresponding results for large classes of groupgraded rings in the second section we modify the construction slightly to handle infinite groups and apply it to a theorem on prime and semiprime infinite crossed products i would like to thank my thesis advisor d s passman for his guidance and encouragement throughout the writing of this paper 1 finite group graded rings and the smash product let g be a multiplicative group an associative ring with identity is said to be g graded if 1 r e r x xec is a direct sum of additive subgroups r x with r x r y c r xy it follows that necessarily 1r e r 1 so that each r x is a unitary r l bimodule if r e r we write r x for the component of r in r x so that r exgcr x r is said to be strongly g graded if r x r y r xy for all x y e g throughout this section r is assumed to be g graded where g is a finite group with igi n let mc r denote the set of n x n matrices over r with the rows and columns indexed by the elements of g if a c mg r we write ax y for the entry in the x y position of a then if a 1 c mc r the matrix product a 8 is given by 2 x x y e xx zaz y z e c if u c g is any subset of g let r u ex eur x in particular r r g now suppose h c g is a subgroup of g we define r h c mc r by r h a e ms r ax y e r xhy l that is r h ex ygcr xhy l e x y where e x y is the matrix unit with 1r in the x y position and zeroes elsewhere note that r g mc r received by the editors september 27 1984 1980 mathematics subject classification primary 16a03 c 1985 american mathematical society 0002 9947 85 1 00 25 per page

the equation utt δu u p for the critical value of p the equation u tt u u p is considered in two and three space dimensions smooth cauchy data of compact support are given at t 0 for the case of three space dimensions john has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if glassey has shown the two dimensional case is similar this paper shows that small data solutions blow up in finite time when p is the critical value in three dimensions and in two

the signless laplacian coefficients and the incidence energy of the graphs without even cycles abstract let n m be the set of all connected graphs of order n with m edges without even cycles where n m 3 2 n 1 we consider the signless laplacian coefficients and the incidence energy in n m two new graph transformations are proposed here among n m it is obtained that there is exactly one minimal element with respect to the quasi ordering according to their signless laplacian coefficients furthermore the graph with the minimal incidence energy is found among n m

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

actions of picard groups on graded rings throughout this paper g denotes a group with identity e if r is a g graded ring we write r g grg and we refer to re as the coefficient ring we denote the category of graded left r modules i e those left r modules m with a g grading m g gmg such that rhmg mhg by r gr the aim of this paper is to present a method to reduce the study of one strongly graded ring r to the study of another strongly graded ring r that is more tractable this reduction process has two aspects reducing the coefficient ring or reducing the grading since we apply this method in our study of module theoretic properties such as semisimplicity for graded rings we require that this reduction process preserve the category of modules r mod and the category of graded modules r gr this leads us to the notion of graded equivalence in this paper a morita context is the usual tuple a b p q where we assume and are isomorphisms a g graded context is a morita context r r p q where r and r are g graded rings rpr and r qr are graded bimodules i e they have a grading which makes them into both graded r modules and graded r modules on the corresponding side and and are graded bimodule homomorphisms see 14 for details about graded modules and homomorphisms given a functor f r gr r gr we say that f is a graded equivalence of categories if any of the following equivalent conditions hold see for example 8

existence and blow up of small amplitude nonlinear waves with a sign changing potential abstract we study the nonlinear wave equation with a sign changing potential in any space dimension if the potential is small and rapidly decaying then the existence of small amplitude solutions is driven by the nonlinear term if the potential induces growth in the linearized problem however solutions that start out small may blow up in finite time

on the laplacian coefficients of signed graphs abstract let g be a signed graph where g is its underlying graph and its sign function defined on edges of g a signed graph the subgraph of is its signed tu subgraph if the signed graph induced by the vertices of consists of trees and or unbalanced unicyclic signed graphs let l d g a be the laplacian of in this paper we express the coefficient of the laplacian characteristic polynomial of based on the signed tu subgraphs of and establish the relation between the laplacian characteristic polynomial of a signed graph with adjacency characteristic polynomials of its signed line graph and signed subdivision graph as an application we identify the signed unicyclic graphs having extremal coefficients of the laplacian characteristic polynomial

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

semilinear schrodinger flows on hyperbolic spaces scattering in h 1 we prove global well posedness and scattering in h1 for the defocusing nonlinear schrodinger equations left begin array ll i partial t delta g u u u 2 sigma u 0 phi end array right on the hyperbolic spaces mathbb h d d 2 for exponents sigma in 0 2 d 2 the main unexpected conclusion is scattering to linear solutions in the case of small exponents for comparison on euclidean spaces scattering in h1 is not known for any exponent sigma in 1 d 2 d and is known to fail for sigma in 0 1 d our main ingredients are certain noneuclidean global in time strichartz estimates and noneuclidean morawetz inequalities

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

on the laplacian coefficients of acyclic graphs abstract let g be a graph of order n and let g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix zhou and gutman recently proved that among all trees of order n the kth coefficient ck is largest when the tree is a path and is smallest for stars a new proof and a strengthening of this result is provided a relation to the wiener index is discussed

finite time blow up to critical semilinear wave equation outside the ball in 3 d abstract in this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3 d blow up result will be established no matter how small the data are we find a radial and positive solution of the corresponding linear problem then our blow up result follows from the idea of zhou and han 2014 what is more we get the upper bound of the lifespan estimate

homological dimension of skew group rings and crossed products abstract in this paper we study the homological dimension of skew group rings and crossed products a sufficient condition for r g a crossed product to have finite right global dimension is given in terms of crossed products over simple artinian factors of r if r is right fbn and left coherent and g is finite some necessary conditions and sufficient conditions for r g a skew group ring of a finite group over a local or semilocal right noetherian ring to have finite right global dimension are also given then in particular if r is commutative noetherian and g is finite we obtain some equivalent conditions for r g a skew group ring to have finite global dimension using work of aljadeff e aljadeff serre s extension theorem for crossed products j london math soc 44 1991 47 54 these results are extended to polycyclic by finite groups

wave equation on riemannian symmetric spaces we consider the wave equation on riemannian symmetric spaces of the noncompact type we prove that the solutions of the homogeneous equation have dispersion properties and we deduce strichartz type estimates for these solutions we use these estimates to treat nonlinear equations with power type as a byproduct we prove the global well posedness for small energy

semisimple strongly graded rings is semisimple meaning in this paper semisimple artinian has been studied byseveral authors the most classical result is maschke s theorem for group rings for crossed products over elds there is a satisfactory answer given by aljadeffand robinson 3 another partial answer for skew group rings was given byalfaro et al 1 a reduction of the problem to crossed products over divisionrings was rst given by jespers and okninski 10 and a more constructiveversion was given by haefner and del rio 8 so in order to give a complete answer tothe problem there is still a gap between crossed products over division rings andcrossed products over elds the rst aim of the paper is to ll this gap showingthat the semisimplicity question for crossed products over division rings reducesto the same question for crossed products over elds in theorem 3 2 we makethis reductionand then in theorem3 3 we put togetherall the piecesof the puzzle a strongly

on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

the group of automorphisms of certain subalgebras of matrix algebras the purpose of this note is to find and characterize the full group of automorphisms of certain prime algebras which are subalgebras of matrix algebras over commutative rings here we extend the methods which we used in 6 to characterize automorphisms of certain upper triangular matrix rings for a central simple k algebra a the group of k automorphisms of a is extensively treated in the literature 5 theorem 6 21 other cases where the automorphism group of a ring is known are the automorphism group of a polynomial ring in two indeterminates over a field and the free algebra in two variables over a field 4 chap 61 finally if a is a prime p z algebra with center c then every c automorphism of a is x inner 7 example 3 91

fourier integral operators on noncompact symmetric spaces of real rank one let x g k be a noncompact symmetric space of real rank one the purpose of this paper is to investigate lp boundedness properties of a certain class of radial fourier integral operators on the space x we will prove that if u is the solution at some fixed time of the natural wave equation on x with initial data f and g and 1 p then u lp x cp f lpbp x 1 g lpbp 1 x we will obtain both the precise behavior of the norm cp and the sharp regularity assumptions on the functions f and g i e the exponent bp that make this inequality possible in the second part of the paper we deal with the analog of e m stein s maximal spherical averages and prove exponential decay estimates of a highly non euclidean nature on the lp norm of supt t 1 f d z where d is a normalized spherical measure

the picard group of a structural matrix algebra abstract we investigate the picard group of a structural matrix or incidence algebra a of a finite preordered set p over a field and we consider five interrelated problems our main technique involves establishing a connection between the group of outer automorphisms out a of a and the group of outer automorphisms of the basic algebra a which is an incidence algebra of the associated partially ordered set p of p we discuss necessary and sufficient conditions for out a to be a natural invariant for the morita equivalence class of a and necessary and sufficient conditions for m n k to be strongly graded by a group g and coefficient ring a containing n primitive orthogonal idempotents

strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation the aim of this article is twofold first we consider the wave equation in the hyperbolic space hi and obtain the counterparts of the strichartz type estimates in this context next we examine the relationship between semilinear hyperbolic equations in the minkowski space and in the hyperbolic space this leads to a simple proof of the recent result of georgiev lindblad and sogge on global existence for solutions to semilinear hyperbolic problems with small data shifting the space time strichartz estimates from the hyperbolic space to the minkowski space yields weighted strichartz estimates in rn x 1r which extend the ones of georgiev lindblad and sogge

descending chain conditions and graded rings abstract the structure of a group graded ring r satisfying certain classical finiteness conditions is described module the homogeneous part of the jacobson radical jgr r it is shown that r jgr r is a finite direct product of matrix rings over group crossed products over division rings in the more general case of a semigroup graded ring r the structure of r module its jacobson radical can be described in terms of finitely many group graded subrings these subrings are shown to inherit the considered finiteness conditions of r as an application we derive results that show when a graded ring is artinian semiprimary or perfect

representation theory of graded artin algebras having initiated the study of graded artin algebras a in 91 here we initiate the study of their representation theory our point of view is to study graded i modules which we believe to be somewhat more tractable than ungraded ones in order to obtain information about all i modules this point of view leads to the introduction in section 1 of the full subcategory mod of mod i consisting of the gradable objects of mod i that is the finitely generated i modules which support a gradation in these terms one of the major results of the paper asserts that mod has finite representation type if and only if mod i has finite representation type thus in section 3 we introduce a number g g 4 designed to measure the size of mod i if g co we show that i has infinite representation type if g co we show that there is a graded artin algebra q of a certain specified form such that i has infinite representation type precisely when 0 has infinite representation type the artin algebra fl has in particular desirable diagrammatic properties and these we will exploit elsewhere we speculate that when g is finite every finitely generated i module is gradable now in case n has finite representation type it is obvious that g is finite we show indeed that for every gradation of a given artin algebra of finite representation type every module is gradable this result and the others cited are chiefly consequences of a result proved in section 4 if a component of the auslander reiten graph of a graded artin algebra contains a gradable module then the component

on the laplacian coefficients of unicyclic graphs abstract let g be a graph of order n and let p g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix generalizing an approach of mohar on graph transformations we show that among all connected unicyclic graphs of order n the k th coefficient c k is largest when the graph is a cycle c n and smallest when the graph is the a s n with an additional edge between two of its pendent vertices a relation to the recently established laplacian like energy of a graph is discussed

the representation theory of fully group graded algebras let g be a finite group k an algebraically closed field of characteristic p and r a fully g graded k algebra of finite dimension the most trivial example of such an algebra is kg this paper attempts to generalize the p modular representation theory of the group algebra kg to the groupgraded algebra r it is shown that most of the theory is the same for r as for kg two areas in which the representation theory of r differs from that of kg are the dimensions of induced modules and the properties of the defect groups of the blocks of r if h is a subgroup of g and v is an r module then the dimension of vr is not necessarily divisible by the dimension of v if b is a block of r and d is a defect group of b then d is a p group as would be expected however it is not necessarily true that d is a sylow intersection nor that d o n d in the first section we define what it means for an algebra to have a g graded structure we show how to give such a structure to the algebra m k of r by r matrices over the field k whenever igl r a number of basic results about fully g graded algebras are then given the second section deals with r modules proofs are given of the mackey decomposition formula the existence of vertices and sources and the green correspondence these three results were already known to hold for r for example they are mentioned in 6 but proofs have not previously been given in the literature they are proved here to make this paper more or less self contained many of the proofs of supporting results for these theorems will not appear as they are easy generalizations of the known proofs for r kg the reader will often be referred to 1 for these proofs the third and last section contains new results the burry carlson puig theorem is generalized and a new definition of the defect groups of a block

on weak equivalences of gradings abstract when one studies the structure e g graded ideals graded subspaces radicals or graded polynomial identities of graded algebras the grading group itself does not play an important role but can be replaced by any other group that realizes the same grading here we come to the notion of weak equivalence of gradings two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component the following question arises naturally when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group it turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading namely a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group the same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset we show that if n 349 then on the full matrix algebra m n f there exists an elementary group grading that is not weakly equivalent to any grading by a finite semi group and if n 3 then any elementary grading on m n f is weakly equivalent to an elementary grading by a finite group

the signless laplacian coefficients and incidence energy of bicyclic graphs abstract let q g x det x i q g i 1 n 1 i i x n i be the characteristic polynomial of the signless laplacian matrix of a graph g of order n this paper investigates how the signless laplacian coefficients i e coefficients of q g x change after some graph transformations these results are used to prove that the set b n of all bicyclic graphs of order n has exactly two minimal elements with respect to the partial ordering of their coefficients furthermore we present a sharp lower bound for the incidence energy of bicyclic graphs of order n and characterize all extremal graphs

group gradings on full matrix rings we study g gradings of the matrix ring mn k k a field and give a complete description of the gradings where all the elements ei j are homogeneous called good gradings among these we determine the ones that are strong gradings or crossed products if g is a finite cyclic group and k contains a primitive g th root of 1 we show how all g gradings of mn k can be produced in particular we give a precise description of all c2 gradings of m2 k and show that for algebraically closed k any such grading is isomorphic to one of the two good gradings

morita equivalences induced by bimodules over hopf galois extensions let h be a hopf algebra and a b be h galois extensions we investigate the category mbha of relative hopf bimodules and the morita equivalences between a and b induced by them

on the signless laplacian coefficients of unicyclic graphs let g be a graph of order n and let qg x i 0n 1 ipi g xn i be the characteristic polynomial of the signless laplacian of g let eg n respectively cg sn g 1 denote the unicyclic graph of order n obtained by a coalescence of a vertex in the cycle cg with an end vertex respectively the center of the path pn g 1 respectively the star sn g 1 it is proved that for k 2 n 1 as g varies over all unicyclic graphs of order n depending on k and n the maximum value of pk g is attained at g cn or e3 n and the minimum value is attained uniquely at g c4 sn 3 or c3 sn 2 except for the resolution of a conjecture on cubic polynomials the uniqueness issue for the maximization problem is also settled

extending irreducible modules an important part of clifford s theory does not work properly for arbitrary fields r to see why we must recall the main steps of this theory we are interested in extensions to rh modules of a fixed h invariant irreducible rn module a where n is a normal subgroup of a finite group h we first pass to the factor algebra u rh 3 modulo the ideal 3 3 rh rh3 generated by the annihilator 3 of 8 in rn if sth is given its natural grading by the factor group g h n see 6 2 below then 3 is a graded ideal so that u is a strongly g graded ring in the sense of 3 or section 1 below whose l component u rn d is a finite dimensional simple ralgebra with i as its irreducible module it follows that the extensions of 8 to rh modules are just the pull backs of the extensions of 2i to 2l modules via the natural epimorphism of rh onto u when 52 is algebraically closed or more generally when i is split over r then 2i is the kronecker product

graded equivalence theory with applications we develop the foundations for graded equivalence theory and apply them to investigate properties such as primeness finite representation type and vertex theory of graded rings the key fact that we prove is that for any two g graded rings r and s such that there is a category equivalence from gr r to gr s that commutes with suspensions then for any subgroup h of g the categories gr h g r and gr h g s of modules graded by the g set of right cosets are also equivalent