exponential decay for a von karman equations with memory in this paper we consider a von karman equations with memory term we show an exponential decay result of solutions under weaker assumption than the ones frequently used in the literature in particular the kernel we are considering is not necessarily exponentially decaying to zero as was assumed before

classifying blocks with abelian defect groups of rank 3 for the prime 2 abstract in this paper we classify all blocks with defect group c 2 n c 2 c 2 up to morita equivalence together with a recent paper of wu zhang and zhou this completes the classification of morita equivalence classes of 2 blocks with abelian defect groups of rank at most 3 the classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field the case considered in this paper is significant because it involves comparison of morita equivalence classes between a group and a normal subgroup of index 2 so requires novel reduction techniques which we hope will be of wider interest we note that this also completes the classification of blocks with abelian defect groups of order dividing 16 up to morita equivalence a consequence is that broue s abelian defect group conjecture holds for all blocks mentioned above

general decay of solutions of a viscoelastic equation abstract in this paper we consider the following viscoelastic equation u t t u 0 t g t u d 0 in a bounded domain and establish a general decay result which is not necessarily of exponential or polynomial type this work generalizes and improves earlier results in the literature

blocks with trivial intersection defect groups we show that each block whose defect groups intersect pairwise trivially either has cyclic or generalised quaternion defect groups or is morita equivalent to one of a given list of blocks of central extensions of automorphism groups of non abelian simple groups in particular we classify all blocks of automorphism groups of non abelian simple groups whose defect groups are non cyclic and intersect pairwise trivially a consequence is that donovans conjecture holds for blocks whose defect groups intersect pairwise trivially

a class of blocks behaving like blocks of p solvable groups abstract we define a class of blocks having similar properties to blocks of p solvable groups and show that a version of the fong swan theorem holds for irreducible brauer characters in such blocks we also show that the height of an irreducible character in such block is bounded by the exponent of the central quotient of a defect group which in particular implies that if further the defect groups are abelian then every irreducible character in the block has height zero

energy decay rates for the semilinear wave equation with memory boundary condition and acoustic boundary conditions in this paper we consider the energy decay rates for the semilinear wave equation with memory boundary condition and acoustic boundary conditions motivated by results of gerbi and said houari 2011 2008 li and zhao 2011 liu and chen 2016 wu etal 2010 and lu etal 2011 we intend to study the energy decay rates for problem 1 1 1 6 by using the perturbed energy method and riemannian geometry method we obtained general energy decay rates

on the focal defect group of a block characters of height zero and lower defect group multiplicities abstract we discuss the focal subgroup of the defect group d of a p block b which we refer to as the focal defect group and denote by d 0 we note that the character group of d d 0 acts in a defect or height preserving fashion on irreducible characters in b and prove that the action on irreducible characters of height zero is semi regular we also prove that all orbits under this action have length divisible by z d d 0 z d as applications we prove that all cartan invariants for b are divisible by z d d 0 z d that if out d is a p group and d 1 then the number of irreducible characters of height zero in b is divisible by p and that if z d d 0 then the block b is of lefschetz type see r knorr g r robinson some remarks on a conjecture of alperin j london math soc 2 39 1 1989 48 60

nilpotent blocks of quasisimple groups for odd primes we investigate the nilpotent blocks of positive defect of the quasisimple groups for odd primes in particular it is shown that every nilpotent block of a quasisimple group has abelian defect groups a conjecture of puig concerning the recognition of nilpotent blocks is also shown for these groups

blocks with frobenius inertial quotients abstract in the paper we prove that a perfect isometry between a block of a finite group and its brauer correspondent can be extended to a perfect isometry between a block of a larger finite group and its brauer correspondent with suitable hypotheses we also obtain a similar result for derived equivalences

on decay and blow up of the solution for a viscoelastic wave equation with boundary damping and source terms abstract in this paper we consider the decay and blow up properties of a viscoelastic wave equation with boundary damping and source terms we first extend the decay result for the case of linear damping obtained by lu et al on a viscoelastic equation with nonlinear boundary damping and source terms global existence and decay of the solution nonlinear analysis real world applications 12 1 2011 295 303 to the nonlinear damping case under weaker assumption on the relaxation function g t then we give an exponential decay result without the relation between g t and g t for the linear damping case provided that g l 1 0 is small enough finally we establish two blow up results one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases the other is for certain solutions with arbitrarily positive initial energy for the linear damping case

an alternating sum for hochschild cohomology of a block inthispaper wepointoutthatwhenp isaprime theadditivestruc ture of the hochschild cohomology ring of a p block of a nite groupg satis es the now familiar type of alternating sum formula using thep subgroupcomplex ofg throughout we work with the tate versionofhochschildcohomology whichcoincideswithordinaryhochschildcoho mologyinpositivedimensions suchanexpression inoursituation allowsthehochschildcohomologyoftheblocktobecomputedfromp localinfor mation the rst such formula for the cohomology of a g module wasobtainedbyp j webb see forexample 6 wealsogiveaformulationofourresultintermsofsubpairs forillustrativepurposes wegiveanappli cationinthesituationwhentheblockiscontrolledbythenormalizerofamaximalsubpair andtheouterautomorphimsofthedefectgroupinducedby the inertial quotient all act without nontrivial xed points this appli cationgeneralizesanearlierresultofkessarandlinckelmann 2 whichdealtwithsuchasituationinthecasethatthedefectgroupisabelian

donovan s conjecture blocks with abelian defect groups and discrete valuation rings we give a reduction to quasisimple groups for donovan s conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring mathcal o consequences are that donovan s conjecture holds for mathcal o blocks with abelian defect groups for the prime two and that using recent work of farrell and kessar for arbitrary primes donovan s conjecture for mathcal o blocks with abelian defect groups reduces to bounding the cartan invariants of blocks of quasisimple groups in terms of the defect a result of independent interest is that in general i e for arbitrary defect groups donovan s conjecture for mathcal o blocks is a consequence of conjectures predicting bounds on the mathcal o frobenius number and on the cartan invariants as was proved by kessar for blocks defined over an algebraically closed field

extensions of nilpotent blocks over arbitrary fields in this paper with a suitable condition we describe the algebraic structure of block extensions of nilpotent blocks over arbitrary fields thus generalize the main result of b kulshammer and l puig on block extensions of nilpotent blocks over algebraically closed fields

refinements of dade s projective conjecture for p solvable groups dade s projective conjecture is known to be true for finite p solvable groups thanks to work of g r robinson but remains open in general work of isaacs and navarro suggested to uno and boltje refinements of this conjecture these refinements were studied for finite p solvable groups by glesser in the present paper inspired by earlier work of turull we propose further refinements of the conjecture that take into account the schur indices and the elements of the brauer group we prove that all these refinements of dade s projective conjecture hold for all finite p solvable groups in particular we obtain that the version of dade s projective conjecture which involves character degree residues modulo p fields of definition and schur indices as well as the full strength of boltje s conjecture both hold for all finite p solvable groups the proof develops a clifford theory for normalizers of chains of p subgroups which allows one to reduce the calculation of the relevant sums to simpler groups

on convexity for energy decay rates of a viscoelastic equation with boundary feedback abstract in this paper we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary we establish an explicit and general decay rate result using some properties of the convex functions our result is obtained without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function

extensions of rickard equivalences broue conjectures that a block of a finite group g with abelian defect group p is rickard equivalent to its brauer correspondent in the normalizer ng p we find a character theoretic criterion on extensions of rickard equivalences between blocks as an application we prove that the conjecture holds for the principal block of a finite group with abelian defect group and cyclic hyperfocal subgroup

blocks with defect group z2n z2n z2m abstract in this paper we prove that a block algebra with defect group z 2 n z 2 n z 2 m where n 2 and m is arbitrary is morita equivalent to its brauer correspondent

second order differential invariants of a family of diffusion equations an equivalence transformation algebra for a class of nonlinear diffusion equations is found after having obtained the second order differential invariants with respect to we get some results which allow us to linearize a subclass of the equations considered

frictional versus viscoelastic damping in a semilinear wave equation in this article we show exponential and polynomial decay rates for the partially viscoelastic nonlinear wave equation subject to a nonlinear and localized frictional damping the equation that models this problem is given by begin eqnarray u tt kappa 0 delta u int 0 t mbox div a x g t s nabla u s ds f u b x h u t 0 mbox in omega times bbb r quad end eqnarray where a b are nonnegative functions a in c 1 overline omega b in l infty omega satisfying the assumption begin eqnarray a x b x geq delta 0 quad forall x in omega end eqnarray and f and h are power like functions we observe that the assumption 0 2 gives us a wide assortment of possibilities from which to choose the functions a x and b x and the most interesting case occurs when one has simultaneous and complementary damping mechanisms taking this point of view into account a distinctive feature of our paper is exactly to consider different and localized damping mechanisms acting in the domain but not necessarily strategically localized dissipations as considered in the prior literature

equivariant alperin robinsonʼs conjecture reduces to almost simple k groups in a recent paper gabriel navarro and pham huu tiep show that the so called alperin weight conjecture can be verified via the classification of the finite simple groups provided any simple group fulfills a very precise list of conditions our purpose here is to show that the equivariant refinement of the alperin s conjecture for blocks formulated by geoffrey robinson in the eighties can be reduced to checking the same statement on any central k extension of any finite almost simple group or of any finite simple group up to verifying an almost necessary condition in appendix a we develop some old arguments that we need in the proof

energy decay rates via convexity for some second order evolution equation with memory and nonlinear time dependent dissipation abstract the stabilization of the following abstract integro differential equation u t a u t 0 t g t s a u s d s q t u t f u t is investigated we establish the general decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g without imposing any restrictive growth assumption on the damping at the origin and strongly weakening the usual assumption of the relaxation function g our approach is based on the multiplier method and make use of some properties of the convex functions these decay results can be applied to various concrete models we shall study some examples to illustrate our result

local control in fusion systems of p blocks of finite groups abstract if p is an odd prime b a p block of a finite group g such that sl 2 p is not involved in n g q e c g q for any b subpair q e then n g z j p controls b fusion where p is a defect group of b this is a block theoretic analogue of glauberman s zj theorem several results of general interest about fusion and blocks are also proved

uniform stabilization of the wave equation by nonlinear boundary feedback the question of uniformly stabilizing the solution of the wave equation y delta y 0 in omega times 0 infty omega is a bounded domain of bf r n by means of a nonlinear feedback law of the following form is studied partial y partial v k x g y on gamma 0 times 0 infty y 0 on gamma 1 times 0 infty gamma 0 gamma 1 being a suitable partition of the boundary of omega and g a continuous nondecreasing function such that g 0 0 we choose k x in l infty k x geqq 0 such that k x vanishes linearly at the interface points x in bar gamma 0 cap bar gamma 1 then if g s behaves like s p 1 s as s to 0 with p 1 and linearly as s to infty it is proved that the energy of every solution decays like t 2 p 1 as t to infty in the case where p 1 the exponential decay rate is proved

fusion category algebras the fusion system f on a defect group p of a block b of a finite group g over a suitable p adic ring o does not in general determine the number l b of isomorphism classes of simple modules of the block we show that conjecturally the missing information should be encoded in a single second cohomology class of the constant functor with value k on the orbit category f c of f centric subgroups q of p of b which glues together the second cohomology classes q of autf q with values in k in k ulshammer puig 13 1 8 we show that if exists there is a canonical quasi hereditary k algebra f b such that alperin s weight conjecture becomes equivalent to the equality l b l f b by work of broto levi oliver 3 the existence of a classifying space of the block b is equivalent to the existence of a certain extension category l of fc by the center functor z if both invariants l exist we show that there is an o algebra l b associated with b having f b as quotient such that alperin s weight conjecture becomes again equivalent to the equality l b l l b furthermore if b has an abelian defect group l b is isomorphic to a source algebra of the brauer correspondent of b

on donovan s conjecture abstract this article contains reduction theorems for some weaker variants of donovan s conjecture which supposes that for every finite group d of prime order p there are only finitely many morita equivalence classes of p blocks of finite groups having defect groups isomorphic to d i when restricting the conjecture to the case of abelian defect groups it can be reduced to only considering p blocks of a class of groups that are close to being direct products of simple groups ii when restricting the conjecture to the case of principal blocks of finite groups having an abelian sylow p subgroup it can be reduced to only considering p blocks of simple groups a weaker version of donovan s conjecture would suppose that there are only finitely many cartan matrices of p blocks of finite groups with defect group isomorphic to a given one iii this conjecture can be reduced to only considering p blocks of quasisimple groups with centers having orders prime to p

some refinements of dade s projective conjecture abstract in this paper a further refinement of dade s projective conjecture due to boltje is presented this new statement includes ideas first published by isaacs and navarro as well as the recent contractibility version of alperin s conjecture introduced by boltje leaning heavily on the work of robinson weaker forms of the conjecture are proved in the case of p solvable groups

towards donovan s conjecture for abelian defect groups abstract we define a new invariant for a p block of a finite group the strong frobenius number which we use to address the problem of reducing donovan s conjecture to normal subgroups of index p as an application we use the strong frobenius number to complete the proof of donovan s conjecture for 2 blocks with abelian defect groups of rank at most 4 and for 2 blocks with abelian defect groups of order at most 64

group graded basic morita equivalences abstract we introduce group graded basic morita equivalences between algebras determined by blocks of normal subgroups and by using the extended brauer quotient we show that they induce graded basic morita equivalences at local levels

affirmative answer to a question of feit as he points out if p is abelian the affirmative answer follows from dade s classification 4 our purpose here is to show that the general case reduces to the abelian one which answers affirmatively feit s question more generally in 4 dade determines the cap group associated with a finite abelian p group showing in particular that this group is finitely generated our main result here cf theorem 2 2 below implies that finite generation is still true in the non abelian case cf corollary 2 4 below this result and the main steps of a proof were already announced in 17 section81 and here we are just discharging ourselves of an old debt actually as we said in 7 section81 our result would follow from a complete classification of endo permutation modules

energy decay rates for von karman system with memory and boundary feedback abstract in this paper we study the von karman plate model with long range memory and boundary nonlinear feedback we establish an explicit and general decay rate result using some properties of the convex functions our result is obtained without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function

2 blocks with abelian defect groups we give a classification up to morita equivalence of 2 blocks of quasi simple groups with abelian defect groups as a consequence we show that donovan s conjecture holds for elementary abelian 2 groups and that the entries of the cartan matrices are bounded in terms of the defect for arbitrary abelian 2 groups we also show that a block with defect groups of the form c2m c2m for m 2 has one of two morita equivalence types and hence is morita equivalent to the brauer correspondent block of the normaliser of a defect group this completes the analysis of the morita equivalence types of 2 blocks with abelian defect groups of rank 2 from which we conclude that donovan s conjecture holds for such 2 groups a further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16 the proof uses the classification of finite simple groups

stability of wave equation with locally viscoelastic damping and nonlinear boundary source abstract in this paper we consider a nonlinear wave equation with nonlinear boundary damping and source terms subject to a localized viscoelastic damping and localized frictional damping we obtain the global existence of solution via potential well method furthermore by introducing suitable lyapunov functionals using multiplier method and constructing convex differential inequality we compare the effect of locally viscoelastic term and frictional damping and establish an explicit and general decay rate result this stability result allows the localized viscoelastic damping and localized frictional damping acting simultaneously or complementarily in the domain which gives the competing relationship of the viscoelastic and frictional damping

existence and decay of solutions of a viscoelastic equation with a nonlinear source abstract in a bounded domain we consider u tt u 0 t g t u d u u where 0 and g is a nonnegative and decaying function we prove a local existence theorem and show for certain initial data and suitable conditions on g and that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

integrating factors adjoint equations and lagrangians integrating factors and adjoint equations are determined for linear and non linear differential equations of an arbitrary order the new concept of an adjoint equation is used for construction of a lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables the method is illustrated by considering several equations traditionally regarded as equations without lagrangians noether s theorem is applied to the maxwell equations

general decay rate estimates for viscoelastic dissipative systems abstract the linear viscoelastic equation is considered we prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity the proofs are based on the multiplier method and on a general lemma about convergent and divergent series for obtaining the uniform decay rates

on some saturated triples and hyperfocal subgroups abstract let b denote a primitive idempotent of the subalgebra k h g in kh of g stable elements where h is a normal subgroup of a finite group g which acts by conjugation on h and k is an algebraically closed field of characteristic p using a generalization of block induction to the case of primitive idempotents in algebras of the form k h g we show that when the hyperfocal subgroup is abelian the fusion system associated to b respectively to the induced stable primitive idempotent of the generalized brauer pair corresponding to the hyperfocal subgroup are the same

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

decay rates of solutions of an anisotropic inhomogeneous n dimensional viscoelastic equation with polynomially decaying kernels we consider the anisotropic and inhomogeneous viscoelastic equation and we prove that the first and second order energy decay polynomially as time goes to infinity when the relaxation function also decays polynomially to zero that is if the kernelg ijkl satisfies dot g ijkl leqq c 0 g ijkl 1 frac 1 p and g ijkl g ijkl 1 frac 1 p in l 1 mathbb r for p 2such that 2 m 1 p then the first and second order energy decay as frac 1 1 t q withq 2 m 1

general decay of the solution energy in a viscoelastic equation with a nonlinear source abstract in a bounded domain we consider u t t u 0 t g t u d u u where 0 and g is a nonnegative and decaying function we prove that for certain class of relaxation functions and certain initial data the rate of decay of energy is similar to that of g this result improves earlier ones in the literature in which only the exponential and polynomial decay rates are considered

nonlinear self adjointness a criterion for linearization of pdes in this communication we show that the property of nonlinear self adjointness is conserved by an invertible point transformation we use this property to give necessary conditions for the existence of an invertible point transformation mapping of a given partial differential equation pde to a linear pde

amalgams blocks weights fusion systems and finite simple groups abstract we associate an iterated amalgam of finite groups to a certain class of fusion systems on finite p groups p a prime in such a way that the p group of the fusion system is a maximal finite p subgroup of the resulting group unique up to conjugacy and furthermore the conjugation action of the resulting usually infinite group on p subgroups induces the original fusion system on the p group 1 in view of earlier work of puig and of broto castellana grodal levi and oliver c broto n castellana j grodal r levi r oliver subgroup families controlling p local finite groups proc london math soc 3 91 2005 325 354 the fusion systems we deal with include all saturated fusion systems the resulting amalgam has free normal subgroups of finite index and we examine the images of the group by its maximal free normal subgroups of finite index these images all contain isomorphic copies of the original p group and are generated by the images of the finite groups used in the amalgamation if there is no non trivial normal p subgroup of the fusion system equivalently if the iterated amalgam constructed has no non trivial normal p subgroup then the generalised fitting subgroup of each of these homomorphic images is a direct product of non abelian simple groups each of order divisible by p and if there is no proper non trivial strongly closed p subgroup in the fusion system then the generalised fitting subgroup of any of these finite groups is characteristically simple we note that the finite dimensional representation theory of this amalgam is almost by construction p locally determined in the case of the fusion system associated to a p block of a finite group we suggest strong links between block theoretic invariants of the above finite epimorphic images of the associated amalgam and of the original block we believe that the results of this paper offer the prospect of at least the beginnings of a structural explanation for some of the current numerical conjectures in block theory e g j l alperin weights for finite groups in proc sympos pure math vol 47 amer math soc providence 1987 pp 369 379 e c dade counting characters in blocks i invent math 109 1 1992 187 210

zj theorems for fusion systems for p an odd prime we generalise the glauberman thompson p nilpotency theorem 5 ch 8 theorem 3 1 to arbitrary fusion systems we define a notion of qd p free fusion systems and show that if f is a qd p free fusion system on some finite p group p then f is controlled by w p for any glauberman functor w generalising glauberman s zj theorem 3 to arbitrary fusion systems

on the p extensions of inertial blocks assume that a block is inertial and has abelian defect groups we prove that a bimodule inducing a basic morita equivalence between this block and its brauer correspondence admits an abelian equivariant p extension

on blocks with trivial source simple modules abstract motivated by an observation in danz and kulshammer 2009 3 we determine the source algebra and therefore all the structure of the blocks without essential brauer pairs where the simple modules of all the brauer corespondents have trivial sources

exponential and polynomial decay for a quasilinear viscoelastic equation abstract we consider the following nonlinear viscoelastic equation u t u t t u u t t 0 t g t s u s d s 0 in a bounded domain for 0 we show that the dissipation induced by the integral term is strong enough to stabilize the solution this result improves an earlier one given by cavalcanti et al in m m cavalcanti v n d cavalcanti j ferreira existence and uniform decay for nonlinear viscoelastic equation with strong damping math methods appl sci 24 2001 1043 1053

asymptotic stability of semigroups associated with linear weak dissipative systems with memory we study the asymptotic behavior of the solutions of a class of linear dissipative integral differential equations we show in the abstract setting a necessary and sufficient condition to get an exponential decay of the solution in the case of the lack of exponential decay we find the polynomial rate of decay of the solution some examples are given

general decay and blow up of solutions for a system of viscoelastic wave equations with nonlinear boundary source terms abstract in this work an initial boundary value problem for a system of viscoelastic wave equations with nonlinear boundary source term of the form u i t t u i u i t t 0 t g i t s u i s d s u i t 0 in 0 t u i x 0 i x u i t x 0 i x in u i x t 0 on 0 0 t u i t t u i 0 t g i t s u i s d s u i t f i u 0 on 1 0 t where i 1 l l 2 is considered in a bounded domain in r n n 1 by the faedo galerkin approximation method we obtain existence and uniqueness of weak solutions under appropriate assumptions on initial data and the relaxation functions we establish general decay and blow up results associated to solution energy estimates for lifespan of solutions are also given

extra special defect groups of order p3 and exponent p2 abstract in this paper we give results on p blocks with the defect groups isomorphic to an extra special group of order p3 we are particularly interested in the number of irreducible ordinary characters and the number of irreducible brauer characters in the block the situation splits naturally into two cases according to the exponent of the extra special group in this paper we concentrate on the exponent p2 case we prove that olsson s conjecture theorem 4 6 and brauer s k b conjecture proposition 5 13 hold for the exponent p2 case we are also able to calculate two important block invariants k b l b and l b m g b 1 1 in this case

the gluing problem in the fusion systems of the symmetric alternating and linear groups a solution to linckelmann s gluing problem in all fusion systems of blocks yields an alternative formulation of alperin s weight conjecture using bredon equivariant cohomology techniques we solve the gluing problem in fusion systems of blocks which are isomorphic to the fusion systems of the symmetric groups or the alternating groups or gld q or sld q at the defining characteristic

asymptotic behavior of the energy for a class of weakly dissipative second order systems with memory a class of second order abstract systems with memory and dirichlet boundary conditions is investigated by suitable liapunov functionals existence of solutions as well as asymptotic behavior are determined in particular when the memory kernel decays exponentially the polynomially decay of the solutions is proved

quasi self adjoint nonlinear wave equations in this paper we generalize the classification of self adjoint second order linear partial differential equation to a family of nonlinear wave equations with two independent variables we find a class of quasi self adjoint nonlinear equations which includes the self adjoint linear equations as a particular case the property of a differential equation to be quasi self adjoint is important e g for constructing conservation laws associated with symmetries of the differential equation

alperin s weight conjecture in terms of equivariant bredon cohomology alperin s weight conjecture 1 admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system 7 of a fusion system of a block which in turn can be interpreted as the equivariant bredon cohomology of a certain functor on the g poset of centric brauer pairs the underlying general constructions of categories and functors needed for this reformulation are described in 1 and 2 respectively and 3 provides a tool for computing the cohomology of the functors arising in 2 taking as starting point the alternating sum formulation of alperin s weight conjecture by knorr robinson 11 the material of the previous sections is applied in 4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors using further work of robinson 15 16 17

nonlinear self adjointness and conservation laws the general concept of nonlinear self adjointness of differential equations is introduced it includes the linear self adjointness as a particular case moreover it embraces the strict self adjointness definition 1 and quasi self adjointness introduced earlier by the author it is shown that the equations possessing nonlinear self adjointness can be written equivalently in a strictly self adjoint form by using appropriate multipliers all linear equations possess the property of nonlinear self adjointness and hence can be rewritten in a nonlinear strictly self adjoint form for example the heat equation ut u 0 becomes strictly self adjoint after multiplying by u 1 conservation laws associated with symmetries are given in an explicit form for all nonlinearly self adjoint partial differential equations and systems

2 blocks whose defect group is homocyclic and whose inertial quotient contains a singer cycle abstract we consider a block b of a finite group with defect group d c 2 m n and inertial quotient e containing a singer cycle an element of order 2 n 1 this implies e e f where e c 2 n 1 f c n and e acts transitively on the elements in d of order 2 we classify the basic morita equivalence classes of b over a complete discrete valuation ring o when m 1 b is basic morita equivalent to the principal block of one of s l 2 2 n f d e or j 1 where j 1 occurs only when n 3 when m 1 b is basic morita equivalent to d e