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

asymptotic properties of high order yule walker estimates of the ar parameters of an arma time series the high order yule walker equations are used to estimate the autoregressive parameters of an autoregressive moving average time series the asymptotic statistical properties of these estimates are derived it is shown that they are asymptotically unbiased and normal the covariance matrix of the limit distribution is derived the special case of estimating the autoregressive parameters of a noise corrupted autoregressive series is also examined

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

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

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

blocks with the hyperfocal subgroup z2n z2n abstract in this paper we calculate the numbers of irreducible ordinary characters and irreducible brauer characters in a block of a finite group g whose associated fusion system over a 2 subgroup p of g which is a defect group of the block has hyperfocal subgroup z 2 n z 2 n for a positive integer number n when the block is controlled by the normalizer n g p and the hyperfocal subgroup is contained in the center of p or when the block is not controlled by n g p and the hyperfocal subgroup is contained in the center of the unique essential subgroup in the fusion system and has order at most 16 in particular alperin s weight conjecture holds in the considered cases

asymptotically stable invariant manifold for coupled nonlinear parabolic hyperbolic partial differential equations abstract this article considers a coupled system of nonlinear parabolic and hyperbolic partial differential equations which arises in the study of wave phenomena which are heat generating or temperature related under appropriate conditions for example high thermal diffusivity it is proved that there exists an invariant manifold for the full system of equations the asymptotic stability of the invariant manifold is also considered moreover it is shown that an equilibrium which is asymptotically stable for flows on the invariant manifold will be asymptotically stable for the full system

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

signal plus noise models in shape classification abstract in this study we show how a signal plus noise formulation can be used to improve the performance of auto regressive shape classifiers in environments with time varying noise characteristics variations of this type may be due to changes in atmospheric conditions or the relative effect of discretization errors this approach is compared to existing methodologies using a number of noisy templates in a cluttered environment typical applications are to aircraft identification and automatic signature validation

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

the gluing problem for some block fusion systems abstract we answer the gluing problem of blocks of finite groups linckelmann 2004 7 4 2 for tame blocks and the principal p block of psl 3 p for p odd in particular we show that the gluing problem for the principal p block of psl 3 p does not have a unique solution when p 1 mod 3

large sample identification and spectral estimation of noisy multivariate autoregressive processes when large samples are available identification and spectral estimation problems of a noisy multivariate autoregressive process are solved independently of the probability law governing the observed data the procedure of solving the problems is divided into 1 estimate parameters by solving pertinent block toeplitz and hankel matrices as well as compute the modified akaike information criterion for an assumed order 2 select the order and the parameter estimators corresponding to the ones which minimize the criterion function 3 compute the asymptotic distribution functions of the parameter estimators 4 estimate power spectral density functions and coherency spectra 5 compute the estimate confidence bounds these five steps are numerically very efficient because the matrix equations related to block hankel and toeplitz matrices are well studied in the literature the procedure is also robust because it is independent of the probability law governing the observed data

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

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

slow manifold for a nonlocal stochastic evolutionary system with fast and slow components abstract this work is devoted to investigating invariant manifolds for a fast slow stochastic evolutionary system with nonlocal diffusion we establish the slow reduction via a random slow manifold which captures slow dynamics of the original stochastic fast slow system a simple example is shown to illustrate this slow reduction method

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

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

upper semicontinuity of attractors and synchronization abstract in this paper we prove that diffusively coupled abstract semilinear parabolic systems synchronize we apply the abstract results obtained to a class of ordinary differential equations and to reaction diffusion problems the technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem explicitly exhibited in the diagonal

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

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

2 blocks with minimal nonabelian defect groups abstract we study numerical invariants of 2 blocks with minimal nonabelian defect groups these groups were classified by redei see redei 1947 41 if the defect group is also metacyclic then the block invariants are known see sambale 43 in the remaining cases there are only two infinite families of interesting defect groups in all other cases the blocks are nilpotent we prove brauer s k b conjecture and olsson s conjecture for all 2 blocks with minimal nonabelian defect groups for one of the two families we also show that alperin s weight conjecture and dade s conjecture are satisfied this paper is a part of the author s phd thesis

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

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

vertices sources and green correspondents of the simple modules for the large mathieu groups we investigate the simple modules for the sporadic simple mathieu groups m22 m23 and m24 as well as those of the automorphism group the covering groups and the bicyclic extensions of m22 in characteristics 2 and 3 we determine the vertices and sources as well as the green correspondents of these simple modules we also find two 3 blocks with elementary abelian defect groups of order 9 in these groups which are morita equivalent to their brauer correspondents

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

the persistence of synchronization under environmental noise it is shown that the synchronization of dissipative systems persists when they are disturbed by additive noise no matter how large the intensity of the noise provided asymptotically stable stationary stochastic solutions are used instead of asymptotically stable equilibria

synchronization by noise we provide sufficient conditions for synchronization by noise i e under these conditions we prove that weak random attractors for random dynamical systems consist of single random points in the case of sde with additive noise these conditions are also essentially necessary in addition we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point as a result synchronization by noise is proven for a large class of sde with additive noise in particular we prove that the random attractor for an sde with drift given by a multidimensional double well potential and additive noise consists of a single random point all examples treated in tearne probab theory relat fields 141 1 2 1 18 2008 are also included

master slave synchronization and invariant manifolds for coupled stochastic systems we deal with abstract systems of two coupled nonlinear stochastic infinite dimensional equations subjected to additive white noise type process this kind of systems may describe various interaction phenomena in a continuum random medium under suitable conditions we prove the existence of an exponentially attracting random invariant manifold for the coupled system and show that this system can be reduced to a single equation with modified nonlinearity this result means that under some conditions we observe nonlinear synchronization phenomena in the coupled system our applications include stochastic systems consisting of i parabolic and hyperbolic equations ii two hyperbolic equations and iii klein gordon and schrodinger equations we also show that the random manifold constructed converges to its deterministic counterpart when the intensity of noise tends to zero

on the glauberman and watanabe correspondences for blocks of finite p solvable groups if g is a finite p solvable group for some prime p a a solvable subgroup of the automorphism group of g of order prime to g such that a stabilises a p block b of g and acts trivially on a defect group p of b then there is a morita equivalence between the block b and its watanabe correspondent w b of c g a given by a bimodule m with vertex p and an endo permutation module as source which on the character level induces the glauberman correspondence and which is an isotypy by watanabe s results

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

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

autoregressive spectral estimation in additive noise the estimation of the spectral density of a discrete time stationary gaussian autoregressive process ar p from a finite set of noise observations is considered a modified spectral estimator based on the high order yule walker equations is considered joint asymptotic normality of this spectral estimator is established a precise asymptotic expression for the covariance matrix of the limiting distribution is obtained the special case of ar 1 plus noise is considered in some detail

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

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

application of the lattice filter to robust estimation of ar and arma models a method for estimating the parameters of an autoregressive moving average arma model is proposed a p q stage lattice whitening filter is used to obtained consistent estimates of the ar parameters of an arma p p model it is shown that a set of ma parameter estimates can also be obtained using this approach with little added computation the method is used for estimation of the ar parameters of an arma process in additive white noise it is shown that the lattice filter is also useful in robust estimation of the ar parameter on an arma process with additive outliers

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

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

asymptotic properties of the high order yule walker estimates of sinusoidal frequencies the asymptotic properties of the high order yule walker hoyw estimator of sinusoidal frequencies are analyzed an explicit formula for the covariance matrix of the hoyw frequency estimation error is derived the effects on estimation accuracy of the number of yw equations the model order and a certain weighting matrix are investigated analytically the optimally weighted hoyw estimator is described the analysis presented together with the already existing empirical evidence provides enough motivation for the use of an overdetermined and high order yule walker estimator

the growth of random walks and levy processes let x i be a sequence of independent identically distributed non degenerate random variables taking values in mathbb r d and s n sum n i 1 x i m n max 1 leqq i leqq n s i define for x 0 g x p x 1 x k x x 2 e x 1 2 1 x 1 leq x m x x 1 e x 1 1 x 1 leq x and h x g x k x m x then if beta sup alpha lim sup x alpha h x 0 delta sup alpha lim inf x alpha h x 0 it is proved that n 1 alpha m n rightarrow 0 for alpha delta while the lim inf is 0 and the lim sup is infty for beta alpha delta some alternative characterizations of the indices beta delta are obtained as well as the analogous results for levy processes

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

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

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