lattice computers for approximating euclidean space in the context of mesh like parallel processing computers for i approximating continuous space and ii analog simulation of the motion of objects and waves in continuous space the present paper is concerned with which mesh like interconnection of processors might be particularly suitable for the task and why processor interconnection schemes based on nearest neighbor connections in geometric lattices are presented along with motivation then two major threads are exploded regarding which lattices would be good the regular lattices for their symmetry and other properties in common with continuous space and the well known root lattices for being in a sense the lattices required for physically natural basic algorithms for motion the main theorem of the present paper implies that the well known lattice a n is the regular lattice having the maximum number of nearest neighbors among the n dimensional regular lattices it is noted that the only n dimensional lattices that are both regular and root are a n and z n z n is the lattice of n cubes the remainder of the paper specifies other desirable properties of a n including other ways it is superior to z n for our purposes

the classification of endo trivial modules this paper settles the classification of all endo trivial modules for a finite p group by treating the remaining case of a p group having maximal elementary abelian subgroups of rank 2 it is shown that the modules constructed by alperin using relative syzygies provide the complete list of endo trivial modules in this case

genotypes of irreducible representations of finite p groups for any characteristic zero coefficient field an irreducible representation of a finite p group can be assigned a roquette p group called the genotype this has already been done by bouc and kronstein in the special cases q and c a genetic invariant of an irrep is invariant under group isomorphism change of coefficient field galois conjugation and under suitable inductions from subquotients it turns out that the genetic invariants are precisely the invariants of the genotype we shall examine relationships between some genetic invariants and the genotype as an application we shall count galois conjugacy classes of certain kinds of irreps

the cutting stock problem and integer rounding an integer programming problem is said to have the integer round up property if its optimal value is given by the least integer greater than or equal to the optimal value of its linear programming relaxation in this paper we prove that certain classes of cutting stock problems have the integer round up property the proof of these results relies upon the decomposition properties of certain knapsack polyhedra

the dade group of a p group let p be a prime number this paper solves the question of the structure of the group d p of endo permutation modules over an arbitrary finite p group p that was open after dade s original papers in 1978 19 20 and it gives a proof of the conjectures proposed in 4 and 10 this leads to a presentation of d p by explicit generators and relations generalizing the presentation obtained by dade when p is abelian

simple modules over green biset functors we present three examples of green biset functors for which their simple modules can be parametrized these are particular cases of a conjecture by serge bouc classifying the simple modules over a green biset functor a that generalizes the classification of simple biset functors we also prove this conjecture under certain hypothesis for a

the modified integer round up property of the one dimensional cutting stock problem a linear integer minimization problem ip has the modified integer round up property mirup if the optimal value of any instance of ip is not greater than the optimal value of the corresponding lp relaxation problem rounded up plus one the aim of this paper is to investigate numerically whether the mirup holds for the one dimensional cutting stock problem the computational experiments carried out with a lot of randomly generated instances of the one dimensional cutting stock problem show that for all instances an integer solution fulfills the mirup moreover in most cases the optimal value equals the round up optimal value of the corresponding lp relaxation similarly the approach proposed to verify the mirup is usable as a new heuristic procedure for solving one dimensional cutting stock problems this heuristic always leads to very good solutions being optimal in the most cases

a filtration of the modular representation functor abstract let f and k be algebraically closed fields of characteristics p 0 and 0 respectively for any finite group g we denote by k r f g k z g 0 f g the modular representation algebra of g over k where g 0 f g is the grothendieck group of finitely generated f g modules with respect to exact sequences the usual operations induction inflation restriction and transport of structure with a group isomorphism between the finitely generated modules of group algebras over f induce maps between modular representation algebras making k r f an inflation functor we show that the composition factors of k r f are precisely the simple inflation functors s c v i where c ranges over all nonisomorphic cyclic p groups and v ranges over all nonisomorphic simple k out c modules moreover each composition factor has multiplicity 1 we also give a filtration of k r f

tornehave morphisms iii the reduced tornehave morphism and the burnside unit functor abstract we shall show that a morphism anticipated by tornehave induces and helps to explain bouc s isomorphism relating a quotient of the burnside unit functor measuring a difference between real and rational representations of finite 2 groups and a quotient of the kernel of linearization measuring a difference between rhetorical and rational 2 biset functors

decomposition approaches for solving the integer one dimensional cutting stock problem with different types of standard lengths abstract this paper considers the integer one dimensional cutting stock problem with different types of standard lengths and the objective of cost minimization decomposition approaches based on the well known column generation technique for exactly solving the continuous relaxation of the problem and specific methods for separately solving the arising residual problems are proposed for evaluating the performance of the different approaches more than 12 000 randomly generated test problems have been solved the computational experiments prove the effectiveness of one of the proposed decomposition procedures with respect to the computing effort and the quality of the solutions obtained

a ghost ring for the left free double burnside ring and an application to fusion systems abstract for a finite group g we define a ghost ring and a mark homomorphism for the double burnside ring b g g of left free g g bisets in analogy to the case of the burnside ring b g the ghost ring has a much simpler ring structure and after tensoring with q one obtains an isomorphism of q algebras as an application of a key lemma we obtain a very general formula for the brauer construction applied to a tensor product of two p permutation bimodules m and n in terms of brauer constructions of the bimodules m and n over a field of characteristic 0 we determine the simple modules of the left free double burnside algebra and prove semisimplicity results for the bifree double burnside algebra these results carry over to results about biset functor categories finally we apply the ghost ring and mark homomorphism to fusion systems on a finite p group we extend a remarkable bijection due to ragnarsson and stancu between saturated fusion systems and certain idempotents of the bifree double burnside algebra over z p to a bijection between all fusion systems and a larger set of idempotents in the bifree double burnside algebra over q

physics like models of computation abstract reversible cellular automata are computer models that embody discrete analogues of the classical physics notions of space time locality and microscopic reversibility they are offered as a step towards models of computation that are closer to fundamental physics

alcahestic subalgebras of the alchemic algebra and a correspondence of simple modules the unified treatment of the five module theoretic notions transfer inflation transport of structure by an isomorphism deflation and restriction is given by the theory of biset functors introduced by bouc in this paper we construct the algebra realizing biset functors as representations the algebra has a presentation similar to the well known mackey algebra we adopt some natural constructions from the theory of mackey functors and give two new constructions of simple biset functors we also obtain a criterion for semisimplicity in terms of the biset functor version of the mark homomorphism the criterion has an elementary generalization to arbitrary finite dimensional algebras over a field

obstructions for gluing biset functors abstract we develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a biset functor defined on the subquotients of a finite group g the obstruction groups for this theory are the reduced cohomology groups of a category d g whose objects are the sections u v of g where 1 v u g and whose morphisms are defined as a generalization of morphisms in the orbit category using this obstruction theory we calculate the obstruction group for some well known p biset functors such as the dade group functor defined on p groups with p odd

endo p permutation modules this paper introduces the notion of endo p permutation kg module i give a characterization of indecomposable endo p permutation modules with vertex p via their source modules which are exactly the indecomposable endo permutation modules whose classes in the dade group d p are g stable in particular when the normalizer of p controls p fusion i give a classification of sources of endo p permutation modules using the recent classification of the dade group

optimality and suboptimality of multiple description vector quantization with a lattice codebook the asymptotic analysis of multiple description vector quantization mdvq with a lattice codebook for sources with smooth probability density functions pdfs is considered in this correspondence goyal et al 2002 observed that as the side distortion decreases and the central distortion correspondingly increases the quantizer cells farther away from the coarse lattice points shrink in a spatially periodic pattern in this correspondence two special classes of index assignments are used along strategic groupings of central quantizer cells to derive a straightforward asymptotic analysis which provides an analytical explanation for the aforementioned observation mdvq with a lattice codebook was shown earlier to be asymptotically optimal in high dimensions with a curious converging property that the side quantizers achieve the space filling advantage of an n dimensional sphere instead of an n dimensional optimal polytope the analysis presented here explains this behavior readily while central quantizer cells on a uniform lattice are asymptotically optimal in high dimensions the present authors have shown that by using nonuniform rather than uniform central quantizer cells the central side distortion product in an mdsq can be reduced by 0 4 db at asymptotically high rate the asymptotic analysis derived here partially unifies these previous results in the same framework though a complete characterization is still beyond reach

a branch and cut and price algorithm for one dimensional stock cutting and two dimensional two stage cutting abstract the one dimensional cutting stock problem 1d csp and the two dimensional two stage guillotine constrained cutting problem 2d 2cp are considered in this paper the gilmore gomory models of these problems have very strong continuous relaxations providing a good bound in an lp based solution approach in recent years there have been several efforts to attack the one dimensional problem by lp based branch and bound with column generation called branch and price and by general purpose c hv a tal g omory cutting planes in this paper we investigate a combination of both approaches i e the lp relaxation at each branch and price node is strengthened by c hv a tal g omory and g omory mixed integer cuts the branching rule is that of branching on variables of the gilmore gomory formulation tests show that for 1d csp general purpose cuts are useful only in exceptional cases however for 2d 2cp their combination with branching is more effective than either approach alone and mostly better than other methods from the literature

equivalences between blocks of p local mackey algebras let g be a finite group and k o k be a p modular system let r o or k there is a bijection between the blocks of the group algebra and the blocks of the so called p local mackey algebra mu 1 r g let b be a block of rg with abelian defect group d let b be its brauer correspondant in n g d it is conjectured by broue that the blocks rgb and rng d b are derived equivalent here we look at equivalences between the corresponding blocks of p local mackey algebras we prove that an analogue of the broue s conjecture is true for the p local mackey algebras in the following cases for the principal blocks of p nilpotent groups and for blocks with defect 1

composition factors of functors abstract let x be a family of finite groups satisfying certain conditions and k be a field we study composition factors radicals and socles of biset and related functors defined on x over k for such a functor m and for a group h in x we construct bijections between some classes of maximal respectively simple subfunctors of m and some classes of maximal respectively simple k out h submodules of m h we use these bijections to relate the multiplicity of a simple functor s h v in m to the multiplicity of v in a certain k out h module related to m h we then use these general results to study the structure of one of the important biset and related functors namely the burnside functor b k which assigns to each group g in x its burnside algebra b k g k z b g where b g is the burnside ring of g we find the radical and the socle of b k in most cases of x and k for example if k is of characteristic p 0 and x is a family of finite abelian p groups we find the radical and the socle series of b k considered as a biset functor on x over k we finally study restrictions of functors to nonfull subcategories for example we find some conditions forcing a simple deflation functor to remain simple as a mackey functor for an inflation functor m defined on abelian groups over a field of characteristic zero we also obtain a criterion for m to be semisimple in terms of the images of inflation and induction maps on m

a comparative study of the arcflow model and the one cut model for one dimensional cutting stock problems abstract we consider the one dimensional cutting stock problem which consists in determining the minimum number of given large stock rolls that has to be cut to satisfy the demands of certain smaller item lengths besides the standard pattern based approach of gilmore and gomory containing an exponential number of variables several pseudo polynomial formulations were proposed in the last decades much research has dealt with arcflow models their relationship to the standard model and possible reduction methods whereas the one cut approach has not attracted that much scientific interest yet in this paper we aim to compare both alternative formulations from a theoretical and numerical point of view as a theoretical main contribution we constructively prove the equivalence of the continuous relaxations of the one cut model the arcflow model and the pattern based model in particular the relationship between the one cut model and the pattern based model has remained an open question since the one cut approach was proposed moreover in order to make a computational comparison of the arcflow model and the one cut model we present how reduction methods partly originating from arcflow considerations can successfully be transferred to the one cut context furthermore we derive relations between the numbers of variables and constraints in both models and investigate their influences in numerical simulations

foncteurs d ensembles munis d une double action resume sommaire 1 introduction 2 notations generalites 3 ensembles munis d une double action 3 1 notations 3 2 formule de mackey 3 3 decomposition d un morphisme 3 4 sous categories 3 4 1 ensembles p q libres 3 4 2 groupes reductifs 4 foncteurs simples 5 une theorie de vortex et de source 6 une seconde theorie de vortex et source 7 le foncteur de burnside 7 1 un homomorphisme deb g dans end g 7 2 les sous foncteurs du foncteur de burnside en caracteristique 0 7 2 1 premier exemple 7 2 2 les constantesmg n 7 2 3 lesb groupes 7 2 4 les sous foncteurs deb 7 2 5 exemples deb groupes 8 exemples de foncteurs simples 8 1 le cas des foncteurs de mackey globaux 8 2 les foncteurssh k 8 3 certains foncteurs simples en caracteristique 0 9 l algebreegg end g egg 9 1 une base naturelle 9 2 multiplication 9 3 semi simplicite 10 le foncteur des representations complexes 10 1 caracteres 10 2 sous foncteurs et decomposition 10 3 semi simplicite

rational p biset functors abstract in this paper i give several characterizations of rational biset functors over p groups which are independent of the knowledge of genetic bases for p groups i also introduce a construction of new biset functors from known ones which is similar to the yoneda construction for representable functors and to the dress construction for mackey functors and i show that this construction preserves the class of rational p biset functors this leads to a characterization of rational p biset functors as additive functors from a specific quotient category of the biset category to abelian groups finally i give a description of the largest rational quotient of the burnside p biset functor when p is odd this is simply the functor r q of rational representations but when p 2 it is a non split extension of r q by a specific uniserial functor which happens to be closely related to the functor of units of the burnside ring

tighter relaxations for the cutting stock problem in the cutting stock problem csp a given order for smaller pieces has to be cut from larger stock material in such a way that the number of stock material needed is minimal based on the classical integer linear programming model the common solution technique consists of solving the corresponding continuous relaxation problem followed by several heuristics which construct integer solutions in many cases an optimal solution can be obtained quickly in this way but for instances which do not possess the integer round up property the optimality of the solution obtained cannot be verified by means of the lp bound in order to overcome this non satisfactory situation two tighter relaxations of the csp are proposed and results of theoretical and numerical investigations are presented

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

the classification of torsion endo trivial modules the main purpose of this paper is to show that if g is a finite p group which is not cyclic quaternion or semi dihedral then the group of endo trivial g modules is detected on restriction to elementary abelian p subgroups of rank 2 consequently the torsion subgroup of the group of endo trivial modules is trivial for any finite p group except in the known cases of cyclic quaternion and semi dihedral groups this requires a large amount of group cohomology recent bounds for the number of bocksteins in serre s theorem carlson s recent theorem expressing serre s theorem in terms of modules and finally the theory of varieties attached to modules a detection theorem for the torsion subgroup of the dade group of all endo permutation modules is deduced from the main theorem the structure of this torsion subgroup can then be fully described when p is odd using the results of bouc thevenaz

the dade group of mackey functors for p groups abstract we introduce endo permutation mackey functors and the mackey dade group of a p group and study their basic properties we exhibit relations between the mackey dade group of a finite p group p and the dade groups of the weyl groups of all subgroups of p as a result we show that the mackey dade group tensored with q over z is the kernel of the linearization map for mackey functors introduced by the author

a remark on the dade group and the burnside group abstract the object of this note is to show that the formula for tensor induction of relative syzygies in the dade group stated in s bouc j reine angew math 523 2000 113 can be viewed as a special case of a functorial homomorphism from the dual b of the burnside group to the subgroup d of the dade group generated by relative syzygies it follows that there exists a short exact sequence of functors 0 r q b d d tors 0 where r q is the functor of rational representations this may be viewed as an improvement from q to z of theorem d of invent math 139 2000 275

connected components of the category of elementary abelian p subgroups we determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p group g for an odd prime p namely it is p if g has rank at least 3 and it is p 1 if g has rank 2 more precisely if g has rank 2 there are exactly 1 2 p 1 or possibly 3 classes for some 3 groups of maximal nilpotency class

strong fusion control and stable equivalences abstract this article is dedicated to the proof of the following theorem let g be a finite group p be a prime number and e be a p block of g assume that the centraliser c g p of an e subpair p e p strongly controls the fusion of the block e and that a defect group of e is either abelian or for odd p has a non cyclic centre then there exists a stable equivalence of morita type between the block algebras o g e and o c g p e p where o is a complete discrete valuation ring of residual characteristic p this stable equivalence is constructed by gluing together a family of local morita equivalences which are induced by bimodules with fusion stable endo permutation sources broue had previously obtained a similar result for principal blocks in relation with the search for a modular proof of the odd z p theorem thus our theorem points towards a block theoretic analogue of the z p theorem which we state in terms of fusion control and morita equivalences

on the ideals and essential algebras of shifted functors of linear representations abstract we present a study on the shifted green biset functors k r f g of linear f representations with coefficients over k for fields k and f of characteristic zero and a finite group g we provide a criterion for the vanishing of their essential algebras and we prove that the condition of uniqueness of minimal groups for simple modules holds for these functors we give a parametrization of a family of simple k r q g modules we also prove the semisimplicity of the category of c r c g modules by means of an equivalence with a category of modules over a semisimple algebra

biset functors and genetic sections for p groups abstract in this note i show that if f is a biset functor defined over finite p groups then for each finite p group p there is a direct summand of f p admitting a natural direct sum decomposition indexed by the irreducible rational representations of p or equivalently by the equivalence classes of origins in p or also by equivalence classes of genetic sections of p this leads to a description of the torsion part of the group of relative syzygies in the dade group of p and to a conjecture on the structure of the torsion part of the whole dade group of p

borel smith functions and the dade group we show that there is an exact sequence of biset functors over p groups 0 cb jb d 0 where cb is the biset functor for the group of borel smith functions b is the dual of the burnside ring functor d is the functor for the subgroup of the dade group generated by relative syzygies and the natural transformation is the transformation recently introduced by the first author in s bouc a remark on the dade group and the burnside group j algebra 279 2004 180 190 we also show that the kernel of mod 2 reduction of is naturally equivalent to the functor b of units of the burnside ring and obtain exact sequences involving the torsion part of d mod 2 reduction of cb and b

rhetorical biset functors rational p biset functors and their semisimplicity in characteristic zero rhetorical biset functors can be defined for any family of finite groups that is closed under subquotients up to isomorphism the rhetorical p biset functors almost coincide with the rational p biset functors we show that over a field with characteristic zero the rhetorical biset functors are semisimple and furthermore they admit a character theory involving primitive characters of automorphism groups of cyclic groups

the group of endo permutation modules the group d p of all endo permutation modules for a finite p group p is a finitely generated abelian group we prove that its torsion free rank is equal to the number of conjugacy classes of non cyclic subgroups of p we also obtain partial results on its torsion subgroup we determine next the structure of q otimes d viewed as a functor which turns out to be a simple functor s e q indexed by the elementary group e of order p 2 and the trivial out e module q finally we describe a rather strange exact sequence relating q otimes d p q otimes b p and q otimes r p where b p is the burnside ring and r p is the grothendieck ring of qp modules

a cutting plane algorithm for the one dimensional cutting stock problem with multiple stock lengths abstract a cutting plane approach combining chvatal gomory cutting planes with column generation is generalized for the case of multiple stock lengths in the one dimensional cutting stock problem appropriate modifications of the column generation procedure and the rounding heuristic are proposed a comparison with the branch and price method for the problem with one stock type and representative test results are reported

an induction theorem for the unit groups of burnside rings of 2 groups abstract let g be a 2 group and b g denote the group of units of the burnside ring of g for each subquotient h k of g there is a generalized induction map from b h k to b g defined as the composition of inflation and multiplicative induction maps we prove that the product of generalized induction maps b h k b g is surjective when the product is taken over the set of all subquotients that are isomorphic to the trivial group or a dihedral 2 group of order 2 n with n 4 as an application we give an algebraic proof for a theorem by tornehave the unit group for the burnside ring of a 2 group aarhus universitet preprint series 1983 84 41 may 1984 which states that tom dieck s exponential map from the real representation ring of g to b g is surjective we also give a sufficient condition for the surjectivity of the exponential map from the burnside ring of g to b g

the functor of rational representations for p groups abstract let k be a field of characteristic p 0 let c p k be the category whose objects are the finite p groups morphisms are the k linear combinations of bisets and composition of morphisms is obtained by k linear extension from the usual product of bisets let f p k denote the category of k linear functors from c p k to the category of k vector spaces this paper investigates the structure of the functor kr q mapping a p group p to kr q p k z r q p as an object of f p k where r q p is the grothendieck group of the category of finite dimensional q p modules the main result is an explicit description of the lattice of all subfunctors of kr q in particular for p odd it is shown that kr q is a uniserial object in f p k for p 2 the lattice of subfunctors of kr q can be described as the lattice of closed subsets of a graph whose vertices are the 2 groups of normal 2 rank 1 in both cases a composition series of kr q is obtained which leads to a formula giving the dimension of the evaluations sq k p of the simple objects sq k of f p k associated to p groups q of normal p rank 1 different from cp this formula can be phrased in terms of rational representations of p but also in terms of the geometry of the lattice of subgroups of p using the notions of basic subgroups and origins for example if p 2 the dimension of s 1 k p is equal to the number of absolutely irreducible q p modules