rings of invariants for modular representations of the klein four group we study the rings of invariants for the indecomposable modular representations of the klein four group for each such representation we compute the noether number and give minimal generating sets for the hilbert ideal and the field of fractions we observe that with the exception of the regular representation the hilbert ideal for each of these representations is a complete intersection

perturbation based successive approximate topology optimization for a displacement minimization problem abstract this paper discusses a new approach for displacement based topology optimization of continuum topology optimization of continuum can be performed with the optimization of material distribution this optimization problem will include a large number of design variables and it may require a high computational cost especially for computation of gradient components with respect to an objective function and determination of an appropriate updating increment a conventional approximate optimization technique may not be effective for this problem because it is a very large scale problem from this viewpoint a new optimization technique which is based on a successive approximation algorithm is proposed the proposed method consists of two main processes 1 a linear approximation for determination of an updating direction and 2 a nonlinear approximation for determining a scale coefficient of the updating vector the first approximation is performed using the perturbation based finite element analysis and the second approximation is performed using the kriging method in order to investigate effectiveness of the proposed method it is applied to some typical examples for topology optimization problem from the numerical results effectiveness and validity of the proposed method are discussed

an investigation concerning optimal design of solid elastic plates abstract we consider the problem of maximizing the integral stiffness of solid elastic plates described by thin plate theory assuming the material volume and plate domain to be given we use the plate thickness function as the design variable and take both maximum and minimum allowable thickness values into account on the basis of a convenient tensorial formulation of the problem where the governing equations are derived by variational analysis and constitute necessary conditions for stationarity we develop an efficient and quite general numerical algorithm by means of which a number of stationary solutions for rectangular and axisymmetric annular plates with various boundary conditions are obtained these numerical results enable us to investigate the optimization problem itself in terms of its major parameters particularly the maximum and minimum values specified for the plate thickness for problems associated with large ratios between these constraint values plate designs with significant integral stiffeners are obtained we find however that these designs are only local optima and that a global optimal plate thickness function does generally neither exist within the class of smooth functions nor within the class of smooth functions with a finite number of discontinuities in order to determine a global optimal solution associated with given thickness constraint values it is therefore necessary to change the optimal design formulation with implications for a number of similar two dimensional optimization problems our results offer valuable indications of the lines along which such changes should be performed

second order sensitivity derivatives in structural analysis haug s 1981 expressions for the calculation of static constraint function second derivatives have resulted in an order of magnitude saving over conventional methods for the calculation of such derivatives a similar but simplified procedure is presented which results in a reduction of computational cost by an additional factor of two the results are presented in a notation that is different from that of haug but more common in other work on structural synthesis and analysis

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

on the optimality of strategies for multiple join relational database systems rely on the join operator to assemble data for answering queries although the order of natural joins here called the strategy for computing the joins does not affect the final result it does determine to a large extent the response time of the query query optimizers therefore try to pick an optimal strategy in practice optimizers usually restrict their search for an optimal strategy to strategies that are linear e g of the form r 1 r 2 r 3 r 4 or that avoid cartesian products or both the purpose of this paper is to examine the conditions under which an optimizer can find an optimal strategy despite having restricted the scope of its search

element connectivity parameterization for topology optimization of geometrically nonlinear structures in the current element density based topology optimization element stiffness is penalized to yield either solid or void or very weak materials but low density elements appear during and even after optimization iterations especially when nonlinear problems involving large deformation are considered low density finite elements cause serious numerical problems as their tangent stiffness matrices lose positive definiteness to completely eliminate the problem caused by low density elements we propose a new method all finite elements are kept solid throughout the optimization process zero length elastic links are introduced to parameterize inter element connectivity the link stiffness is penalized in this approach the design variables are defined on the links and vary from 0 and 1 corresponding to the unconnected and rigidly connected states since the finite elements used to discretize the analysis domain always remain solid typical numerical problems encountered by the standard element density based formulation disappear to implement the present method several issues such as the handling of the mass constraint and raster imaging are also investigated

on the hilbert series of hochschild cohomology of block algebras we show that the degrees and relations of the hochschild cohomology of a p block algebra of a finite group over an algebraically closed field of prime characteristic p are bounded in terms of the defect groups of the block and that for a fixed defect d there are only finitely many hilbert series of hochschild cohomology algebras of blocks of defect d the main ingredients are symonds proof of benson s regularity conjecture and the fact that the hochschild cohomology of a block is finitely generated as a module over block cohomology which is an invariant of the fusion system of the block on a defect group

least weight design of perforated elastic plates ii abstract the paper investigates implications of some recent mathematical developments in the fields of shape optimization and relaxation of variational problems considering the least weight design of perforated elastic plates in either flexure or plane stress for a prescribed compliance it is shown that at low rib densities microstructures consisting of a combination of first and second order infinitesimal ribs is superior to those consisting of purely first order infinitesimal ribs moreover it is indicated that thin ribs of infinite length thickness ratio do not contribute significantly to the stiffness in a direction normal to their plane on the basis of this conclusion a simple specific cost function is derived and then it is used in the design of circular uniformly loaded perforated plates with zero value of poisson s ratio as a basis for comparison several intuitively selected topographies are optimized for the case of simply supported plates and in part ii of this study a variational analysis is used to obtain the optimal solutions for plates with simply supported clamped or loaded edges

using semi joins to solve relational queries the semi join is a relational algebraic operation that selects a set of tuples in one relation that match one or more tuples of another relation on the joining domains semi joins have been used as a basic ingredient in query processing strategies for a number of hardware and software database systems however not all queries can be solved entirely using semi joins in this paper the exact class of relational queries that can be solved using semi joins is shown it is also shown that queries outside of this class may not even be partially solvable using short semi join programs in addition a linear time membership test for this class is presented

avoiding cartesian products for multiple joins computing the natural join of a set of relations is an important operation in relational database systems the ordering of joins determines to a large extent the computation time of the join since the number of possible orderings could be very large query optimizers first reduce the search space by using various heuristics and then try to select an optimal ordering of joins avoiding cartesian products is a common heuristic for reducing the search space but it cannot guarantee optimal ordering in its search space because the cheapest cartesian product free cpf for short ordering could be significantly worse than an optimal non cpf ordering by a factor of an arbitrarily large number in this paper we use programs consisting of joins semijoins and projections for computing the join of some relations and we introduce a novel algorithm that derives programs from cpf orderings of joins we show that there exists a cpf ordering from which our algorithm derives a program whose cost is within a constant factor of the cost of an optimal ordering thus our result demonstrates the effectiveness of avoiding cartesian products as a heuristic for restricting the search space of orderings of joins

the module structure of a group action on a polynomial ring a finiteness theorem consider a group acting on a polynomial ring over a finite field we study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries for example any indecomposable module that appears as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite there are also applications to invariant theory giving a priori bounds on the degrees of the generators

topology optimization of continuum structures with local stress constraints we introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria we first consider relevant stress criteria for porous composite materials initially by studying the stress states of the so called rank 2 layered materials then on the basis of the theoretical study of the rank 2 microstructures we propose an empirical model that extends the power penalized stiffness model also called simp for solid isotropic microstructure with penalization for inter mediate densities in a second part solution aspects of topology problems are considered to deal with the so called singularity phenomenon of stress constraints in topology design an constraint relaxation of the stress constraints is used we describe the mathematical programming approach that is used to solve the numerical optimization problems and show results for a number of example applications 1998 john wiley sons ltd

structural optimization using kriging approximation an optimization method using kriging approximation is applied to a structural optimization problem the method involves two main processes the first is a space estimation process that uses the kriging method and the second is an optimization process the use of the kriging method makes it easier to perform the approximation optimization as an example of the estimation performed as part of structural optimization a response surface for layout optimization of beam reinforcement is estimated to evaluate the applicability of the kriging method kriging estimation is compared with neural network approximation as a numerical example the optimization of a stiffened cylinder for an eigenfrequency problem is illustrated the obtained results clearly show the applicability of the method

the noether bound in invariant theory of finite groups abstract let r be a commutative ring v a finitely generated free r module and g glr v a finite group acting naturally on the graded symmetric algebra a sym v let ag denote the minimal number m such that the ring ag of invariants can be generated by finitely many elements of degree at most m furthermore let h g be a normal subgroup such that the index g h is invertible in r in this paper we prove the inequality a g a h g h for h 1 and g invertible in r we obtain noether s bound ag g which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order g is invertible in r

coinvariants and the regular representation of a cyclic p group we consider an indecomposable representation of a cyclic p group z p r over a field of characteristic p we show that the top degree of the corresponding ring of coinvariants is less than frac r 2 3r p r 2 this bound also applies to the degrees of the generators for the invariant ring of the regular representation

a homogenization method for shape and topology optimization abstract shape and topology optimization of a linearly elastic structure is discussed using a modification of the homogenization method introduced by bendsoe and kikuchi together with various examples which may justify validity and strength of the present approach for plane structures

topology optimization by penalty top method in traditional structural topology optimization to the material properties of continuum finite elements of fixed form and coupling are varied to find the optimal topology that satisfies the design problem we develop an alternative fundamental formulation where the design space search is dependent on the coupling and the goal of the topology optimization by penalty top method is to determine the optimal finite element coupling constraints by this approach seemingly disparate topology design problems e g the design of structural supports topology optimization for fluid mechanics problems and topology optimization by the element connectivity parameterization ecp method can be understood as related formulations under a common topology optimization umbrella and more importantly this general framework can be applied to new design problems for example in modern multibody dynamics synthesis the geometric form of finite elements of fixed material properties and interconnectivity are varied to find the optimal topology that satisfies the mechanism design problem the a priori selection of coupling e g by revolute or translational joints severely limits the design space search the top method addresses this limitation in a novel way we develop the methodology and apply the top method to the diverse design problems discussed above

structure theorems over polynomial rings given a polynomial ring r over a field k and a finite group g we consider a finitely generated graded rg module s we regard s as a kg module and show that various conditions on s are equivalent such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of d karagueuzian p symonds the module structure of a group action on a polynomial ring a finiteness theorem preprint http www ma umist ac uk pas preprints

degrees of acyclicity for hypergraphs and relational database schemes database schemes winch intuitively are collecuons of table skeletons can be wewed as hypergraphs a hypergraph is a generalization of an ordinary undirected graph such that an edge need not contain exactly two nodes but can instead contain an arbitrary nonzero number of nodes a class of acychc database schemes was recently introduced a number of basic desirable propemes of database schemes have been shown to be equivalent to acyclicity this shows the naturalness of the concept however unlike the situation for ordinary undirected graphs there are several natural noneqmvalent notions of acyclicity for hypergraphs and hence for database schemes various desirable properties of database schemes are constdered and it is shown that they fall into several equivalence classes each completely characterized by the degree of acycliclty of the scheme the results are also of interest from a purely graph theoretic viewpomt the original notion of aeyclicity has the countermtmtive property that a subhypergraph of an acychc hypergraph can be cyclic this strange behavior does not occur for the new degrees of acyelicity that are considered

on the castelnuovo mumford regularity of the cohomology ring of a group we prove benson s regularity conjecture that the regularity of the cohomology of a finite group is zero we also prove some of his generalizations to other classes of groups dave benson in 2 conjectured that for any finite groupg and any prime p the castelnuovomumford regularity of the cohomology ring h g fp is zero he showed that reg h g fp 0 and succeeded in proving equality when the difference between the dimension and the depth is at most two the purpose of this paper is to prove benson s regularity conjecture as a corollary of the following result theorem 0 1 if a compact lie group g acts on a smooth manifold m with finite dimensional mod p homology then reg h g m fp dimm dimg by taking g to be finite m to be a point and using benson s inequality we obtain a proof of benson s original conjecture corollary 0 2 for any finite group g we have reg h g fp 0 we record here one easily stated consequence proposition 0 3 for any non trivial finite group the cohomology ring h g fp is generated by elements in degree at most g 1 and the relations between them as a graded commutative algebra are generated in degrees at most 2 g 1 this bound is weak although it can be improved somewhat at the cost of a more complicated formulation but previously no such bound was known we go on to prove some other conjectures of benson on the regularity of the cohomology of other classes of groups 3 specifically for compact lie groups and virtual poincare duality groups the proof uses standard techniques in equivariant cohomology based on work of quillen 30 and a paper of duflot 14 david green and simon king have verified benson s conjecture computationally for all groups of order less than 256 20 there is survey of the properties group cohomology from the point of view of commutative algebra by benson 3 1 castelnuovo mumford regularity we will work in the category of z graded rings and modules so maps will respect the grading elements will be homogeneous etc without specific mention for a module m 1991 mathematics subject classification primary 20j06 secondary 57s15 partially supported by a grant from the leverhulme trust

group actions on rings and the cech complex abstract we have previously shown that when a finite group g acts on a polynomial ring s in n variables over a finite field k only finitely many isomorphism classes of indecomposable k g modules occur as summands of s we have also shown that the regularity of the invariant subring s g is at most zero which has various consequences for example s g is generated in degrees at most n g 1 provided n g 2 both of these theorems depend on the structure theorem of karagueuzian and the author which is proved by means of a long and complicated calculation the aim of this paper is to prove these results using a more conceptual method

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

topology and detail geometry optimization for beam structures using homotopy modeling this paper describes a novel design concept and an algorithm for topology and detail geometry design of a beam structure topology layout cross sectional shape and size of each element are treated as a design variable homotopy theory is used to classify a topological type of a beam structure in the design concept not topology itself of a structure but a detail geometry design space defined by topology is transformed by introducing the novel concept of an algebraic topology expression and the topology metamorphosis algorithm active topology transformation becomes possible a numerical example of topology layout and cross sectional optimization of reinforcement for a beam structure is illustrated

the second main theorem vector for the modular regular representation of c2 we study the ring of invariants for a finite dimensional representation v of the group c2 of order 2 in characteristic 2 let denote a generator of c2 and x1 y1 xm ym a basis of v then xi xi and yi yi xi to our knowledge this ring for any prime p was first studied by david richman 12 in 1990 he gave a first main theorem for v2 c2 that is he proved that the ring of invariants when p 2 is generated by xi ni yi2 xiyi tr a 2 a m where a 0 1 m ya y1a1y2a2 ymam and tr a ya y1 x1 a1 y2 x2 a2 ym xm am in this paper we prove the second main theorem for v2 c2 that is we show that all relations between these generators are generated by relations of type i i axitr a i 0 and of type ii tr a tr b l ixi lnltr i l j k ni l jxj ltr l k for all m we also derive relations of type iii which are simpler and can be used in place of the relations of type ii

topology optimization of regions of darcy and stokes flow this paper treats the topology optimization problem of obtaining an optimal layout of regions of darcy and stokes flow where the objective is the total potential power functional representing aver

on generators of the hilbert ideal for cyclic groups in modular invariant theory abstract let g be a cyclic group of order p 2 and v be a faithful indecomposable representation of g over a field f of characteristic p we show that the hilbert ideal of the invariant ring is generated by polynomials of degree at most g whenever dim v 4 p or dim v p 2 2 p proving a conjecture of derksen and kemper in this particular case

invariants of finite groups over fields of characteristicp abstract let v denote a finite dimensional k vector space and let g denote a finite group of k linear automorphisms of v let v m denote the direct sum of m copies of v and let g act on the symmetric algebra k v m of v m by the diagonal action on v m a result of noether implies that if char k 0 then k v m g can be generated as a k algebra by polynomials whose degrees are g no matter how large m is this paper proves that this result no longer holds when the characteristic of k divides g more precisely it is proved in this case that there is a positive number depending only on g and char k such that every set of k algebra generators of k v m g contains a generator whose degree is m

on applying kriging based approximate optimization to inaccurate data abstract this paper describes a new approach for the use of kriging based approximate optimization with inaccurate data a conventional kriging system is applicable to several kinds of complex systems however there are some difficulties in applying a kriging based system to a noisy system therefore this paper proposes a new approach for kriging based estimation with inaccurate data the principal idea is summarized as an improvement of a semivariogram and determining process of its parameters by applying the proposed method to some numerical examples the applicability of the kriging system to an approximate optimization system of inaccurate data is investigated the applicability and problems in applying the kriging system to inaccurate data are discussed and numerical results illustrate the validity and effectiveness of the proposed method

on vector invariants over finite fields abstract let f denote a finite field and let s m n f denote a set of generators of the invariants of sl n f acting on mn component vectors this paper proves that if m n 1 then s m n f must contain a generator whose degree is greater than or equal to m n 2 f 1 similar results are obtained for the vector invariants of other groups of matrices with entries in f

penalty finite element methods for the analysis of stokesian flows a study of a class of finite element methods for the analysis of stokes problem based on the use of exterior penalty formulations is described the effects of selective reduced integration i e the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly are investigated error estimates are derived and the numerical stability of these methods as depicted by a special babuska brezzi condition is explored in some detail the results of several numerical experiments with these methods are also given

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

symmetric powers and modular invariants of elementary abelian p groups abstract let e be an elementary abelian p group of order q p n let w be a faithful indecomposable representation of e with dimension 2 over a field k of characteristic p and let v s m w with m q we prove that the rings of invariants k v e are generated by elements of degree q and relative transfers this extends recent work of wehlau 18 on modular invariants of cyclic groups of order p if m p we prove that k v e is generated by invariants of degree 2 q 3 extending a result of fleischmann sezer shank and woodcock 6 for cyclic groups of order p our methods are primarily representation theoretic and along the way we prove that for any d q with d m q s d v is projective relative to the set of subgroups of e with order m and that the sequence s d v d 0 is periodic with period q modulo summands which are projective relative to the same set of subgroups these results extend results of almkvist and fossum 1 on cyclic groups of prime order

on the desirability of acyclic database schemes a class of database schemes called acychc was recently introduced it is shown that this class has a number of desirable properties in particular several desirable properties that have been studied by other researchers m very different terms are all shown to be eqmvalent to acydicity in addition several equivalent charactenzauons of the class m terms of graphs and hypergraphs are given and a smaple algorithm for determining acychclty is presented also given are several eqmvalent characterizations of those sets m of multivalued dependencies such that m is the set of murlvalued dependencies that are the consequences of a given join dependency several characterizations for a conflict free in the sense of lien set of muluvalued dependencies are provided

theoretical aspects of the internal element connectivity parameterization approach for topology optimization the internal element connectivity parameterization i ecp method is an alternative approach to overcome numerical instabilities associated with low stiffness element states in non linear problems in i ecp elements are connected by zero length links while their link stiffness values are varied therefore it is important to interpolate link stiffness properly to obtain stably converging results the main objective of this work is two fold 1 the investigation of the relationship between the link stiffness and the stiffness of a domain discretizing patch by using a discrete model and a homogenized model and 2 the suggestion of link stiffness interpolation functions the effects of link stiffness penalization on solution convergence are then tested with several numerical examples the developed homogenized i ecp model can also be used to physically interpret an intermediate design variable state