compatibility of local and global langlands correspondences we prove the compatibility of local and global langlands correspondences for gln which was proved up to semisimplification in ht more precisely for the ndimensional l adic representation rl of the galois group of an imaginary cm field l attached to a conjugate self dual regular algebraic cuspidal automorphic representation of gln al which is square integrable at some finite place we show that frobenius semisimplification of the restriction of rl to the decomposition group of a place v of l not dividing l corresponds to v by the local langlands correspondence if v is square integrable for some finite place v 6 l we deduce that rl is irreducible we also obtain conditional results in the case v l

ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

adaptive computations using material forces and residual based error estimators on quadtree meshes quadtree is a hierarchical data structure that is well suited for h adaptive mesh refinement due to the presence of hanging nodes classical shape functions are non conforming on quadtree meshes in this paper we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes to this end the recently proposed construction of polygonal basis functions is adapted to quadtree elements a fast technique for calculating stiffness matrix on quadtree meshes is introduced residual based error estimators and material force technique are used to estimate the error on quadtree meshes the performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary value problems

weight monodromy conjecture for p adically uniformized varieties the aim of this paper is to prove the weight monodromy conjecture deligne s conjecture on the purity of monodromy filtration for varieties p adically uniformized by the drinfeld upper half spaces of any dimension the ingredients of the proof are to prove a special case of the hodge standard conjecture and apply a positivity argument of steenbrink m saito to the weight spectral sequence of rapoport zink as an application by combining our results with the results of schneider stuhler we compute the local zeta functions of p adically uniformized varieties in terms of representation theoretic invariants we also consider a p adic analogue by using the weight spectral sequence of mokrane

state complexity of combined operations we study the state complexity of combined operations two particular combined operations are studied star of union and star of intersection it is shown that the state complexity of a combined operation is not necessarily similar to the combination of the individual state complexities of the participating operations

on the numerical modelling of quasi static crack growth in linear elastic fracture mechanics in this paper we present a strategy for the simulation of a propagating crack under mixed mode linear elastic conditions using a discontinuous finite element method a key issue is to accurately compute the incremental change in the kink angle of the propagating crack during subsequent steps we have chosen to work with a domain formulation of the material force vector as a criteria for the propagation direction we describe the theoretical background together with the numerical implementation in detail and show some results for different loading conditions copyright 2005 john wiley sons ltd

strategies for computing goal oriented a posteriori error measures in non linear elasticity we investigate the characteristics and performance of goal oriented a posteriori error measures for a class of non linear elasticity models while restriction is made to small strain theory the chosen error measure of the displacement field can be global or local probing the chosen quantity in a specific spatial point the error is computable with the aid of the solution of a dual problem whose data depend on the error measure the main thrust of the paper is to evaluate the performance of a few different approximation strategies for computing the dual solution the chosen strategies are compared in terms of accuracy ease of implementation reliability and cost efficiency a well known numerical example the cook s membrane is used for the numerical evaluations

the inverse galois problem for mathrm psl _ 2 mathbb f _ p we show that the simple group psl2 fp occurs as the galois group of an extension of the rationals for all primes p 5 we obtain our galois extensions by studying the galois action on the second etale cohomology groups of a specific elliptic surface

a quasi static discontinuous galerkin configurational force crack propagation method for brittle materials summary this paper presents a framework for r adaptive quasi static configurational force cf brittle crack propagation cast within a discontinuous galerkin dg symmetric interior penalty sipg finite element scheme cracks are propagated in discrete steps with a staggered algorithm along element interfaces which align themselves with the predicted crack propagation direction the key novelty of the work is the exploitation of the dg face stiffness terms existing along element interfaces to propagate a crack in a mesh independent r adaptive quasi static fashion driven by the cf at the crack tip this adds no new degrees of freedom to the data structure additionally as dg methods have element specific degrees of freedom a geometry driven p adaptive algorithm is also easily included allowing for more accurate solutions of the cf on a moving crack front further for non determinant systems we introduce an average boundary condition that restrains rigid body motion leading to a determinant system to the authors knowledge this is first time such a boundary condition has been described the proposed formulation is validated against single and multiple crack problems with single and mixed mode cracks demonstrating the predictive capabilities of the method this article is protected by copyright all rights reserved

application of material forces to hyperelastostatic fracture mechanics i continuum mechanical setting the concern of this work is a consequent exploitation of the notion of material forces for the application within hyperelastostatic fracture mechanics contrary to physical forces material forces act on the material manifold thus essentially representing the tendency of defects like cracks or inclusions to move relative to the ambient material based on the formulation of the appropriate quasi static balance laws in the material space we aim at a fresh look onto classical aspects of hyperelastostatic fracture mechanics operating throughout within the geometrically nonlinear setting we emphasize on the one hand the duality of the direct and the inverse motion description and on the other hand we re establish the classical path integrals from elementary equilibrium considerations in the material space

the monadic quantifier alternation hierarchy over grids and graphs the monadic second order quantifier alternation hierarchy over the class of finite graphs is shown to be strict the proof is based on automata theoretic ideas and starts from a restricted class of graph like structures namely finite two dimensional grids considering grids where the width is a function of the height we prove that the difference between the levels k 1 and k of the monadic hierarchy is witnessed by a set of grids where this function is k 1 fold exponential we then transfer the hierarchy result to the class of directed or undirected graphs using an encoding technique called strong reduction it is notable that one can obtain sets of graphs which occur arbitrarily high in the monadic hierarchy but are already definable in the first order closure of existential monadic second order logic we also verify that these graph properties even belong to the complexity class nlog which indicates a profound difference between the monadic hierarchy and the polynomial hierarchy

on the limits of the communication complexity technique for proving lower bounds on the size of minimal nfa s in contrast to the minimization of deterministic finite automata dfa s the task of constructing a minimal nondeterministic finite automaton nfa for a given nfa is pspace complete moreover there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal nfa s since one is unable to efficiently estimate the size of a minimal nfa in an efficient way one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal nfa for a given regular language here we consider the robust and most successful lower bound proof technique that is based on communication complexity in this paper it is proved that even a strong generalization of this method fails for some concrete regular languages to fail is considered here in a very strong sense there is an exponential gap between the size of a minimal nfa and the achievable lower bound for a specific sequence of regular languages the generalization of the concept of communication protocols is also strong here it is shown that cutting the input word into 2 o n 1 4 pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata it seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head

l independence of the trace of monodromy let x be a proper smooth variety over a local field k of mixed characteristics and let l be a prime number different from the characteristic of the residue field of k let ik be the inertia subgroup of gk gal k k our main result is the l independence of the alternating sum of traces of g ik on h x ql and its comparison with the traces on p adic cohomology 0 introduction let k be a complete discrete valuation field with finite residue field fph gk the absolute galois group of k ik the inertia subgroup of gk and wk the weil group of k recall that the weil group is a subgroup of gk defined by the following exact sequence 0 ik wk u z z hz 0 0 ik gk gal fp fp gal fph fp 0 where the map u is defined by g 7 f for the geometric frobenius f x 7 x in gal fp fp and f denotes the separable closure of f for any field f we define a subset w k of wk to be w k g wk u g 0 let x be a variety over k throughout the paper a variety x over a field k means a reduced irreducible scheme x separated and of finite type over k throughout this paper x means the scalar extension x k k we denote by l a prime number p we consider the traces of the action of elements of w k or wk on the compact support etale cohomology h c x ql lim n h c x z lz zl ql let us recall the following classical conjecture conjecture s t for any variety x over k and g w k tr g hi c x ql is a rational integer which is independent of the choice of l remark if x is a d dimensional proper smooth variety the conjecture above holds for i 0 1 2d 1 2d sga7 1 if x has good reduction the above conjecture is true for any i due to the weil conjecture proved by p deligne de in this paper we shall prove the following weak versions of the conjecture typeset by ams tex 1

a computational framework of three dimensional configurational force driven brittle crack propagation abstract we consider a variational formulation of quasi static brittle fracture and develop a new finite element based computational framework for propagation of cracks in three dimensional bodies we outline a consistent thermodynamical framework for crack propagation in elastic solids and show that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global clausius planck inequality in the sense of coleman s method consequently the crack propagation direction associated with the classical griffith criterion is identified by the material configurational force which maximizes the local dissipation at the crack front the variational formulation is realized numerically by a standard spatial discretization with finite elements which yields a discrete formulation of the global dissipation in terms configurational nodal forces therefore the constitutive setting of crack propagation in the space discretized finite element context is naturally related to discrete nodes of a typical finite element mesh in a consistent way with the node based setting the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface facets of the mesh the crucial step for the success of this procedure is its embedding into an r adaptive crack facet reorientation procedure based on configurational force based indicators in conjunction with crack front constraints we propose a staggered solution procedure that results in a sequence of positive definite discrete subproblems with successively decreasing overall stiffness providing a robust algorithmic setting in the postcritical range the predictive capabilitiy of the proposed formulation is demonstrated by means of representative numerical simulations

on configurational forces in the context of the finite element method the theory of configurational forces is briefly recast together with the underlying balance laws it is shown that in the case of a homogeneous body without body forces the additional balance laws are identically satisfied if the standard force balance holds in approximate solutions for example obtained by finite elements the equilibrium is not satisfied exactly thus configurational forces occur on discretization nodes an implementation of the configurational force balance into the finite element scheme is presented the use of configurational forces is discussed with three main aspects it is demonstrated how configurational forces can be used to check and to improve the finite element solution examples from fracture mechanics and problems with material inhomogeneities are discussed copyright 2001 john wiley sons ltd

descriptional and computational complexity of finite automata a survey finite automata are probably best known for being equivalent to right linear context free grammars and thus for capturing the lowest level of the chomsky hierarchy the family of regular languages over the last half century a vast literature documenting the importance of deterministic nondeterministic and alternating finite automata as an enormously valuable concept has been developed in the present paper we tour a fragment of this literature mostly we discuss developments relevant to finite automata related problems like for example i simulation of and by several types of finite automata ii standard automata problems such as fixed and general membership emptiness universality equivalence and related problems and iii minimization and approximation we thus come across descriptional and computational complexity issues of finite automata we do not prove these results but we merely draw attention to the big picture and some of the main ideas involved

communication complexity method for measuring nondeterminism in finite automata while deterministic finite automata seem to be well understood surprisingly many important problems concerning nondeterministic finite automata nfa s remain open one such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata in this paper the concept of communication complexity is applied in order to achieve progress in this problem area the main results are as follows 1 deterministic communication complexity provides lower bounds on the size of nfa s with bounded unambiguity applying this fact the proofs of several results about nfa s with limited ambiguity can be simplified and presented in a uniform way 2 there is a family of languages konk2 with an exponential size gap between nfa s with polynomial leaf number ambiguity and nfa s with ambiguity k this partially provides an answer to the open problem posed by b ravikumar and o ibarra 1989 siam j comput 18 1263 1282 and h leung 1998 siam j comput 27 1073 1082

application of material forces to hyperelastostatic fracture mechanics ii computational setting the concern of this work is a novel algorithmic treatment of hyperelastostatic fracture mechanics problems consistent to the notion of material forces within the geometrically nonlinear setting of continuum mechanics to this end we consider the continuum mechanics of material forces as outlined in part i of this work p steinmann int j solid struct 37 7371 7391 which act contrary to the common physical forces on the material manifold or rather in the material space in the sequel it is proposed to discretize the corresponding quasi static balance of pseudo momentum by a standard galerkin finite element procedure as a result we obtain global discrete node point quantities the material node point forces which prove to be of the same qualitative and quantitative importance for the assessment of fracture mechanics problems as the classical j integral

extensions of goal oriented error estimation methods to simulations of highly nonlinear response of shock loaded elastomer reinforced structures this paper describes extensions of goal oriented methods for a posteriori error estimation and control of numerical approximation to a class of highly nonlinear problems in computational solid mechanics an updated lagrangian formulation of the dynamical large deformation response of structures composed of strain rate sensitive elastomers and elastoplastic materials is developed to apply the theory of goal oriented error estimation a backward in time dual formulation of these problems is derived and residual error estimators for meaningful quantities of interest are established the target problem class is that of axisymmetric deformations of layered elastomer reinforced shells of revolution subjected to shock loading extensive numerical results on solutions of representative problems are given it is shown that extensions of the theory of goal oriented error estimation can be developed and applied effectively to a class of highly nonlinear multi physics problems in solid and structural mechanics

reduced basis approximation and a posteriori error estimation for stress intensity factors we develop reduced basis approximations rigorous a posteriori error bounds and offline online computational procedures for the accurate fast and reliable prediction of stress intensity factors or strain energy release rates for mode i linear elastic crack problems relevant to fracture mechanics we demonstrate the efficiency and rigor of our numerical method in several examples

state complexity of operations on two way finite automata over a unary alphabet the paper determines the number of states in two way deterministic finite automata 2dfa over a one letter alphabet sufficient and in the worst case necessary to represent the results of basic language theoretic operations on 2dfas with a certain number of states it is proved that i intersection of an m state 2dfa and an n state 2dfa requires between m n and m n 1 states ii union of an m state 2dfa and an n state 2dfa between m n and 2m n 4 states iii kleene star of an n state 2dfa g n o n 2 states where g n e 1 o 1 n l n n is the maximum value of lcm p 1 p k for p i

adaptive strategies and error control for computing material forces in fracture mechanics the concept of material forces pertains to a variation of the inverse motion map while the placement field is kept fixed from the weak formulation of the self equilibrating eshelby material stress tensor it turns out that the classical j integral formulations in fracture mechanics are just special cases due to the choice of particular weight functions in this contribution we discuss a posteriori error control of the material forces as part of an adaptive strategy to reduce the discretization error to an acceptable level the data of the dual problem involves the quite non conventional tangent stiffness of the material eshelby stress tensor with respect to a variation of the physical strain field the suggested strategy is applied to the common fracture mechanics problem of a single edged crack whereby different strategies for computing the j integral are compared we also consider the case in which the crack edges are not parallel i e a notch

limitations of lower bound methods for deterministic nested word automata finite automata operating on nested words were introduced by alur and madhusudan in 2006 while nested word automata retain many of the desirable properties of ordinary finite automata there is no known efficient minimization algorithm for deterministic nested word automata and interestingly state complexity bounds for nested word automata turn out to differ significantly from the corresponding bounds for ordinary finite automata consequently lower bounds for the state complexity of nested word languages need to rely on fooling set type techniques we discuss limitations of the techniques and show that even in the deterministic case the bounds given by the lower bound methods may be arbitrarily far away from the actual state complexity of the nested word language

ambiguity and structural ambiguity of symmetric difference nfas abstract okhotin showed an exponential trade off in the conversion from nondeterministic unary automata to unambiguous nondeterministic unary automata we show that the trade off in the case of unary symmetric difference automata to finitely structurally ambiguous unary symmetric difference automata is linear with constant 1 in the number of states in particular for every n state unary nondeterministic symmetric difference automaton there is an equivalent finitely structurally ambiguous n state unary symmetric difference nondeterministic automaton we also consider the complexity of deciding unambiguity for k deterministic finite automata and investigate the interplay between ambiguity and structural ambiguity

partial orders on words minimal elements of regular languages and state complexity abstract the classical partial orders on strings prefix suffix subsegment subsequence lexical and dictionary order can be generalized to the case where the alphabet itself has a partial order this was done by higman for the subsequence order and by kundu for the prefix order higman proved that for any language l the set min l of minimal elements in l with respect to the generalized subsequence order is finite kundu proved that for any regular language l the set min l of minimal elements in l with respect to the generalized prefix order is also regular here we extend his result to the other orders and give upper bounds for the number of states of the finite automata recognizing min l the main contribution of this paper however is the proof of lower bounds the upper bounds are shown to be tight in particular if l is recognized by a deterministic finite automaton with n states then any deterministic or even nondeterministic finite automaton recognizing min l needs exponentially many states in n here min is taken with respect to a generalized prefix suffix or subsegment order with a partially ordered alphabet of 4 letters whose hasse diagram contains just one edge or with respect to the ordinary subsequence order we also give a new proof of a theorem of sakoda and sipser about the complementation of nondeterministic finite automata

theory and numerics for finite deformation fracture modelling using strong discontinuities a general finite element approach for the modelling of fracture is presented for the geometrically non linear case the kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements thus the deformation map is defined in terms of one continuous and one discontinuous portion considered as mutually independent giving rise to a weak formulation of the equilibrium consisting of two coupled equations in addition two different fracture criteria are considered firstly a principle stress criterion in terms of the material mandel stress in conjunction with a material cohesive zone law relating the cohesive mandel traction to a material displacement jump associated with the direct discontinuity secondly a criterion of griffith type is formulated in terms of the material crack driving force mcdf with the crack propagation direction determined by the direction of the force corresponding to the direction of maximum energy release apart from the material modelling the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria copyright 2005 john wiley sons ltd

nondeterministic complexity in subclasses of convex languages abstract we study the nondeterministic state complexity of basic regular operations on the classes of prefix suffix factor and subword free closed and convex regular languages and on the classes of right left two sided and all sided ideal regular languages for the operations of concatenation intersection union reversal star and complementation we get tight upper bounds for all considered classes except for complementation on factor and subword convex languages most of our witnesses are described over optimal alphabets the description of a proper suffix convex language over a five letter alphabet meeting the upper bound 2 n for complementation and obtaining an asymptotically tight bound n for complementation of unary prefix free languages are among the most interesting results of this paper

goal oriented a posteriori error estimates in linear elastic fracture mechanics goal oriented a posteriori error estimators are presented in this paper for the error obtained while approximately evaluating the j integral in linear elastic fracture mechanics using the finite element method the j integral provides a criterion for crack propagation of a pre existing crack therefore its reliable and accurate evaluation is of considerable importance in engineering applications to prevent failure of structures this can be achieved by goal oriented adaptive finite element methods based on the well established strategy of solving an auxiliary dual problem in order to control the error of a non linear functional here the j integral in this paper two principally different goal oriented adaptive strategies are presented the first strategy rests upon equilibrated residual error estimators based on local neumann problems whereas the second one makes use of averaging techniques the paper is concluded by a numerical example that illustrates our theoretical results

finite automata and unary languages abstract we prove that o e n log n states are sufficient to simulate an n state 1nfa recognizing a unary language by a 1dfa the lower bound is the same similar tight bounds are shown for the simulation of a 2dfa by a 1dfa and a 1nfa we also show that o n 2 states are sufficient and necessary to simulate an n state 1nfa recognizing a unary language by a 2dfa

optimal simulation of self verifying automata by deterministic automata self verifying automata are a special variant of finite automata with a symmetric kind of nondeterminism we study the conversion of self verifying automata to deterministic automata from a descriptional complexity point of view the main result is the exact cost in terms of the number of states of such a simulation

goal oriented r adaptivity based on variational arguments in the physical and material spaces abstract the error in a local quantity of interest depends on the error in the corresponding dual solution or generalized green s function this fact is used in dual weighted based goal oriented error estimation techniques and h or p adaptive algorithms this contribution is concerned with a novel approach for goal oriented r adaptivity based on energy minimization principles for the primal and the dual problems the minimization of the energy of the primal problem as well as the minimization of the energy of the dual problem with respect to a design function lead to the primal and dual material residuals respectively these residuals are error indicators for the discretization error on the current mesh we propose a goal oriented mesh optimization algorithm which ends in an optimized mesh with respect to a chosen quantity of interest by means of selected computational examples we demonstrate the capability of the proposed framework

nondeterministic state complexity of star free languages we investigate the nondeterministic state complexity of several operations on finite automata accepting star free and unary star free languages it turns out that in most cases exactly the same tight bounds as for general regular languages are reached this nicely complements the results recently obtained by brzozowski and liu 2011 8 for the operation problem of star free and unary star free languages accepted by deterministic finite automata

minimal nfa problems are hard finite automata fa s are of fundamental importance in theory and in applications the following basic minimization problem is studied given a dfa deterministic fa find a minimum equivalent no

computing upper and lower bounds for the j integral in two dimensional linear elasticity we present an a posteriori method for computing rigorous upper and lower bounds for the j integral in two dimensional linear elasticity the j integral which is typically expressed as a contour integral is recast as a quadratic continuous functional of the displacement involving only area integration by expanding the quadratic output about an approximate finite element solution the output is expressed as a known computable quantity plus linear and quadratic functionals of the solution error the quadratic component is bounded by the energy norm of the error scaled by a continuity constant which is determined explicitly the linear component is expressed as an inner product of the errors in the displacement and in a computed adjoint solution and bounded by an appropriate combination of the energy norms of the error in the displacement and the adjoint upper bounds for the energy norm of the error are obtained by using a complementary energy approach requiring the computation of equilibrated stress fields the method is illustrated with two fracture problems in plane strain elasticity an important feature of the method presented is that the computed bounds are rigorous with respect to the exact weak solution of the elasticity equations