anisotropic mesh adaptation for crack detection in brittle materials the quasi static brittle fracture model proposed by g francfort and j j marigo can be gamma approximated at each time evolution step by the ambrosio tortorelli functional in this paper we focus on a modification of this functional which includes additional constraints via penalty terms to enforce the irreversibility of the fracture as well as the applied displacement field second we build on this variational model an adapted discretization to numerically compute the time evolving minimizing solution we present the derivation of a novel a posteriori error estimator driving the anisotropic adaptive procedure the main properties of these automatically generated meshes are to be very fine and strongly anisotropic in a very thin neighborhood of the crack but only far away from the crack tip while they show a highly isotropic behavior in a neighborhood of the crack tip instead as a consequence of these properties the resulting discretizations follow very closely the propagation of the fracture

algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

shape optimization of shell structures based on nurbs description using automatic differentiation the consolidation of the link among four fields in computational mathematics and mechanics is the main objective of this work surfaces based on nurbs non uniform rational b spline mathematical optimization the finite element method fem in structural analysis and automatic differentiation ad are applied in shape optimization of shells this problem is performed taking into account the fact that material and mechanical characteristics influence both the structural shape and the thickness variation in order to obtain the best performance with respect to a specific criterium some techniques were implemented to modify the shell geometry conserving the same parameterization without a new finite element mesh generation the shape modification is carried out using an optimization code based on the data obtained by a finite element analysis and gradients evaluation in this work the optimization procedure is performed using an sqp sequential quadratic programming algorithm where the variables are the control points in homogeneous coordinates the knot vectors and the thickness the functional value is determined by the fem and gradients are evaluated using ad some examples are analyzed and discussed as a consequence of the shape optimization shells with high structural performance and aesthetically beautiful shapes can be obtained copyright 2011 john wiley sons ltd

bounds on universal sequences universal sequences for graphs a concept introduced by aleliunas m sc thesis university of toronto toronto ontario canada january 1978 and aleliunas et al proc 20th annual symposium on foundation of computer science 1979 pp 218 223 are studied by letting u d n denote the minimum length of a universal sequence for d regular undirected graphs with n nodes the latter paper has proved the upper bound u d n o d 2 n 3 log n using a probabilistic argument here a lower bound of u 2 n omega n log n is proved from which u d n omega n log n for all d is deduced also for complete graphs u n 1 n omega n log 2 n log log n an explicit construction of universal sequences for cycles d 2 of length n o log n is given

average case intractability vs worst case intractability we show that not all sets in np or other levels of the polynomial time hierarchy have efficient average case algorithms unless the arthur merlin classes ma and am can be derandomized to np and various subclasses of p poly collapse to p furthermore other complexity classes like p pp and pspace are shown to be intractable on average unless they are easy in the worst case

3d tetrahedral unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric in this paper we present a 3d tetrahedral unstructured and anisotropic mesh generator that is not based on the delaunay frontal or octree method instead it proceeds by local optimizations and uses an anisotropic shape criterion to fit a metric field then we introduce a new 3d metric field that tightens the mesh around interfaces when the calculation domain is divided in several subdomains and a 3d metric field that places enough elements through each subdomain thickness without introducing too many nodes in the other directions finally we show some applications for material forming geometries

locating p poly optimally in the extended low hierarchy the low hierarchy in np and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes and almost all of them could be located optimally therein however until now the exact location of p poly remained open

on hard instances instance complexity was introduced by orponen ko schoning and watanabe 1994 as a measure of the complexity of individual instances of a decision problem comparing instance complexity to kolmogorov complexity they introduced the notion of p hard instances and conjectured that every set not in p has p hard instances whereas this conjecture is still unsettled fortnow and kummer 6 proved that np hard sets have p hard instances unless p np the unbounded version of the conjecture was proven wrong by kummer 1995 we introduce a slightly weaker notion of hard instances in the unbounded version we characterize the classes of recursive enumerable resp recursive sets by hard instances in bounded versions we characterize the class p hard instances are shown to be stronger than complexity cores introduced by lynch 1975 nevertheless np hard sets must have super polynomially dense hard instances unless p np

on relativized exponential and probabilistic complexity classes an oracle x is constructed such that the exponential complexity class ep x 2 equals the probabilistic class r r x this shows that it will be difficult to prove that ep 2 is different from r r although it seems very unlikely that these two classes are equal the result subsumes several known results about relativized computations i the existence of relativized polynomial hierarchies extending two levels long t 1978 dissertation purdue univ lafayette ind heller h 1984 a siam j comput 13 717 725 heller h 1984 b math systems theory 17 71 84 ii the existence of an oracle x such that bpp x p x 2 stockmeyer l 1983 proc 15th stoc pp 118 126 iii the existence of an oracle x such that np x is polynomially turing reducible to a sparse set wilson c 1983 proc 24th focs pp 329 334 immerman n and mahaney s 1983 conference on computational complexity theory santa barbara march 21 25 the result shows possible inclusion relations for nonrelativized complexity classes and points out that certain results about probabilistic complexity classes and about polynomial size circuits cannot be improved unless methods are applied which do not relativize

circuit lower bounds for merlin arthur classes we show that for each k 0 mathsf ma 1 mathsf ma with 1 bit of advice does not have circuits of size n k this implies the first superlinear circuit lower bounds for the promise versions of the classes mathsf ma mathsf am and mathsf zpp parallel mathsf np we extend our main result in several ways for each k we give an explicit language in mathsf ma cap mathsf coma 1 which does not have circuits of size n k we also adapt our lower bound to the average case setting i e we show that mathsf ma 1 cannot be solved on more than 1 2 1 n k fraction of inputs of length n by circuits of size n k furthermore we prove that mathsf ma does not have arithmetic circuits of size n k for any k as a corollary to our main result we obtain that derandomization of mathsf ma o 1 implies the existence of pseudorandom generators computable using o 1 bits of advice

on sparse hard sets for counting classes abstract in this paper we study one word decreasing self reducible sets which are introduced by lozano and toran 1991 these are the usual self reducible sets with the peculiarity that the self reducibility machine makes at most one query and this is lexicographically smaller than the input we show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one word decreasing self reducible using this fact we can prove that for any class k chosen from pp np c p mod2p mod3 p it holds that 1 if there is a sparse pbtt hard set for k then k p and 2 if there is a sparse snbtt hard set for k then k np co np this generalizes the result of ogiwara and watanabe 1991 to the mentioned complexity classes

anisotropic mesh adaptation for the generalized ambrosio tortorelli functional with application to brittle fracture abstract quasi static crack propagation in brittle materials is modeled via the ambrosio tortorelli approximation the crack is modeled by a smooth phase field defined on the whole computational domain since the crack is confined to a thin layer the employment of anisotropic adapted grids is shown to be a really effective tool in containing computational costs we extend the anisotropic error analysis applied to the classical ambrosio tortorelli approximation by artina et al to the generalized ambrosio tortorelli functional where a unified framework for several elasticity laws is dealt with as well as a non convex fracture energy can be accommodated after deriving an anisotropic a posteriori error estimator we devise an algorithm which alternates optimization and mesh adaptation both anti plane and plane strain configurations are numerically checked

nonlinear finite element computation of the equilibrium stability and motion of the extensional beam and ring abstract discrete gauss integration of the element total potential energy is applied to the formation of a cubic cubic c1 extensible completely nonlinear curved beam finite element the versatility accuracy effectiveness and robustness of the element and the newton raphson technique used to solve the nonlinear algebraic stiffness equation set up with it is numerically demonstrated by computations of the nonlinear equilibrium stability and motion of beams and rings

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

nonuniform proof systems a new framework to describe nonuniform and probabilistic complexity classes the concept of non uniform proof systems is introduced this notion allows a uniform description of non uniform complexity classes 10 probabilistic classes e g bpp 8 15 27 32 am 2 3 and language classes defined by simultaneous non uniform and nondeterministic time bounds non uniform proof systems provide a better understanding of many results concerning these classes particularly their connections to uniform complexity measures we give an uniform approach to lowness results 19 20 for various complexity classes for instance we show that co np poly np is contained in the third level of the low hierarchy and that np c np co np poly implies that the polynomial time hierarchy collapses to its second level see also 1 finally some evidence is given that the low hierarchy cannot be extended beyond its third level by the current techniques

sparse sets lowness and highness we develop the notions of generalized lowness for sets in ph the union of the polynomial time hierarchy and of generalized highness for arbitrary sets also we develop the notions of extended lowness and extended highness for arbitrary sets these notions extend the decomposition of np into low sets and high sets developed by schoning 15 and studied by ko and schoning 9 we show that either every sparse set in ph is generalized high or no sparse set in ph is generalized high further either every sparse set is extended high or no sparse set is extended high in both situations the former case corresponds to the polynomial time hierarchy having only finitely many levels while the latter case corresponds to the polynomial time hierarchy extending infinitely many levels

output functional control for nonlinear equations driven by anisotropic mesh adaption the navier stokes equations the contribution of this paper is twofold firstly moving from the very well known dual weighted residual dwr method we set up a theoretical framework for a goal oriented a posteriori analysis of nonlinear partial differential equations accounting for different approximations of the primal and dual problems nonhomogeneous dirichlet boundary conditions even different on passing from the primal to the dual problem the error due to data approximation and the effect of a possible stabilization secondly moving from this framework and employing anisotropic interpolation error estimates a sound anisotropic mesh adaption procedure is devised for the numerical approximation of the navier stokes equations by continuous piecewise linear finite elements the resulting adaptive procedure is thoroughly addressed and validated on some relevant test cases

state variable planning under structural restrictions algorithms and complexity abstract computationally tractable planning problems reported in the literature so far have almost exclusively been defined by syntactical restrictions to better exploit the inherent structure in problems it is probably necessary to study also structural restrictions on the underlying state transition graph the exponential size of this graph though makes such restrictions costly to test hence we propose an intermediate approach using a state variable model for planning and defining restrictions on the separate state transition graphs for each state variable we identify such restrictions which can tractably be tested and we present a planning algorithm which is correct and runs in polynomial time under these restrictions the algorithm has been implemented and it outperforms graphplan on a number of test instances in addition we present an exhaustive map of the complexity results for planning under all combinations of four previously studied syntactical restrictions and our five new structural restrictions this complexity map considers both the optimal and non optimal plan generation problem

new collapse consequences of np having small circuits we show that if a self reducible set has polynomial size circuits then it is low for the probabilistic class zpp np as a consequence we get a deeper collapse of the polynomial time hierarchy ph to zpp np under the assumption that np has polynomial size circuits this improves on the well known result in karp and lipton proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 stating a collapse of ph to its second level sigmap 2 under the same assumption furthermore we derive new collapse consequences under the assumption that complexity classes like up fewp and c p have polynomial size circuits finally we investigate the circuit size complexity of several language classes in particular we show that for every fixed polynomial s there is a set in zpp np which does not have o s n size circuits

an adaptive finite element approximation of a variational model of brittle fracture the energy of the francfort marigo model of brittle fracture can be approximated in the sense of gamma convergence by the ambrosio tortorelli functional in this work we formulate and analyze two adaptive finite element algorithms for the computation of its local minimizers for each algorithm we combine a newton type method with residual driven adaptive mesh refinement we present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the ambrosio tortorelli functional

a general procedure for the direct computation of turning and bifurcation points we address the formulation and numerical implementation of general purpose algorithme suitable for the direct computation of stability points in non linear solid mechanics the effectiveness of the resulting algorithm is illustrated by numerical simulation that include bifurcation and loss of stability in elastoplasticity non linear rods and non linear three dimensional shells

computation times of np sets of different densities abstract in this paper we study the computational complexity of sets of different densities in np we show that the deterministic computation time for sets in np can depend on their density if and only if there is a collapse or partial collapse of the corresponding higher nondeterministic and deterministic time bounded complexity classes we show also that for np sets of different densities there exist complete sets of the corresponding density under polynomial time turing reductions finally we show that these results can be interpreted as results about the complexity of theorem proving and proof presentation in axiomatized mathematical systems this interpretation relates fundamental questions about the complexity of our intellectual tools to basic structural problems about p np conp and pspace discussed in this paper

quasistatic crack growth in nonlinear elasticity in this paper we prove a new existence result for a variational model of crack growth in brittle materials proposed in 19 we consider the case of n dimensional nonlinear elasticity for an arbitrary n 1 with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads both depending on time

reliability and efficiency of an anisotropic zienkiewicz zhu error estimator abstract in this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the poisson problem supplied with mixed boundary conditions the error estimator may be classified as a residual based one but its novelty is twofold firstly it employs anisotropic estimates of the interpolation error for linear triangular finite elements and secondly it makes use of the zienkiewicz zhu recovery procedure to approximate the gradient of the exact solution finally we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy and we include some numerical test cases to assess the robustness of the proposed numerical algorithm

application of the implicitly updated arnoldi method with a complex shift and invert strategy in mhd the implicitly updated arnoldi method introduced by sorensen with an internal qr iteration is a very useful eigenvalue solver for nonsymmetric eigenvalue problems to make this method rigorous in finding internal eigenvalues a complex shift and invert strategy is used therefore a complex variant of the method has been constructed and the method has been compared with a lanczos method as implemented by cullum et al for a practical problem in magnetohydrodynamics

existence and convergence for quasi static evolution in brittle fracture this paper investigates the mathematical well posedness of the variational model of quasi static growth for a brittle crack proposed by francfort and marigo in 15 the starting point is a time discretized version of that evolution which results in a sequence of minimization problems of mumford and shah type functionals the natural weak setting is that of special functions of bounded variation and the main difficulty in showing existence of the time continuous quasi static growth is to pass to the limit as the time discretization step tends to 0 this is performed with the help of a jump transfer theorem which permits under weak convergence assumptions for a sequence un of sbv functions to its bv limit u to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence un in particular it is shown that the notion of minimizer of a mumford and shah type functional for its own jump set is stable under weak convergence assumptions furthermore our analysis justifies numerical methods used for computing the time continuous quasi static evolution 2003 wiley periodicals inc

the superconvergent patch recovery spr and adaptive finite element refinement abstract the error estimation processes which have been developed by the authors depend on the accuracy with which stresses gradients can be recovered the recovery methods available currently such as the l 2 projection and averaging are inaccurate and an improved process is needed this paper presents a very much improved recovery process yielding superconvergent values throughout the domain details of this method named the superconvergent patch recovery are presented with test results showing its efficiency examples of adaptive h based refinement are given showing high local and overall efficiency indices for error estimation

the computational complexity of propositional strips planning abstract i present several computational complexity results for propositional strips planning i e strips planning restricted to ground formulas different planning problems can be defined by restricting the type of formulas placing limits on the number of pre and postconditions by restricting negation in pre and postconditions and by requiring optimal plans for these types of restrictions i show when planning is tractable polynomial and intractable np hard in general it is pspace complete to determine if a given planning instance has any solutions extremely severe restrictions on both the operators and the formulas are required to guarantee polynomial time or even np completeness for example when only ground literals are permitted determining plan existence is pspace complete even if operators are limited to two preconditions and two postconditions when definite horn ground formulas are permitted determining plan existence is pspace complete even if operators are limited to zero preconditions and one postcondition one of the interesting tractable problems is if each operator is restricted to positive preconditions and one postcondition only ground literals the blocks world problem slightly modified is a subproblem of this restricted planning problem these results in combination with previous analyses are not encouraging for domain independent planning

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

competing provers yield improved karp lipton collapse results via competing provers we show that if a language a is self reducible and has polynomial size circuits then s2a s2 building on this we strengthen the kamper afk theorem namely we prove that if np np conp poly then the polynomial hierarchy collapses to s2np conp we also strengthen yap s theorem namely we prove that if np conp poly then the polynomial hierarchy collapses to s2np under the same assumptions the best previously known collapses were to zppnp and zppnpnp respectively siam journal on computing 28 1 1998 311 journal of computer and system sciences 52 3 1996 421 building on proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 journal of computer and system sciences 39 1989 21 theoretical computer science 85 2 1991 305 theoretical computer science 26 3 1983 287 it is known that s2 zppnp proceedings of the 42nd ieee symposium on foundations of computer science ieee computer society press silver spring md 2001 pp 620 629 that result and its relativized version show that our new collapses indeed improve the previously known results the kamper afk theorem and yap s theorem are used in the literature as bridges in a variety of results ranging from the study of unique solutions to issues of approximation and so our results implicitly strengthen those results

design sensitivity analysis of non linear structures in regular and critical states abstract a general design sensitivity formulation is introduced for non linear elastic structures reaching critical equilibrium states a discretized formulation is considered first and the sensitivity analysis is discussed for regular and critical states including analysis of post critical behavior next a variational approach is presented for beam and surface structures for which generalized stress and strain operator formulation is applied the formulae for sensitivity of bifurcation loads are derived in terms of adjoint fields which are shown to be equivalent to post buckling fields in the case of symmetric bifurcation

planning for conjunctive goals abstract the problem of achieving conjunctive goals has been central to domain independent planning research the nonlinear constraint posting approach has been most successful previous planners of this type have been complicated heuristic and ill defined i have combined and distilled the state of the art into a simple precise implemented algorithm tweak which i have proved correct and complete i analyze previous work on domain independent conjunctive planning in retrospect it becomes clear that all conjunctive planners linear and nonlinear work the same way the efficiency and correctness of these planners depends on the traditional add delete list representation for actions which drastically limits their usefulness i present theorems that suggest that efficient general purpose planning with more expressive action representations is impossible and suggest ways to avoid this problem

robustness of probabilistic computational complexity classes under definitional perturbations various types of probabilistic algorithms play an increasingly important role in computer science especially in computational complexity theory probabilistic polynomial time complexity classes are defined and compared to each other emphasizing some structural relationships to the known complexity classes p np pspace the classes r and zpp corresponding to the so called las vegas polynomial time bounded algorithms are given special attention it is shown that many definitions that arise naturally from different types of algorithms are equivalent in defining the same class r or zpp resp these robustness results justify finally the tractability of the above probabilistic classes

counting classes are at least as hard as the polynomial time hierarchy in this paper it is shown that many natural counting classes such as pp c p and text mod k text p are at least as computationally hard as ph the polynomial time hierarchy in the following sense for each bf k of the counting classes above every set in bf k ph is polynomial time randomized many one reducible to a set in bf k with two sided exponentially small error probability as a consequence of the result it is seen that all the counting classes above are computationally harder than ph unless ph collapses to a finite level some other consequences are also shown

a phase field model for rate independent crack propagation robust algorithmic implementation based on operator splits the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field following our recent work c miehe f welschinger m hofacker thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations international journal for numerical methods in engineering doi 10 1002 nme 2861 on phase field type fracture we propose in this paper a new variational framework for rate independent diffusive fracture that bases on the introduction of a local history field it contains a maximum reference energy obtained in the deformation history which may be considered as a measure for the maximum tensile strain obtained in history it is shown that this local variable drives the evolution of the crack phase field the introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology in particular it allows for the construction of a new algorithmic treatment of diffusive fracture here we propose an extremely robust operator split scheme that successively updates in a typical time step the history field the crack phase field and finally the displacement field a regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added the proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids we demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples

complexity of finite horizon markov decision process problems controlled stochastic systems occur in science engineering manufacturing social sciences and many other cntexts if the systems is modeled as a markov decision process mdp and will run ad infinitum the optimal control policy can be computed in polynomial time using linear programming the problems considered here assume that the time that the process will run is finite and based on the size of the input there are mny factors that compound the complexity of computing the optimal policy for instance there are many factors that compound the complexity of this computation for instance if the controller does not have complete information about the state of the system or if the system is represented in some very succint manner the optimal policy is provably not computable in time polynomial in the size of the input we analyze the computational complexity of evaluating policies and of determining whether a sufficiently good policy exists for a mdp based on a number of confounding factors including the observability of the system state the succinctness of the representation the type of policy even the number of actions relative to the number of states in almost every case we show that the decision problem is complete for some known complexity class some of these results are familiar from work by papadimitriou and tsitsiklis and others but some such as our pl completeness proofs are surprising we include proofs of completeness for natural problems in the as yet little studied classes np pp

the approximate solution of defining equations for generalized turning points a set of defining equations for turning points is shown to produce a test function that can be computed iteratively to double accuracy an inexact newton method is then derived that gives a method for the efficient calculation of turning points applications are given to discretized boundary value problems using conjugate gradients

on polynomial time bounded truth table reducibility of np sets to sparse sets it is proved that if text p ne text np then there exists a set in text np that is not polynomial time bounded truth table reducible in short leqq text btt text p reducible to any sparse set in other words it is proved that no sparse leqq text btt text p hard set exists for text np unless text p text np by using the technique proving this result the intractability of several number theoretic decision problems i e decision problems defined naturally from number theoretic problems is investigated it is shown that for these number theoretic decision problems if it is not in text p then it is not leqq text btt text p reducible to any sparse set

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

anisotropic error estimates for elliptic problems moving from the anisotropic interpolation error estimates derived in 12 we provide here both a priori and a posteriori estimates for a generic elliptic problem the a priori result is deduced by following the standard finite element theory for the a posteriori estimate the analysis extends to anisotropic meshes the theory presented in 3 5 numerical test cases validate the derived results

circuit size lower bounds and non reducibility to sparse sets as remarked in cook towards a complexity theory of synchronous parallel computation univ of toronto 1980 a nonlinear lower bound on the circuit size of a language in p or even in np is not known the best known published lower bound seems to be due to paul proceedings 7th acm symposium on theory of computing 1975 in this paper it is shown first that for each nonnegative integer k there is a language l k in 2 2 of the meyer and stockmeyer proceedings 13th ieee symposium on switching and automata theory 1972 hierarchy which does not have o n k size circuits using the same techniques one is able to prove several similar results for example it is shown that for each nonnegative integer k there is a language l k in np that does not have o n k size uniform circuits this follows as a corollary of a stronger result shown in the paper this result like the others to follow is not provable by direct diagonalization it thus points to the most interesting feature of the techniques used hereby using the polynomial time hierarchy they are able to prove results about np that cannot seem to proved by direct diagonalization finally it is noted that existence of small circuits is in suitable contexts equivalent to being reducible to sparse sets using this one is able to prove for example that for any time constructible superpolynomial function f n ntime f n contains a language which is not many to one p time reducible to any sparse set

the shrinking property for np and conp we study the shrinking and separation properties two notions well known in descriptive set theory for np and conp and show that under reasonable complexity theoretic assumptions both properties do not hold for np and the shrinking property does not hold for conp in particular we obtain the following results 1 np and conp do not have the shrinking property unless ph is finite in general s n p and p n p do not have the shrinking property unless ph is finite this solves an open question posed by selivanov 1994 33 2 the separation property does not hold for np unless email protected conp 3 the shrinking property does not hold for np unless there exist np hard disjoint np pairs existence of such pairs would contradict a conjecture of even et al 1984 6 4 the shrinking property does not hold for np unless there exist complete disjoint np pairs moreover we prove that the assumption np conp is too weak to refute the shrinking property for np in a relativizable way for this we construct an oracle relative to which email protected conp np conp and np has the shrinking property this solves an open question posed by blass and gurevich 1984 3 who explicitly ask for such an oracle

symmetric space bounded computation abstract a symmetric turing machine is one such that the yields relation between configurations is symmetric the space complexity classes for such machines are found to be intermediate between the corresponding deterministic and nondeterministic space complexity classes certain natural problems are shown to be complete for symmetric space complexity classes and the relationship of symmetry to determinism and nondeterminism is investigated

a model for the quasi static growth of brittle fractures existence and approximation results we give a precise mathematical formulation of a variational model for the irreversible quasi static evolution of brittle fractures proposed by g a francfort and j j marigo and based on griffith s theory of crack growth in the two dimensional case we prove an existence result for the quasi static evolution and show that the total energy is an absolutely continuous function of time although we cannot exclude the possibility that the bulk energy and the surface energy may present some jump discontinuities this existence result is proved by a time discretization process where at each step a global energy minimization is performed with the constraint that the new crack contains all cracks formed at the previous time steps this procedure provides an effective way to approximate the continuous time evolution

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

an adjoint variable method for sensitivity analysis of non linear elastic systems summary an adjoint variable method for design sensitivity analysis of non linear elastic systems is presented the method uses domain parameterization and a mutual form of the hu washizu energy principle and extends results reported in a recent work for linear elastic systems to non linear elasticity non linearities due to finite deformations and non linear hyperelastic constitutive models are considered in contrast to other methods for non linear sensitivity analysis the present formulation can be applied with force displacement or mixed approximate solution methods the mutual energy expression used in the adjoint sensitivity derivation is developed from a non linear extension of the hu washizu energy functional and yields a linear governing equation for the adjoint system this has important ramifications for the computational cost of a sensitivity analyses of non linear systems excluding the cost of determining the response of the system the cost of a sensitivity analysis for a non linear system is essentially the same as that for a linear system finite element implementation of the resulting sensitivity expressions is discussed and two numerical examples are presented the first example involves large deformations of a mooney rivlin body while the second involves design sensitivity analysis for mixed solution methods

an anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics petroleum engineering and image processing standard numerical methods can produce spurious oscillations when they are used to solve those problems a common approach to avoid this difficulty is to design a proper numerical scheme and or a proper mesh so that the numerical solution validates the discrete counterpart dmp of the maximum principle satisfied by the continuous solution a well known mesh condition for the dmp satisfaction by the linear finite element solution of isotropic diffusion problems is the non obtuse angle condition that requires the dihedral angles of mesh elements to be non obtuse in this paper a generalization of the condition the so called anisotropic non obtuse angle condition is developed for the finite element solution of heterogeneous anisotropic diffusion problems the new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem several variants of the new condition are obtained based on one of them two metric tensors for use in anisotropic mesh generation are developed to account for dmp satisfaction and the combination of dmp satisfaction and mesh adaptivity numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

expressive equivalence of planning formalisms abstract a concept of expressive equivalence for planning formalisms based on polynomial transformations is defined it is argued that this definition is reasonable and useful both from a theoretical and from a practical perspective if two languages are equivalent then theoretical results carry over and more practically we can model an application problem in one language and then easily use a planner for the other language in order to cope with the problem of exponentially sized solutions for planning problems an even stronger concept of expressive equivalence is introduced using the novel esp reduction four different formalisms for propositional planning are then analyzed namely two variants of strips ground tweak and the sas formalism although these may seem to exhibit different degrees of expressive power it is proven that they are in fact expressively equivalent under esp reduction this means that neither negative goals partial initial states nor multi valued state variables increase the expressiveness of standard propositional strips

towards efficient universal planning a randomized approach one of the most widespread approaches to reactive planning is schoppers universal plans we propose a stricter definition of universal plans which guarantees a weak notion of soundness not present in the original definition and isolate three different types of completeness that capture different behaviors exhibited by universal plans we show that universal plans which run in polynomial time and are of polynomial size cannot satisfy even the weakest type of completeness unless the polynomial hierarchy collapses by relaxing either the polynomial time or the polynomial space requirement the construction of universal plans satisfying the strongest type of completeness becomes trivial as an alternative approach we study randomized universal planning by considering a randomized version of completeness and a restricted but nontrivial class of problems we show that there exists randomized universal plans running in polynomial time and using polynomial space which are sound and complete for the restricted class of problems we also report experimental results on this approach to planning showing that the performance of a randomized planner is not easily compared to that of a deterministic planner

complexity classes of equivalence problems revisited to determine if two lists of numbers are the same set we sort both lists and see if we get the same result the sorted list is a canonical form for the equivalence relation of set equality other canonical forms arise in graph isomorphism algorithms to determine if two graphs are cospectral have the same eigenvalues we compute their characteristic polynomials and see if they are equal the characteristic polynomial is a complete invariant for cospectrality finally an equivalence relation may be decidable in p without either a complete invariant or canonical form blass and gurevich 1984 asked whether these conditions on equivalence relations having an fp canonical form having an fp complete invariant and being in p are distinct they showed that this question requires non relativizing techniques to resolve we extend their results and give new connections to probabilistic and quantum computation