computational complexity of holant problems we propose and explore a novel alternative framework to study the complexity of counting problems called holant problems compared to counting constraint satisfaction problems csp it is a refinement with a more explicit role for the constraint functions both graph homomorphism and csp can be viewed as special cases of holant problems we prove complexity dichotomy theorems in this framework our dichotomy theorems apply to local constraint functions which are symmetric functions on boolean input variables and evaluate to arbitrary real or complex values we discover surprising tractable subclasses of counting problems which could not easily be specified in the csp framework when all unary functions are assumed to be free mathrm holant problems the tractable ones consist of functions that are degenerate or of arity at most two or holographic transformations of fibonacci gates when only two special unary functions the constant zero and constant one functions are assumed to be free mathrm holant c problems we further identify three special families of tractable cases then we prove that all other cases are p hard the main technical tool we use and develop is holographic reductions another technical tool used in combination with holographic reductions is polynomial interpolations

integral mappings and the principle of local reflexivity for noncommutative l 1 spaces the operator space analogue of the strong form of the principle of local re exivity is shown to hold for any von neumann algebra predual and thus for any c algebraic dual this is in striking contrast to the situation for c algebras since for example k h does not have that property the proof uses the kaplansky density theorem together with a careful analysis of two notions of integrality for mappings of operator spaces

a collapse theorem for holographic algorithms with matchgates on domain size at most 4 holographic algorithms with matchgates are a novel approach to design polynomial time computation they use kasteleyn s algorithm for perfect matchings and more importantly a holographic reduction the two fundamental parameters of a holographic reduction are the domain size k of the underlying problem and the basis size a holographic reduction transforms the computation to matchgates by a linear transformation that maps to a tensor product space of a linear space of dimension 2 we prove a sharp basis collapse theorem which shows that for domain size 3 and 4 all non trivial holographic reductions can be expressed with a basis of size 1 or 2 respectively the main proof techniques are matchgate identities and a group property of matchgate signatures

on subalgebras of the car algebra abstract every nuclear unital separable c algebra a is unitally and completely isometrically ismorphic to the range of a unital completely positive projection on the car algbra b m 2 m 2 let c be a nuclearly imbeddable separable unital c algebra in the sense of voiculescu vo13 then there exists a c subalgebra e of the car algebra b such that c is isomorphic to a quotient c algebra of e we include one of the possible proofs that exact c algebras are nuclearly imbeddable proofs of g covariant analogs are outlined for compact g some related results are obtained on non selfadjoint operator algebras the unital separable nuclear c algebras not of type i are mutually cb isomorphic and have cb distance 256 separable lindenstrauss spaces are isometric to quotients of the car algebra by a sum of closed left and a closed right ideal

an effective dichotomy for the counting constraint satisfaction problem bulatov proceedings of the 35 th international colloquium on automata languages and programming part 1 lecture notes in comput sci 5125 springer new york 2008 pp 646 661 gave a dichotomy for the counting constraint satisfaction problem csp a problem from csp is characterized by a constraint language gamma a fixed finite set of relations over a finite domain d an instance of the problem uses these relations to constrain an arbitrarily large finite set of variables bulatov showed that the problem of counting the satisfying assignments of instances of any problem from csp is either in polynomial time fp or is p complete his proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field we give an elementary proof of bulatov s dichotomy based on succinct representations which we call frames of a class of highly structured relations which we call strongly rectangular we show that these are precisely the relation

a discontinuous galerkin formulation of a model of gradient plasticity at finite strains abstract the work presented here constitutes an extension to the finite strain regime of a discontinuous galerkin based strain gradient plasticity formulation presented in djoko et al j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis comput methods appl mech engrg 196 2007 3881 3897 j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 2 algorithms and numerical analysis comput methods appl mech engrg 197 2007 1 21 the focus here is on algorithmic and computational aspects of the formulation at finite strains the adoption of a logarithmic hyperelastic plastic formulation preserves the essential features of the infinitesimal formulation this key ingredient allows the predictor corrector solution algorithms developed for the infinitesimal gradient formulation to be extended readily to the finite strain regime the use of low order elements is essential to contain the computational expense of the formulation but these elements are prone to locking the method of enhanced assumed strains for geometrically nonlinear problems is utilised to circumvent this limitation the form of the consistent tangent modulus is derived for the case of gradient plasticity two numerical examples are presented to illustrate aspects of the approximation scheme and the algorithm as well as features of the model of gradient plasticity

localization properties of strain softening gradient plasticity models part i strain gradient theories this paper compares and evaluates strain gradient extensions of the conventional plasticity theory attention is focused on the ability of individual formulations to act as localization limiters i e to regularize the boundary value problem in the presence of softening and to prevent localization of plastic strain increments into a set of zero measure to keep the presentation simple and to highlight the essential properties of the investigated models only the static rate independent response in the small strain range and in the one dimensional setting is considered these restrictions permit an analytical or semi analytical treatment of the problem while the basic characteristics of the solutions remain valid in the general multi dimensional case the onset of localization is characterized as a bifurcation from a uniform state the subsequent evolution of the localized process zone and of the shape of the strain profile is studied numerically it is shown that certain pathologies e g expansion of the plastic region accompanied by stress locking may arise at later stages of localization a similar analysis of models with gradients of internal variables is presented in a companion paper

clifford gates in the holant framework abstract we show that the clifford gates and stabilizer circuits in the quantum computing literature which admit efficient classical simulation are equivalent to affine signatures under a unitary condition the latter is a known class of tractable functions under the holant framework

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

approximate counting for complex weighted boolean constraint satisfaction problems constraint satisfaction problems or csps have been extensively studied in for instance artificial intelligence database theory graph theory and statistical physics from a practical viewpoint it is beneficial to approximately solve those csps when one tries to approximate the total number of truth assignments that satisfy all boolean valued constraints for unweighted boolean csps there is a known trichotomy theorem by which all such counting problems are neatly classified into exactly three categories under polynomial time randomized approximation preserving reductions in contrast we obtain a dichotomy theorem of approximate counting for complex weighted boolean csps provided that all complex valued unary constraints are freely available to use it is the expressive power of free unary constraints that enables us to prove such a stronger complete classification theorem this discovery makes a step forward in the quest for the approximation complexity classification of all counting csps to deal with complex weights we employ proof techniques of factorization and arity reduction along the line of solving holant problems moreover we introduce a novel notion of t constructibility that naturally induces approximation preserving reducibility our result also gives an approximation analogue of the dichotomy theorem on the complexity of exact counting for complex weighted boolean csps

a dichotomy theorem for constraint satisfaction problems on a 3 element set the constraint satisfaction problem csp provides a common framework for many combinatorial problems the general csp is known to be np complete however certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes there is therefore a fundamental research direction aiming to separate those subclasses of the csp that are tractable and those which remain np complete schaefer gave an exhaustive solution of this problem for the csp on a 2 element domain in this article we generalise this result to a classification of the complexity of the csp on a 3 element domain the main result states that every subproblem of the csp is either tractable or np complete and the criterion separating them is that conjectured in bulatov et al 2005 and bulatov and jeavons 2001b we also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure finally we exhibit a polynomial time algorithm which for a given set of allowed constraints outputs if this set gives rise to a tractable problem class to obtain the main result and the algorithm we extensively use the algebraic technique for the csp developed in jeavons 1998b bulatov et al 2005 and bulatov and jeavons 2001b

weak exactness for c algebras and application to condition ao abstract kirchberg introduced weak exactness for von neumann algebras as an analogue of exactness for c algebras ozawa found useful characterizations of weak exactness and he proved that weak exactness for group von neumann algebras is equivalent to exactness of the groups we generalize this concept to inclusions of c algebras in von neumann algebras and study some fundamental properties and relationship to kirchberg s weak exactness we then give some permanence properties which are similar to those of exact groups in the last section we study a similar condition to ozawa s condition ao with our weak exactness and generalize ozawa s theorem for bi exact groups as a corollary we give new examples of prime factors

spin systems on k regular graphs with complex edge functions let k 1 be an integer and let h h 0 0 h 0 1 h 1 0 h 1 1 be a complex valued symmetric function on domain 0 1 i e where h 0 1 h 1 0 we introduce a new technique called a syzygy and prove a dichotomy theorem for the following class of problems specified by k and h given an arbitrary k regular graph g v e where the function h is attached to each edge compute z g email protected s v 0 1 u v eh s u s v z is known as the partition function of the spin system also known as counting graph homomorphisms on domain size two and is a special case of holant problems the dichotomy theorem gives a complete classification of the computational complexity of this problem depending on k and h the dependence on k and h is explicit

the complexity of the counting constraint satisfaction problem the counting constraint satisfaction problem csp h over a finite relational structure h can be expressed as follows given a relational structure g over the same vocabulary determine the number of homomorphisms from g to h in this article we characterize relational structures h for which csp h can be solved in polynomial time and prove that for all other structures the problem is p complete

localization study of a regularized variational damage model abstract the paper presents a detailed analysis and extended formulation of a rate independent regularized damage model proposed by mielke and roubicek 2006 localization properties are studied in the context of a simple one dimensional problem but the results reveal the fundamental features of the basic model and of its modified versions the initial bifurcation from a uniform solution is described analytically while the complete failure process is studied numerically modifications of the regularizing term and of the dissipation distance are introduced and their effect on the global response is investigated it is shown that with a proper combination of model parameters a realistic shape of the load displacement diagram can be achieved and pathological effects such as extremely brittle response or expansion of the damage zone accompanied by stress locking can be eliminated

the complexity of approximating bounded degree boolean csp the degree of a csp instance is the maximum number of times that any variable appears in the scopes of constraints we consider the approximate counting problem for boolean csp with bounded degree instances for constraint languages containing the two unary constant relations 0 and 1 when the maximum allowed degree is large enough at least 6 we obtain a complete classification of the complexity of this problem it is exactly solvable in polynomial time if every relation in the constraint language is affine it is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of 0 1 and binary implication otherwise there is no fpras unless np rp for lower degree bounds additional cases arise where the complexity is related to the complexity of approximately counting independent sets in hypergraphs

an operator space approach to condition c abstract we show that there exists a natural embedding from the tensor product v w of the biduals of operator spaces v and w into the bidual v w of the injective tensor product of v and w which is separately weak continuous from this we define condition c for operator spaces

a complexity dichotomy for partition functions with mixed signs partition functions also known as homomorphism functions form a rich family of graph invariants that contain combinatorial invariants such as the number of k colorings or the number of independent sets of a graph and also the partition functions of certain spin glass models of statistical physics such as the ising model building on earlier work by dyer and greenhill random structures algorithms 17 2000 pp 260 289 and bulatov and grohe theoret comput sci 348 2005 pp 148 186 we completely classify the computational complexity of partition functions our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or p complete partition functions are described by symmetric matrices with real entries and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or p complete while in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases for partition functions described by hadamard matrices these turn out to be central in our proofs we obtain a simple algebraic tractability criterion which says that the tractable cases are those representable by a quadratic polynomial over the field mathbb f 2

localization properties of strain softening gradient plasticity models part ii theories with gradients of internal variables the second part of this paper compares and evaluates enhancements of the conventional plasticity theory by gradients of internal variables attention is focused on their performance as localization limiters both explicit and implicit gradient formulations are considered it is shown that certain models suffer by serious mathematical deficiencies that would complicate their numerical implementation some other models are appropriate only at early stages of the softening process but later exhibit locking accompanied by a spurious expansion of the localized plastic zone the comparative study indicates that a convenient and robust tool for regularized modeling of the entire localization process is provided by the implicit gradient approach combined with a suitable form of the hardening softening law

the expressibility of functions on the boolean domain with applications to counting csps an important tool in the study of the complexity of constraint satisfaction problems csps is the notion of a relational clone which is the set of all relations expressible using primitive positive formulas over a particular set of base relations post s lattice gives a complete classification of all boolean relational clones and this has been used to classify the computational difficulty of csps motivated by a desire to understand the computational complexity of weighted counting csps we develop an analogous notion of functional clones and study the landscape of these clones one of these clones is the collection of log supermodular lsm functions which turns out to play a significant role in classifying counting csps in the conservative case where all nonnegative unary functions are available we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone containing all functions thus any counting csp that contains a single nontrivial non lsm function is computationally as hard to approximate as any problem in p furthermore we show that any nontrivial functional clone in a sense that will be made precise contains the binary function implies as a consequence in the conservative case all nontrivial counting csps are as hard to approximate as bis the problem of counting independent sets in a bipartite graph given the complexity theoretic results it is natural to ask whether the implies clone is equivalent to the clone of lsm functions we use the mobius transform and the fourier transform to show that these clones coincide precisely up to arity 3 it is an intriguing open question whether the lsm clone is finitely generated finally we investigate functional clones in which only restricted classes of unary functions are available

weak exactness for dual operator spaces we introduce a new tensor product circle times sigma lambda and study the weak condition c which is also called weak exactness for dual operator spaces our definition of weak condition c is equivalent to kirchberg s notion of weak exactness in the case of von neumann algebras we also study the connection of weak exact w tros with their linking von neumann algebras and study the structure of exact respectively nuclear w tros c 2007 elsevier inc all rights reserved

an evaluation of higher order plasticity theories for predicting size effects and localisation conventional plasticity theories are unable to capture the observed increase in strength of metallic structures with diminishing size they also give rise to ill posed boundary value problems at the onset of material softening in order to overcome both deficiencies a range of higher order plasticity theories have been formulated in the literature the purpose of this paper is to compare existing higher order theories for the prediction of a size effect and the handling of localisation effects to this end size effect predictions for foils in bending are compared with existing experimental data furthermore a study of one dimensional harmonic incremental solutions from a uniform reference state allows one to assess the nature of material localisation as predicted by these competing higher order theories these analyses show that only one of the theories considered the fleck hutchinson strain gradient plasticity theory based upon the toupin mindlin strain gradient framework fleck n a hutchinson j w 1997 strain gradient plasticity adv appl mech 33 295 361 allows one to describe both phenomena the other theories show either nonphysical size effects or a pathologically localised post peak response

dispersion and isogeometric analyses of second order and fourth order implicit gradient enhanced plasticity models implicit gradient plasticity models incorporate higher order spatial gradients via an additional helmholtz type equation for the plastic multiplier so far the enrichment has been limited to second order spatial gradients resulting in a formulation that can be discretised using c0 continuous finite elements herein an implicit gradient plasticity model is formulated that includes a fourth order gradient term as well a comparison between the localisation properties of both the implicit gradient plasticity formulations and the explicit second order gradient plasticity model is made using a dispersion analysis the higher order continuity requirement for the fourth order implicit gradient plasticity model has been met by exploiting the higher order continuity property of isogeometric analysis which uses nonuniform rational b splines as shape functions instead of lagrange polynomials the discretised variables displacements and plastic multiplier may require different orders of interpolation an issue that is also addressed numerical results show that both formulations can be used as a localisation limiter but that quantitative differences occur and a different evolution of the localisation band is obtained for 2 dimensional problems

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

a micromechanics based approach for the derivation of constitutive elastic coefficients of strain gradient media a micromechanics based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented at the local scale all phases of the composite obey the classic equations of three dimensional elasticity but since the assumption of strict separation of scales is not verified the macroscopic behavior is described by the equations of strain gradient elasticity the methodology uses the series expansions at the local scale for which higher order terms which are generally neglected in standard homogenization framework are kept in order to take into account the microstructural effects an energy based micro macro transition is then proposed for upscaling and constitutes in fact a generalization of the hill mandel lemma to the case of higher order homogenization problems the constitutive relations and the definitions for higher order elasticity tensors are retrieved by means of the state law associated to the derived macroscopic potential as an illustration purpose we derive the closed form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite for handling the problems with an arbitrary microstructure a fft based computational iterative scheme is proposed in the last part of the paper its efficiency is shown in the particular case of composites reinforced by long fibers

model reductions for inference generality of pairwise binary and planar factor graphs we offer a solution to the problem of efficiently translating algorithms between different types of discrete statistical model we investigate the expressive power of three classes of model those with binary variables with pairwise factors and with planar topology as well as their four intersections we formalize a notion of simple reduction for the problem of inferring marginal probabilities and consider whether it is possible to simply reduce marginal inference from general discrete factor graphs to factor graphs in each of these seven subclasses we characterize the reducibility of each class showing in particular that the class of binary pairwise factor graphs is able to simply reduce only positive models we also exhibit a continuous spectral reduction based on polynomial interpolation which overcomes this limitation experiments assess the performance of standard approximate inference algorithms on the outputs of our reductions

inapproximability of the tutte polynomial the tutte polynomial of a graph g is a two variable polynomial t g x y that encodes many interesting properties of the graph we study the complexity of the following problem for rationals x and y take as input a graph g and output a value which is a good approximation to t g x y jaeger et al have completely mapped the complexity of exactly computing the tutte polynomial they have shown that this is p hard except along the hyperbola x 1 y 1 1 and at four special points we are interested in determining for which points x y there is a fully polynomial randomised approximation scheme fpras for t g x y under the assumption rp np we prove that there is no fpras at x y if x y is in one of the half planes x np there is no fpras at the point x y 0 email protected when l 2 is a positive integer thus there is no fpras for counting nowhere zero l flows for l 2 this is an interesting consequence of our work since the corresponding decision problem is in p for example for l 6 although our main concern is to distinguish regions of the tutte plane that admit an fpras from those that do not we also note that the latter regions exhibit different levels of intractability at certain points x y for example the integer points on the x axis or any point in the positive quadrant there is a randomised approximation scheme for t g x y that runs in polynomial time using an oracle for an np predicate on the other hand we identify a region of points x y at which even approximating t g x y is as hard as p

the complexity of weighted boolean csp with mixed signs we give a complexity dichotomy for the problem of computing the partition function of a weighted boolean constraint satisfaction problem such a problem is parameterized by a set c of rational valued functions which generalize constraints each function assigns a weight to every assignment to a set of boolean variables our dichotomy extends previous work in which the weight functions were restricted to being non negative we represent a weight function as a product of the form 1 sg where the polynomial s determines the sign of the weight and the non negative function g determines its magnitude we show that the problem of computing the partition function the sum of the weights of all possible variable assignments is in polynomial time if either every function in c can be defined by a pure affine magnitude with a quadratic sign polynomial or every function can be defined by a magnitude of product type with a linear sign polynomial in all other cases computing the partition function is fp p complete

quantum circuits that can be simulated classically in polynomial time a model of quantum computation based on unitary matrix operations was introduced by feynman and deutsch it has been asked whether the power of this model exceeds that of classical turing machines we show here that a significant class of these quantum computations can be simulated classically in polynomial time in particular we show that two bit operations characterized by 4 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time this contrasts with the known universality of two bit operations and demonstrates that efficient quantum computation of restricted classes is reconcilable with the polynomial time turing hypothesis therefore it is possible that the techniques introduced bring the quantum computational model within the realm of algebraic complexity theory in a manner consistent with one view of quantum physics the wave function is simulated deterministically and randomization arises only in the course of making measurements the results generalize the quantum model in that they do not require the matrices to be unitary in a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read twice boolean formulae all our results are based on the use of gates that are defined in terms of their graph matching properties

on injectivity and nuclearity for operator spaces an injective operator space v which is dual as a banach space has the form er 1 e where r is an injective von neumann algebra and where e is a projection in r this is used to show that an operator space v is nuclear if and only if it is locally reflexive and v ast ast is injective it is also shown that any exact operator space is locally reflexive

complexity classification of the six vertex model abstract we prove a complexity dichotomy theorem for the six vertex model for every setting of the parameters of the model we prove that computing the partition function is either solvable in polynomial time or p hard the dichotomy criterion is explicit

holographic algorithms by fibonacci gates abstract we introduce fibonacci gates as a polynomial time computable primitive and develop a theory of holographic algorithms based on these gates the fibonacci gates play the role of matchgates in valiant s theory valiant 2008 19 they give rise to polynomial time computable counting problems on general graphs while matchgates mainly work over planar graphs only we develop a signature theory and characterize all realizable signatures for fibonacci gates for bases of arbitrary dimensions we prove a basis collapse theorem we apply this theory to give new polynomial time algorithms for certain counting problems we also use this framework to prove that some slight variations of these counting problems are p hard holographic algorithms with fibonacci gates prove to be useful as a general tool for classification results of counting problems dichotomy theorems cai et al 2009 7

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

strongly non local gradient enhanced finite strain elastoplasticity this paper presents a gradient plasticity formulation developed in a strongly non local format and applicable to ductile failure for finite strain plasticity the obtained formulation is an extension of a framework proposed by simo computer methods in applied mechanics and engineering 1988 66 199 into the softening and localization regime the presented model inherits the well established regularization properties of its strongly non local enrichment and leads to an implicit computational scheme with a consistent tangent operator numerical examples are given which illustrate the strongly non local character of the formulation the influence of the type of intrinsic length scale considered as well as several topics on plastic failure initiation and evolution into the geometrically non linear regime copyright 2003 john wiley sons ltd

an implicit tensorial gradient plasticity model formulation and comparison with a scalar gradient model many rate independent models for metals utilize the gradient of effective plastic strain to capture size dependent behavior this enhancement sometimes termed as explicit gradient formulation requires higher order tractions to be imposed on the evolving elasto plastic boundary and the resulting numerical framework is complicated an implicit scalar gradient model was thus developed in peerlings peerlings r h j 2007 on the role of moving elastic plastic boundaries in strain gradient plasticity model simul mater sci eng 15 109 120 that has only c0 continuity requirements and its implementation is straightforward however both explicit and implicit scalar gradient models can be problematic when the effective plastic strains do not have smooth profiles to address this limitation an implicit tensorial gradient model is proposed in this paper based on the generalized micromorphic framework it is also demonstrated that the scalar and tensorial implicit gradient models give similar results when the effective plastic strains fluctuate smoothly

a dichotomy theorem for the approximate counting of complex weighted bounded degree boolean csps we determine the computational complexity of approximately counting the total weight of variable assignments for every complex weighted boolean constraint satisfaction problem or csp with any number of additional unary i e arity 1 constraints particularly when degrees of input instances are bounded from above by a fixed constant all degree 1 counting csps are obviously solvable in polynomial time when the instance s degree is more than two we present a dichotomy theorem that classifies all counting csps admitting free unary constraints into exactly two categories this classification theorem extends to complex weighted problems an earlier result on the approximation complexity of unweighted counting boolean csps of bounded degree the framework of the proof of our theorem is based on a theory of signature developed from valiant s holographic algorithms that can efficiently solve seemingly intractable counting csps despite the use of arbitrary complex weight our proof of the classification theorem is rather elementary and intuitive due to an extensive use of a novel notion of limited t constructibility for the remaining degree 2 problems in contrast they are as hard to approximate as holant problems which are a generalization of counting csps

local properties of ternary rings of operators and their linking c algebras abstract we show that some local properties such as nuclearity exactness and local reflexivity of ternary rings of operators tros are closely related to the local properties of their linking c algebras we also show some equivalent conditions for nuclear tros and show that haagerup s decomposition property for completely bounded maps and pisier s norm can be naturally generalized to tros

deterministic polynomial time approximation algorithms for partition functions and graph polynomials in this paper we show a new way of constructing deterministic polynomial time approximation algorithms for computing complex valued evaluations of a large class of graph polynomials on bounded degr

a complete dichotomy rises from the capture of vanishing signatures we prove a complexity dichotomy theorem for holant problems over an arbitrary set of complex valued symmetric constraint functions mathcal f on boolean variables this extends and unifies all previous dichotomies for holant problems on symmetric constraint functions taking values without a finite modulus we define and characterize all symmetric vanishing signatures they turn out to be essential to the complete classification of holant problems the dichotomy theorem has an explicit tractability criterion expressible in terms of holographic transformations a holant problem defined by a set of constraint functions mathcal f is solvable in polynomial time if it satisfies this tractability criterion and is p hard otherwise the tractability criterion can be intuitively stated as follows a set mathcal f is tractable if 1 every function in mathcal f has arity at most two or 2 mathcal f is transformable to an affine type or 3 mathcal f is transformable to a product type o

gradient dependent plasticity formulation and algorithmic aspects a plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure hardening parameter but also on the laplacian thereof the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity to properly solve the set of non linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements for appropriate boundary conditions this formulation can also be derived from a variational principle accordingly the theory is complete

a critical comparison of nonlocal and gradient enhanced softening continua continuous models of material degradation may cease to produced meaningful results in the presence of high strain gradients these gradients may occur for instance in the propagation of waves with high wave numbers and at stress concentrators adding nonlocal or gradient terms to the constitutive modelling may enhance the ability of the models to describe such situations the effect of adding nonlocal or gradient terms and the relation between these enhancements are examined in a continuum damage setting a nonlocal damage model and two different gradient damage models are considered in one of the gradient models higher order deformation gradients enter the equilibrium equations explicitly while in the other model the gradient influence follows in a more implicit way from an additional partial differential equation the latter implicit gradient formulation can be rewritten in the integral format of the nonlocal model and can therefore be regarded as truly nonlocal this is not true for the explicit formulation in which the nonlocality is limited to an infinitesimal volume this fundamental difference between the formulations results in quite different behaviour in wave propagation localisation and at crack tips this is shown for the propagation of waves in the models their localisation properties and the behaviour at a crack tip the responses of the nonlocal model and the implicit gradient model agree remarkably well in these situations while the explicit gradient formulation shows an entirely different and sometimes nonphysical response

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

beyond csp a dichotomy for counting weighted eulerian orientations with ars abstract we define and explore a notion of unique prime factorization for constraint functions and use this as a new tool to prove a complexity classification for counting weighted eulerian orientation problems with arrow reversal symmetry ars we prove that all such problems are either polynomial time computable or p hard we show that the class of weighted eulerian orientation problems subsumes all weighted counting constraint satisfaction problems csp on boolean variables more significantly we establish a novel connection between csp and counting weighted eulerian orientation problems that is global in nature this connection is based on a structural determination of all half weighted affine linear subspaces over z 2 which is proved using mobius inversion

a variational principle for gradient plasticity abstract we elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by aifantis and co workers the generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher order spatial gradients of the equivalent plastic strain in the yield condition first it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands due to the presence of higher order gradients additional boundary conditions for the equivalent plastic strain are required this question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work we demonstrate here that upon assuming a certain type of additional boundary conditions the structural symmetries of the gradient dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain the variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given

reflection positivity rank connectivity and homomorphism of graphs it is shown that a graph parameter can be realized as the number of homomorphisms into a fixed weighted graph if and only if it satisfies two linear algebraic conditions reflection positivity and exponential rank connectivity in terms of statistical physics this can be viewed as a characterization of partition functions of vertex models

the complexity of weighted boolean csp this paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted boolean constraint satisfaction problem the problem is parameterized by a finite set mathcal f of nonnegative functions that may be used to assign weights to the configurations feasible solutions of a problem instance classical constraint satisfaction problems correspond to the special case of 0 1 valued functions we show that computing the partition function i e the sum of the weights of all configurations is text sf fp text sf p complete unless either 1 every function in mathcal f is of product type or 2 every function in mathcal f is pure affine in the remaining cases computing the partition function is in p

expressiveness of matchgates matchgates have been used to simulate classically in polynomial time assemblies of quantum gates in this paper it is shown that every matchgate for 2 input 2 output functions has to obey a certain set of five polynomial identities it is also shown that no matchgate can realize a nontrivial control gate for any number of inputs on the other hand it is proved that classical boolean formulae can be expressed as matchcircuits of polynomial size

from continuous to discontinuous failure in a gradient enhanced continuum damage model abstract a computational framework for the description of the combined continuous discontinuous failure in a regularised strain softening continuum is proposed the continuum is regularised through the introduction of gradient terms into the constitutive equations at the transition to discrete failure the problem fields are enhanced through a discontinuous interpolation based on the partition of unity paradigm of finite element shape functions the inclusion of internal discontinuity surfaces where boundary conditions are applied without modifications of the original finite element mesh avoids the unrealistic damage growth typical of this class of regularised continuum models combined models allow for the analysis of the entire failure process from diffuse microcracking to localised macrocracks the discretisation procedure is described in detail and numerical examples illustrate the performance of the combined continuous discontinuous approach

fe formulation of a nonlocal plasticity theory a nonlocal continuum plasticity theory is presented the nonlocal field introduced here is defined as a certain weighted average of the corresponding local field taken over all the material points in the body hereby a quantity with the dimension of length occurs as a material parameter when this so called internal length is equal to zero the local classical plasticity theory is regained in the present model the yield function will depend on a nonlocal field the consistency condition and the integration algorithm result in integral equations for determination of the field of plastic multipliers the integral equations are classified as fredholm equations of the second kind and the existence of a solution will be commented upon after discretization a matrix equation is obtained and an algorithm for finding the solution is proposed for a generalized von mises material a plane boundary value problem is solved with a fe method since the nonlocal quantities are integrals c0 continuous elements are sufficient the solution strategy is split into a displacement estimate for equilibrium and the integration of constitutive equations in the numerical simulations shear band formation is analysed and the results display mesh insensitivity

holographic algorithm with matchgates is universal for planar csp over boolean domain we prove a complexity classification theorem that classifies all counting constraint satisfaction problems csp over boolean variables into exactly three classes 1 polynomial time solvable 2

approximation complexity of complex weighted degree two counting constraint satisfaction problems constraint satisfaction problems have been studied in numerous fields with practical and theoretical interests in recent years major breakthroughs have been made in a study of counting constraint satisfaction problems or csps in particular a computational complexity classification of bounded degree csps has been discovered for all degrees except for two where the degree of an input instance is the maximal number of times that each input variable appears in a given set of constraints despite the efforts of recent studies however a complexity classification of degree 2 csps has eluded from our understandings this paper challenges this open problem and gives its partial solution by applying two novel proof techniques t 2 constructibility and parametrized symmetrization which are specifically designed to handle arbitrary constraints under randomized approximation preserving reductions we partition entire constraints into four sets and we classify the approximation complexity of all degree 2 csps whose constraints are drawn from two of the four sets into two categories problems computable in polynomial time or problems that are at least as hard as sat our proof exploits a close relationship between complex weighted degree 2 csps and holant problems which are a natural generalization of complex weighted csps