relaxation of three solenoidal wells and characterization of extremal three phase h measures we fully characterize quasiconvex hulls for three arbitrary solenoidal divergence free wells in dimension three with this aim we establish weak lower semicontinuity of certain functionals with integrands restricted to generic two dimensional planes and convex in up to three rank 2 directions within the planes within the framework of the theory of compensated compactness the latter represents an example when the differential constraints fail the constant rank condition but nevertheless the so called convexity still implies lower semicontinuity and fancyscript a quasiconvexity which essentially means that rank 2 convexity implies s quasiconvexity that is quasiconvexity in the sense of the divergence free differential constraints on the planes the proof employs a version of muller s estimates of haar wavelet projections in terms of the riesz transform the above semicontinuity result is then applied to the three solenoidal wells problem via analogs of sverak s nontrivial quasiconvex functions and connectedness properties of the rank 2 envelopes as another application of the semicontinuity result we obtain a geometric result of a more general nature characterization of certain extremal three point h measures for three phase mixtures of three characteristic functions in dimension three we also discuss the applicability of the results to problems with other differential constrains in particular to three linear elastic wells and further generalizations

regularity of inverses of sobolev deformations with finite surface energy abstract let u be a sobolev w 1 p map from a bounded open set r n to r n we assume u to satisfy some invertibility properties that are natural in the context of nonlinear elasticity namely the topological condition inv and the orientation preserving constraint det d u 0 these deformations may present cavitation which is the phenomenon of void formation we also assume that the surface created by the cavitation process has finite area if p n 1 we show that a suitable defined inverse of u is a sobolev map a partial result is also given for the critical case p n 1 the proof relies on the techniques used in the study of cavitation

elastic scalar invariants in the theory of defective crystals in the context of a continuum theory of crystals with defects the elastic scalar invariants are functions of the lattice vectors and their spatial gradients which remain invariant under elastic ch

a recognition and parsing algorithm for arbitrary conjunctive grammars conjunctive grammars are basically context free grammars with an explicit set intersection operation added to the formalism of rules this paper presents a cubic time recognition and parsing algorithm for this family of grammars which is applicable to an arbitrary conjunctive grammar without any initial transformations the algorithm is in fact an extension of the context free recognition and parsing algorithm due to graham harrison and ruzzo and it retains the cubic time complexity of its prototype it is shown that for the case of linear conjunctive grammars this algorithm can be modified to work in quadratic time and use linear space the given algorithm is then applied to solve the membership problem for conjunctive grammars in polynomial time and subsequently to prove the problem s p completeness as well as p completeness of the membership problem for linear conjunctive grammars

sequential machine characterizations of trellis and cellular automata and applications we look at a simple but general model of a systolic system called a trellis automaton ta a ta is equivalent in computational power to a one dimensional unbounded cellular automaton ca a model of parallel computation which has been studied extensively in the literature different varieties of ta s are equivalent to corresponding variations of ca s we present for the first time sequential machine characterizations of ta s ca s the sequential machines are useful and powerful tools for investigating properties of ta s ca s they ar easy to program because unlike the parallel models one does not have to deal with the problem of synchronization several applications are given in particular we prove a new speed up theorem which is stronger than what has previously been shown

bootstrapping variables in algebraic circuits we show that for the blackbox polynomial identity testing pit problem it suffices to study circuits that depend only on the first extremely few variables one needs only to consider size s degree s circuits that depend on the first log c s variables where c is a constant and composes a logarithm with itself c times thus the hitting set generator hsg manifests a bootstrapping behavior a partial hsg against very few variables can be efficiently grown to a complete hsg a boolean analog or a pseudorandom generator property of this type is unheard of our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially w r t variables this is repeated c times in an efficient way pushing the envelope further we show that i a quadratic time blackbox pit for 6 913 variate degree s size s polynomials will lead to a near complete derandomization of pit and ii a blackbox pit for n variate degree s size s circuits in s n time for 1 2 will lead to a near complete derandomization of pit in contrast s n time is trivial our second idea is to study depth 4 circuits that depend on constantly many variables we show that a polynomial time computable o s 1 49 degree hsg for trivariate depth 4 circuits bootstraps to a quasipolynomial time hsg for general polydegree circuits and implies a lower bound that is a bit stronger than that of kabanets and impagliazzo kabanets v impagliazzo r 2003 proceedings of the thirty fifth annual acm symposium on theory of computing stoc 03

uniform rearrangement in defective crystals we show that an abstract notion of defectiveness yields the slip mechanism of classic plasticity theory by reasoning in purely kinematical terms and we catalogue the canonical forms of slip that occur in this way there appears in consequence a precise version of the idea of elastic plastic decomposition

concatenation free languages the expressive capacity of three different types of regular expressions without concatenation is studied in particular we consider alphabetic concatenation free expressions which are ordinary regular expressions without concatenation simple concatenation free expressions where the set of literals is a finite set of words instead of letters and concatenation free expressions where additionally complementation operations are possible characterizations of the corresponding language classes are obtained in particular a characterization of unary concatenation free languages by the boolean closure of certain sets of languages is shown the characterizations are then used to derive a strict hierarchy that is in turn strictly included in the family of regular languages the closure properties of the families are investigated furthermore the position of the family of concatenation free languages in the subregular hierarchy is considered and settled for the unary case in particular there are concatenation free languages that do not belong to any of the families in the hierarchy moreover except for comets all the families considered in the subregular hierarchy are strictly included in the family of concatenation free languages

on nontrivial solutions of a dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue let be a bounded domain in rn n 2 with smooth boundary let p r r be a continuous function such that p 0 0 and limt p t t p exist we discuss the existence of nontrivial solutions of the dirichlet problem u p u in u 0 on our hypotheses relate the position of the point p p in the plane with the extended spectrum of they seem to allow noticeably greater freedom for the numbers p than is usually found in the literature in the sense that the interval with end points p is here permitted to contain in its interior an eigenvalue of arbitrarily multiplicity of it also appears that we significantly weaken some technical conditions on p of another paper which treats the crossing of a simple eigenvalue

a minimization problem involving variation of the domain keywords calculus of variations existence and regularity of minimizers deformation of cristals energy functional reference caa article 1992 003doi 10 1002 cpa 3160450704view record in web of science record created on 2008 11 25 modified on 2017 05 12

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

nonlocal approximation of nonisotropic free discontinuity problems we prove that a class of free discontinuity problems with nonisotropic bulk and surface energy densities is approximated in the sense of gamma convergence by suitable families of nonlocal integral functionals some connections with the wulff set are also pointed out

language equations with complementation expressive power consider a system of language equations of the form xi i x1 xn 1 i n where every i may contain the operations of concatenation and complementation these systems have been studied in language equations with complementation decision problems a okhotin o yakimova theoretical computer science 376 2007 112 126 this paper investigates the family of languages representable by unique solutions of such systems a method for proving nonrepresentability of particular languages is developed several natural subfamilies of this family are compared to each other and to the main known families of formal languages their position in the hierarchy is established

discrete order preserving semigroups and stability for periodic parabolic differential equations on etudie la stabilite globale des solutions des equations paraboliques dans une forme non auto adjointe avec des coefficients periodiques dans le temps la structure mathematique est celle des semigroupes preservant l ordre a temps discret s n n e n sur un reseau de banach

systolic trellis automata stability decidability and complexity systolic trellis automata are models of hexagonally connected and triangularly shaped systolic arrays this paper studies the problems of stability decidability and complexity for them the original definition of systolic trellis automata requires that an input string is fed to a specific row of processors here it is shown that given a homogeneous trellis automaton we can construct an equivalent one stable or superstable which allows to feed the input string to any sufficiently long row of processors moreover some closure and decidability results for trellis automata are established and the computational complexity of languages accepted by trellis automata is investigated

bulk and interfacial energy densities for structured deformations of continua to provide a basis for the application of techniques in the calculus of variations to non classical deformations of continua these structured deformations of continua are suitable for describing deformations of materials whose kine matics warrants analysis at both the macroscopic and microscopic levels the motivation for this work lies in the study of equilibrium configura tions of crystals with defects in a defective crystal the macroscopic defor mation together with the referential bravais lattice configuration do not suace to describe fully the configuration of a deformed body phenomena such as slips vacancies and dislocations may be present in the deformed bravais lattice basis thus preventing the use of the cauchy born hypoth esis as described below in a perfect crystal it is postulated that the crystal lattice consists of identical atoms located at all positions vectors

state complexity of unambiguous operations on finite automata abstract the paper determines the number of states in finite automata necessary to represent unambiguous variants of the union concatenation and kleene star operations on formal languages for the disjoint union of languages represented by an m state and an n state deterministic finite automata dfa the state complexity is m n 1 for the unambiguous concatenation it is known to be m 2 n 1 2 n 2 daley et al orthogonal concatenation language equations and state complexity j ucs 2010 and this paper shows that this number of states is necessary already over a binary alphabet for the unambiguous star the state complexity function is determined to be 3 8 2 n 1 in the case of a unary alphabet disjoint union requires up to 1 2 m n states in a dfa unambiguous concatenation has state complexity m n 2 and unambiguous star requires n 2 states in the worst case for nondeterministic finite automata as well as for unambiguous finite automata the complexity for disjoint union is m n for unambiguous concatenation square and star the resulting complexities are m n 2n and n 1 respectively and all of them are witnessed by binary languages in the unary nondeterministic or unambiguous case the corresponding complexities are m n for disjoint union m n 1 and 2 n 1 for unambiguous concatenation and square respectively and n 1 for unambiguous star

injectivity almost everywhere for weak limits of sobolev homeomorphisms abstract let r n be an open set and let f w 1 p r n be a weak sequential limit of sobolev homeomorphisms then f is injective almost everywhere for p n 1 both in the image and in the domain for p n 1 we construct a strong limit of homeomorphisms such that the preimage of a point is a continuum for every point in a set of positive measure in the image and the topological image of a point is a continuum for every point in a set of positive measure in the domain

conjunctive grammars with restricted disjunction it is shown that every conjunctive language is generated by a conjunctive grammar from a special subclass in which every nonterminal a has at most one rule of the general form a a email protected n while the rest of the rules for a must be of the type a w where w is a terminal string for context free grammars a similar property does not hold s a greibach et al 1992 3 if it is furthermore required that each rule a w has a nonempty w then a substantial subfamily of conjunctive languages can be generated yet it remains unknown whether such grammars are as powerful as conjunctive grammars of the general form

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

a complete list of invariants for defective crystals the classical theory of continuous distributions of dislocations has traditionally focused on the burgers vectors and the dislocation density tensor as descriptions of defectiveness we prove that generally there is an infinite number of tensor densities with similarly descriptive properties and that there is a functional basis for this list which strictly includes the burgers vectors and dislocation density moreover the changes of state which preserve these densities turn out to represent slip in certain surfaces associated with crystal geometry so that the basic mechanism of plasticity emerges naturally from abstract ideas which neither anticipate nor involve the kinematics of particular types of crystal defects

on a class of invariant functionals a characterization of a class of functionals invariant under isochoric changes of domain is obtained this class contains strictly the null lagrangians

l a l b decidability results from complete formal systems abstract the equivalence problem for deterministic pushdown automata is shown to be decidable we exhibit a complete formal system for deducing equivalent pairs of deterministic rational boolean series on the alphabet associated with a dpda m we then extend the result to deterministic pushdown transducers from a free monoid into an abelian group a general algebraic and logical framework inspired by harrison et al theoret comput sci 9 1979 173 205 conrcelle theoret comput sci 6 1978 255 279 and meitus kybernetika 5 1989 14 25 in russian is developed

on generalized language equations a system of generalized language equations over an alphabet a is a set of n equations in n variables xi gi x1 xn i 1 n where the gi are functions from p a n into p a i 1 n p a denoting the set of all languages over a furthermore the gi are expressible in terms of set operations concatenations and stars which involve the variable xi as well as certain mixed languages in this note we investigate existence and uniqueness of solutions of a certain subclass of generalized language equations furthermore we show that a solution is regular if all fixed languages are regular

area time efficient architecture of fft based montgomery multiplication the modular multiplication operation is the most time consuming operation for number theoretic cryptographic algorithms involving large integers such as rsa and diffie hellman implementations reveal that more than 75 percent of the time is spent in the modular multiplication function within the rsa for more than 1 024 bit moduli there are fast multiplier architectures to minimize the delay and increase the throughput using parallelism and pipelining however such designs are large in terms of area and low in efficiency in this paper we integrate the fast fourier transform fft method into the mclaughlin s framework and present an improved fft based montgomery modular multiplication mmm algorithm achieving high area time efficiency compared to the previous fft based designs we inhibit the zero padding operation by computing the modular multiplication steps directly using cyclic and nega cyclic convolutions thus we reduce the convolution length by half furthermore supported by the number theoretic weighted transform the fft algorithm is used to provide fast convolution computation we also introduce a general method for efficient parameter selection for the proposed algorithm architectures with single and double butterfly structures are designed obtaining low area latency solutions which we implemented on xilinx virtex 6 fpgas the results show that our work offers a better area latency efficiency compared to the state of the art fft based mmm architectures from and above 1 024 bit operand sizes we have obtained area latency efficiency improvements up to 50 9 percent for 1 024 bit 41 9 percent for 2 048 bit 37 8 percent for 4 096 bit and 103 2 percent for 7 680 bit operands furthermore the operating latency is also outperformed with high clock frequency for length 64 transform and above

unrestricted complementation in language equations over a one letter alphabet recently the complete solution of systems of language equations over a one letter alphabet has been derived where the operators are union concatenation and star this paper addresses the question of adding the complementation operator it is known that if no stars are present and if the concatenation operator is restricted such equations need not have solutions but if they do they can be solved explicitly if the constants are regular in which case the solutions are also regular in this paper we study language equations over a one letter alphabet where unrestricted concatenation interacts with complementation union and star we define a procedure for determining a regular solution of the general type of equation assuming that all constants are regular discuss its properties and show examples we then prove that in general equations with complementation and complementation and concatenation in which all constants are regular may have nonregular solutions this shows that language equations can be used to obtain non context free languages from regular languages using only operators under which the regular languages are closed

configurational stress yield and flow in rate independent plasticity the role of configurational stress in yield and plastic flow is discussed for a macroscopic model of rate independent finite strain plasticity the model is based on the traditional elastic plastic decomposition of the deformation gradient on integral balance laws and on thermodynamically restricted rate independent constitutive relations its formulation emphasizes the intermediate configuration in both the development of constitutive relations and the expression of balance laws in addition to the usual balance laws a couple balance is included to represent the action of plastic couples in the intermediate configuration in particular it is shown that the internal couple decomposes into a non dissipative configurational stress and a dissipative couple that resists plastic flow the couple balance thus determines a relation between the configurational stress and the plastic flow resistance a relation that can be interpreted as a generalized yield condition a dissipation function is introduced and a maximum dissipation criterion is used to obtain additional constitutive restrictions which lead to a counterpart in the intermediate configuration of the classical normality conditions the versatility of the framework is illustrated by applying it to rigid plastic flow in which case a nonlinear generalization of the classical levy von mises theory is obtained

computational completeness of equations over sets of natural numbers systems of finitely many equations of the form f x 1 x n email protected x 1 x n are considered in which the unknowns x i are sets of natural numbers while the expressions f j may contain singleton constants and the operations of union and pairwise addition s t m n email protected s email protected t it is shown that the family of sets representable by unique least greatest solutions of such systems is exactly the family of recursive r e co r e respectively sets of numbers basic decision problems for these systems are located in the arithmetical hierarchy the same results are established for equations with addition and intersection

material symmetry and singularities in solids we examine the relations between the symmetry group of a crystalline or solid body and the line singularities in the deformation or microstructural field which the material can support a method is developed which in some relevant cases allows us to describe a singularity by means of a suitable derham current supported on the defect submanifold this current measures the obstruction to the extension of the local deformation or microstructure to a globally defined field the procedure described here yields the basis of a formulation of weak field equations for a defective material and clarifies the structure of the compatibility relations for virtually any class of material symmetry

language equations over a one letter alphabet with union concatenation and star a complete solution abstract this paper develops a theory of language equations over a one letter alphabet where the operations involved in the equations are union unrestricted concatenation and star several key theorems are proven and tied together to provide an algorithm which enables one to solve any equation of this type an application of this result is a constructive proof of the well known fact that any context free grammar with only one terminal always generates a regular language

on the closure properties of linear conjunctive languages linear conjunctive grammars are conjunctive grammars in which the body of each conjunct contains no more than a single nonterminal symbol they can at the same time be thought of as a special case of conjunctive grammars and as a generalization of linear context free grammars that provides an explicit intersection operation although the set of languages generated by these grammars is known to include many important noncontext free languages linear conjunctive languages are still all square time and several practical algorithms have been devised to handle them which makes this class of grammars quite suitable for use in applications in this paper we investigate the closure properties of the language family generated by linear conjunctive grammars the main result is its closure under complement which implies that it is closed under all set theoretic operations we also consider several cases in which the concatenation of two linear conjunctive languages is certain to be linear conjunctive in addition it is demonstrated that linear conjunctive languages are closed under quotient with finite languages not closed under quotient with regular languages and not closed under free homomorphism

a lusin type theorem for gradients abstract we prove that for every borel vector field f there exists a function u of class c 1 whose gradient du agrees with f outside a set of arbitrary small measure

on the unique satisfiability problem unique sat is the problem of deciding whether a given boolean formula has exactly one satisfying truth assignment this problem is a typical moreover complete representative of a natural class of problems about unique solutions all these problems belong to the class dife l1 l2 l1 lz np studied by papadimitriou and yannakakis we consider the relationship between these two classes particularly whether unique sat is dife complete it is if np co np we construct an oracle relative to which unique sat is not complete for dif and another oracle relative to which unique sat is complete for dif e whereas np v co np

on the number of nonterminals in linear conjunctive grammars the number of nonterminals in a linear conjunctive grammar is considered as a descriptional complexity measure of this family of languages it is proved that a hierarchy collapses and for every linear conjunctive grammar there exists and can be effectively constructed a linear conjunctive grammar that accepts the same language and contains exactly two nonterminals this yields a partition of linear conjunctive languages into two nonempty disjoint classes of those with nonterminal complexity 1 and 2 the basic properties of the family of languages for which one nonterminal suffices are established nonterminal complexity of grammars in the linear normal form is also investigated

on solving star equations abstract a star equation in x is a language equation x f x where f is a function mapping languages to languages which can be expressed using the operations union concatenation from the left by a constant language and unrestricted star we show existence of solutions as well as uniqueness under certain additional conditions also the equation is studied whether a star equation has a regular solution if all its constant languages are regular

fracture surfaces and the regularity of inverses for bv deformations motivated by nonlinear elasticity theory we study deformations that are approximately differentiable orientation preserving and one to one almost everywhere and in addition have finite surface energy this surface energy mathcal e was used by the authors in a previous paper and has connections with the theory of currents in the present paper we prove that mathcal e measures exactly the area of the surface created by the deformation this is done through a proper definition of created surface which is related to the set of discontinuity points of the inverse of the deformation in doing so we also obtain an sbv regularity result for the inverse

the moving frame and defects in crystals abstract i outline a special theory of defective crystals where the microstructure is represented by fields of lattice vectors and their spatial derivatives since these fields are assumed to be smooth in the model it is a central issue to be precise about the sense in which such fields can represent defects and that is done by introducing elastic invariant integrals of which the most elementary examples are the burger s integrals and the dislocation density it turns out that the notion of slip has a natural place in the analysis of these elastic invariant integrals and moreover that the formulation invites one to draw results from cartan s theory of equivalence of vector fields and also from the theory of lie groups indeed it is a remarkable fact that one can identify material points in a crystal that has constant dislocation density tensor with an appropriate lie group as a consequence of which one finds that such crystals have a self similarity which generalises the classic idea of generating a perfect crystal lattice by translation of a particular unit cell generally it seems that there is much to be done in adapting known mathematical results to this context

existence of minimizers for a variational problem in two dimensional nonlinear magnetoelasticity we prove the existence of energy minimizing configurations for a two dimensional variational model of magnetoelastic materials capable of large deformations the model is based on an energy functional which is the sum of the nonlocal self energy the energy stored in the magnetic field generated by the body and permeating the whole ambient space and of the local anisotropy energy which is not weakly lower semicontinuous a further feature of the model is the presence of a non convex constraint on one of the unknowns the magnetization which is a unit vector field

generalized language equations with multiple solutions abstract a generalized language equation x f x is an equation in a variable x over an alphabet a where the language function f is expressible in terms of arbitrary set operations including complementation and concatenation from the left by a constant i e a language in previous work these equations were studied provided an easily verifiable condition was met in this case solutions always exist and are always unique in the present paper the condition is dropped as a consequence our equations may have zero one or more up to uncountably infinitely many solutions we give a representation theorem for these equations and use it to classify them for equations in one variable a complete classification is given including parametric representations of all solutions for systems of m equations in m variables the notion of 1 reducible substitution graph is introduced and employed to reduce the problem of existence of solutions of the entire system to the same problem for a single equation the paper concludes with a discussion of the implications if one assumes that all constants are regular languages and poses some open problems

characterizations and computational complexity of systolic trellis automata abstract systolic trellis automata are simple models for vlsi we characterize the computing power of these models in terms of turing machines the characterizations are useful in proving new results as well as giving simpler proofs of known results we also derive lower and upper bounds on the computational complexity of the models

parsing by matrix multiplication generalized to boolean grammars the well known parsing algorithm for context free grammars due to valiant 1975 25 is analyzed and extended to handle the more general boolean grammars which are context free grammars augmented with conjunction and negation operators in the rules the algorithm reduces construction of a parsing table to computing multiple products of boolean matrices of various sizes its time complexity on an input string of length n is o bmm n logn where bmm n is the number of operations needed to multiply two boolean matrices of size nxn which is o n w with w 2 373 as per the current knowledge a parse tree can be constructed in time mm n log o 1 n where mm n is the complexity of multiplying two integer matrices by applying a known efficient procedure for determining witnesses for boolean matrix multiplication the algorithm has a succinct proof of correctness and is ready to be implemented

language equations with complementation decision problems systems of language equations of the form xi i x1 xn 1 i n are studied here every i may contain the operations of concatenation and complementation the properties of having solutions and of having a unique solution are given mathematical characterizations as decision problems the former is np complete while the latter is pspace hard and is in co re and its decidability remains in general open uniqueness becomes decidable in the case of a unary alphabet where it is us complete and in the case of linear concatenation where it is l complete

unstable periodic solutions of a parabolic problem in the presence of non well ordered lower and upper solutions abstract we prove the solvability of the dirichlet periodic problem for a semilinear parabolic equation assuming the existence of a lower solution and an upper solution which do not satisfy the ordering condition our results yield the existence of unstable solutions localized by means of and requiring weak regularity of the coefficients of the equation

relaxation of bulk and interfacial energies in this paper we obtain an integral representation for the relaxation inbv p of the functional u mapsto int limits omega f x nabla u x dx int limits sum u varphi x u x v x dh n 1 x with respect to thebv weak convergence

covariance in plasticity covariance is imposed to the second law of thermodynamics and consequences are shown for elastic perfectly plastic bodies in this setting pointwise balances of standard and configurational action

least and greatest solutions of equations over sets of integers systems of equations with sets of integers as unknowns are considered with the operations of union intersection and addition of sets s t m n m s n t these equations were recently studied by the authors representing hyper arithmetical sets by equations over sets of integers theory of computing systems 51 2012 196 228 and it was shown that the class of sets representable by their unique solutions is exactly the class of hyper arithmetical sets in this paper it is demonstrated that greatest solutions of such equations represent exactly the 1 1 sets in the analytical hierarchy and all those sets can already be represented by systems in the resolved form x i i x 1 x n least solutions of such resolved systems represent exactly the recursively enumerable sets

on the expressive power of univariate equations over sets of natural numbers equations of the form email protected x are considered where the unknown x is a set of natural numbers the expression f x may contain the operations of set addition defined as s t m n email protected s email protected t union intersection as well as ultimately periodic constants an equation with a non periodic solution of exponential growth rate is constructed at the same time it is demonstrated that no sets with super exponential growth rate can be represented it is also shown that restricted classes of these equations cannot represent sets with super linearly growing complements nor sets that are additive bases of order 2 the results have direct implications on the power of unary conjunctive grammars with one nonterminal symbol

some remarks on the continuum theory of defects in solids a succinct account of the classical continuum theory of defects is given and some critical remarks on its relevance for the mechanics of the model are discussed

parsing boolean grammars over a one letter alphabet using online convolution consider context free grammars generating strings over a one letter alphabet for the membership problem for such grammars stated as given a grammar g and a string a n determine whether a n is generated by g only a naive o g n 2 time algorithm is known this paper develops a new algorithm solving this problem which is based upon fast multiplication of integers works in time g nlog 3n 2 o l o g n and is applicable to context free grammars augmented with boolean operations known as boolean grammars for unambiguous grammars the running time of the algorithm is reduced to g nlog 2n 2 o l o g n the algorithm is based upon a simplification of the online integer multiplication algorithm by fischer and stockmeyer m j fischer l j stockmeyer fast on line integer multiplication journal of computer and system sciences 9 3 1974 317 331

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

faster polynomial multiplication over finite fields polynomials over finite fields play a central role in algorithms for cryptography error correcting codes and computer algebra the complexity of multiplying such polynomials is still a major open problem let p be a prime and let mp n denote the bit complexity of multiplying two polynomials in fp x of degree less than n for n large compared to p we establish the bound mp n o n log n 8log n log p where log n min k n log k log n 1 stands for the iterated logarithm this improves on the previously best known bound mp n o n log n log log n log p which essentially goes back to the 1970s

injectivity and self contact in nonlinear elasticity let be a bounded open connected subset of 3 with a sufficiently smooth boundary the additional condition det dx vol is imposed on the admissible deformations of a hyperelastic body whose reference configuration is we show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate we also extend j ball s theorems on existence to this case by establishing the existence of a minimizer of the energy in the space w1 p 3 p 3 that is injective almost everywhere

nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators periodic problems where t is a given positive number a is the subdifferential of a timedependent lower semicontinuous convex function from h into 0 a with qf f co and where b t is a possibly nonmonotone multivalued nonlinear operator from d b t c h into h such that d c d b t m for all t e 0 t the case when b t is identically zero has been investigated by many authors see e g bknilan and b is 2 nagai 13 and yamada 20 as for the case when b t is not identically zero nor monotone however it seems that very little study has been devoted to this kind of problem on the other hand the cauchy problem for 1 1 has been studied by several authors i e attouch and damlamian 11 biroli 3 koi and watanabe 9 and the author 14 151 the main objective of the present paper is to pursue and develop the method employed in 151 for the periodic problem p p this paper is composed of five sections section 2 contains some preliminaries in section 3 we shall formulate main results in two distinctive types of theorems roughly speaking according to whether the external force f t can be taken arbitrarily large or not in section 4 we shall prove them by much the same method based on a schauder tychonoff type fixed point theorem as in 151 in section 5 it will be exemplified that our results give a 248 0022 0396184 3 00

on the existence of prime decompositions we investigate factorizations of regular languages in terms of prime languages a language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps we give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a prime decomposition we show that there exist non regular unary languages that do not have prime decompositions we also consider infinite factorizations of unary languages

faster integer multiplication for more than 35 years the fastest known method for integer multiplication has been the schonhage strassen algorithm running in time o n log n log log n under certain restrictive conditions there is a corresponding omega n log n lower bound all this time the prevailing conjecture has been that the complexity of an optimal integer multiplication algorithm is theta n log n we take a major step towards closing the gap between the upper bound and the conjectured lower bound by presenting an algorithm running in time n log n 2 o log n the running time bound holds for multitape turing machines the same bound is valid for the size of boolean circuits

implicit language equations existence and uniqueness of solutions we introduce a new type of language equation namely implicit equations and derive a number of properties which clearly demonstrate that the behavior of implicit equations differs substantially from that of conventional or explicit ones explicit language equations express each variable as a regular expression in terms of their variables and constant languages implicit language equations express constant languages as regular expressions in terms of their variables and constant languages an algorithm is given for determining a solution of a given system of implicit language equations if one exists provided all constant languages are regular and the operations permitted in the regular expressions are unrestricted union and concatenation from the left by a constant a characterization of the uniqueness of a solution of such a system and a method for determining all solutions are also provided

formal languages over gf 2 abstract variants of the union and concatenation operations on formal languages are investigated in which boolean logic in the definitions that is conjunction and disjunction is replaced with the operations in the two element field gf 2 conjunction and exclusive or union is thus replaced with symmetric difference whereas concatenation gives rise to a new gf 2 concatenation operation which is notable for being invertible all operations preserve regularity and for a pair of languages recognized by an m state and an n state dfa their gf 2 concatenation is recognized by a dfa with m 2 n states and this number of states is in the worst case necessary similarly the state complexity of gf 2 inverse is 2 n 1 next a new class of formal grammars based on gf 2 operations is defined and it is shown to have the same computational complexity as ordinary grammars with union and concatenation in particular simple parsing in time o n 3 fast parsing in the time of matrix multiplication and parsing in nc2

input driven languages are linear conjunctive linear conjunctive grammars define the same family of languages as one way real time cellular automata a okhotin on the equivalence of linear conjunctive grammars to trellis automata rairo ita 2004 and this family is known to be incomparable with the context free languages v terrier on real time one way cellular array theoret comput sci 1995 this paper demonstrates the containment of the languages accepted by input driven pushdown automata a k a visibly pushdown automata in the family of linear conjunctive languages which is established by a direct simulation of an input driven automaton by a one way real time cellular automaton on the other hand it is shown that the language families defined by the unambiguous grammars the lr k grammars and the ll k grammars are incomparable with the linear conjunctive languages

orlicz sobolev nematic elastomers abstract we extend the existence theorems in barchiesi et al 2017 for models of nematic elastomers and magnetoelasticity to a larger class in the scale of orlicz spaces these models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration and a functional corresponding to the nematic energy or the exchange and magnetostatic energies in magnetoelasticity where the energy density is integrated over the deformed configuration in order to obtain the desired compactness and lower semicontinuity we show that the regularity requirement that maps create no new surface can still be imposed when the gradients are in an orlicz class with an integrability just above the space dimension minus one we prove that the fine properties of orientation preserving maps satisfying that regularity requirement namely being weakly 1 pseudomonotone h 1 continuous a e differentiable and a e locally invertible are still valid in the orlicz sobolev setting

multiplicity result for semilinear dissipative hyperbolic equations let zq z r and r be the set of all positive integers integers nonnegative reals and reals respectively and let v r n g 1 be a bounded domain with smooth boundary v which is assumed to be of class c 2 let q s 0 2p v and l q be the space of measurable lebesgue square integrable real valued functions on q with usual inner product 2 5 5 and corresponding norm 2 1 1 by h v we mean the completion of c v with respect to the norm 0 0 5 5 defined by 1

set constraints in some equational theories set constraints are relations between sets of ground terms over a given alphabet they give a natural formalism for many problems in program analysis type inference order sorted unification and constraint logic programming in this paper we start studies of set constraints in the environment given by equational specifications we show that in the case of associativity i e in free monoids as well as in the case of associativity and commutativity i e in commutative monoids the problem of consistency of systems of set constraints is undecidable in linear nonerasing shallow theories the consistency of systems of positive set constraints is nexptime complete and in linear shallow theories the problem for positive and negative set constraints is decidable

γ convergence approximation of fracture and cavitation in nonlinear elasticity our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by henao and mora corral arch rational mech anal 197 619 655 2010 the total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation it is a free discontinuity problem since the crack set and the set of new surface are unknowns of the problem the expression of the functional involves a volume integral and two surface integrals and this fact makes the problem numerically intractable in this paper we propose an approximation in the sense of convergence by functionals involving only volume integrals which makes a numerical approximation by finite elements feasible this approximation has some similarities to the modica mortola approximation of the perimeter and the ambrosio tortorelli approximation of the mumford shah functional but with the added difficulties typical of nonlinear elasticity in which the deformation is assumed to be one to one and orientation preserving

multiplicity results for a semilinear boundary value problem with the nonlinearity crossing higher eigenvalues on etudie le probleme aux valeurs limites au g u h x u 0 ou a est un operateur differentiel lineaire auto adjoint uniformement elliptique d ordre 2

unambiguous conjunctive grammars over a one symbol alphabet abstract it is demonstrated that unambiguous conjunctive grammars over a unary alphabet a have non trivial expressive power and that their basic properties are undecidable the key result is that for every base of positional notation k 11 and for every one way real time cellular automaton operating over the alphabet of base k digits between k 9 4 and k 1 2 the language of all strings a n with the base k representation of the form 1 w 1 where w is accepted by the automaton is described by an unambiguous conjunctive grammar another encoding is used to simulate a cellular automaton in a unary language containing almost all strings these constructions are used to show that for every fixed unambiguous conjunctive language l 0 testing whether a given unambiguous conjunctive grammar generates l 0 is undecidable

absolutely continuous functions of several variables abstract a class of absolutely continuous functions ofnvariables is introduced the absolute continuity implies continuity weak differentiability with gradient inln differentiability almost everywhere area coarea and degree formulae this gives a unified approach to some previously known conditions for the mentioned results based on monotonicity finite dilatation or higher integrability of the gradient

global invertibility for orientation preserving sobolev maps via invertibility on or near the boundary by a result of ball proc r soc edinb sect a math 88 315 328 1981 https doi org 10 1017 s030821050002014x a locally orientation preserving sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension as shown here for any dimension the conclusions of ball s theorem and related results can be reached while completely avoiding the problem of homeomorphic extension for suitable domains it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps an application in nonlinear elasticity is the existence of homeomorphic minimizers with finite distortion whose boundary values are not fixed as a tool in the proofs strictly orientation preserving maps and their global invertibility properties are studied from a purely topological point of view

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

note on lusin n condition and the distributional determinant abstract let r n be an open set we show that for a continuous mapping f w 1 n 1 r n with j f l 1 the validity of the lusin n condition implies that the distributional jacobian equals to the pointwise jacobian

local invertibility in sobolev spaces with applications to nematic elastomers and magnetoelasticity we define a class of deformations in w 1 p omega mathbb r n p n 1 with a positive jacobian that do not exhibit cavitation we characterize that class in terms of the non negativity of the topological degree and the equality det det that the distributional determinant coincides with the pointwise determinant of the gradient maps in this class are shown to satisfy a property of weak monotonicity and as a consequence they enjoy an extra degree of regularity we also prove that these deformations are locally invertible moreover the neighbourhood of invertibility is stable along a weak convergent sequence in w 1 p and the sequence of local inverses converges to the local inverse we use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps we apply those results to prove the existence of minimizers in some models for nematic elastomers and magnetoelasticity

efficient computation of the characteristic polynomial of a threshold graph an efficient algorithm is presented to compute the characteristic polynomial of a threshold graph threshold graphs were introduced by chvatal and hammer as well as by henderson and zalcstein in 1977 a threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex which is a vertex adjacent to all the other vertices threshold graphs are special kinds of cographs which themselves are special kinds of graphs of clique width 2 we obtain a running time of o n log 2 n for computing the characteristic polynomial while the previously fastest algorithm ran in quadratic time we improve the running time drastically in the case where there is a small number of alternations between 0 s and 1 s in the sequence defining a threshold graph a simple recurrence equation for the determinant of a weighted threshold graph matrix is presented the characteristic polynomial is computed in o n log 2 n arithmetic operations the algorithm is more efficient if the alternations in the defining sequence are o n as the numbers may be of length n log n the bit complexity is investigated too

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians