characterizations of locally testable linear and affine invariant families the linear or affine invariance is the property of a function family that is closed under linear or affine transformations on the domain and closed under linear combinations of functions respectively both the linear and affine invariant families of functions are generalizations of many symmetric families for instance the low degree polynomials kaufman and sudan 2007 21 introduced the notions of constraint and characterization to characterize the locally testable affine and linear invariant families of functions over finite fields of constant size in this article it is shown that for any finite field f of size q and characteristic p and its arbitrary extension field k of size q if an affine invariant family k n f has a k local constraint then it is k locally testable for k k 2 q pq 2 q p 4 and that if a linear invariant family k n f has a k local characterization then it is k locally testable for k 2k 2 q pq 4 q p 1 consequently for any prime field f of size q any positive integer k we have that for any affine invariant family over field f the four notions of the constraint the characterization the formal characterization and the local testability are equivalent modulo a poly k q of the corresponding localities and that for any linear invariant family the notions of the characterization the formal characterization and the local testability are equivalent modulo a poly k q of the corresponding localities the results significantly improve and are in contrast to the characterizations in 21 which have locality exponential in q even if the field k is prime in the research above a missing result is a characterization of linear invariant function families by the more natural notion of constraint for this we show that a single strong local constraint is sufficient to characterize the local testability of a linear invariant boolean function family and that for any finite field f of size q greater than 2 there exists a linear invariant function family over f such that it has a strong 2 local constraint but is not q d q 1 1 locally testable the proof for this result provides an appealing approach toward more negative results in the theme of characterization of locally testable algebraic properties which is rare and of course significant

distributional chaos revisited in their famous paper schweizer and smital introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval further their approach was extended to the general compact metric space case in this article we provide an example which shows that the definition of distributional chaos and as a result li yorke chaos may be fulfilled by a dynamical system with intuitively regular dynamics embedded in 3 next we state strengthened versions of distributional chaos which as we show are present in systems commonly considered to have complex dynamics we also prove that any interval map with positive topological entropy contains two invariant subsets x y c i such that f x has positive topological entropy and f y displays distributional chaos of type 1 but not conversely

distributional chaos for operators on banach spaces abstract four notions of distributional chaos namely dc1 dc2 dc 2 1 2 and dc3 are studied within the framework of operators on banach spaces it is known that for general dynamical systems dc1 dc2 dc 2 1 2 dc3 we show that dc1 and dc2 coincide in our context which answers a natural question in contrast there exist dc 2 1 2 operators which are not dc2 under the condition that there exists a dense set x 0 x such that t n x 0 for any x x 0 dc3 operators are shown to be dc1 moreover we prove that any upper frequently hypercyclic operator is dc 2 1 2 finally several examples are provided to distinguish between different notions of distributional chaos li yorke chaos and irregularity

li yorke and distributionally chaotic operators abstract we study li yorke chaos and distributional chaos for operators on banach spaces more precisely we characterize li yorke chaos in terms of the existence of irregular vectors sufficient computable criteria for distributional and li yorke chaos are given together with the existence of dense scrambled sets under some additional conditions we also obtain certain spectral properties finally we show that every infinite dimensional separable banach space admits a distributionally chaotic operator which is also hypercyclic

combining vns with constraint programming for solving anytime optimization problems abstract this paper presents an hybrid search method for solving on line optimization problems that are modelled using the vcsp valued constraint satisfaction problems framework to each constraint is associated a valuation representing the cost to pay when this constraint will be violated by a solution our method vns lds cp uses principles of vns variable neighborhood search and combines a partial tree search limited discrepancy search with constraint propagation in order to compute local optima experiments on the celar benchmarks demonstrate significant improvements on other competing methods lns cp gr lobjois l lemaitre m verfaillie g 2000 large neighbourhood search using constraint propagation and greedy reconstruction for valued csp resolution in proceedings of the ecai2000 workshop on modelling and solving problems with constraints another hybrid method using vcsp s and two standard versions of simulated annealing li y h 1997 directed annealing search in constraint satisfaction and optimization ph d thesis imperial college of science department of computing quick and medium moreover vns lds cp clearly satisfies the key properties of anytime algorithms finally vns lds cp has been successfully applied to a real life on line resource allocation problem in computer networks

mean li yorke chaos in banach spaces abstract we investigate the notion of mean li yorke chaos for operators on banach spaces we show that it differs from the notion of distributional chaos of type 2 contrary to what happens in the context of topological dynamics on compact metric spaces we prove that an operator is mean li yorke chaotic if and only if it has an absolutely mean irregular vector as a consequence absolutely cesaro bounded operators are never mean li yorke chaotic dense mean li yorke chaos is shown to be equivalent to the existence of a dense or residual set of absolutely mean irregular vectors as a consequence every mean li yorke chaotic operator is densely mean li yorke chaotic on some infinite dimensional closed invariant subspace a dense mean li yorke chaos criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained moreover we construct an example of an invertible hypercyclic operator t such that every nonzero vector is absolutely mean irregular for both t and t 1 several other examples are also presented finally mean li yorke chaos is also investigated for c 0 semigroups of operators on banach spaces

solving weighted csp by maintaining arc consistency recently a general definition of arc consistency ac for soft constraint frameworks has been proposed by t schiex proc cp 2000 singapore 2000 pp 411 424 in this paper we specialize this definition to weighted csp and introduce two o ed3 enforcing algorithms then we refine the definition and introduce a stronger form of arc consistency ac along with two o n2d2 ed3 algorithms as in the csp case an important application of ac is to combine it with search we empirically demonstrate that a branch and bound algorithm that maintains either ac or ac is a state of the art general solver for weighted csp our experiments cover binary max csp and max sat problems

clique inference process for solving max csp abstract few works on problems csp max csp and weighted csp was carried out in the field of combinatorial optimization whereas this field contains many algorithmic tools which can be used for the resolution of these problems in this paper we introduce the binary clique concept clique which expresses incompatibilities between values of two csp variables we propose a linear formulation for any binary clique and we present several ways to exploit it in order to compute lower bounds or to solve max csp we also show that the binary clique concept can be exploited in the weighted csp framework the obtained theoretical and experimental results are very interesting and they open new prospects to exploit the combinatorial optimization algorithmic tools for the resolution of csp max csp and weighted csp

distributional chaos for backward shifts we provide sufficient conditions which give uniform distributional chaos for backward shift operators we also compare distributional chaos with other well studied notions of chaos for linear operators like devaney chaos and hypercyclicity and show that devaney chaos implies uniform distributional chaos for weighted backward shifts but there are examples of backward shifts which are uniformly distributionally chaotic and not hypercyclic

distributional chaos for operators with full scrambled sets in this article we answer in the negative the question of whether hypercyclicity is sufficient for distributional chaos for a continuous linear operator we even prove that the mixing property does not suffice moreover we show that an extremal situation is possible there are hypercyclic and non hypercyclic operators such that the whole space consists except zero of distributionally irregular vectors

maintaining reversible dac for max csp abstract we introduce an exact algorithm for maximizing the number of satisfied constraints in an overconstrained csp max csp the algorithm which can also solve weighted csp probabilistic csp and other similar problems is based on directed arc inconsistency counts dac the usage of dac increases the lower bound of branch and bound based algorithms for max csp improving their efficiency originally dac were defined following a static variable ordering in this paper we relax this condition showing how dac can be defined from a directed constraint graph these new graph based dac can be effectively used for lower bound computation interestingly any directed constraint graph of the considered problem is suitable for dac computation so the selected graph can change dynamically during search aiming at optimizing the exploitation of directed arc inconsistencies in addition directed arc inconsistencies are maintained during search propagating the effect of value pruning with these new elements we present the pfc maintaining reversible dac algorithm pfc mrdac a natural successor of pfc dac for max csp we provide experimental evidence for the superiority of pfc mrdac on random and real overconstrained csp instances including problems with weighted constraints

checking approximate computations of polynomials and functional equations a majority of the results on self testing and correcting deal with programs which purport to compute the correct results precisely we relax this notion of correctness and show how to check programs that compute only a numerical approximation to the correct answer the types of programs that we deal with are those computing polynomials and functions defined by certain types of functional equations we present results showing how to perform approximate checking self testing and self correcting of polynomials settling in the affirmative a question raised by p gemmell et al proceedings of the 23rd acm symposium on theory of computing 1991 pp 32 42 r rubinfeld and m sudan proceedings of the third annual acm siam symposium on discrete algorithms orlando fl 1992 pp 23 43 r rubinfeld and m sudan siam j comput 25 1996 pp 252 271 we obtain this by first building approximate self testers for linear and multilinear functions we then show how to perform approximate checking self testing and self correcting for those functions that satisfy addition theorems settling a question raised by r rubinfeld siam j comput 28 1999 pp 1972 1997 in both cases we show that the properties used to test programs for these functions are both robust in the approximate sense and stable finally we explore the use of reductions between functional equations in the context of approximate self testing our results have implications for the stability theory of functional equations

algebraic testing and weight distributions of codes we study the testing problem that is the problem of determining maybe probabilistically if a function to which one has oracle access satisfies a given property we propose a framework in which to formulate and carry out the analysis of several known tests this framework establishes a connection between testing and the theory of weight distributions of codes we illustrate this connection by giving a coding theoretic interpretation of several tests that fall under the label of low degree tests we also show how the connection naturally suggests a new way of testing for linearity over finite fields we derive from the macwilliams theorems a general result the duality testing lemma and use it to analyze the simpler tests that fall into our framework in contrast to other analyses of tests the ones we present elicit the fact that a test s probability of rejecting a fuction depends on how far away the function is from every function that satisfies the property of interest

the complexity of soft constraint satisfaction over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types one very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations known as polymorphisms in this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints we launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation which we call a multimorphism we show that many tractable sets of soft constraints both established and novel can be characterised by the presence of particular multimorphisms we also show that a simple set of np hard constraints has very restricted multimorphisms finally we use the notion of multimorphism to give a complete classification of complexity for the boolean case which extends several earlier classification results for particular special cases

cyclic consistency a local reduction operation for binary valued constraints valued constraint satisfaction provides a general framework for optimisation problems over finite domains it is a generalisation of crisp constraint satisfaction allowing the user to express preferences between solutions consistency is undoubtedly the most important tool for solving crisp constraints it is not only a family of simplification operations on problem instances it also lies at the heart of intelligent search techniques g kondrak p van beek artificial intelligence 89 1997 365 387 and provides the key to solving certain classes of tractable constraints p g jeavons d a cohen m c cooper artificial intelligence 101 1998 251 265 arc consistency was generalised to valued constraints by sacrificing the uniqueness of the arc consistency closure m c cooper t schiex artificial intelligence in press the notion of 3 cyclic consistency introduced in this paper again sacrifices the unique closure property in order to obtain a generalisation of path consistency to valued constraints which is checkable in polynomial time in max csp 3 cyclic consistency can be established in polynomial time and even guarantees a local form of optimality the space complexity of 3 cyclic consistency is optimal since it creates no new constraints

measures of chaos and a spectral decomposition of dynamical systems on the interval let f 0 1 0 1 be continuous for x y e 0 1 the upper and lower distance distribution functions fx y and fxy are defined for any t 0 as the lim sup and lim inf as n oc of the average number of times that the distance ifi x fi y i between the trajectories of x and y is less than t during the first n iterations the spectrum of f is the system f of lower distribution functions which is characterized by the following properties 1 the elements of f are mutually incomparable 2 for any f e f there is a perfect set pf 8 0 such that fuv f and fuv 1 for any distinct u v e pf 3 if s is a scrambled set for f then there are f g in f and a decomposition s sf u sg sg may be empty such that fuv f if u v e sf and fuv g if u v e sg our principal results are 1 if f has positive topological entropy then f is nonempty and finite and any f e f is zero on an interval 0 e where e 0 and hence any pf is a scrambled set in the sense of li and yorke 2 if f has zero topological entropy then f f where f 1 it follows that the spectrum of f provides a measure of the degree of chaos of f in addition a useful numerical measure is the largest of the numbers fo 1 f t dt where f e f

arc consistency for soft constraints the notion of arc consistency plays a central role in constraint satisfaction r dechter constraint processing morgan kaufmann san mateo ca 2003 it is known since the introduction of valued and semi ring constraint networks in 1995 that the notion of local consistency can be extended to constraint optimisation problems defined by soft constraint frameworks based on an idempotent cost combination operator this excludes non idempotent operators such as which define problems which are very important in practical applications such as max csp where the aim is to minimise the number of violated constraints in this paper we show that using a weak additional axiom satisfied by most existing soft constraints proposals it is possible to define a notion of soft arc consistency that extends the classical notion of arc consistency and this even in the case of non idempotent cost combination operators a polynomial time algorithm for enforcing this soft arc consistency exists and its space and time complexities are identical to that of enforcing arc consistency in csps when the cost combination operator is strictly monotonic for example max csp a directional version of arc consistency first introduced by m c cooper reduction operations in fuzzy or valued constraint satisfaction fuzzy sets and systems 134 3 2003 311 342 is potentially even stronger than the non directional version since it allows non local propagation of penalties we demonstrate the utility of directional arc consistency by showing that it not only solves soft constraint problems on trees but that it also implies a form of local optimality which we call arc irreducibility

on the robustness of functional equations in this paper we study the general question of how characteristics of functional equations influence whether or not they are robust we isolate examples of properties which are necessary for the functional equations to be robust on the other hand we show other properties which are sufficient for robustness we then study a general class of functional equations which are of the form forall x y f f x y f x y f x f y 0 where f is an algebraic function we give conditions on such functional equations that imply robustness our results have applications to the area of self testing correcting programs we show that self testers and self correctors can be found for many functions satisfying robust functional equations including algebraic functions of trigonometric functions such as tan x 1 over 1 cot x ax over 1 ax cosh x

the principal measure of a quantum harmonic oscillator at the end of his paper oprocha 2006 j phys a math gen 39 14 559 65 oprocha said that it is difficult and interesting to know the exact value of the principal measure of an annihilation operator of the unforced quantum harmonic oscillator motivated by this in this paper we prove that this exact value is equal to 1 besides we point out that exhibits distributional chaos for any 0 2

partial constraint satisfaction abstract a constraint satisfaction problem involves finding values for variables subject to constraints on which combinations of values are allowed in some cases it may be impossible or impractical to solve these problems completely we may seek to partially solve the problem in particular by satisfying a maximal number of constraints standard backtracking and local consistency techniques for solving constraint satisfaction problems can be adapted to cope with and take advantage of the differences between partial and complete constraint satisfaction extensive experimentation on maximal satisfaction problems illuminates the relative and absolute effectiveness of these methods a general model of partial constraint satisfaction is proposed