chaotic functions with zero topological entropy on caracterise la classe m c i i des applications chaotiques en ce sens on montre que m contient certaines appliquations qui ont a la fois une entropie topologique nulle et des attracteurs infinis on montre que le complement de m consiste en applications qui ont seulement des trajectoires approchables par des cycles

parametric duality and kernelization lower bounds and upper bounds on kernel size determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms theoretically it has been proved that a parameterized problem is kernelizable if and only if it is fixed parameter tractable practically applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance i e a kernel has led to very efficient algorithms and now goes hand in hand with the design of practical algorithms for solving mathcal np hard problems well known examples of such parameterized problems include the vertex cover problem which is kernelizable to a kernel of size bounded by 2k and the planar dominating set problem which is kernelizable to a kernel of size bounded by 335k in this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems in terms of our lower bound results we show for example that unless mathcal p mathcal np planar vertex cover does not have a problem kernel of size smaller than 4k 3 and planar independent set and planar dominating set do not have kernels of size smaller than 2k in terms of our upper bound results we further reduce the upper bound on the kernel size for the planar dominating set problem to 67 k improving significantly the 335 k previous upper bound given by alber fellows and niedermeier j acm 51 2004 pp 363 384 this latter result is obtained by introducing a new set of reduction and coloring rules which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel the paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem

lower bounds for the happy coloring problems in this paper we study the maximum happy vertices and the maximum happy edges problems mhv and mhe for short very recently the problems attracted a lot of attention and were studied in agrawal 17 aravind et al 16 choudhari and reddy 18 misra and reddy 17 main focus of our work is lower bounds on the computational complexity of these problems established lower bounds can be divided into the following groups np hardness of the above guarantee parameterization kernelization lower bounds answering questions of misra and reddy 17 exponential lower bounds under the set cover conjecture and the exponential time hypothesis and inapproximability results moreover we present an mathcal o ell k randomized algorithm for mhv and an mathcal o 2 k algorithm for mhe where ell is the number of colors used and k is the number of required happy vertices or edges these algorithms cannot be improved to subexponential taking proved lower bounds into account

the parameterised complexity of list problems on graphs of bounded treewidth we consider the parameterised complexity of several list problems on graphs with parameter treewidth or pathwidth in particular we show that list edge chromatic number and list total chromatic number are fixed parameter tractable parameterised by treewidth whereas list hamilton path is w1 hard even parameterised by pathwidth these results resolve two open questions of fellows fomin lokshtanov rosamond saurabh szeider and thomassen 2011

constant length substitution systems and countable distributionally scrambled sets abstract in this paper we study the compact dynamical systems which are on the edge of distributional chaos that is whose distributionally scrambled sets are finite or countable first we show that a constant length substitution system without eventually periodic substitution sequence has only finite distributionally scrambled sets then we give some constant length substitution systems whose distributionally scrambled sets may have any given finite cardinal number at last we provide a compact dynamical system generated by some constant length substitution systems whose distributionally scrambled set has at most countably many elements

dominating sets in planar graphs branch width and exponential speed up we introduce a new approach to design parameterized algorithms on planar graphs which builds on the seminal results of robertson and seymour on graph minors graph minors provide a list of powerful theoretical results and tools however the widespread opinion in the graph algorithms community about this theory is that it is of mainly theoretical importance in this paper we show how deep min max and duality theorems from graph minors can be used to obtain exponential speed up to many known practical algorithms for different domination problems our use of branch width instead of the usual tree width allows us to obtain much faster algorithms by using this approach we show that the k dominating set problem on planar graphs can be solved in time o 215 13 sqrt k n3

distributional chaos for linear operators abstract we characterize distributional chaos for linear operators on frechet spaces in terms of a computable condition dcc and also as the existence of distributionally irregular vectors a sufficient condition for the existence of dense uniformly distributionally irregular manifolds is presented which is very general and can be applied to many classes of operators distributional chaos is also analyzed in connection with frequent hypercyclicity and the particular cases of weighted shifts and composition operators are given as an illustration of the previous results

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

a note on distributional chaos with respect to a sequence abstract the aim of this note is to use methods developed by kuratowski and mycielski to prove that some more common notions in topological dynamics imply distributional chaos with respect to a sequence in particular we show that the notion of distributional chaos with respect to a sequence is only slightly stronger than the definition of chaos due to li and yorke namely positive topological entropy and weak mixing both imply distributional chaos with respect to a sequence which is not the case for distributional chaos as introduced by schweizer and smital

a polynomial algorithm for the k cut problem for fixed k the k cut problem is to find a partition of an edge weighted graph into k nonempty components such that the total edge weight between components is minimum this problem is np complete for an arbitrary k and its version involving fixing a vertex in each component is np hard even for k 3 we present a polynomial algorithm for k fixed that runs in onki 2 3k 2 4tn m steps where tn m is the running time required to find the minimum s t cut on a graph with n vertices and m edges

a note on the map with the whole space being a scrambled set abstract in this paper we prove that there is no compact system with the whole space being a schweizer smital scrambled set and the huang ye homeomorphism w huang x d ye homeomorphisms with the whole compacta being scrambled sets ergod theory dynam systems 21 2001 77 91 has no schweizer smital pair moreover we present a simple noncompact system whose schweizer smital scrambled set can be the whole space

distributionally chaotic systems of type 2 and rigidity abstract in this paper we deal with uniformly rigid systems obtained by a method introduced by katznelson and weiss and show that such systems never contain dc2 pairs on the other hand we introduce a modification of this technique that leads to a uniformly rigid system with dc2 pairs we also show that every dynamical system contains a pair of distinct points which is not dc2

chaos on hyperspaces abstract let f be a continuous self map of a compact metric space x the transformation f induces in a natural way a self map f defined on the hyperspace k x of all nonempty closed subsets of x we study which of the most usual notions of chaos for dynamical systems induced by f are inherited by f and vice versa we consider distributional chaos li yorke chaos chaos devaney chaos topological chaos positive topological entropy specification property and their variants this answers questions stated independently by roman flores and banks

improved integrality gap upper bounds for traveling salesperson problems with distances one and two abstract we prove new integrality gap upper bounds for the traveling salesperson problem with distances one and two 1 2 tsp we obtain these bounds by investigating the structure of the support graph of optimal solutions to the subtour elimination linear programming relaxation with one additional cutting plane inequality for undirected 1 2 tsp our main results are as follows all instances have an integrality gap of at most 5 4 instances admitting half integral solutions have integrality gap at most 7 6 instances admitting subcubic solutions of cost at most the order of the instance have integrality gap at most 10 9 even without the cutting plane this bound is tight and holds in particular for basic solutions in the fractional 2 matching polytope with cost at most the order of the instance for directed 1 2 tsp instances we show an integrality gap upper bound of 3 2 for general instances and of 5 4 for instances admitting half integral solutions we prove these bounds by providing local search algorithms that in polynomial time find 2 matchings with few components in the support of the solution we show that the run times of our algorithms cannot be considerably improved under standard complexity theoretic assumptions we show that finding improved tsp solutions via local search is intractable for edge change parameterized by the size of the neighborhoods even for instances with distances one and two this strengthens a result of daniel marx

tight lower bounds for certain parameterized np hard problems based on the framework of parameterized complexity theory we derive tight lower bounds on the computational complexity for a number of well known np hard problems we start by proving a general result namely that the parameterized weighted satisfiability problem on depth t circuits cannot be solved in time n o k m o 1 where n is the circuit input length m is the circuit size and k is the parameter unless the t 1 st level w t 1 of the w hierarchy collapses to fpt by refining this technique we prove that a group of parameterized np hard problems including weighted sat hitting set set cover and feature set cannot be solved in time n o k m o 1 where n is the size of the universal set from which the k elements are to be selected and m is the instance size unless the first level w 1 of the w hierarchy collapses to fpt we also prove that another group of parameterized problems which includes weightedq sat for any fixed q 2 clique independent set and dominating set cannot be solved in time n o k unless all search problems in the syntactic class snp introduced by papadimitriou and yannakakis are solvable in subexponential time note that all these parameterized problems have trivial algorithms of running time either n km o 1 or o n k

parameterized and subexponential time complexity of satisfiability problems and applications we study the parameterized and the subexponential time complexity of the weighted and the unweighted satisfiability problems on bounded depth normalized boolean circuits we establish relations between the subexponential time complexity of the weighted and the unweighted satisfiability problems and use them to derive relations among the subexponential time complexity of several np hard problem we then study the role of certain natural structural parameters of the circuit in characterizing the parameterized and the subexponential time complexity of the circuit satisfiability problems under consideration we obtain threshold functions on some circuit structural parameters including the depth the number of gates the fan in and the maximum number of variable occurrences that lead to tight characterizations of the parameterized and the subexponential time complexity of the circuit satisfiability problems under consideration

devaney chaos li yorke chaos and multi dimensional li yorke chaos for topological dynamics abstract let t x x written t x be a topological semiflow flow on a uniform space x with t a multiplicative topological semigroup group not necessarily discrete we then prove if t x is non minimal topologically transitive with dense almost periodic points then it is sensitive to initial conditions as a result of this devaney chaos sensitivity to initial conditions for this very general setting let r x be a c 0 semiflow on a polish space then we show if r x is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior then it is multi dimensional li yorke chaotic that is there is a uncountable set x such that for any k 2 and any distinct points x 1 x k one can find two time sequences s n t n with s n x 1 x k x 1 x k x k and t n x 1 x k p p x k consequently devaney chaos multi dimensional li yorke chaos moreover let x be a non singleton polish space then we prove any weakly mixing c 0 semiflow r x is densely multi dimensional li yorke chaotic any minimal weakly mixing topological flow t x with t abelian is densely multi dimensional li yorke chaotic any weakly mixing topological flow t x is densely li yorke chaotic we in addition construct a completely li yorke chaotic minimal sl 2 r acting flow on the compact metric space r our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup group t

chaotic sets of shift and weighted shift maps abstract in this paper li yorke chaotic sets generated by shift and weighted shift maps are studied the characterization of li yorke chaotic sets by p scrambled sets maximal scrambled sets and orbit invariants are proved for the general shift maps for weighted shift maps on infinite dimensional spaces the necessary and sufficient conditions for li yorke chaotic are proved both in the abstract sequence setting and in the eigenfunction setting besides a constructive proof is provided for the devaney chaos of weighted shift maps on the schwartz space

distributional chaos caused by snap back repeller in metric space abstract in this paper we extend the conception of snap back repeller of marotto s to general metric space and prove that a continuous map with snap back repeller is distributionally chaotic

recurrence properties of hypercyclic operators we generalize the notions of hypercyclic operators mathfrak u frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics namely mathcal a hypercyclicity we then state an mathcal a hypercyclicity criterion inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion and we show that this criterion characterizes the mathcal a hypercyclicity for weighted shifts we also investigate which density properties can the sets n x u n in mathbb n t nx in u have for a given hypercyclic operator and we study the new notion of reiteratively hypercyclic operators

algebraic fiber bundles abstracr when x is a finite simplicial complex and g is any of a certain class of groups a classification of g principal bundles over x in terms of projective modules over a ring r g x is given this generalizes swan s classification of vector bundles and uses the results of mulvey often r can be taken to be noetherian in this case spec r is usually reducible with cohomologically trivial irreducible components information is derived concerning the nature of projective modules over such rings and some results are obtained indicating how such information reflects information about x

distributional chaos in a sequence abstract in this paper we prove a sufficient condition for the continuous map of a compact metric space for being distributively chaotic in a sequence as an application it is proved that a continuous map of an interval is chaotic in the li yorke sense if and only if it is distributively chaotic in a sequence

chaotic sets of continuous and discontinuous maps abstract this paper discusses li yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps scrambled sets and maximal scrambled sets are introduced to characterize li yorke chaotic sets the orbit invariant for a scrambled set is discussed some properties about maximality equivalence and uniqueness of maximal scrambled sets are also discussed it is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps such as the gauss map interval exchange transformations and a class of planar piecewise isometries are studied finally some open problems on scrambled sets are listed and remarked

incremental list coloring of graphs parameterized by conservation incrementally k list coloring a graph means that a graph is given by adding vertices step by step and for each intermediate step we ask for a vertex coloring such that each vertex has one of the colors specified by its associated list containing some of in total k colors we introduce the conservative version of this problem by adding a further parameter c n specifying the maximum number of vertices to be recolored between two subsequent graphs differing by one vertex the conservation parameter c models the natural quest for a modest evolution of the coloring in the course of the incremental process instead of performing radical changes we show that even on bipartite graphs the problem is np hard for k 3 and w 1 hard for an unbounded number of colors when parameterized by c in contrast also on general graphs the problem becomes fixed parameter tractable with respect to the combined parameter k c we prove that the problem has an exponential size kernel with respect to k c and there is no polynomial size kernel unless np conp poly furthermore we investigate the parameterized complexity on various subclasses of perfect graphs we show fixed parameter tractability for the combined parameter treewidth and number k of colors finally we provide empirical findings on the practical relevance of our approach in terms of an effective graph coloring heuristic

the specification property and infinite entropy for certain classes of linear operators abstract we study the specification property and infinite topological entropy for two specific types of linear operators translation operators on weighted lebesgue function spaces and weighted backward shift operators on sequence f spaces it is known from the work of bartoll martinez gimenez murillo arcila and peris that for weighted backward shift operators the existence of a single non trivial periodic point is sufficient for specification we show this also holds for translation operators on weighted lebesgue function spaces this implies in particular that for these operators the specification property is equivalent to devaney chaos we show that these forms of chaos unsurprisingly imply infinite topological entropy but that surprisingly the converse does not hold

parameterized complexity of critical node cuts we consider the following graph cut problem called critical node cut cnc given a graph g on n vertices and two positive integers k and x determine whether g has a set of k vertices whose removal leaves g with at most x connected pairs of vertices we analyze this problem in the framework of parameterized complexity that is we are interested in whether or not this problem is solvable in f n o 1 time i e whether or not it is fixed parameter tractable for various natural parameters we consider four such parameters the size k of the required cut the upper bound x on the number of remaining connected pairs the lower bound y on the number of connected pairs to be removed the treewidth w of g we determine whether or not cnc is fixed parameter tractable for each of these parameters we determine this also for all possible aggregations of these four parameters apart from w k moreover we also determine whether or not cnc admits a polynomial kernel for all these parameterizations that is whether or not there is an algorithm that reduces each instance of cnc in polynomial time to an equivalent instance of size o 1 where is the given parameter

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

recurrence and topological entropy of translation operators abstract in this paper we provide several characterizations on weak recurrence of translation operators on weighted lebesgue spaces and on continuous function spaces it is shown that the translation operator admits a nonzero nonwandering point if and only if it has a nonzero recurrent point if and only if it is hypercyclic that the translation operator admits a nonzero almost periodic point if and only if it is devaney chaotic moreover every hypercyclic translation operator has infinite topological entropy but the converse is not true finally we remark that there exists a linear operator which has the specification property on the whole space

an isomorphism between subexponential and parameterized complexity theory we establish a close connection between sub exponential time complexity and parameterized complexity by proving that the so called miniaturization mapping is a reduction preserving isomorphism between the two theories

nondeterminism within p classes of machines using very limited amounts of nondeterminism are studied the p np question is related to questions about classes lying within p complete sets for these classes are given

linear operators with infinite entropy abstract we examine the chaotic behavior of certain continuous linear operators on infinite dimensional banach spaces and provide several equivalent characterizations of when these operators have infinite topological entropy for example it is shown that infinite topological entropy is equivalent to non zero topological entropy for translation operators on weighted lebesgue function spaces in particular finite non zero entropy is impossible for this class of operators which answers a question raised by yin and wei

stable sets and mean li yorke chaos in positive entropy systems abstract it is shown that in a topological dynamical system with positive entropy there is a measure theoretically rather big set such that a multivariant version of mean li yorke chaos happens on the closure of the stable or unstable set of any point from the set it is also proved that the intersections of the sets of asymptotic tuples and mean li yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively

data reductions fixed parameter tractability and random weighted d cnf satisfiability data reduction is a key technique in the study of fixed parameter algorithms in the ai literature pruning techniques based on simple and efficient to implement reduction rules also play a crucial role in the success of many industrial strength solvers understanding the effectiveness and the applicability of data reduction as a technique for designing heuristics for intractable problems has been one of the main motivations in studying the phase transition of randomly generated instances of np complete problems in this paper we take the initiative to study the power of data reductions in the context of random instances of a generic intractable parameterized problem the weighted d cnf satisfiability problem we propose a non trivial random model for the problem and study the probabilistic behavior of the random instances from the model we design an algorithm based on data reduction and other algorithmic techniques and prove that the algorithm solves the random instances with high probability and in fixed parameter polynomial time o d knm where n is the number of variables m is the number of clauses and k is the fixed parameter we establish the exact threshold of the phase transition of the solution probability and show that in some region of the problem space unsatisfiable random instances of the problem have parametric resolution proof of fixed parameter polynomial size also discussed is a more general random model and the generalization of the results to the model

distributional chaos for the forward and backward control traffic model the interest in car following models has increased in the last years due to its connection with vehicle to vehicle communications and the development of driverless cars some non linear models such as the gazes herman rothery model were already known to be chaotic we consider the linear forward and backward control traffic model for an infinite number of cars on a track we show the existence of solutions with a chaotic behaviour by using some results of linear dynamics of c0 semigroups in contrast we also analyse which initial configurations lead to stable solutions

differentiation in modules over topological algebras in recent years there has been a systematic development of the theory of a bundles that is vector bundles where the field of coefficients is replaced by a topological algebra a usually an involutive one the above extension is justified by its applications in pure mathematics differential topology 15 as well as in relativistic quantum mechanics via e g the serre swan theorem 22 bundles with coefficients in a commutative unital banach algebra or yet a c algebra were considered by t k kandelaki 6 71 k fujii s and a s miseenko and yu p solov ev 15 161 furthermore a mallios studied a bundles with a being a locally m convex algebra with unit l 12 note also that certain results are still valid for more general topological algebras 13 sect 53 or even topological rings 21 the results already obtained are mainly within the framework of topological a bundles the basic difficulty in the development of a theory of differential a bundles is the lack of a suitable differential calculus in the libres the latter being topological modules over a one might of course use the existing methods of differentiation in the underlying topological vector spaces cf for instance 18 however besides the intrinsic weaknesses of these methods regarding the higher order chain rule continuity of differentiable maps differentiability of the composition map and evaluation map which are necessary for differential geometric considerations on bundles the additional algebraic operations scalar multiplication by elements of a involution should be taken into account because they are essentially involved in the structure of a bundles cf 11 183 for the same reason incompatibility with involution the 255 0022 247x 92 55 00

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

distributionally scrambled invariant sets in a compact metric space abstract the paper solves a question posed by oprocha on the existence of invariant distributionally chaotic scrambled sets we show among other things that a continuous map f acting on compact metric space x d with a weak specification property fixed point and infinitely many mutually distinct periods has a dense mycielski i e c dense set of type f invariant distributionally scrambled set as a consequence we describe a class of maps with a distributionally scrambled invariant set of full lebesgue measure in the case when x is a k dimensional cube

a function for which the limit points are not contained in the closure of the periodic points we develop a c 1 function f 1 6 1 1 6 1 for which f p f this answers a query from block and coppel 1992

an initial study of time complexity in infinite domain constraint satisfaction abstract the constraint satisfaction problem csp is a widely studied problem with numerous applications in computer science and artificial intelligence for infinite domain csps there are many results separating tractable and np hard cases while upper and lower bounds on the time complexity of hard cases are virtually unexplored hence we initiate a study of the worst case time complexity of such csps we analyze backtracking algorithms and determine upper bounds on their time complexity we present asymptotically faster algorithms based on enumeration techniques and we show that these algorithms are applicable to well studied problems in for instance temporal reasoning finally we prove non trivial lower bounds applicable to many interesting csps under the assumption that certain complexity theoretic assumptions hold the gap between upper and lower bounds is in many cases surprisingly small which suggests that our upper bounds cannot be significantly improved

continuous vector bundles over topological algebras ii on presente des analogies de la theorie des fibres a vectoriels avec la theorie classique des fibres vectoriels complexes continus

a deterministic 2 2 k 1 n algorithm for k sat based on local search local search is widely used for solving the propositional satisfiability problem papadimitriou 1991 showed that randomized local search solves 2 sat in polynomial time recently schoning 1999 proved that a close algorithm for k sat takes time 2 2 k n up to a polynomial factor this is the best known worst case upper bound for randomized 3 sat algorithms cf also recent preprint by schuler et al we describe a deterministic local search algorithm for k sat running in time 2 2 k 1 n up to a polynomial factor the key point of our algorithm is the use of covering codes instead of random choice of initial assignments compared to other weakly exponential algorithms our algorithm is technically quite simple we also describe an improved version of local search for 3 sat the improved algorithm runs in time 1 481n up to a polynomial factor our bounds are better than all previous bounds for deterministic k sat algorithms

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

a note on schweizer smital chaos abstract in this paper we consider a continuous map f x x where x is a compact metric space and prove that for any positive integer n f is schweizer smital chaotic if and only if f n is too

an improved deterministic local search algorithm for 3 sat we slightly improve the pruning technique presented in dantsin et al theoret comput sci 289 2002 69 to obtain an o 1 473n deterministic algorithm for 3 sat

a chaotic function with a distributively scrambled set of full lebesgue measure abstract misiurewicz proved that there exists a continuous map of the interval 0 1 onto itself for which there exists a scrambled set of full lebesgue measure in this paper we form a continuous interval map which has a distributively scrambled set of full lebesgue measure in which each point has dense orbit this contains misiurewicz s result since any distributively scrambled set must be scrambled but the converse is not generally true

on distributionally chaotic and null systems abstract by a topological dynamical system we mean a pair x f where x is a compactum and f is a continuous self map on x a system is said to be null if its topological sequence entropies are zero along all strictly increasing sequences of natural numbers we show that there exists a null system which is distributionally chaotic this system admits open distributionally scrambled sets and its collection of all maximal distributionally scrambled sets has the same cardinality as the collection of all subsets of the phase space finally such system can even exist on continua

on miniaturized problems in parameterized complexity theory we introduce a general notion of miniaturization of a problem that comprises the different miniaturizations of concrete problems considered so far we develop parts of the basic theory of miniaturizations using the appropriate logical formalism we show that the miniaturization of a definable problem in w t lies in w t too in particular the miniaturization of the dominating set problem is in w 2 furthermore we investigate the relation between f k no k time and subexponential time algorithms for the dominating set problem and for the clique problem

distributional chaos in dendritic and circular julia sets abstract if x and y belong to a metric space x we call x y a dc1 scrambled pair for f x x if the following conditions hold 1 for all t 0 lim sup n 1 n 0 i n d f i x f i y t 1 and 2 for some t 0 lim inf n 1 n 0 i n d f i x f i y t 0 if d x is an uncountable set such that every x y d form a dc1 scrambled pair for f we say f exhibits distributional chaos of type 1 if there exists t 0 such that condition 2 holds for any distinct points x y d then the chaos is said to be uniform a dendrite is a locally connected uniquely arcwise connected compact metric space in this paper we show that a certain family of quadratic julia sets one that contains all the quadratic julia sets which are dendrites and many others which contain circles has uniform dc1 chaos

a characterization of nonchaotic continuous maps of the interval stable under small perturbations recent results of the second author show that every continuous map of the interval to itself either has every trajectory approximable by cycles some times this is possible even in the case when the trajectory is not asymptotically periodic or is e chaotic for some e 0 in certain cases the first property is stable under small perturbations this means that a perturbed map can be chaotic but the chaos must be small whenever the perturbation is small in other words there are nonchaotic maps without chaos explosions in the paper we give a characterization of these maps along with some consequences using the main result it is possible to prove that generically the nonchaotic maps are stable asymptotic properties of trajectories have been intensively considered since the middle of the sixties when excellent papers by a n sarkovskii appeared the main reason is applicability of these results in mathematical modelling important are maps with trajectories having regular asymptotical behavior like asymptotical periodicity however this property can be easily destroyed by ar bitrarily small perturbations of the maps and the corresponding unstable models in connection with this there appeared some results guaranteeing the stability in 15 it is shown that every map having no periodic orbits of period dif ferent from 1 2 2 for some n with per nowhere dense is stable this result was generalized 16 to the maps with per nowhere dense which have all trajectories asymptotically periodic these maps can have cycles of period 2 for every n but no other cycles and are characterized e g by the property that per is closed 12 cf also 10

devaney s chaos implies distributional chaos in a sequence abstract let x be a complete metric space without isolated points and let f x x be a continuous map in this paper we prove that if f is transitive and has a periodic point of period p then f is distributionally chaotic in a sequence particularly chaos in the sense of devaney is stronger than distributional chaos in a sequence

operators with the specification property abstract we study a version of the specification property for linear dynamics operators having the specification property are investigated and relationships with other well known dynamical notions such as mixing devaney chaos and frequent hypercyclicity are obtained

dynamical systems over the space of upper semicontinuous fuzzy sets given a locally compact separable metric space e a perfect transformation defined on e induces a zadeh extension which is a transformation from the space of all upper semicontinuous fuzzy sets defined on e to itself here the latter space is equipped with the hit or miss topology in this setting each upper semicontinuous fuzzy set is identified with its hypograph a closed subset in the product space of e and 0 1 this approach does not require the relevant fuzzy sets to hold a compact support and it also overcomes the drawback of the traditional level set method further it is proved that the zadeh extension with this setting is continuous in the hit or miss topology and dynamical properties of zadeh extension regarding iteration have been explored

on open problems concerning distributional chaos for triangular maps abstract we show that in the class t of the triangular maps x y f x g x y of the square there is a map of type 2 with non minimal recurrent points which is not dc3 we also show that every dc1 continuous map of a compact metric space has a trajectory which cannot be weakly approximated by trajectories of compact periodic sets these two results make possible to answer some open questions concerning classification of maps in t with zero topological entropy and contribute to an old problem formulated by a n sharkovsky

constructing np intermediate problems by blowing holes with parameters of various properties the search for natural np intermediate problems is one of the holy grails within computational complexity ladner s original diagonalization technique for generating np intermediate problems blowing holes has a serious shortcoming it creates problems with a highly artificial structure by arbitrarily removing certain problem instances in this article we limit this problem by generalizing ladner s method to use parameters with various characteristics this allows one to define more fine grained parameters resulting in np intermediate problems where we only blow holes in a controlled subset of the problem we begin by fully characterizing the problems that admit np intermediate subproblems for a broad and natural class of parameterizations and extend the result further such that structural csp restrictions based on parameters that are hard to compute such as tree width are covered thereby generalizing a result by grohe for studying certain classes of problems including csps parameterized by constraint languages we consider more powerful parameterizations first we identify a new method for obtaining constraint languages such that csp are np intermediate the sets can have very different properties compared to previous constructions by for instance bodirsky grohe and provides insights into the algebraic approach for studying the complexity of infinite domain csps second we prove that the propositional abduction problem parameterized by constraint languages admits np intermediate problems this settles an open question posed by nordh zanuttini we provide a framework to obtain np intermediate problems based on diagonalization we categorize different types of parameters that are usable in our framework we construct np intermediate constraint satisfaction problems we resolve the open question whether boolean abduction admits intermediate problems

on parameterized exponential time complexity in this paper we study the notion of parameterized exponential time complexity we show that a parameterized problem can be solved in parameterized 2 o f k p n time if and only if it is solvable in time o 2 d f k q n for any constant d 0 where p and q are polynomials we then illustrate how this equivalence can be used to show that special instances of parameterized np hard problems are as difficult as the general instances for example we show that the planar dominating set problem on degree 3 graphs can be solved in 2 o k p n parameterized time if and only if the general planar dominating set problem can apart from their complexity theoretic implications our results have some interesting algorithmic implications as well

mean li yorke chaos for random dynamical systems abstract in this paper we studied the complexity of some random dynamical systems with positive topological entropy the random dynamical systems are usually generated by stochastic partial differential equations spdes and contain randomness in many ways we proved that there exists mean li yorke chaotic phenomenon in some random dynamical systems with positive entropy

vector bundles and k theory over topological algebras functional analysis techniques mainly pertaining to the theory of intinitedimensional representations have been recently applied mostly by soviet mathematicians to compute hermitian k groups related to the classification of quadratic forms with coefftcients in group rings see for instance the relevant survey articles by miseenko 59 801 in this concern it has been pointed out that without loss of generality one may profitably consider instead as coefficient ring an abstract c algebra 59 p 931 in a similar framework namely that of vector bundles with coefftcients in c algebras topological invariants of elliptic operators related to the classical atiyah singer formula have also been studied by miieenko in 60 see also 61 i whereas kasparov has given in 42 a similar study by still applying c algebra techniques cf the comments at the end of this paper besides vector bundles over banach algebras and in particular c algebras commutative or not have already been applied by kandelaki 37 381 and independently by fujii 27 one of their main results being a generalization of the serre swan theorem 41 for complex vector bundles cf theorem 4 2 in this paper furthermore connections with the theory of extensions of c algebras 12 bdf theory have also been considered by kandelaki 39 401 on the other hand topological in particular locally m convex algebras necessarily not normed ones have been recently used in connection for instance with applications to relativistic quantum theory 18 191 thus the extension of the preceding considerations to vector bundles with coefftcients in topological algebras more general than normed ones and in particular in locally m convex ones 55 581 is naturally revealed now this task is further feasible as a consequence of the generalization within the topological algebras theory framework of previous relevant work concerning applications of algebraic topology and complex analysis to banach algebras theory 24 53 55 621 452 0022 247x 83 3 00

factor maps and invariant distributional chaos abstract the main aim of this article is to show that maps with the specification property have invariant distributionally scrambled sets and that this kind of scrambled set can be transferred from factor to extension under finite to one factor maps this solves some open questions in the literature of the topic we also show how our method can be applied in practice taking as example poincare map of time periodic nonautonomous planar differential equation z v t z 1 e i t z 2 z 3 n where and n are sufficiently small positive real numbers

generalized specification property and distributionally scrambled sets abstract suppose the dynamical system x f has generalized specification property in this paper we research the existence of the invariant mycielski distributionally scrambled sets for dynamical system x f moreover some properties about measure center are discussed and a dichotomy theorem is proved

half integrality lp branching and fpt algorithms a recent trend in parameterized algorithms is the application of polytope tools to fixed parameter tractable fpt algorithms e g cygan et al focs 2011 52nd annual symposium on foundations of computer science ieee 2011 pp 150 159 narayanaswamy et al stacs 2012 symposium on theoretical aspects of computer science 2012 pp 338 349 although this approach has yielded significant speedups for a range of important problems it requires the underlying polytope to have very restrictive properties including half integrality and nemhauser trotter style persistence properties to date these properties are essentially known to hold only for two classes of polytopes covering the cases of vertex cover nemhauser and trotter math program 8 1975 pp 232 248 and node multiway cut garg et al j alg 50 2004 pp 49 61 taking a slightly different approach we view half integrality as a discrete relaxation of a problem e g a relaxation of the search space from 0 1 v to 0

chaotic semigroups from second order partial differential equations abstract we give general conditions on given parameters to ensure devaney and distributional chaos for the solution c 0 semigroup corresponding to a class of second order partial differential equations we also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups in the case of chaos we prove that the c 0 semigroup admits a strongly mixing measure with full support we also give concrete examples of partial differential equations such as the telegraph equation whose solutions satisfy these properties

fixed parameter tractability definability and model checking in this article we study parameterized complexity theory from the perspective of logic or more specifically descriptive complexity theory we propose to consider parameterized model checking problems for various fragments of first order logic as generic parameterized problems and show how this approach can be useful in studying both fixed parameter tractability and intractability for example we establish the equivalence between the model checking for existential first order logic the homomorphism problem for relational structures and the substructure isomorphism problem our main tractability result shows that model checking for first order formulas is fixed parameter tractable when restricted to a class of input structures with an excluded minor on the intractability side for every t ge 0 we prove an equivalence between model checking for first order formulas with t quantifier alternations and the parameterized halting problem for alternating turing machines with t alternations we discuss the close connection between this alternation hierarchy and downey and fellows w hierarchy on a more abstract level we consider two forms of definability called fagin definability and slicewise definability that are appropriate for describing parameterized problems we give a characterization of the class fpt of all fixed parameter tractable problems in terms of slicewise definability in finite variable least fixed point logic which is reminiscent of the immerman vardi theorem characterizing the class ptime in terms of definability in least fixed point logic

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

exact algorithms for treewidth and minimum fill in we show that the treewidth and the minimum fill in of an n vertex graph can be computed in time mathcal o 1 8899 n our results are based on combinatorial proofs that an n vertex graph has mathcal o 1 7087 n minimal separators and mathcal o 1 8135 n potential maximal cliques we also show that for the class of asteroidal triple free graphs the running time of our algorithms can be reduced to mathcal o 1 4142 n

tight complexity bounds for fpt subgraph problems parameterized by the clique width we give tight algorithmic lower and upper bounds for some double parameterized subgraph problems when the clique width of the input graph is one of the parameters

on the compressibility of mathcal np instances and cryptographic applications we study compression that preserves the solution to an instance of a problem rather than preserving the instance itself our focus is on the compressibility of mathcal np decision problems we consider mathcal np problems that have long instances but relatively short witnesses the question is whether one can efficiently compress an instance and store a shorter representation that maintains the information of whether the original input is in the language or not we want the length of the compressed instance to be polynomial in the length of the witness and polylog in the length of original input such compression enables succinctly storing instances until a future setting will allow solving them either via a technological or algorithmic breakthrough or simply until enough time has elapsed in this paper we first develop the basic complexity theory of compression including reducibility completeness and a stratification of mathcal np with respect to compression we then show that compressibility say of sat would have vast implications for cryptography including constructions of one way functions and collision resistant hash functions from any hard on average problem in mathcal np and cryptanalysis of key agreement protocols in the bounded storage model when mixed with time complexity based cryptography

a measure conquer approach for the analysis of exact algorithms for more than 40 years branch reduce exponential time backtracking algorithms have been among the most common tools used for finding exact solutions of np hard problems despite that the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds motivated by this we use an approach that we call measure conquer as an attempt to step beyond such limitations the approach is based on the careful design of a nonstandard measure of the subproblem size this measure is then used to lower bound the progress made by the algorithm at each branching step the idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit and hence lead to a significantly better worst case time analysis in order to show the potentialities of measure conquer we consider two well studied np hard problems minimum dominating set and maximum independent set for the first problem we consider the current best algorithm and prove thanks to a better measure a much tighter running time bound for it for the second problem we describe a new simple algorithm and show that its running time is competitive with the current best time bounds achieved with far more complicated algorithms and standard analysis our examples show that a good choice of the measure made in the very first stages of exact algorithms design can have a tremendous impact on the running time bounds achievable