stable sets and epsilon stable sets in positive entropy systems in this paper the chaoticity appearing in the stable and unstable sets of a dynamical system with positive entropy is investigated it is shown that in any positive entropy system there is a measure theoretically rather big set such that the closure of the stable or unstable set of any point from the set contains a weak mixing set moreover the bowen entropy of these weak mixing sets are also estimated at the same time it is proved that the topological entropy of any topological system can be calculated in terms of the dispersion of the pre images of epsilon stable sets which answers an open question posed by d fiebig u r fiebig and z h nitecki ergod th dynam sys 23 1785 1806 2003

complex dynamics of cellular automata rule 119 abstract in this paper the dynamical behaviors of cellular automata rule 119 are studied from the viewpoint of symbolic dynamics in the bi infinite symbolic sequence space 2 it is shown that there exists one bernoulli measure global attractor of rule 119 which is also the nonwandering set of the rule moreover it is demonstrated that rule 119 is topologically mixing on the global attractor and possesses the positive topological entropy therefore rule 119 is chaotic in the sense of both li yorke and devaney on the global attractor it is interesting that rule 119 a member of wolfram s class ii which was said to be simple as periodic before actually possesses a chaotic global attractor in 2 finally it is noted that the method presented in this work is also applicable to studying the dynamics of other rules especially the 112 bernoulli shift rules therein

local and pointwise error estimates of the local discontinuous galerkin method applied to the stokes problem we prove local and pointwise error estimates for the local discontinuous galerkin method applied to the stokes problem in two and three dimensions by using techniques originally developed by a schatz math comp 67 1998 877 899 to prove pointwise estimates for the laplace equation we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries

eigenvectors of the multi component generalization of the six vertex model abstract the most general multi component solution of the star triangle equations under an ice type restriction is considered the transfer matrix of this model is diagonalized and the eigenvectors are written down explicitly using the quantum inverse scattering formalism the eigenvectors are shown to have simple symmetry properties under an exchange of arguments

sharkovsky s program for the classification of triangular maps is almost completed abstract for a continuous map of the interval there are more than 50 conditions characterizing zero topological entropy some are applicable to the class of triangular maps x y f x g x y of the square but only a few of them are equivalent in this more general setting in 1989 a n sharkovsky posed the problem of proving or disproving all possible implications between them during last 20 years 32 conditions were considered and most of the work was done only 45 relations out of 992 remained not clear in this paper we give a survey of known results provide two new examples disproving another 26 possible implications and spell out the remaining 19 open problems all but one concern distributional chaos

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 high order elliptic pde based level set reinitialisation method using a discontinuous galerkin discretisation abstract in this paper an efficient high order accurate level set reinitialisation method is proposed based on the elliptic reinitialisation method basting and kuzmin 2013 1 which is discretised spatially using the discontinuous galerkin dg symmetric interior penalty method sipg in order to achieve this a number of improvements have been made to the elliptic reinitialisation method including reformulation of the underlying minimisation problem driving the solution adoption of a lagrange multiplier approach for enforcing a dirichlet boundary condition on the implicit level set interface and adoption of a narrow band approach numerical examples confirm the high order accuracy of the resultant method by demonstrating experimental orders of convergence congruent with optimal convergence rates for the sipg method that is h p 1 and h p in the l 2 and dg norms respectively furthermore the degree to which the level set function satisfies the eikonal equation improves proportionally to h p and the often ignored homogeneous dirichlet boundary condition on the interface is shown to be satisfied accurately with a rate of convergence of at least h 2 for all polynomial orders

a crank nicolson discontinuous finite volume element method for a coupled non stationary stokes darcy problem abstract in this paper we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a crank nicolson discretization in time and we obtain the solution at the first time step by applying a first order backward euler method to solve the coupling of time dependent stokes darcy problem the proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head whereas the pressure is approximated by the piecewise constant elements then the unique solvability of the approximate solution for the discrete problem is derived in addition the optimal error estimates of the full discretization in standard l 2 norm and broken h 1 norm are obtained finally a series of numerical experiments are provided to illustrate the features of the proposed method such as the optimal accuracy orders adaptive mesh refinement mass conservation capability to conveniently deal with complicated geometries with different types of boundary conditions and the applicability to the problems with realistic parameters in particular the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain it encompasses both fractures with high permeability which act as the prior flow conduit and fractures with low permeability as a flow barrier

the quantum symmetric xxz chain at delta 1 2 alternating sign matrices and plane partitions we consider the ground state wavefunction of the quantum symmetric antiferromagnetic xxz chain with open and twisted boundary conditions at along with the ground state wavefunction of the corresponding o n loop model at n 1 based on exact results for finite size systems sums involving the wavefunction components and in some cases the largest component itself are conjectured to be directly related to the total number of alternating sign matrices and plane partitions in certain symmetry classes

pointwise error estimates for discontinuous galerkin methods with lifting operators for elliptic problems in this article we prove some weighted pointwise estimates for three discontinuous galerkin methods with lifting operators appearing in their corresponding bilinear forms we consider a dirichlet problem with a general second order elliptic operator

the finite size su 3 perk schultz model with deformation parameter q exp 2iπ 3 from extensive numeric diagonalizations of the su 3 perk schultz hamiltonian with a special value of the anisotropy and different boundary conditions we have observed simple regularities for a significant part of its eigenspectrum in particular the ground state energy and nearby excitations belong to this part of the spectrum our simple formulae describing these regularities recall apart from some selection rules the eigenspectrum of the free fermion model based on the numerical observations we formulate several conjectures using explicit solutions of the associated nested bethe ansatz equations guessed from an analysis of the functional equations of the model we provide evidence for some of them

chaos induced by regular snap back repellers abstract this paper is concerned with chaos induced by regular snap back repellers one new criterion of chaos induced by strictly coupled expanding maps in compact sets of metric spaces is established by employing this criterion the nondegenerateness assumption in the marotto theorem established in 1978 is weakened in addition it is proved that a regular snap back repeller and a regular homoclinic orbit to a regular expanding fixed point in finite dimensional spaces imply chaos in the sense of li yorke an illustrative example is provided with computer simulations

the importance of being odd in this letter i mainly consider a finite xxz spin chain with periodic boundary conditions and an odd number of sites this system is described by the hamiltonian hxxz j 1n jx j 1x jy j 1y jz j 1z as it turns out the ground state energy is proportional to the number of sites e 3n 2 for a special value of the asymmetry parameter 1 2 the trigonometric polynomial q u the zeros of which are parameters of the ground state bethe eigenvector is explicitly constructed this polynomial of degree n n 1 2 satisfies the baxter t q equation using the second independent solution of this equation that corresponds to the same eigenvalue of the transfer matrix it is possible to find a derivative of the ground state energy w r t the asymmetry parameter this derivative is closely connected with the correlation function jz j 1z 1 2 3 2n2 this correlation function is related to the average number of spin strings for the ground state nstring n 1 n i would like to stress that all the above simple formulae are not applicable to the case of an even number of sites which is usually considered

expansive measures for flows abstract we extend the concept of expansive measure 2 from homeomorphism to flows we prove for continuous flows on compact spaces that every expansive measure has no singularities in the support is aperiodic is expansive with respect to time t maps but not conversely remains expansive under topological equivalence vanishes along the orbits and is natural under suspensions we apply these properties to prove that there are no expansive flows in the sense of 26 of any closed surface

continuum modeling of supply chain networks using discontinuous galerkin methods using a connectivity matrix we establish a continuum modeling approach with partial differential equations of conservation laws for simulating materials flow in supply chain networks a number of existing and new constitutive relationships for modeling velocity are summarized or proposed to effectively treat strong advection components within the modeling system we apply discontinuous galerkin dg methods for solving production flow in a supply chain network in addition a number of dg properties are analyzed for treating network flow in particular a nearly optimal error estimate is obtained using a new estimating technique that utilizes two physical meaningful assumptions on the connectivity matrix numerical examples are provided to simulate a single node a serial supply chain and an entire network as well as to investigate the influence of influx variation and node shut down to the profiles of work in progress wip and outflux it is shown that the proposed modeling approach is applicable to a large number of scenarios including re entrant lines and the proposed dg algorithm is robust and accurate for predicting wip and outflux behaviors

almost periodically forced circle flows abstract we study general dynamical and topological behaviors of minimal sets in skew product circle flows in both continuous and discrete settings with particular attentions paying to almost periodically forced circle flows when a circle flow is either discrete in time and unforced i e a circle map or continuous in time but periodically forced behaviors of minimal sets are completely characterized by classical theory the general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies the topological complexity of the forcing space and the possible loss of mean motion property on one hand we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of denjoy sets of circle maps in the sense that they can be almost automorphic cantorian and everywhere non locally connected but on the other hand we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of denjoy theory for instance an almost periodically forced circle flow can be positively transitive and its minimal sets can be li yorke chaotic and non almost automorphic as an application of our results we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic sl 2 r valued continuous or discrete cocycle continuous almost periodically forced circle flows are among the simplest non monotone multi frequency dynamical systems they can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through mather theory in the damping free cases they also naturally arise in 2d almost periodic floquet theory as well as in climate models discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear schrodinger equations with almost periodic potentials they have been widely used as models for studying strange non chaotic attractors and intermittency phenomena during the transition from order to chaos hence the study of these flows is of fundamental importance to the understanding of multi frequency driven dynamical irregularities and complexities in non monotone dynamical systems

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

the xxz spin chain at δ 1 2 bethe roots symmetric functions and determinants a number of conjectures have been given recently concerning the connection between the antiferromagnetic xxz spin chain at 1 2 and various symmetry classes of alternating sign matrices here we use the integrability of the xxz chain to gain further insight into these developments in doing so we obtain a number of new results using baxter s q function for the xxz chain for periodic twisted and open boundary conditions these include expressions for the elementary symmetric functions evaluated at the ground state solution of the bethe roots in this approach schur functions play a central role and enable us to derive determinant expressions which appear in certain natural double products over the bethe roots when evaluated these give rise to the numbers counting different symmetry classes of alternating sign matrices

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

dynamic parrondo s paradox abstract this paper is focused on studying parrondo s paradox in non linear dynamics specifically how the periodic combination of the individual maps f and g can give rise to chaos or order we construct dynamical systems exhibiting the paradox for several notions of chaos derived from topological dynamics the effect of altering the order of the combination i e considering f g or g f is analyzed as well as the robustness of the parrondo effect under small perturbations conditions for avoiding parrondian dynamics are also obtained placing special emphasis on the notion of chaos given by turbulence

chaos induced by heteroclinic cycles connecting repellers and saddles in locally compact metric spaces abstract this paper is concerned with chaos induced by heteroclinic cycles connecting repellers and saddles for maps in locally compact metric spaces one criterion of chaos for such types of maps is established in locally compact metric spaces by employing the coupled expansion theory where the map is only continuous in some domains of interest in particular one criterion of chaos in the euclidean space r n is also established the maps in the criteria are proved to have a positive topological entropy and to be chaotic in the sense of li yorke an illustrative example is provided with computer simulations

independence in topological and c dynamics we develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods one of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties we also reframe our theory of dynamical independence in terms of tensor products and thereby expand its scope to c dynamics

positive topological entropy and δ weakly mixing sets abstract the notion of weakly mixing set is introduced and studied it is proved that weakly mixing sets share many properties with weakly mixing sets in particular if a dynamical system has positive topological entropy then the collection of weakly mixing sets is residual in the closure of the collection of entropy sets in the hyperspace the existence of weakly mixing sets in a topological dynamical system admitting an ergodic invariant measure which is not measurable distal is obtained these results generalize several well known results and also answer several open questions

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

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

pointwise error estimates of the local discontinuous galerkin method for a second order elliptic problem in this paper we derive some pointwise error estimates for the local discontinuous galerkin ldg method for solving second order elliptic problems in r n n 2 our results show that the pointwise errors of both the vector and scalar approximations of the ldg method are of the same order as those obtained in the l 2 norm except for a logarithmic factor when the piecewise linear functions are used in the finite element spaces moreover due to the weighted norms in the bounds these pointwise error estimates indicate that when at least piecewise quadratic polynomials are used in the finite element spaces the errors at any point z depend very weakly on the true solution and its derivatives in the regions far away from z these localized error estimates are similar to those obtained for the standard conforming finite element method

chaotic properties of mappings on a probability space sensitive dependence on initial conditions is widely understood as being the central idea of chaos we first give sufficient conditions both topological and ergodic on an endomorphism to ensure the sensitivity property then a strong sensitivity concept is introduced sufficient conditions on a transformation implying strong sensitivity are given we also provide bounds for the strong sensitivity constant

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

spin chains and combinatorics in this paper we continue the investigation of finite xxz spin chains with periodic boundary conditions and odd number of sites initiated in our previous article stroganov yu g 2001 j phys a math gen 34 l179 85 as it turned out for a special value of the asymmetry parameter 1 2 the hamiltonian of the system has an eigenvalue which is exactly proportional to the number of sites e 3n 2 using mathematica we have found explicitly the corresponding eigenvectors for n 17 the obtained results support the conjecture our paper that this special eigenvalue corresponds to the ground state vector we make a lot of conjectures concerning the correlations of the model many remarkable relations between the wavefunction components are noted it turns out for example that the ratio of the largest component to the least one is equal to the number of the alternating sign matrices

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

pointwise error estimates of discontinuous galerkin methods with penalty for second order elliptic problems in this paper discontinuous galerkin methods with penalty for solving second order elliptic problems are considered error estimates are studied for these methods in particular optimal localized pointwise error estimates are established on quasi uniform grids in rn n ge 2

free fermion branches in some quantum spin models extensive numerical analysis of the eigenspectra of the suq n invariant perk schultz hamiltonian shows some simple regularities for a significant part of the eigenspectrum inspired by those results we have found two sets of solutions of the associated nested bethe ansatz equations the first set is obtained at a special value of the anisotropy q exp i n 1 n and describes in particular the ground state and nearby excitations as a sum of free fermion quasienergies the second set of solutions provides the energies in the sectors whose number ni of particles of distinct species i 0 n 1 are less than or equal to unity except for one of the species for this last set we obtain the eigenspectra of a free fermion model for arbitrary values of the anisotropy

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

graph theoretic structure of maps of the cantor space abstract in this paper we develop unifying graph theoretic techniques to study the dynamics and the structure of spaces h 0 1 n and c 0 1 n the space of homeomorphisms and the space of self maps of the cantor space respectively using our methods we give characterizations which determine when two homeomorphisms of the cantor space are conjugate to each other we also give a new characterization of the comeager conjugacy class of the space h 0 1 n the existence of this class was established by kechris and rosendal and a specific element of this class was described concretely by akin glasner and weiss our characterization readily implies many old and new dynamical properties of elements of this class for example we show that no element of this class has a li yorke pair implying the well known glasner weiss result that there is a comeager subset of h 0 1 n each element of which has topological entropy zero our analogous investigation in c 0 1 n yields a surprising result there is a comeager subset of c 0 1 n such that any two elements of this set are conjugate to each other by an element of h 0 1 n our description of this class also yields many old and new results concerning dynamics of a comeager subset of c 0 1 n

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

syndetically proximal pairs abstract for continuous self maps of compact metric spaces we study the syndetically proximal relation and in particular we identify certain sufficient conditions for the syndetically proximal cell of each point to be small we show that any interval map f with positive topological entropy has a syndetically scrambled cantor set and an uncountable syndetically scrambled set invariant under some power of f in the process of proving this we improve a classical result about interval maps and establish that if f is an interval map with positive topological entropy and m 2 then there is n n such that the one sided full shift on m symbols is topologically conjugate to a subsystem of f 2 n the classical result gives only semi conjugacy

asymptotic pairs stable sets and chaos in positive entropy systems abstract we consider positive entropy g systems for certain countable discrete infinite left orderable amenable groups g we will use local analysis to study the connection between the existence of asymptotic pairs chaotic sets and stable sets examples are given for the case of integer lattice groups the heisenberg group and the groups of integral unipotent upper triangular matrices

chaotic and topological properties of β transformations article history in this paper we prove that the transformations are chaotic in the sense of both li yorke and devaney the topological and metric properties of the sets of points with dense or non dense orbits are investigated we also prove the result that the set of points with non dense orbits under the transformation is of full hausdorff dimension for any 1

on fuzzy entropy and topological entropy of fuzzy extensions of dynamical systems the goal of this paper is to establish connections between two entropy concepts namely the fuzzy entropy of a fuzzy set which appears in fuzzy set theory and the topological entropy of zadeh s extensions of continuous maps defined on metric compact spaces which is a topological dynamics concept we develop several concepts that allow us to simultaneously address the two entropy concepts except for stating several basic properties we are partially capable of specifying in which classes of fuzzy sets dynamically complex behavior can appear by using these instruments such instruments are necessary for a proper understanding of the behavior of fuzzy systems generated from crisp discrete dynamical systems

new conjecture for the suq n perk schultz models we present a new conjecture for the suq n perk schultz models this conjecture extends a conjecture presented in our article alcaraz f c and stroganov yu g j phys a math gen 35 6767 87

emptiness formation probability of the xxz spin heisenberg chain at δ using a multiple integral representation for the correlation functions we compute the emptiness formation probability of the xxz spin heisenberg chain at anisotropy we prove that it is expressed in terms of the number of alternating sign matrices

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

on the distal and asymptotic sets for β transformations abstract in this paper we study the distal and asymptotic sets of a given point in the transformation dynamical system and prove that the hausdorff dimension of the distal set of any given point is one and the hausdorff dimension of the asymptotic set of any given point is zero

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

δ weakly mixing subset in positive entropy actions of a nilpotent group abstract the notion of weakly mixing subsets is introduced for countable torsion free discrete group actions it is shown that for a finitely generated torsion free discrete nilpotent group action positive topological entropy implies the existence of weakly mixing subsets and while there exists a finitely generated torsion free discrete solvable group action which has positive topological entropy but without any weakly mixing subsets

invariant manifolds global attractors and almost periodic solutions of nonautonomous difference equations the article is devoted to the study of quasi linear nonautonomous difference equations invariant manifolds compact global attractors almost periodic and recurrent solutions and chaotic sets first we prove that such equations admit an invariant continuous section an invariant manifold then we obtain the conditions for the existence of a compact global attractor and characterize its structure third we derive a criterion for the existence of almost periodic and recurrent solutions of the quasi linear nonautonomous difference equations finally we prove that quasi linear maps with chaotic base admit a chaotic compact invariant set the obtained results are applied while studying triangular maps invariant manifolds compact global attractors almost periodic and recurrent solutions and chaotic sets

superconvergence property of an over penalized discontinuous galerkin finite element gradient recovery method a polynomial preserving recovery method is introduced for over penalized symmetric interior penalty discontinuous galerkin solutions to a quasi linear elliptic problem as a post processing method the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns as well as general meshes satisfying condition by means of the averaging technique we prove the polynomial preserving recovery method for averaged solutions is superconvergent satisfying similar estimates as those for conforming finite element methods we deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator consequently the a posteriori error estimator based on the recovered gradient is asymptotically exact extensive numerical results consistent with our analysis are presented

on proximality with banach density one abstract let x t be a topological dynamical system a pair of points x y x 2 is called banach proximal if for any e 0 the set n z d t n x t n y e has banach density one we study the structure of the banach proximal relation a useful tool is the notion of the support of a topological dynamical system we show that a dynamical system is strongly proximal if and only if every pair in x 2 is banach proximal a subset s of x is banach scrambled if every two distinct points in s form a banach proximal pair but not asymptotic we construct a dynamical system with the whole space being a banach scrambled set even though the banach proximal relation of the full shift is of first category it has a dense mycielski invariant banach scrambled set we also show that for an interval map it is li yorke chaotic if and only if it has a cantor banach scrambled set

dynamical compactness and sensitivity abstract to link the auslander point dynamics property with topological transitivity in this paper we introduce dynamically compact systems as a new concept of a chaotic dynamical system x t given by a compact metric space x and a continuous surjective self map t x x observe that each weakly mixing system is transitive compact and we show that any transitive compact m system is weakly mixing then we discuss the relationships between it and other several stronger forms of sensitivity we prove that any transitive compact system is li yorke sensitive and furthermore multi sensitive if it is not proximal and that any multi sensitive system has positive topological sequence entropy moreover we show that multi sensitivity is equivalent to both thick sensitivity and thickly syndetic sensitivity for m systems we also give a quantitative analysis for multi sensitivity of a dynamical system

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

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

uniqueness and attractivity of the carrying simplex for discrete time competitive dynamical systems abstract for the discrete time dissipative competitive dynamical system on r n it is proved that generically there is a unique n 1 dim invariant carrying simplex attracting all nontrivial orbits which solved the conjecture raised by h l smith in 1986

global and cocycle attractors for non autonomous reaction diffusion equations the case of null upper lyapunov exponent abstract in this paper we obtain a detailed description of the global and cocycle attractors for the skew product semiflows induced by the mild solutions of a family of scalar linear dissipative parabolic problems over a minimal and uniquely ergodic flow we consider the case of null upper lyapunov exponent for the linear part of the problem then two different types of attractors can appear depending on whether the linear equations have a bounded or an unbounded associated real cocycle in the first case e g in periodic equations the structure of the attractor is simple whereas in the second case which occurs in aperiodic equations the attractor is a pinched set with a complicated structure we describe situations when the attractor is chaotic in measure in the sense of li yorke besides we obtain a non autonomous discontinuous pitchfork bifurcation scenario for concave equations applicable for instance to a linear dissipative version of the chafee infante equation

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

on lyapunov exponent and sensitivity sensitive dependence on initial conditions is widely understood as being the central idea of chaos for a large class of transformations of the interval we prove that positiveness of the lyapunov exponent implies the sensitivity property we also provide bounds for the sensitivity constant

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

chaos among self maps of the cantor space glasner and weiss have shown that a generic homeomorphism of the cantor space has zero topological entropy hochman has shown that a generic transitive homeomorphism of the cantor space has the property that it is topologically conjugate to the universal odometer and hence far from being chaotic in any sense we show that a generic self map of the cantor space has zero topological entropy moreover we show that a generic self map of the cantor space has no periodic points and hence is not devaney chaotic nor devaney chaotic on any subsystem however we exhibit a dense subset of the space of all self maps of the cantor space each element of which has infinite topological entropy and is devaney chaotic on some subsystem

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy

on pairwise sensitivity we define the concept of pairwise sensitivity with respect to initial conditions for the endomorphisms on lebesgue metric spaces the idea is that the orbits of almost every pair of nearby initial points for the product of the invariant measure of a pairwise sensitive map may diverge from a positive quantity independent of the initial points we study the relationships between pairwise sensitivity and weak mixing pairwise sensitivity and positiveness of metric entropy and we compute the largest sensitivity constant

symmetric and nonsymmetric discontinuous galerkin methods for reactive transport in porous media for solving reactive transport problems in porous media we analyze three primal discontinuous galerkin dg methods with penalty namely symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg a cut off operator is introduced in dg to treat general kinetic chemistry error estimates in l2 h1 are established which are optimal in h and nearly optimal in p we develop a parabolic lift technique for sipg which leads to h optimal and nearly p optimal error estimates in the l2 l2 and negative norms numerical results validate these estimates we also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces

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

an optimal order l2 error estimate for nonsymmetric discontinuous galerkin methods for a parabolic equation in multiple space dimensions we analyze the nonsymmetric discontinuous galerkin methods nipg and iipg for linear elliptic and parabolic equations with a spatially varied coefficient in multiple spatial dimensions we consider d linear approximation spaces on a uniform rectangular mesh but our results can be extended to smoothly varying rectangular meshes using a blending or boolean interpolation we obtain a superconvergence error estimate in a discrete energy norm and an optimal order error estimate in a semi discrete norm for the parabolic equation the l 2 optimality for the elliptic problem follows directly from the parabolic estimates numerical results are provided to validate our theoretical estimates we also discuss the impact of penalty parameters on convergence behaviors of nipg

quasi optimal a priori interface error bounds and a posteriori estimates for the interior penalty method in this work we show quasi optimal interface error estimates for solutions obtained by the symmetric interior penalty discontinuous galerkin method it is proved that the numerical solution restricted to an interface converges with order lvert ln h rvert h k 1 under suitable regularity requirements where the logarithmic factor is only present in the lowest order case i e k 1 for this case we also derive and analyze two a posteriori error estimators which demonstrate that the jump terms of the discrete fluxes are not essential to obtain local efficiency and reliability we support our analysis by numerical results and demonstrate that the interface approximation can be locally postprocessed to obtain discrete solutions of order h k 1 2 in the energy norm

conditional entropy and fiber entropy for amenable group actions abstract in this paper we introduce the notions of topological conditional entropy and fiber entropy for a given factor map between two amenable group actions and prove three variational principles for conditional entropy and fiber entropy moreover as an application of our variational principle we prove that the countable to one extension and the distal extension have zero conditional topological entropy in particular the topological entropy of a distal system is zero

topological entropy of fuzzified dynamical systems a discrete dynamical system is given by a compact metric space x and any continuous self map defined on x this discrete dynamical system can be naturally extended to the space of fuzzy sets on x in this paper we study relations between the sizes of the topological entropies of the original dynamical system and of its fuzzy counterpart among other things we present a constructive proof of the fact that even very weak assumptions on the crisp discrete dynamical system ensure infinite topological entropy of the fuzzy system however we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system

the wavefunctions for the free fermion part of the spectrum of the suq n quantum spin models we conjecture that the free fermion part of the eigenspectrum observed recently for the suq n perk schultz spin chain hamiltonian in a finite lattice with q exp i n 1 n is a consequence of the existence of a special simple eigenvalue for the transfer matrix of the auxiliary inhomogeneous suq n 1 vertex model which appears in the nested bethe ansatz approach we prove that this conjecture is valid for the case of the suq 3 spin chain with periodic boundary condition in this case we obtain a formula for the components of the eigenvector of the auxiliary inhomogeneous 6 vertex model q exp 2i 3 which permits us to find one by one all components of this eigenvector and consequently to find the eigenvectors of the free fermion part of the eigenspectrum of the suq 3 spin chain similarly as in the known case of the suq 2 case at q exp i2 3 our numerical and analytical studies induce some conjectures for special rates of correlation functions

analysis of mixed interior penalty discontinuous galerkin methods for the cahn hilliard equation and the hele shaw flow this paper proposes and analyzes two fully discrete mixed interior penalty discontinuous galerkin dg methods for the fourth order nonlinear cahn hilliard equation both methods use the backward euler method for time discretization and interior penalty dg methods for spatial discretization they differ from each other on how the nonlinear term is treated one of them is based on fully implicit time stepping and the other uses energy splitting time stepping the primary goal of the paper is to prove the convergence of the numerical interfaces of the dg methods to the interface of the hele shaw flow this is achieved by establishing error estimates that depend on epsilon 1 only in some low polynomial orders instead of exponential orders similar to x feng and a prohl numer math 74 2004 pp 47 84 the crux is to prove a discrete spectrum estimate in the discontinuous galerkin finite element space however the validity of such a result is not obvious because the dg space is not a subspa