good parameters for a class of node sets in quasi monte carlo integration for 2 s 12 we determine good parameters in a general construction of node sets for s dimensional quasi monte carlo integration recently introduced by the third author some of the parameters represent optimal choices in this construction and lead to improvements on node sets obtained by earlier techniques

scrambled sets of continuous maps of 1 dimensional polyhedra let k be a 1 dimensional simplicial complex in r3 without isolated vertexes x iki be the polyhedron of k with the metric dk induced by k and f x x be a continuous map in this paper we prove that if k is finite then the interior of every scrambled set of f in x is empty we also show that if k is an infinite complex then there exist continuous maps from x to itself having scrambled sets with nonempty interiors and if x r or r then there exist c maps of x with the whole space x being a scrambled set

imbedded lattice rules for multidimensional integration this paper is concerned with imbedded sequences of lattice rules the theoretical properties of certain finite imbedded sequences are developed and an efficient search method for finding good imbedded sequences of this kind is discussed an algorithm for computing these sequences is presented and some numerical results using the search method and this algorithm are given

chaotic dynamics of continuous time topological semi flows on polish spaces abstract differently from lyapunov exponents li yorke devaney and others that appeared in the literature we introduce the concept chaos for a continuous semi flow f r x x on a polish space x with a metric d which is useful in the theory of ode and is invariant under topological equivalence of semi flows our definition is weaker than devaney s one since here f may have neither fixed nor periodic elements but it implies repeatedly observable sensitive dependence on initial data there is an 0 such that for any x x there corresponds a dense g set s u x in x satisfying lim sup t d f t x f t y y s u x this sensitivity is obviously stronger than guckenheimer s one that requires only d f t x f t y for some moment t 0 and some y arbitrarily close to x

on weak product recurrence and synchronization of return times abstract the paper is devoted to a study of product recurrence first we prove that notions of p s pr and p u b d pr are exactly the same as product recurrence completing that way results of p dong s shao x ye product recurrent properties disjointness and weak disjointness israel j math 188 1 2012 463 507 and consequently extending the characterization of return times of distal points which originated from the works of furstenberg we also study the structure of the set of return times of weakly mixing sets as a consequence we obtain new sufficient conditions for s pr and also find a short proof that weakly mixing systems are disjoint with all minimal distal systems in particular our proof does not involve furstenberg s structure theorem of minimal distal systems

scrambled net variance for integrals of smooth functions hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter one version randomized t m s nets has the property that the integral estimates are unbiased and that the variance is o 1 n for any square integrable integrand stronger assumptions on the integrand allow one to find rates of convergence this paper shows that for smooth integrands over s dimensions the variance is of order n 3 log n 3 1 compared to n 1 for ordinary monte carlo thus the integration errors are of order n 3 2 log n s 1 2 in probability this compares favorably with the rate n 1 log n s 1 for unrandomized t m s nets

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

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

a construction of polynomial lattice rules with small gain coefficients in this paper we construct polynomial lattice rules which have in some sense small gain coefficients using a component by component approach the gain coefficients as introduced by owen indicate to what degree the method improves upon monte carlo we show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component by component decays at a rate of n 2 1 for all 0 assuming that the function under consideration has bounded variation of order for some 0 1 and where n denotes the number of quadrature points an analogous result is obtained for korobov polynomial lattice rules it is also established that these rules are almost optimal for the function space considered in this paper furthermore we discuss the implementation of the component by component approach and show how to reduce the computational cost associated with it finally we present numerical results comparing scrambled polynomial lattice rules and scrambled digital nets

on the power of adaptive information for functions with singularities we study from a probabilistic viewpoint the problem of locating singularities of functions using function evaluations we show that under the assumption of a wiener like probability distribution on the class of singular functions an adaptive algorithm can locate a singular point accurately with only a small probability of failure as an application we show that an integration algorithm that adaptively locates a singular point is probabilistically superior to nonadaptive algorithms

the asymptotic efficiency of randomized nets for quadrature an l 2 type discrepancy arises in the average and worst case error analyses for multidimensional quadrature rules this discrepancy is uniquely defined by k x y which serves as the covariance kernel for the space of random functions in the average case analysis and a reproducing kernel for the space of functions in the worst case analysis this article investigates the asymptotic order of the root mean square discrepancy for randomized 0 m s nets in base b for moderately smooth k x y the discrepancy is o n 1 log n s 1 2 and for k x y with greater smoothness the discrepancy is o n 3 2 log n s 1 2 where n b m is the number of points in the net numerical experiments indicate that the t m s nets of faure niederreiter and sobol do not necessarily attain the higher order of decay for sufficiently smooth kernels however niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions

linear orderings and the full periodicity kernel for the n star abstract we show that baldwin s characterization of the set of periods of continuous self maps of the n star can be expressed in terms of a finite number of linear orderings additionally we study the minimal sets of periods which force a continuous self map of the n star to have periodic points of all periods

path integral approach to multiple scattering multiple scattering of a wave in a system of scatterers with random uncorrelated positions is studied with path integrals the edwards gulyaev expression for the position averaged green s function is used to find the density expansion of the complex optical potential the expansion is in terms of exact medium propagators and scattering matrices in the medium the first term is the coherent potential approximation a source dependent generalization of the path integral is used to derive a functional equation for the optical potential this leads to a hierarchy for correlation functions that involves exact medium propagators and scattering matrices the simplest truncations yield new integral equations that are generalizations of the coherent potential approximation and are compatible with the density expansion

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

optimal quadrature for haar wavelet spaces this article considers the error of the scrambled equidistribution quadrature rules in the worst case random case and average case settings the underlying space of integrands is a hilbert space of multidimensional haar wavelet series hwav the asymptotic orders of the errors are derived for the case of the scrambled t m s nets and t s sequences these rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands hwav

product recurrence and distal points recurrence is studied in the context of actions of compact semigroups on compact spaces an important case is the action of the stone tech compactification of an acting group if the semigroup e acts on the space x and f is a closed subsemigroup of e then x in x is said to be frecurrent if px x for some p e f and product f recurrent if whenever y is an f recurrent point in some space y on which e acts the point x y in the product system is f recurrent the main result is that under certain conditions a point is product f recurrent if and only if it is a distal point

quadrature error bounds with applications to lattice rules reproducing kernel hilbert spaces are used to derive error bounds and worst case integrands for a large family of quadrature rules in the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature however the theory developed here does not require periodicity and is not restricted to lattice rules an analysis of variance anova decomposition is employed in defining the inner product it is shown that imbedded rules are superior when integrating functions with large high order anova effects

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

functional equations for path integrals we consider the density matrices that arise in the statistical mechanics of the electron phonon systems in the path integral representation the phonon coordinates can be eliminated this leads to an action that depends on pairs of points on a path that depends explicitly on time differences and that contains the phonon occupation numbers the integral is reduced to a standard form by scaling to the thermal length we use the technique of integration by parts and add specially chosen generating functionals to the action we set down functional derivative equations for the source dependent density matrix and for the mass operator this allows us to develop a series of approximations for the operator in terms of exact propagators the crudest approximation is a coherent potential approximation applicable at a general temperature

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

distributional chaos and irregular recurrence abstract for a continuous map x x of a compact metric space we study relations between distributional chaos and the existence of a point which is quasi weakly almost periodic but not weakly almost periodic we provide an example showing that the existence of such a point does not imply the strongest version of distributional chaos dc1 using this we prove that even in the class of triangular maps of the square there are no relations to dc1 this result among others contributes to the solution of a problem formulated by a n sharkovsky in the eighties

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

extensible lattice sequences for quasi monte carlo quadrature integration lattices are one of the main types of low discrepancy sets used in quasi monte carlo methods however they have the disadvantage of being of fixed size this article describes the construction of an infinite sequence of points the first bm of which forms a lattice for any nonnegative integer m thus if the quadrature error using an initial lattice is too large the lattice can be extended without discarding the original points generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice the spectral test used for finding pseudorandom number generators is one important example of such a discrepancy the performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems

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

can adaption help on the average we study adaptive information for approximation of linear problems in a separable hilbert space equipped with a probability measure it is known that adaption does not help in the worst case for linear problems we prove that adaption also doesnot help on the average that is there exists nonadaptive information which is as powerful as adaptive information this result holds for orthogonally invariant measures we provide necessary and sufficient conditions for a measure to be orthogonally invariant examples of orthogonally invariant measures include gaussian measures and in the finite dimensional case weighted lebesgue measures

on scrambled sets for chaotic functions some recent research has raised questions concerning the possible sizes of scrambled sets for chaotic functions we answer these questions by showing that a scrambled set can have full measure but cannot be residual although a scrambled set can be second category in every interval we also indicate relationships that exist between chaotic functions and transitive functions

average case complexity of multivariate integration we study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical wiener sheet measure to derive the average case complexity one needs to obtain optimal sample points we prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years this relation enables us to show that optimal sample points can be derived from hammersley points extending the result of roth and using the recent result of wasilkowski we conclude that the average case complexity is e 1 lne 1 d 1 2

statistical properties of generalized discrepancies when testing that a sample of n points in the unit hypercube 0 1 d comes from a uniform distribution the kolmogorov smirnov and the cramer von mises statistics are simple and well known procedures to encompass these measures of uniformity hickernell introduced the so called generalized p discrepancies these discrepancies can be used in numerical integration through monte carlo and quasi monte carlo methods design of experiments uniformity testing and goodness of fit tests the aim of this paper is to derive the statistical asymptotic properties of these statistics under monte carlo sampling in particular we show that under the hypothesis of uniformity of the sample of points the asymptotic distribution is a complex stochastic integral with respect to a pinned brownian sheet on the other hand if the points are not uniformly distributed then the asymptotic distribution is gaussian

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

representation of functions as walsh series to different bases and an application to the numerical integration of high dimensional walsh series we will prove the following theorem on walsh series and we will derive from this theorem an effective and constructive method for the numerical integration of walsh series by number theoretic methods further concrete computer calculations are given theorem for base b 2 dimension s 1 and o 1 c 0 b s n c a e r let i e c be the class of all functions f 0 l s c which are representable by absolutely convergent walsh series to base b with walsh coefficients w h hs with the following property w h hs 2 provided that a 1 s where

integration of nonperiodic functions of two variables by fibonacci lattice rules abstract two dimensional lattice rules are applied to continuous functions over the unit square which do not have a continuous periodic extension it is shown that provided lattice points at vertices and edges are treated appropriately certain functions including all bilinear functions are integrated exactly whenever the lattice contains a possibly rotated square unit cell the fibonacci lattice with denominators f k for the nodes is then shown to have a square unit cell if and only if k is odd numerical experiments for fibonacci rules and copies of fibonacci rules confirm that there are significant different between the odd k and even k cases

partition function of a particle subject to gaussian noise we study the averaged partition function for a quantum particle subjected to gaussian noise using the path integral representation the noise is characterized by a covariance function with a strength and a range it falls off rapidly with distance but the analytic form at short distances and the dimensionality are important the remaining parameter is the thermal length of the particle for a finite range we study the behavior of the partition function over the entire domain of strengths and thermal lengths the techniques used are successively more accurate upper and lower bounds that include contributions from configurations involving traps particular attention is paid to a self consistent field analysis lower bound and to a nonlocal quadratic action bound we also study the white noise limit i e vanishing range with finite values of the other parameters in one dimension the white noise limit leads to convergent results in three or higher dimensions the divergent terms can be isolated and computed in two dimensions the degree of divergences changes at a finite value of the product of the strength and thermal length squared

alternative sampling methods for estimating multivariate normal probabilities we study the performance of alternative sampling methods for estimating multivariate normal probabilities through the ghk simulator the sampling methods are randomized versions of some quasi monte carlo samples halton niederreiter niederreiter xing sequences and lattice points and some samples based on orthogonal arrays latin hypercube orthogonal array and orthogonal array based latin hypercube samples in general these samples turn out to have a better performance than monte carlo and antithetic monte carlo samples improvements over these are large for low dimensional 4 and 10 cases and still significant for dimensions as large as 50

a multivariate fast discrete walsh transform with an application to function interpolation for high dimensional problems such as approximation and integration one cannot afford to sample on a grid because of the curse of dimensionality an attractive alternative is to sample on a low discrepancy set such as an integration lattice or a digital net this article introduces a multivariate fast discrete walsh transform for data sampled on a digital net that requires only o n log n operations where n is the number of data points this algorithm and its inverse are digital analogs of multivariate fast fourier transforms this fast discrete walsh transform and its inverse may be used to approximate the walsh coefficients of a function and then construct a spline interpolant of the function this interpolant may then be used to estimate the function s effective dimension an important concept in the theory of numerical multivariate integration numerical results for various functions are presented

on chaotic minimal center of attraction of a lagrange stable motion for topological semi flows abstract let f r x x be a topological semi flow on a polish space x in 1977 karl sigmund conjectured that if there is a point x in x such that the motion f x has just x as its minimal center of attraction then all such x form a residual subset of x in this paper we first present a positive solution to this conjecture and then apply it to the study of chaotic dynamics occurring inside or near the minimal center of attraction of a motion f x

nonmetrizable topological dynamics and ramsey theory applying ideas from topological dynamics in compact metric spaces to the stone cech compactification of a discrete semigroup several new proofs of old results and some new results in ramsey theory are obtained in particular two ultrafilter proofs of van der waerden s theorem are given an ultrafilter approach to central sets sets which are combinatorially rich is developed this enables us to show that for any partition of the positive integers one cell is both additively and multiplicatively central also a fortuitous answer to a question of ellis is obtained

chaotic oscillations of a linear hyperbolic pde with a general nonlinear boundary condition abstract this article establishes a theorem that guarantees the occurrence of chaotic oscillations in a system governed by a linear hyperbolic partial differential equation pde with a nonlinear boundary condition nbc compared with the nbcs in all previous related literature such an nbc is more general both the left end and the right end of the system parameter interval for the occurrence of chaotic oscillations are precisely characterized the chaotic results obtained are further applied to two specific nbcs and the telegraph equation finally numerical examples verify the effectiveness of theoretical prediction

efficient weighted lattice rules with applications to finance good lattice rules are an important type of quasi monte carlo algorithm they are known to have good theoretical properties in the sense that they can achieve an error bound or optimal error bound that is independent of the dimension for weighted function spaces with suitably decaying weights to use the theory of weighted function spaces for practical applications one has to determine what weights should be used after explaining that lattice rules based on figures of merit with classical weights classical lattice rules may not give good results for high dimensional problems we propose a matching strategy which chooses the weights in such a way that the typical functions of a weighted korobov space are similar in the sense of similar relative sensitivity indices to those of the given function the matching strategy relates the weights of the spaces to the sensitivity indices of the given function we apply the korobov construction and the component by component construction of lattice rules with the suitably chosen weights to the valuation of high dimensional financial derivative securities and find that such weighted lattice rules improve dramatically on the results for the classical lattice rules moreover they are competitive with other types of low discrepancy sequences we also demonstrate that lattice rules combined with variance reduction techniques and dimension reduction techniques increase the efficiency significantly we show that an acceptance rejection approach to the construction of lattice rules can reduce the construction cost with almost no loss of accuracy

average case complexity of linear multivariate problems we study the average case complexity of a linear multivariate problem lmp defined on functions of d variables we consider two classes of information the first std consists of function values and the second all of all continuous linear functionals tractability of lmp means that the average case complexity is o 1 e p with p independent of d we prove that tractability of an lmp in std is equivalent to tractability in all although the proof is no constructive we provide a simple condition to check tractability in all we also address the optimal design problem for an lmp by using a relation to the worst case setting we find the order of the average case complexity and optimal sample points for multivariate function approximation

a new method for the asymptotic evaluation of a class of path integrals a general method of calculating the asymptotic behavior of a class of wiener path integrals is given these integrals are averages of functionals of the local time the technique is essentially a variation on the well known replica method now widely used in condensed matter physics combined with the laplace method for evaluating integrals containing a large parameter the leading term is given and from the construction one sees that the error is typically of order 1 t

nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition abstract this paper studies the nonisotropic chaotic oscillations of the initial boundary value problem of one dimensional wave equation with a mixing transport term it separately considers that the boundary condition at the right end of the wave equation is a superlinear type and linear perturbation of such type each causing the total energy of the underlying system to rise and fall due to the interaction with a mixing transport term for each type of boundary condition the occurrence of nonisotropic chaotic oscillations is rigorously proved numerical examples verify the effectiveness of theoretical prediction

latin hyperrectangle sampling for computer experiments latin hypercube sampling lhs is a popular method for evaluating the expectation of functions in computer experiments however when the expectation of interest is taken with respect to a nonuniform distribution the usual transformation to the probability space can cause relatively smooth functions to become extremely variable in areas of low probability consequently the equal probability cells inherent in hypercube methods often tend to sample an insufficient proportion of the total points in these areas this article introduces latin hyperrectangle sampling lhrs a generalization of lhs that allows for nonequal cell probabilities to address this problem a number of examples are given illustrating the improvement of the proposed methodology over lhs with respect to the variance of the resulting estimators extensions to orthogonal array based lhs stratified lhs and scrambled nets are also described

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

monte carlo variance of scrambled net quadrature hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter this paper studies the variance of one such hybrid scrambled nets by applying a multidimensional multiresolution wavelet analysis to the integrand the integrand is assumed to be measurable and square integrable but not necessarily of bounded variation in simple monte carlo every nonconstant term of the multiresolution contributes to the variance of the estimated integral for scrambled nets certain low dimensional and coarse terms do not contribute to the variance for any integrand in l2 the sampling variance tends to zero faster under scrambled net quadrature than under monte carlo sampling as the number of function evaluations n tends to infinity some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the monte carlo variance uniformly over all integrands f latin hypercube sampling is a special case of scrambled net quadrature

on the exponent of discrepancies we study various discrepancies with arbitrary weights in the l 2 norm over domains whose dimension is proportional to d we are mostly interested in large d the exponent p of discrepancy is defined as the smallest number for which there exists a positive number c such that for all d and e there exist ce p points with discrepancy at most e we prove that for the most standard case of discrepancy anchored at zero the exponent is at most 1 41274 which slightly improves the previously known bound 1 47788 for discrepancy anchored at and for quadrant discrepancy at we prove that the exponent is at most 1 31662 for 1 2 1 2 for unanchored discrepancy we prove that the exponent is at most 1 27113 the previous bound was 1 28898 it is known that for all these discrepancies the exponent is at least 1

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments

strong tractability of integration using scrambled niederreiter points we study the randomized worst case error and the randomized error of scrambled quasi monte carlo qmc quadrature as proposed by owen the function spaces considered in this article are the weighted hilbert spaces generated by haar like wavelets and the weighted sobolev hilbert spaces conditions are found under which multivariate integration is strongly tractable in the randomized worst case setting and the randomized setting respectively the e exponents of strong tractability are found for the scrambled niederreiter nets and sequences the sufficient conditions for strong tractability for sobolev spaces are more lenient for scrambled qmc quadratures than those for deterministic qmc net quadratures

uncertainty quantification by alternative decompositions of multivariate functions this article advocates factorized and hybrid dimensional decompositions fdd hdd as alternatives to analysis of variance dimensional decomposition add for second moment statistical analysis of

upper bounds for density matrices using path integrals we discuss upper bounds for density matrices for a particle that occur in potential theory and as a result of averaging over other degrees of freedom the weiner path integral representation is used a basic technique of symanzik is generalized in a variety of ways to a series of sharper upper bounds there are distinct ways of applying the technique to the two parameter integrals that describe averaged density matrices a single parameter application of the symanzik technique leads to a temperature dependent potential problem the nature of the bounds is illustrated for two parameter integrals by studying a soluble quadratic action a one dimensional delta correlation function action and shell correlation functions the dependence on dimension is studied in the latter case

lattice methods for multiple integration theory error analysis and examples this paper is concerned with numerical integration over the unit cube in s dimensions of functions that have reasonably smooth periodic extensions in each dimension the concept of a lattice rule is introduced and error bounds are developed in terms of the dual lattice examples of lattice rules discussed in the paper are the rectangle rule the number theoretic good lattice methods of korobov and others the body centred cubic rule and a generalization of the latter to a family w nr the w nr rules appear to have interesting extrapolation properties both for fixed r and for r n

optimal polynomials for t m s nets and numerical integration of multivariate walsh series existence theorems for good parameters in a construction of low discrepancy point sets based on rational functions over finite fields are proved a special class of these point sets is introduced and good parameters in the construction of these special point sets are tabulated in this way point sets in the s dimensional unit cube for 3 leq s leq 15 are obtained with very small discrepancy these special point sets yield excellent numerical results when applied to the quasi monte carlo integration of multivariate walsh series

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

local time representation of path integrals we derive a local time path integral representation for a generic one dimensional time independent system in particular we show how to rephrase the matrix elements of the bloch density matrix as a path integral over x dependent local time profiles the latter quantify the time that the sample paths x t in the feynman path integral spend in the vicinity of an arbitrary point x generalization of the local time representation that includes arbitrary functionals of the local time is also provided we argue that the results obtained represent a powerful alternative to the traditional feynman kac formula particularly in the high and low temperature regimes to illustrate this point we apply our local time representation to analyze the asymptotic behavior of the bloch density matrix at low temperatures further salient issues such as connections with the sturm liouville theory and the rayleigh ritz variational principle are also discussed

integration and approximation of multivariate functions average case complexity with isotropic wiener measure we study the average case complexity of multivariate integration and l 2 function approximation for the class f c 0 1 d of continuous functions of d variables the class f is endowed with the isotropic wiener measure brownian motion in levy s sense furthermore for both problems only function values are used as data

the mean square discrepancy of scrambled t s sequences the discrepancy arises in the worst case error analysis for quasi monte carlo quadrature rules low discrepancy sets yield good quadrature rules this article considers the mean square discrepancies for scrambled lam t m s nets and t s sequences in base b it is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple monte carlo sampling if the reproducing kernel defining the discrepancy satisfies a lipschitz condition with respect to one of its variables separately then the asymptotic order of the root mean square discrepancy is o n 1 log n s 1 2 for scrambled nets if the reproducing kernel satisfies a lipschitz condition with respect to both of its variables then the asymptotic order of the root mean square discrepancy is o n 3 2 log n s 1 2 for scrambled nets for an arbitrary number of points taken from a t s sequence the root mean square discrepancy appears to be no better than o n 1 log n s 1 2 regardless of the smoothness of the reproducing kernel

on the numerical integration of walsh series by number theoretic methods in analogy to the theory of good lattice points for the numerical integration of rapidly converging fourier series a theory for the fast numerical integration of walsh series is developed the basis for this theory is provided by a class of very well distributed point sets in the s dimensional unit cube the so called t m s nets

asymptotic optimality of regular sequence designs we study linear estimators for the weighted integral of a stochastic process the process may only be observed on a finite sampling design the error is defined in a mean square sense and the process is assumed to satisfy sacks ylvisaker regularity conditions of order r n o we show that sampling at the quantiles of a particular density already yields asymptotically optimal estimators hereby we extend the results of sacks and ylvisaker for regularity r o or 1 and we confirm a conjecture by eubank smith and smith

variance reduction via lattice rules this is a review article on lattice methods for multiple integration over the unit hypercube with a variance reduction viewpoint it also contains some new results and ideas the aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science these models can usually be reformulated as integration problems over the unit hypercube with a large sometimes infinite number of dimensions we examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low dimensional subspaces the criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators numerical examples illustrate the effectiveness of the approach

quasi random sequences and their discrepancies quasi random also called low discrepancy sequences are a deterministic alternative to random sequences for use in monte carlo methods such as integration and particle simulations of transport processes the error in uniformity for such a sequence of n points in the s dimensional unit cube is measured by its discrepancy which is of size log n s n 1 for large n as opposed to discrepancy of size log log n 1 2 n 1 2 for a random sequence i e for almost any randomly chosen sequence several types of discrepancies one of which is new are defined and analyzed a critical discussion of the theoretical bounds on these discrepancies is presented computations of discrepancies are presented for a wide choice of dimension s number of points n and different quasi random sequences in particular for moderate or large s there is an intermediate regime in which the discrepancy of a quasi random sequence is almost exactly the same as that of a randomly chosen sequence a simplified p

a computer search of rank 2 lattice rules for multidimensional quadrature for certain lattice rules of rank 2 it has been shown that a unique representation exists in a form suitable for computer evaluation the present paper describes computer searches of such rules reports results and identifies rules that appear promising for the numerical evaluation of practical multidimensional integrals

lattice rules for multiple integration and discrepancy upper and lower bounds for the discrepancy of nodes in lattice rules for multidimensional numerical integration are established in this way the applicability of lattice rules is extended to nonperiodic integrands

regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence abstract inverse limits of nilsystems in topologically dynamical systems were characterized by host kra and maass recently namely for each d n a certain generalization of the regionally proximal relation was introduced and for a distal minimal system it was shown that such a relation is an equivalence one which determines the maximal d step nilfactor one of the main results in this article is to show that the above results hold for a general minimal system a combinatorial consequence is also deduced which is the topological correspondence of the result obtained by host and kra for positive upper banach density subsets using ergodic methods

existence of chaotic oscillations in second order linear hyperbolic pdes with implicit boundary conditions abstract this paper establishes rigorously mathematical theorems that guarantee the existence of chaotic oscillations in the systems of second order linear hyperbolic pdes it separately considers the systems with nonlinear explicit boundary conditions ebcs and nonlinear implicit boundary conditions ibcs as well as those with such ibcs subjected to small perturbations where ibcs include ebcs as special cases but the latter cannot in general be expressed by the former numerical examples are demonstrated to illustrate the effectiveness of theoretical results

lattice integration rules of maximal rank formed by copying rank 1 rules for integration of periodic functions over the s dimensional unit cube theoretical and empirical evidence suggests that certain lattice rules of maximal rank are more effective as judged by a standard test than the widely used rules of rank 1 in a rank 1 rule all the points are generated by taking multiples of a single rational vector modulo 1 in the manner suggested by korobov the rules in question are formed by taking n s copies with small values of n principally n 2 of rules of rank 1 there is also empirical evidence that by the same criterion these maximal rank rules are preferable in dimensions greater than 2 to the rank 2 rules found by sloan and walsh

chaotic dynamics of minimal center of attraction of discrete amenable group actions abstract let g be a discrete infinite amenable group which acts from the left on a compact metric space x in this paper we study the chaotic dynamics exhibited inside and near a minimal center of attraction of g x relative to any folner net in g

the asymptotic evaluation of a class of path integrals ii the asymptotic behavior of a class of wiener like path integrals functions of the local time is determined these integrals are of interest in themselves and also arise very naturally in the theory of disordered systems we show that by making use of a grassman algebra i e a set of anticommuting variables the earlier treatment of this problem can be greatly simplified in particular the previous use of replica trick which involves a difficult to justify analytic continuation in the number of field components is thus avoided

randomized polynomial lattice rules for multivariate integration and simulation lattice rules are among the best methods to estimate integrals in a large number of dimensions they are part of the quasi monte carlo set of tools a theoretical framework for a class of lattice rules defined in a space of polynomials with coefficients in a finite field is developed in this paper a randomized version is studied implementations and criteria for selecting the parameters are discussed and examples of its use as a variance reduction tool in stochastic simulation are provided certain types of digital net constructions as well as point sets constructed by taking all vectors of successive output values produced by a tausworthe random number generator are special cases of this method

error bounds for the method of good lattice points new error bounds are obtained for the method of good lattice points for multidimensional quadrature when m the number of quadrature points is prime one of these bounds reduces the constant in niederreiter s asymptotic error bound if the dimension exceeds 2 together they give very much smaller numerical bounds for all values of m