singular hyperbolic attractors are chaotic we prove that a singular hyperbolic attractor of a 3 dimensional flow is chaotic in two different strong senses first the flow is expansive if two points remain close at all times possibly with time reparametrization then their orbits coincide second there exists a physical or sinai ruelle bowen measure supported on the attractor whose ergodic basin covers a full lebesgue volume measure subset of the topological basin of attraction moreover this measure has absolutely continuous conditional measures along the center unstable direction is a u gibbs state and is an equilibrium state for the logarithm of the jacobian of the time one map of the flow along the strong unstable direction this extends to the class of singular hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic or axiom a attractors for flows in particular these results can be applied i to the flow defined by the lorenz equations ii to the geometric lorenz flows iii to the attractors appearing in the unfolding of certain resonant double homoclinic loops iv in the unfolding of certain singular cycles and v in some geometrical models which are singular hyperbolic but of a different topological type from the geometric lorenz models in all these cases the results show that these attractors are expansive and have physical measures which are u gibbs states

marcus versus stratonovich for systems with jump noise the famous ito stratonovich dilemma arises when one examines a dynamical system with a multiplicative white noise in physics literature this dilemma is often resolved in favour of the stratonovich prescription because of its two characteristic properties valid for systems driven by brownian motion i it allows physicists to treat stochastic integrals in the same way as conventional integrals and ii it appears naturally as a result of a small correlation time limit procedure on the other hand the marcus prescription ieee trans inform theory 24 164 1978 stochastics 4 223 1981 should be used to retain i and ii for systems driven by a poisson process levy flights or more general jump processes in present communication we present an in depth comparison of the ito stratonovich and marcus equations for systems with multiplicative jump noise by the examples of a real valued linear system and a complex oscillator with noisy frequency the kubo anderson oscillator we compare solutions obtained with the three prescriptions

statistical properties for flows with unbounded roof function including the lorenz attractor for geometric lorenz attractors including the classical lorenz attractor we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors we also obtain the functional central limit theorem and moment estimates as well as iterated versions of these results a consequence is deterministic homogenisation convergence to a stochastic differential equation for fast slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two

from rates of mixing to recurrence times via large deviations abstract a classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties for example the existence of invariant measures with stochastic like behaviour such as large deviations or decay of correlations such geometric structures are generally highly non trivial and thus a natural question is the extent to which this approach can be applied in this paper we show that in many cases stochastic like behaviour itself implies that the system has certain non trivial geometric properties which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration as a by product of our techniques we also obtain some new results on large deviations for certain classes of systems which include viana maps and multidimensional piecewise expanding maps

general rough integration levy rough paths and a levy kintchine type formula we consider rough paths with jumps in particular the analogue of lyons extension theorem and rough integration are established in a jump setting offering a pathwise view on stochastic integration against cadlag processes a class of levy rough paths is introduced and characterized by a sub ellipticity condition on the left invariant diffusion vector fields and a certain integrability property of the carnot caratheodory norm with respect to the levy measure on the group using hunt s framework of lie group valued levy processes examples of levy rough paths include a standard multi dimensional levy process enhanced with a stochastic area as constructed by d williams the pure area poisson process and brownian motion in a magnetic field an explicit formula for the expected signature is given

equidistribution for nonuniformly expanding dynamical systems and application to the almost sure invariance principle let t m mt m m be a nonuniformly expanding dynamical system such as logistic or intermittent map let v m rdv m rd be an observable and vn n 1k 0v tkvn k 0n 1v tk denote the birkhoff sums given a probability measure on m we consider vn as a discrete time random process on the probability space m m in smooth ergodic theory there are various natural choices of such as the lebesgue measure or the absolutely continuous t invariant measure they give rise to different random processes we investigate relation between such processes we show that in a large class of measures it is possible to couple redefine on a new probability space every two processes so that they are almost surely close to each other with explicit estimates of closeness the purpose of this work is to close a gap in the proof of the almost sure invariance principle for nonuniformly hyperbolic transformations by melbourne and nicol

the strong p variation of martingales and orthogonal series let 1 p and letx x n n 0 be a sequence of scalars the strongp variation ofx denoted byw p x is defined as w p x sup left left x 0 p sum limits k 1 infty x n k x n k 1 p right 1 p right where the supremum runs over all increasing sequences of integers 0 n 0 n 1 n 2 let 1 p 2 and letm m n n 0 be a martingale inl p our main results are as follows if sigma mathbb e m n m n 1 p infty thenw p m is finite a s and we have mathbb e w p m p leqq c mathbb e m 0 p sum limits n geqq 1 mathbb e m n m n 1 p for some constantc depending only onp on the other hand let n be an arbitrary orthonormal system of functions inl 2 considerx x n n 0 inl 2 and lets n 0 x i i ands s n n 0 we prove that if x n p 1 p 2 thenw p s t for a e t and w p s 2 c x n p 1 p for some constantc each of these results is an extension of a result proved by bretagnolle for sums of independent mean zero r v s the casep 2 in also discussed our proofs use the real interpolation method of lions peetre they admit extensions in the banach space valued case provided suitable assumptions are imposed on the banach space

excessive gap technique in nonsmooth convex minimization in this paper we introduce a new primal dual technique for convergence analysis of gradient schemes for nonsmooth convex optimization as an example of its application we derive a primal dual gradient method for a special class of structured nonsmooth optimization problems which ensures a rate of convergence of order o 1 k where k is the iteration count another example is a gradient scheme which minimizes a nonsmooth strongly convex function with known structure with rate of convergence o 1 k2 in both cases the efficiency of the methods is higher than the corresponding black box lower complexity bounds by an order of magnitude

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

deterministic homogenization for fast slow systems with chaotic noise consider a fast slow system of ordinary differential equations of the form x a x y e 1b x y y e 2g y where it is assumed that b averages to zero under the fast flow generated by g we give conditions under which solutions x to the slow equations converge weakly to an it o diffusion x as e 0 the drift and diffusion coefficients of the limiting stochastic differential equation satisfied by x are given explicitly our theory applies when the fast flow is anosov or axiom a as well as to a large class of nonuniformly hyperbolic fast flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on the fast flow

large deviations for nonuniformly hyperbolic systems we obtain large deviation estimates for a large class of nonuniformly hyperbolic systems namely those modelled by young towers with summable decay of correlations in the case of exponential decay of correlations we obtain exponential large deviation estimates given by a rate function in the case of polynomial decay of correlations we obtain polynomial large deviation estimates and exhibit examples where these estimates are essentially optimal in contrast with many treatments of large deviations our methods do not rely on thermodynamic formalism hence for holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton as well as polynomial estimates in situations where there is not a unique equilibrium measure

the 3 d unconditionally stable associated hermite finite difference time domain method this article discussed the implementation of 3 d associated hermite ah finite difference time domain fdtd method and its application in the analysis of lumped parameters modeling based on the paralleling in order scheme the basic formula of the 3 d ah fdtd method incorporated with complex frequency shifted cfs perfectly matched layer pml absorption boundary was derived then by using the kronecker product scheme the coefficient matrix of the final matrix equation was fast generated and by using the permutation technology its bandwidth was compressed updating the coefficient matrix and source term of the matrix equation was implemented when dealing with the lumped parameters to solve the final matrix to obtain the ah expanding coefficients the direct or indirect methods mainly by the iterative method could be used finally several numerical results were presented to validate the accuracy and efficiency of the proposed method

smoothing techniques for computing nash equilibria of sequential games we develop first order smoothing techniques for saddle point problems that arise in finding a nash equilibrium of sequential games the crux of our work is a construction of suitable prox functions for a certain class of polytopes that encode the sequential nature of the game we also introduce heuristics that significantly speed up the algorithm and decomposed game representations that reduce the memory requirements enabling the application of the techniques to drastically larger games an implementation based on our smoothing techniques computes approximate nash equilibria for games that are more than four orders of magnitude larger than what prior approaches can handle finally we show near linear further speedups from parallelization

an explicit and unconditionally stable fdtd method for the analysis of general 3 d lossy problems the root cause of the instability of an explicit finite difference time domain fdtd method is quantitatively identified for the analysis of general lossy problems where both dielectrics and conductors can be lossy and inhomogeneous based on the root cause analysis an efficient algorithm is developed to eradicate the root cause of instability and subsequently achieve unconditional stability in an explicit fdtd based simulation of general lossy problems numerical experiments have demonstrated the unconditional stability accuracy and efficiency of the proposed method

an improved paralleling in order solving scheme for ah fdtd method using eigenvalue transformation this paper describes a modification of the unconditionally stable finite difference time domain fdtd algorithm based on the associated hermite ah functions the original method is involved with solving a banded nested matrix equation to calculate the ah expanding coefficients using lower upper lu decomposition which is the main cost for cpu time and memory storage here the calculation of expansion coefficients is modified as a paralleling in order solving scheme by using the eigenvalue transformation which greatly enhances the computational efficiency and reduces the memory storage the numerical results demonstrate the performance of this improvement

superdiffusive limits for deterministic fast slow dynamical systems we consider deterministic fast slow dynamical systems on mathbb r m times y of the form begin aligned left begin array ll x k 1 n x k n n 1 a big x k n big n 1 alpha b big x k n big v y k y k 1 f y k end array right end aligned where alpha in 1 2 under certain assumptions we prove convergence of the m dimensional process x n t x lfloor nt rfloor n to the solution of the stochastic differential equation begin aligned mathrm d x a x mathrm d t b x diamond mathrm d l alpha end aligned where l alpha is an alpha stable levy process and diamond indicates that the stochastic integral is in the marcus sense in addition we show that our assumptions are satisfied for intermittent maps f of pomeau manneville type

central limit theorem and stable laws for intermittent maps in the setting of abstract markov maps we prove results concerning the convergence of renormalized birkhoff sums to normal laws or stable laws they apply to one dimensional maps with a neutral fixed point at 0 of the form x x1 for 0 1 in particular for 1 2 we show that the birkhoff sums of a holder observable f converge to a normal law or a stable law depending on whether f 0 0 or f 0 0 the proof uses spectral techniques introduced by sarig and wiener s lemma in non commutative banach algebras

differential equations driven by rough paths with jumps abstract we develop the rough path counterpart of ito stochastic integration and differential equations driven by general semimartingales this significantly enlarges the classes of ito forward stochastic differential equations treatable with pathwise methods a number of applications are discussed

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

primal dual first order methods with mathcal o 1 epsilon iteration complexity for cone programming in this paper we consider the general cone programming problem and propose primal dual convex smooth and or nonsmooth minimization reformulations for it we then discuss first order methods suitable for solving these reformulations namely nesterov s optimal method nesterov in doklady an sssr 269 543 547 1983 math program 103 127 152 2005 nesterov s smooth approximation scheme nesterov in math program 103 127 152 2005 and nemirovski s prox method nemirovski in siam j opt 15 229 251 2005 and propose a variant of nesterov s optimal method which has outperformed the latter one in our computational experiments we also derive iteration complexity bounds for these first order methods applied to the proposed primal dual reformulations of the cone programming problem the performance of these methods is then compared using a set of randomly generated linear programming and semidefinite programming instances we also compare the approach based on the variant of nesterov s optimal method with the low rank method proposed by burer and monteiro math program ser b 95 329 357 2003 math program 103 427 444 2005 for solving a set of randomly generated sdp instances

efficient implementation for the ah fdtd method with iterative procedure and cfs pml an efficient implementation for the associated hermite finite difference time domain ah fdtd method with iterative procedure is proposed which is an extension of previous works the original and the paralleling in order based ah fdtd methods by introducing the alternating direction implicit method the direct calculation of five point matrix equation is replaced with iterative solving of tridiagonal matrices equations for the 2 d case only two tridiagonal matrixes need to be calculated in a full iterative cycle in addition the complex frequency shifted cfs perfectly matched layer pml is extended to the ah fdtd method and implemented together with this efficient solution scheme numerical examples show the superior computational performance of the proposed iterative procedure by comparison with the conventional fdtd method the adi fdtd method and previous ah fdtd methods the effectiveness of cfs pml in ah fdtd is also verified

explicit coupling argument for non uniformly hyperbolic transformations the transfer operator corresponding to a uniformly expanding map enjoys good spectral properties we verify that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map for non uniformly expanding maps with a uniformly expanding induced map we obtain explicit estimates for mixing rates exponential stretched exponential polynomial that again depend continuously on the constants for the induced map together with data associated with the inducing time finally for non uniformly hyperbolic transformations we obtain the corresponding estimates for rates of decay of correlations

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

advanced statistical properties of dispersing billiards a new approach to statistical properties of hyperbolic dynamical systems emerged recently it was introduced by l s young and modified by d dolgopyat it is based on coupling method borrowed from probability theory we apply it here to one of the most physically interesting models sinai billiards it allows us to derive a series of new results as well as make significant improvements in the existing results first we establish sharp bounds on correlations including multiple correlations then we use our correlation bounds to obtain the central limit theorem clt the almost sure invariance principle asip the law of iterated logarithms and integral tests

the weighted majority algorithm we study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials with a prediction to be made in each and the goal of the learner is to make few mistakes we are interested in the case where the learner has reason to believe that one of some pool of known algorithms will perform well but the learner does not know which one a simple and effective method based on weighted voting is introduced for constructing a compound algorithm in such a circumstance we call this method the weighted majority algorithm we show that this algorithm is robust in the presence of errors in the data we discuss various versions of the weighted majority algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool for example given a sequence of trials if there is an algorithm in the pool a that makes at most m mistakes then the weighted majority algorithm will make at most c log a m mistakes on that sequence where c is fixed constant

limit theorems in the stadium billiard we prove that the birkhoff sums for almost every relevant observable in the stadium billiard obey a non standard limit law more precisely the usual central limit theorem holds for an observable if and only if its integral along a one codimensional invariant set vanishes otherwise a open image in new window normalization is needed as one of the two key steps in the argument we obtain a limit theorem that holds in young towers with exponential return time statistics in general an abstract result that seems to be applicable to many other situations

almost sure invariance principle for nonuniformly hyperbolic systems we prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems discrete time systems and flows are covered by this result in particular the result applies to the planar periodic lorentz flow with finite horizon statistical limit laws such as the central limit theorem the law of the iterated logarithm and their functional versions are immediate consequences

subexponential decay of correlations we describe a method for proving subexponential lower bounds for correlations functions and apply it to study decay of correlations for maps with countable markov partitions one result is that ls young s upper estimates y2 are optimal in many situations our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on banach spaces which we apply to the iterates of the transfer operator

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation abstract over the last 10 years or so advanced statistical properties including exponential decay of correlations have been established for certain classes of singular hyperbolic flows in three dimensions the results apply in particular to the classical lorenz attractor however many of the proofs rely heavily on the smoothness of the stable foliation for the flow in this paper we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the stable foliation these properties include existence of srb measures central limit theorems and associated invariance principles as well as results on mixing and rates of mixing the properties hold equally for singular hyperbolic flows in higher dimensions provided the center unstable subspaces are two dimensional

an unconditionally stable one step arbitrary order leapfrog adi fdtd method and its numerical properties an one step arbitrary order leapfrog alternating direction implicit finite difference time domain adi fdtd method is presented its unconditional stability is analytically proven and numerically verified its numerical properties are shown to be identical to those of the conventional adi fdtd and lod fdtd methods however its computational expenditure is found to be the lowest in other words in comparison with the adi fdtd and lod fdtd methods the one step arbitrary order leap frog adi fdtd method retains identical numerical modeling accuracy but with higher computational efficiency

homogenization for deterministic maps and multiplicative noise a recent paper of melbourne stuart a note on diffusion limits of chaotic skew product flows nonlinearity 24 2011 1361 1367 gives a rigorous proof of convergence of a fast slow deterministic system to a stochastic differential equation with additive noise in contrast to other approaches the assumptions on the fast flow are very mild in this paper we extend this result from continuous time to discrete time moreover we show how to deal with one dimensional multiplicative noise this raises the issue of how to interpret certain stochastic integrals it is proved that the integrals are of stratonovich type for continuous time and neither

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

a new unconditionally stable scheme for fdtd method using associated hermite orthogonal functions an unconditionally stable solution using associated hermite ah functions is proposed for the finite difference time domain fdtd method the electromagnetic fields and their time derivatives in time do main maxwell s equations are expanded by these orthonormal basis func tions by applying galerkin temporal testing procedure to these expanded equations the time variable can be eliminated from the calculations a set of implicit equations is derived to calculate the magnetic filed expansion coefficients of all orders of ah functions for the temporal variable and the electrical field coefficients can be obtained respectively with the ap propriate translation and scale parameters we can find a minimum order basis functions subspace to approach a particular electromagnetic field the numerical results have shown that the proposed method can reduce the cpu time to 0 59 of the traditional fdtd method while maintaining good accuracy

smooth approximation of stochastic differential equations consider an ito process x satisfying the stochastic differential equation dx a x dt b x dw where a b are smooth and w is a multidimensional brownian motion suppose that wn has smooth sample paths and that wn converges weakly to w a central question in stochastic analysis is to understand the limiting behavior of solutions xn to the ordinary differential equation dxn a xn dt b xn dwn the classical wong zakai theorem gives sufficient conditions under which xn converges weakly to x provided that the stochastic integral b x dw is given the stratonovich interpretation the sufficient conditions are automatic in one dimension but in higher dimensions the correct interpretation of b x dw depends sensitively on how the smooth approximation wnis chosen in applications a natural class of smooth approximations arise by setting wn t n 1 2 nt0v sds where t is a flow generated e g by an ordinary differential equation and v is a mean zero observable under mild conditions on t we give a definitive answer to the interpretation question for the stochastic integral b x dw our theory applies to anosov or axiom a flows t as well as to a large class of nonuniformly hyperbolic flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on t the methods used in this paper are a combination of rough path theory and smooth ergodic theory

rate of convergence in the weak invariance principle for deterministic systems we obtain the first results on convergence rates in the prokhorov metric for the weak invariance principle functional central limit theorem for deterministic dynamical systems our results hold for uniformly expanding hyperbolic axiom a systems as well as nonuniformly expanding hyperbolic systems such as dispersing billiards henon like attractors viana maps and intermittent maps as an application we obtain convergence rates for deterministic homogenization in multiscale systems

robust transitive singular sets for 3 flows are partially hyperbolic attractors or repellers inspired by lorenz remarkable chaotic flow we describe in this paper the structure of all c 1 robust transitive sets with singularities for flows on closed 3 manifolds they are partially hyperbolic with volume expanding central direction and are either attractors or repellers in particular any c 1 robust attractor with singularities for flows on closed 3 manifolds always has an invariant foliation whose leaves are forward contracted by the flow and has positive lyapunov exponent at every orbit showing that any c 1 robust attractor resembles a geometric lorenz attractor

rates in almost sure invariance principle for dynamical systems with some hyperbolicity we prove the almost sure invariance principle with rate o n varepsilon for every varepsilon 0 for holder continuous observables on nonuniformly expanding and nonuniformly hyperbolic transformations with exponential tails examples include gibbs markov maps with big images axiom a diffeomorphisms dispersing billiards and a class of logistic and henon maps the best previously proved rate is o n 1 4 log n 1 2 log log n 1 4 as a part of our method we show that nonuniformly expanding transformations are factors of markov shifts with simple structure and natural metric similar to the classical young towers the factor map is lipschitz continuous and probability measure preserving for this we do not require the exponential tails

random walks and levy processes as rough paths we consider random walks and levy processes in a homogeneous group g for all p 0 we completely characterise almost all g valued levy processes whose sample paths have finite p variation and give sufficient conditions under which a sequence of g valued random walks converges in law to a levy process in p variation topology in the case that g is the free nilpotent lie group over mathbb r d so that processes of finite p variation are identified with rough paths we demonstrate applications of our results to weak convergence of stochastic flows and provide a levy khintchine formula for the characteristic function of the signature of a levy process at the heart of our analysis is a criterion for tightness of p variation for a collection of cadlag strong markov processes