on the inverse problem of statistical physics from irreversible semigroups to chaotic dynamics abstract we show that all measure preserving stationary markov processes arise as projections of kolmogorov dynamical systems which have positive entropy production and are prototypes of chaos this result not only contributes to the clarification of the relation of dynamics with stochastic processes but also shows the physical significance of the misra prigogine courbage theory of irreversibility in the more general context of the inverse problem of statistical physics because we want positivity preserving transformations our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of positive dilations

an explicit approach to the λ operator and the h theorem in the prigogine theory of irreversibility abstract a reformulation of the prigogine theory of irreversible dynamical systems is given in contrast to the previous works a stochastic integral with respect operator valued martingales is employed here this approach provides a mathematically rigorous and simple construction of the operator for dynamical flows it has an explicit random character we derive also the boltzmann h theorem explicitly from the properties of the semigroup only finally the form of the boltzmann kinetic equation for the irreversible dynamics is found

on k flows and irreversibility here we give the proof of a general theorem concerning irreversibility that was stated earlier by misra and prigogine in terms of a unitary group describing a deterministic dynamical evolution and a related markov semigroup describing an associated coarse grained probabilistic evolution it is shown that the original dynamics are necessarily those of ak flow thus a reversible dynamics which permits such an intertwining to an irreversible description must possess a high degree of instability

time vs ensemble averages for nonstationary time series abstract we analyze whether sliding window time averages applied to stationary increment processes converge to a limit in probability the question centers on averages correlations and densities constructed via time averages of the increment x t t x t t x t e g x t t ln p t t p t in finance and economics where p t is a price and the assumption is that the increment is distributed independently of t we apply tchebychev s theorem to the construction of statistical ensembles and then show that the convergence in probability condition is not satisfied when applied to time averages of functions of stationary increments we further show that tchebychev s theorem provides the basis for constructing approximate ensemble averages and densities from a single historic time series where as in fx markets the series shows a definite statistical periodicity the convergence condition is not satisfied strongly enough for densities and certain averages but is well satisfied by specific averages of direct interest rates of convergence cannot be established independently of specific models however our analysis shows how to decide which empirical averages to avoid and which ones to construct

rigged hilbert spaces associated with misra prigogine courbage theory of irreversibility abstract it is proved that in the misra prigogine courbage theory of irreversibility using the internal time superoperator fixing its associated non unitary transformation amounts to rigging the corresponding hilbert liouville space more precisely it is demonstrated that any determines three canonical riggings of the liouville space l first one with a hilbert space with a norm greater than the relative one from l a second one with a hilbertian space which is a kothe space if is compact and is a nuclear space if has certain nuclear properties and finally a third one with a smaller hilbertian space with a still stronger topology which is nuclear if n is hilbert schmidt for some positive integer n in contrast any rigging of this type originated in a dynamical system having an internal time superoperator defines a in a canonical way

rates of convergence to equilibrium in the prigogine misra courbage theory of irreversibility the rates of convergence to equilibrium in the prigogine misra courbage theory of irreversibility as developed by goldstein misra and courbage are examined it is found that arbitrarily slow convergence to equilibrium should be present in fact in a certain precise sense it should be the most abundant behavior this is compared with the common beliefs in kinetic theory

from irreversible markov semigroups to chaotic dynamics abstract we answer qualitatively the inverse coarse graining problem of statistical physics namely which microscopic dynamics give rise to a given physically observed markov semigroup as a result of exact coarse graining we prove the fact that all markov chains arise as projections of dynamical systems in larger spaces and show that in particular the irreversible markov chains arise as projections of chaotic systems of kolmogorov type this result generalizes our previous results on the misra prigogine courbage semigroups because we want positivity preserving transformations our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of positive dilations

from probabilistic descriptions to deterministic dynamics abstract we answer the question of dilating a markov semigroup to a deterministic evolution in a larger space we show that certain markov processes may arise as projections of kolmogorov dynamical systems this result shows in particular the physical significance of the misra prigogine courbage theory of irreversibility because we want positivity preserving transformations in both case our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of the natural extension of exact systems

on the approach to statistical equilibrium in an infinite particle lattice dynamic model a deterministic reversible model dynamic system of infinite interacting particles on a lattice is studied the model is a modification by de haan of the kac ring model the time evolution of the statistical states can be described by a hierarchy for the reduced distribution functions it is shown that the system is isomorphic to a bernoulli shift the isomorphism is given in terms of the infinite process of observation of the states of one particle in a fixed site the approach to equilibrium is studied and time asymmetric initial states that approach equilibrium are constructed it is shown that these states contain nonvanishing correlations between infinitely distant particles the bernoulli property allows the construction of a markovian irreversible description of the model realized by a nonunitary transformation the action of this transformation on the statistical states the correlations and the rate of the approach to equilibrium is studied

an equivalent markov model for burst errors in digital channels a hidden markov model for burst errors is specified by a probability transition matrix p an initial probability vector p and the state dependent probability of error matrix b several procedures are available for estimating p p and b from a given error observation sequence however even with some restrictions on the structure of the underlying markov models the estimation procedures are computationally intensive particularly when the observation sequence contains long strings of identical symbols we show that under some mild assumptions a markov model with an arbitrary transition matrix p is equivalent to a markov model with a unique block diagonal transition matrix spl lambda we also present a computationally very efficient algorithm for estimating spl lambda from a set of observation using a modified baum welch 1972 algorithm

scaling in currency exchange we study the scaling behavior in currency exchange rates our results suggest that they satisfy scaling with an exponent close to 0 5 but that it differs qualitatively from that of a simple random walk indeed price variations cannot be considered as independent variables and subtle correlations are present furthermore we introduce a novel statistical analysis for economic data which makes the physical properties of a signal more evident and eliminates the systematic effects of time periodicity

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

almost zeros of a set of polynomials of r s the notion of an exact zero of a sot of polynomials p of r s is extended to that of an almost zero the almost zeros are defined as minima of a norm function of p and their invariance under a certain equivalence relation defined on p is established the region of the complex plane which contains the prime almost zero global minimum is defined and necessary and sufficient conditions characterizing the set of almost zeros are given the role of almost zeros of p in the zero assignment problem of the combinants of p is investigated under the assumption of non assignability for p it is shown that the almost zeros act as nearly fixed zeros of the combinants every almost zero z is the centre of a disc d z r which contains at least one zero of all combinants of p the size of the radius r of this disc provides a measure of the ability of the almost zero to attract the zeros of the combinants of p a number of results for estimating upper bounds for the radius r associated with subfamilies of the ge

matrix pencil methodologies for computing the greatest common divisor of polynomials hybrid algorithms and their performance the computation of the greatest common divisor gcd of several polynomials is a problem that emerges in many fields of applications the gcd computation has a non generic nature and thus its numerical computation is a hard problem in this paper we examine the family of matrix pencil methods for gcd computation and investigate their performance as far as their complexity error analysis and their effectiveness for evaluating approximate solutions the relative merits of the various variants of such methods are examined for the different cases of sets of polynomials with varying number of elements and degree the developed algorithms combine symbolical and numerical programming and this is what we define here as hybrid computations the combination of numerical operations with symbolical programming can improve the nature of the methods and guarantees the stability of the algorithm furthermore it emphasizes the significance of hybrid computations in complex problems such as the computation of gcd all m

pole assignment of strictly proper and proper linear systems by constant output feedback a new unifying approach for the study of pole assignment by constant output feedback cpap for strictly proper and proper linear systems is presented the multilinear nature of cpap is reduced to a linear problem and to the standard multilinear problem of decomposability of multivectors the solvability of cpap thus becomes a problem of finding real intersections of a linear variety with the grassmann variety of a projective space an alternative proof to the ml n m i n are the numbers of outputs inputs states necessary and sufficient condition for generic pole assignability of strictly proper systems by complex output feedback is given the above result is extended to the ml n 1 condition for the case of proper systems stronger necessary conditions for generic pole assignment by a real output feedback are obtained by using the recently introduced invariant the plucker matrix pn such conditions are that pn must have full rank and ml n strictly proper case or ml n 1 proper case it is sh

measure preserving markov processes densely similar to endomorphisms abstract we characterize the contractions densely similar to the adjoint of some isometry we deduce a sufficient condition in order to have the evolution of a measure preserving markov process densely similar to the evolution of an endomorphism on the set of l2 states this study is in the scope of the reciprocal of the misra prigogine construction in k systems

the pmc processes associated to the 1 2 1 2 bernoulli shift are also bernoulli it is proven that the prigogine misra courbage pmc processes associated to the 1 2 1 2 bernoulli shift are bernoulli shifts the bernoulli partitions are constructed explicitly by using the decomposition of the transition kernels of the pmc processes on the fibers of the stable manifold of the transformed point the generating property of these partitions is proved for any two symbol bernoulli shifts

stochastic quasi newton molecular simulations we report a new and efficient factorized algorithm for the determination of the adaptive compound mobility matrix b in a stochastic quasi newton method s qn that does not require additional potential evaluations for one dimensional and two dimensional test systems we previously showed that s qn gives rise to efficient configurational space sampling with good thermodynamic consistency c d chau g j a sevink and j g e m fraaije j chem phys 128 244110 2008 potential applications of s qn are quite ambitious and include structure optimization analysis of correlations and automated extraction of cooperative modes however the potential can only be fully exploited if the computational and memory requirements of the original algorithm are significantly reduced in this paper we consider a factorized mobility matrix b jj t and focus on the nontrivial fundamentals of an efficient algorithm for updating the noise multiplier j the new algorithm requires o n 2 multiplications per time step instead of the o n 3 multiplications in the original scheme due to choleski decomposition in a recursive form the update scheme circumvents matrix storage and enables limited memory implementation in the spirit of the well known limited memory broyden fletcher goldfarb shanno l bfgs method allowing for a further reduction of the computational effort to o n we analyze in detail the performance of the factorized fsu and limited memory l fsu algorithms in terms of convergence and multiscale sampling for an elementary but relevant system that involves multiple time and length scales finally we use this analysis to formulate conditions for the simulation of the complex high dimensional potential energy landscapes of interest

non unitary transformation of conservative to dissipative evolutions the authors briefly review in a general context the theory of non unitary transformations between conservative dynamics with internal time and dissipative evolutions developed previously in the context of statistical mechanics and apply the results to show the existence of a similarity transformation between the wave equation and the dissipative telegraphist equation

on the estimation of the order of a markov chain and universal data compression the authors estimate the order of a finite markov source based on empirically observed statistics the performance criterion adopted is to minimize the probability of underestimating the model order while keeping the overestimation probability exponent at a prescribed level a universal asymptotically optimal test in the sense just defined is proposed for the case where a given integer is known to be the upper bound of the true order for the case where such a bound is unavailable an alternative rule based on the lempel ziv data compression algorithm is shown to be asymptotically optimal also and computationally more efficient

characterizations of intrinsically random dynamical systems abstract we show that intrinsically random dynamical systems with the prigogine operator of the form of a random laplace transform can be characterized as kolmogorov flows k flows we also obtain a spectral characterization in the language of the weyl commutation relation as a consequence we conclude that the dynamical system is intrinsically random if and only if its liouvillian and time operators form a schrodinger couple

intrinsical randomness of kolmogorov ℤd actions on a lebesgue space we introduce a concept of an intrinsically weak and strong randomness of a d action on a lebesgue space and we show that kolmogorov d actions are intrinsically weak random and bernoulli d actions are intrinsically strong random

estimating the number of states of a finite state source the problem of estimating the number of states of a finite alphabet finite state source is investigated an estimator is developed that asymptotically attains the minimum probability of understanding the number of states among all estimators with a prescribed exponential decay rate of overestimation probability the proposed estimator relies on the lempel ziv data compression algorithm in an intuitively appealing manner

grassmann invariants almost zeros and the determinantal zero pole assignment problems of linear multivariable systems the strucxpects of the rationxr spaces x c x fwhere x c is the columxof the transfer function matrix g s and x f is the space associated with right matrix fracxcriptions of g s are investigated for x f gcanonixqb s grassmgesentax g x c and g x f arxd and are shown xmplete basis free invariants for x c and x f respectively the almost zeros az and almost decoupling zeros adz of g s gdefinex local minima of a normgn defix g x c and g x f respectively the computation and certain aspects of the distribution in the complex plane of azs and adzs are examined the role of azs and adzs in the determinantal zero and pole assignment problems respectively is examined next two important families of systems are defined the strongly zero non assignable szna and the strongly pole non assignable spna systems for szna and spna systems minimal radius discs dem e z r em e z and dem f zb centred at an az and adz respectively are defined it is shown that dem e r em e z contains at least one zer

a nonequilibrium entropy for dynamical systems it is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure these are chosen by considering the totality of future possibilities for observation on the system it is shown that this entropy is necessarily a nondecreasing function of the timet there is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal under suitable conditions which are stated explicitly this entropy approaches the equilibrium entropy ast and the fine grained entropy ast in particular the conditions can always be satisfied if the system is ak system as in the sinai billiard models some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition and at the same time elucidate the connection between our concept of entropy and the entropy invariant of kolmogorov and sinai

stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations abstract we discuss two fundamental aspects of fokker planck equations that are nonlinear with respect to probability densities first we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions second we demonstrate that nonlinear fokker planck equations can be interpreted as linear fokker planck equations describing nonlinear families of markov diffusion processes we exploit this finding in order to derive complete hierarchies of probability densities from nonlinear fokker planck equations

matrix representation of the shifting operation and numerical properties of the eres method for computing the greatest common divisor of sets of many polynomials the extended row equivalence and shifting eres method is a matrix based method developed for the computation of the greatest common divisor gcd of sets of many polynomials in this paper we present the formulation of the shifting operation as a matrix product which allows us to study the fundamental theoretical and numerical properties of the eres method by introducing its complete algebraic representation then we analyse in depth its overall numerical stability in finite precision arithmetic numerical examples and comparison with other methods are also presented

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

chapman kolmogorov equation for non markovian shift invariant measures we study the class c of probability measures invariant with respect to the shift transformation on k z where k is a finite set of integers which satisfies the chapman kolmogorov equation for a given stochastic matrix we construct a dense subset of measures in c distinct from the markov measure when is irreduable and aperiodic these measures are ergodic but not weary mixing we show that the set of measures with infinite memory is g dense in c and that the markov measure is the unique measure which maximizes the kolmogorov sinai k s entropy in c we give examples of ergodic measures in c with zero entropy

martingales nonstationary increments and the efficient market hypothesis abstract we discuss the deep connection between nonstationary increments martingales and the efficient market hypothesis for stochastic processes x t with arbitrary diffusion coefficients d x t we explain why a test for a martingale is generally a test for uncorrelated increments we explain why martingales look markovian at the level of both simple averages and 2 point correlations but while a markovian market has no memory to exploit and cannot be beaten systematically a martingale admits memory that might be exploitable in higher order correlations we also use the analysis of this paper to correct a misstatement of the fair game condition in terms of serial correlations in fama s paper on the emh we emphasize that the use of the log increment as a variable in data analysis generates spurious fat tails and spurious hurst exponents

martingales detrending data and the efficient market hypothesis we discuss martingales detrending data and the efficient market hypothesis for stochastic processes x t with arbitrary diffusion coefficients d x t beginning with x independent drift coefficients r t we show that martingale stochastic processes generate uncorrelated generally nonstationary increments generally a test for a martingale is therefore a test for uncorrelated increments a detrended process with an x dependent drift coefficient is generally not a martingale and so we extend our analysis to include the class of x t dependent drift coefficients of interest in finance we explain why martingales look markovian at the level of both simple averages and 2 point correlations and while a markovian market has no memory to exploit and presumably cannot be beaten systematically it has never been shown that martingale memory cannot be exploited in 3 point or higher correlations to beat the market we generalize our markov scaling solutions presented earlier and also generalize the martingale formulation of the efficient market hypothesis emh to include x t dependent drift in log returns we also use the analysis of this paper to correct a misstatement of the fair game condition in terms of serial correlations in fama s paper on the emh we end with a discussion of levy scharacterization of brownian motion and prove that an arbitrary martingale is topologically inequivalent to a wiener process

nonunitary similarity transformation of conservative to dissipative evolutions intertwining without time operator reversible evolutions are usually expressed in terms of unitary groups on separable hilbert spaces whereas irreversible ones are described by contraction semigroups in the theory of nonunitary similarity transformations intertwining unitary groups and contraction semigroups proposed initially in the context of statistical mechanics as part of an exact theory of irreversibility the unitary groups with such intertwining property have been qualified by the existence of an internal time operator this work tackles the question of existence of internal time operators for unitary groups with the intertwining property equivalent conditions to the existence of internal time operators for such unitary groups are given on the basis of the sz nagy foias harmonic analysis of operators on hilbert spaces north holland amsterdam 1970 dilation theory and the theory of shift invariant subspaces these conditions permit us to solve the inverse intertwining problem in the negative there are unitary groups with

generalised resultants dynamic polynomial combinants and the minimal design problem the theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems dap which provides the unifying description of the dynamic as well as static pole and zero dynamic assignment problems in linear systems the assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original dap this paper demonstrates the origin of dynamic polynomial combinants from linear systems examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree and examines their spectral assignment central to this study is the link of dynamic combinants to the theory of generalised resultants which provide the matrix representation of the dynamic combinants the paper considers the case of coprime set of polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above an

chaotic dynamics markovian coarse graining and information abstract the conditions for mapping chaotic dynamics into a markov chain through finite coarse graining are derived a chapman kolmmogorov equation giving rise to an h theorem is deduced in which the transition probability matrix is expressed entirely in terms of the properties of the underlying deterministic dynamics and of the phase space partition the validity of the conditions ensuring the mapping is verified on simple models of dissipative as well as conservative dynamical systems the implications of the results in the problem of irreversibility and the relation between deterministic chaos information and complexity are discussed

computation of the gcd of polynomials using gaussian transformations and shifting abstract a new numerical method for the computation of the greatest common divisor gcd of an m set of polynomials of r s pm d of maximal degree d is presented this method is based on a recently developed theoretical algorithm karcanias 1987 that uses elementary transformations and shifting operations the present algorithm takes into account the non generic nature of gcd and thus uses steps which minimize the introduction of additional errors and defines the gcd in an approximate sense for a given set pm d with a basis matrix pm the method defines first the most orthogonal uncorrupted base pr from the rows of pm where r rank pm m by applying successively gaussian transformations and shifting on the basis matrix pr e rr d 1 we produce each time a new basis matrix pz with z rank pz r the method terminates when the rank of pz is approximately equal to 1 the coefficient vector of the gcd is then defined as a row of the unit rank matrix pz the method defines the exact degree of t

on lambda and internal time operators abstract an extended construction of transformations leading from conservative to dissipative evolutions is presented and new sets of admissible states are determined then is presented a probabilistic approach to the concept of internal time operator in this probabilistic context it is shown that is the expected value of the new time operator the scaling function of its tail distribution and that the choice of a profile function reduces to at most three possibilities

reversible dynamics and the macroscopic rate law for a solvable kolmogorov system the three bakers reaction we investigate a piecewise linear area preserving mapt describing two coupled baker transformations on two squares with coupling parameter 0 c 1 the resulting dynamical system is kolmogorov for anyc 0 for rational values ofc we construct a generating partition on whicht induces a markov chain this markov structure is used to discuss the decay of correlation functions exponential decay is found for a class of functions related to the partition explicit results are given forc 2 n the macroscopic analog of our model is a leaking process between two badly stirred containers according to the markov analysis the corresponding progress variable decays exponentially but the rate coefficients characterizing this decay are not those determined from the one way flux across the cell boundary the validity of the macroscopic rate law is discussed

a comment on the paper stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations by t d frank abstract the purpose of this comment is to correct mistaken assumptions and claims made in the paper stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations by t d frank t d frank stochastic feedback non linear families of markov processes and nonlinear fokker planck equations physica a 331 2004 391 our comment centers on the claims of a non linear markov process and a non linear fokker planck equation first memory in transition densities is misidentified as a markov process second the paper assumes that one can derive a fokker planck equation from a chapman kolmogorov equation but no proof was offered that a chapman kolmogorov equation exists for the memory dependent processes considered a non linear markov process is claimed on the basis of a non linear diffusion pde for a 1 point probability density we show that regardless of which initial value problem one may solve for the 1 point density the resulting stochastic process defined necessarily by the conditional probabilities the transition probabilities is either an ordinary linearly generated markovian one or else is a linearly generated non markovian process with memory we provide explicit examples of diffusion coefficients that reflect both the markovian and the memory dependent cases so there is neither a non linear markov process nor a non linear fokker planck equation for a conditional probability density the confusion rampant in the literature arises in part from labeling a non linear diffusion equation for a 1 point probability density as non linear fokker planck whereas neither a 1 point density nor an equation of motion for a 1 point density can define a stochastic process in a closely related context we point out that borland misidentified a translation invariant 1 point probability density derived from a non linear diffusion equation as a conditional probability density finally in the appendix a we present the theory of fokker planck pdes and chapman kolmogorov equations for stochastic processes with finite memory

intrinsic irreversibility of kolmogorov dynamical systems we give the mathematical details and various extensions of the results stated in previous work of courbage and prigogine intrinsic random systems are deterministic and conservative dynamical systems for which we can associate two dissipative markov processes through a one to one change of representation the first leading to equilibrium for t and the second for t the microscopic formulation of the second principle of thermodynamics permits to lift the degeneracy by the exclusion of all states that do not approach equilibrium for t the set of admitted initial conditions d is then characterized by a non equilibrium entropy functional which is infinite for rejected initial states and takes finite values for admitted initial conditions thus rejected initial states correspond to an infinite amount of information to realize this selection rule we consider general probability measures on phase space that are not necessarily absolutely continuous and we extend the theory of transition to markov processes to such measures owing to the non invariance of d under the time inversion the evolution of these states in the new representation can only be given by one of the two possible markov processes

necessary and sufficient conditions for zero assignment by constant squaring down the problem of zero assignment by constant squaring down czap is studied for minimal systems described by a transfer function g s rm l s m l using tools from exterior algebra and algebraic geometry conditions for its solvability are given in terms of the numbers m l the forney dynamical order and the rank of the plucker matrix p for the cases l 1 or l m 1 it is shown that g s is completely zero assignable cza if and only if 1 for the cases l 1 m 1 it is proved that g s is cza or generically zero assignable gza under a complex squaring down if and only if l m l 1 and 1 the latter conditions also provide necessary conditions for the existence of a real solution for a generic g s sufficient conditions for solvability of czap by a real squaring down are that l m l 1 and that the number g a0 al 1 1 a0 al 1 i j ai aj is odd for some set a0 al 1 satisfying 1 l 1i 0ai l l 1 2 and 0 a0 a1 al 1 m 1 apart from the existence results the present approach also allows the computation of the solutions whenever they exist finally it is proved that the only fixed zeros of czap are the zeros of g s and a procedure for computing the solutions of czap whenever such solutions exist based on an optimization problem is given

from dynamical systems to the langevin equation abstract it is shown that some dynamical systems which in microscopic time are fully deterministic exhibit a long time stochastic behavior governed by the langevin equation the limit in which this becomes a precise statement is obtained by an appropriate scaling of time and force the results are exemplified by a computer experiment

the role of stable manifolds and information in the prigogine theory of irreversibility the image of dirac measures x by the operator of the construction of prigogine and collaborators is shown to be concentrated in the stable manifold xst x and its density function is studied for bernoulli shifts the valuev exp h t whereh t is the kolmogorov entropy appears as a critical point for the behavior of it is also proved that no loss of information is involved by passing from the dynamical system to the markov process when vx 1 2 the discussion is based on the introduction of an invariant for markov systems that generalizes the usual kolmogorov entropy for dynamical systems

anomalous scaling of stochastic processes and the moses effect the state of a stochastic process evolving over a time t is typically assumed to lie on a normal distribution whose width scales like t frac 1 2 however processes in which the probability distribution is not normal and the scaling exponent differs from frac 1 2 are known the search for possible origins of such anomalous scaling and approaches to quantify them are the motivations for the work reported here in processes with stationary increments where the stochastic process is time independent autocorrelations between increments and infinite variance of increments can cause anomalous scaling these sources have been referred to as the joseph effect and the noah effect respectively if the increments are nonstationary then scaling of increments with t can also lead to anomalous scaling a mechanism we refer to as the moses effect scaling exponents quantifying the three effects are defined and related to the hurst exponent that characterizes the overall scaling of the stochastic process methods of time series analysis that enable accurate independent measurement of each exponent are presented simple stochastic processes are used to illustrate each effect intraday financial time series data are analyzed revealing that their anomalous scaling is due only to the moses effect in the context of financial market data we reiterate that the joseph exponent not the hurst exponent is the appropriate measure to test the efficient market hypothesis

evolution semigroups and time operators on banach spaces we present new general methods to obtain shift representation of evolution semigroups defined on banach spaces we introduce the notion of time operator associated with a generalized shift on a banach space and give some conditions under which time operators can be defined on an arbitrary banach space we also tackle the problem of scaling of time operators and obtain a general result about the existence of time operators on banach spaces satisfying some geometric conditions the last part of the paper contains some examples of explicit constructions of time operators on function spaces

normal factorisation of polynomials and computational issues abstract this paper introduces the notion of normal factorisation of polynomials and then presents a new numerical algorithm for the computation of this factorisation this procedure avoids computation of roots of polynomials and it is based on the use of algorithms determining the greatest common divisor of polynomials in this paper the general aspects of the algorithm are discussed and a symbolic implementation is given the theoretical algorithm provides the basis for a numerical procedure the general aspects of which are also discussed here the advantage of such a factorisation is that it handles the determination of multiplicities and produces factors of lower degree and with distinct roots for such polynomials robust numerical techniques based on finding roots may then be used if it is desired to work out the usual irreducible factorisation a detailed description of the implementation of the algorithm is presented the problem considered here is an integral part of computations for algebraic control problems

invariance properties and characterization of the greatest common divisor of a set of polynomials on m sets of polynomials from r s of degree d x1d4c5 m d the notion of extended r equivalence emd under x1d4c5 m d is characterized by a complete invariant and a canonical form it is shown that the greatest common divisor g c d of x1d4c5 m d is an invariant of x1d4c5 m d for proper sets x1d4c5 m d it is also shown that the g c d is invariant under the combined action of e r e transformations and shifting operations finally a systematic procedure for the computation of the g c d of x1d4c5 mdis given this procedure is based on the use of elementary row operations on real matrices and shifting operations

decentralized determinantal assignment problem fixed and almost fixed modes and zeros the decentralized determinantal assignment problem ddap is defined as the unifying description for the study of pole and zero assignment problems under decentralized output state feedback dof dsf and decentralized squaring down dsd respectively ddap is reduced to a linear problem of zero assignment of polynomial combinants and a multilinear problem of restricted decomposability of multivectors the decentralization characteristic dc and the decentralized polynomial grassmann representative d s gr of ddap are defined the fixed zero polynomial of ddap is then determined as the zero polynomial of d s gr the canonical d s gr cd r s gr and the decentralized plucker matrix dpm of ddap are introduced and necessary conditions for arbitrary assignment of the non fixed zeros are given in terms of the dpm the family of strongly zero non assignable sna systems is defined and for such systems the notion of the fixed zero is extended to that of the almost fixed zero afz an

markov approximations and decay of correlations for anosov flows we develop markov approximations for very general suspension flows based on this we obtain a stretched exponential bound on time correlation functions for 3 d anosov flows that verify uniform nonintegrability of foliations these include contact anosov flows and geodesic flows on compact surfaces of variable negative curvature our bound on correlations is stable under small smooth perturbations

entropy functionals associated to shifts dynamical systems abstract we construct entropy functionals which discriminate the measures which converge to equilibrium in the case of shifts dynamical systems in particular for k shifts it is proved that some functionals associated to them decrease monotonically to zero

monotonic entropy growth for a nonlinear model of random exchanges we present a proof of the monotonic entropy growth for a nonlinear discrete time model of a random market this model based on binary collisions also may be viewed as a particular case of ulam s redistribution of energy problem we represent each step of this dynamics as a combination of two processes the first one is a linear energy conserving evolution of the two particle distribution for which the entropy growth can be easily verified the original nonlinear process is actually a result of a specific coarse graining of this linear evolution when after the collision one variable is integrated away this coarse graining is of the same type as the real space renormalization group transformation and leads to an additional entropy growth the combination of these two factors produces the required result which is obtained only by means of information theory inequalities

markov processes hurst exponents and nonlinear diffusion equations with application to finance abstract we show by explicit closed form calculations that a hurst exponent h 1 2 does not necessarily imply long time correlations like those found in fractional brownian motion fbm we construct a large set of scaling solutions of fokker planck partial differential equations pdes where h 1 2 thus markov processes which by construction have no long time correlations can have h 1 2 if a markov process scales with hurst exponent h 1 2 then it simply means that the process has nonstationary increments for the scaling solutions we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient d x t is specified as an example we generate a class of student t like densities from the class of quadratic diffusion coefficients notably the tsallis density is one member of that large class the tsallis density is usually thought to result from a nonlinear diffusion equation but instead we explicitly show that it follows from a markov process generated by a linear fokker planck equation and therefore from a corresponding langevin equation having a tsallis density with h 1 2 therefore does not imply dynamics with correlated signals e g like those of fbm a short review of the requirements for fbm is given for clarity and we explain why the usual simple argument that h 1 2 implies correlations fails for markov processes with scaling solutions finally we discuss the question of scaling of the full green function g x t x t of the fokker planck pdes

nearest common root of a set of polynomials a structured singular value approach abstract the paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients in particular we seek the minimum magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root it is shown that the problem is equivalent to the solution of a structured singular value problem arising in robust control for which numerous techniques are available it is also shown that the method can be extended to the calculation of an approximate gcd of fixed degree by introducing the notion of the generalized structured singular value of a matrix the work generalizes previous results by the authors involving the calculation of the approximate gcd of two polynomials although the general case considered here is considerably harder and relies on a matrix dilation approach and several preliminary transformations

faithfulness property of the transition from bernoulli systems to irreversible markov processes abstract in a previous paper misra prigogine and courbage have shown that each bernoulli dynamical system may be associated through positivity preserving similarity transformations to a stochastics markov process here we give generalized baker transformarmations are necessarily distinct processes associated to two distinct generalized baker transformation are necessarily distinct

le chatelier s principle in replicator dynamics the le chatelier principle states that physical equilibria are not only stable but they also resist external perturbations via short time negative feedback mechanisms a perturbation induces processes tending to diminish its results the principle has deep roots e g in thermodynamics it is closely related to the second law and the positivity of the entropy production here we study the applicability of the le chatelier principle to evolutionary game theory i e to perturbations of a nash equilibrium within the replicator dynamics we show that the principle can be reformulated as a majorization relation this defines a stability notion that generalizes the concept of evolutionary stability we determine criteria for a nash equilibrium to satisfy the le chatelier principle and relate them to mutualistic interactions game theoretical anticoordination showing in which sense mutualistic replicators can be more stable than say competing ones there are globally stable nash equilibria where the le chatelier principle is violated even locally in contrast to the thermodynamic equilibrium a nash equilibrium can amplify small perturbations though both types of equilibria satisfy the detailed balance condition

martingale option pricing we show that our earlier generalization of the black scholes partial differential equation pde for variable diffusion coefficients is equivalent to a martingale in the risk neutral discounted stock price previously the equivalence of black scholes to a martingale was proven for the case of the gaussian returns model by harrison and kreps but we prove it for much a much larger class of returns models where the returns diffusion coefficient depends irreducibly on both returns x and time t that option prices blow up if fat tails in logarithmic returns x are included in market return is also proven

scaling in the norwegian stock market the main objective of this paper is to investigate the validity of the much used assumptions that stock market returns follow a random walk and are normally distributed for this purpose the concepts of chaos theory and fractals are applied two independent models are used to examine price variations in the norwegian and us stock markets the first model used is the range over standard deviation or r s statistic which tests for persistence or antipersistence in the time series both the norwegian and us stock markets show significant persistence caused by long run memory components in the series in addition an average non periodic cycle of four years is found for the us stock market these results are not consistent with the random walk assumption the second model investigates the distributional scaling behaviour of the high frequency price variations in the norwegian stock market the results show a remarkable constant scaling behaviour between different time intervals this means that there is no intrinsic time scale for the dynamics of stock price variations the relationship can be expressed through a scaling exponent describing the development of the distributions as the time scale changes this description may be important when constructing or improving pricing models such that they coincide more closely with the observed market behaviour the empirical distributions of high frequency price variations for the norwegian stock market is then compared to the levy stable distribution with the relevant scaling exponent found by using the r s and distributional scaling analysis good agreement is found between the levy profile and the empirical distribution for price variations less than 6 standard deviations covering almost three orders of magnitude in the data for probabilities larger than 6 standard deviations there seem to be an exponential fall off from the levy profile in the tails which indicates that the second moment may be finite

description of a class of markov processes equivalent tok shifts it results from recent works of prigogine and collaborators that one can construct a nonunitary operator which realizes an equivalence between the positive actions of a reversible dynamical system and an irreversible markov process going to equilibrium we consider here this construction and we prove that a fork shifts the transition probability of the associated markov process is concentrated in the stable manifold of the transformed point by the shift with a point mass concentrated on the deterministic trajectory and b for bernoulli shifts the measures which go to equilibrium are the same for the deterministic system and the markov process

squaring down and the problems of almost zeros for continuous time systems abstract in this paper we construct precompensators which square down a system such that the resulting square system has the same unstable zeros and system gains we derive the minimal gain of the precompensator to achieve such a square system if the gain of this precompensator is too large then we derive an explicit trade off between the gain of the precompensator and the number of unstable zeros we allow the precompensator to introduce in the system

nonunitary transformations and hydrodynamic limit in the one dimensional hard point gas abstract the ergodic theoretical formalism of nonunitary transformations is compared with the derivation of kinetic equations by the hydrodynamic scaling limit the one dimensional hard point system is used as an illustration both methods rest on the k mixing property of the physical systems considered we show that the specific form of the operator is important only on the microscopic time scale on the macroscopic scale it acts like a projector with broken time reversal symmetry we also discuss the connection between the kolmogorov sinai k s entropy of the model dynamical system the relative entropy production rate in the nonunitary formalism and the macroscopic entropy production rate

multidimensional gaussian sums arising from the distribution of birkhoff sums in zero entropy dynamical systems a duality formula of the hardy and littlewood type for multidimensional gaussian sums is proved in order to estimate the asymptotic long time behaviour of the distribution of the birkhoff sums sn of a sequence generated by a skew product dynamical system on the torus with zero lyapounov exponents the sequence taking the values 1 is a pairwise independent but not independent otherwise ergodic sequence with infinite range dependence the model corresponds to the motion of a particle on an infinite cylinder hopping backward and forward along its axis with a transversal acceleration parameter we show that when the parameter is rational then all the moments of the normalized sums but the second are unbounded with respect to n while for irrational with bounded continuous fraction representation all these moments are finite and bounded with respect to n

nonlocality of the misra prigogine courbage semigroup we show that the markov semigroups constructed by misra prigogine and courbage through nonunitary similarity transformations of kolmogorov systems are not implementable by local point transformations i e they are not the frobenius perron semigroups associated with noninvertible point transformations in contrast with the semigroups obtained by coarse graining projections our result is a straightforward generalization of the proof of the nonlocality of the similarity transformation given by goldstein misra and courbage and also of the previous illustration by misra and prigogine for the baker transformation and completes the characterization of the misra prigogine courbage semigroups

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

variable diffusion in stock market fluctuations abstract we analyze intraday fluctuations in several stock indices to investigate the underlying stochastic processes using techniques appropriate for processes with nonstationary increments the five most actively traded stocks each contains two time intervals during the day where the variance of increments can be fit by power law scaling in time the fluctuations in return within these intervals follow asymptotic bi exponential distributions the autocorrelation function for increments vanishes rapidly but decays slowly for absolute and squared increments based on these results we propose an intraday stochastic model with linear variable diffusion coefficient as a lowest order approximation to the real dynamics of financial markets and to test the effects of time averaging techniques typically used for financial time series analysis we find that our model replicates major stylized facts associated with empirical financial time series we also find that ensemble averaging techniques can be used to identify the underlying dynamics correctly whereas time averages fail in this task our work indicates that ensemble average approaches will yield new insight into the study of financial markets dynamics our proposed model also provides new insight into the modeling of financial markets dynamics in microscopic time scales

hurst exponents markov processes and fractional brownian motion there is much confusion in the literature over hurst exponents recently we took a step in the direction of eliminating some of the confusion one purpose of this

complexity of strings in the class of markov sources shannon s self information of a string is generalized to its complexity relative to the class of finite state machine fsm defined sources unlike an earlier generalization the new one is valid for both short and long strings the definition is justified in part by a theorem stating that asymptotically the mean complexity provides a tight lower bound for the mean length of all so called regular codes this also generalizes shannon s noiseless coding theorem for a large subclass of fsm sources a simple algorithm is described for computing the complexity

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

non trivial stably free modules over crossed products we consider the class of crossed products of noetherian domains with universal enveloping algebras of lie algebras for algebras from this class we give a sufficient condition for the existence of projective non free modules this class includes weyl algebras and universal envelopings of lie algebras for which this question known as the non commutative serre s problem has been extensively studied previously it turns out that the method of lifting of non trivial stably free modules from simple ore extensions can be applied to crossed products after an appropriate choice of filtration the motivating examples of crossed products are provided by the class of relativistic internal time algebras originating in non equilibrium physics

approach to equilibrium for locally expanding maps in ℝ k by using a well known technique from classical statistical mechanics of one dimensional lattice spin systems we prove existence of an absolutely continuous invariant asymptotic measure for certain locally expanding mapst of the unit cube in k we generalize herewith in a certain sense the results of lasota and yorke on piecewise expanding maps of the unit interval to higher dimensions we show a kuzmin type theorem for these systems from which exponential approach to equilibrium and strong mixing properties follow

on intrinsic randomness of dynamical systems we discuss the problem of nonunitary equivalence via positivity preserving similarity transformations between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes dynamical systems admitting such nonunitary equivalence with stochastic markov processes are said to beintrinsically random in a previous work it was found that the so called bernoulli systems discrete time are intrinsically random in this sense this result is extended here by showing that a more general class of dynamical systems the so calledk systems andk flows are intrinsically random the connection of intrinsic randomness with local instability of motion is briefly discussed we also show that markov processes associated through nonunitary equivalence tononisomorphic k flows are necessarily non isomorphic