measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

on multiscale entropy analysis for physiological data abstract we perform an analysis of cardiac data using multiscale entropy as proposed in costa et al multiscale entropy analysis of complex physiological time series phys rev lett 89 2002 068102 we reproduce the signatures of the multiscale entropy for the three cases of young healthy hearts atrial fibrillation and congestive heart failure we show that one has to be cautious how to interpret these signatures in terms of the underlying dynamics in particular we show that different dynamical systems can exhibit the same signatures depending on the sampling time and that similar systems may have different signatures depending on the time scales involved besides the total amount of data we identify the sampling time the correlation time and the period of possible nonlinear oscillations as important time scales which have to be involved in a detailed analysis of the signatures of multiscale entropies we illustrate our ideas with the lorenz equation as a simple deterministic chaotic system

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

multivariate multiscale fractional order weighted permutation entropy of nonlinear time series abstract in this letter multivariate multiscale fractional permutation entropy mmfpe and multivariate weighted multiscale fractional permutation entropy mwmfpe have been proposed to provide insights for the study of time series when measuring the dynamics of complex systems the mmfpe and mwmfpe methods are sensitive to the signal evolution meanwhile they can provide some analysis of complexity over multiple time series as well as multiple channel signals we perform these methods on synthetic tri variate time series to explore some of the interesting properties especially for negative information and information deception it can be seen that more complex system is more likely to be deceptive the amplitude information of time series which is taken into account in the mwmfpe can weaken this deception the methods are also employed to the closing prices and trade volume of financial stock markets from different areas according to the mwmfpe results the indices can be divided into three groups 1 cac40 hsi nasdaq s p500 2 n225 and 3 shencheng implying that it has a capacity to distinguish these financial stock market

weighted multiscale renyi permutation entropy of nonlinear time series abstract in this paper based on renyi permutation entropy rpe which has been recently suggested as a relative measure of complexity in nonlinear systems we propose multiscale renyi permutation entropy mrpe and weighted multiscale renyi permutation entropy wmrpe to quantify the complexity of nonlinear time series over multiple time scales first we apply mpre and wmpre to the synthetic data and make a comparison of modified methods and rpe meanwhile the influence of the change of parameters is discussed besides we interpret the necessity of considering not only multiscale but also weight by taking the amplitude into account then mrpe and wmrpe methods are employed to the closing prices of financial stock markets from different areas by observing the curves of wmrpe and analyzing the common statistics stock markets are divided into 4 groups 1 dji s p500 and hsi 2 nasdaq and ftse100 3 dax40 and cac40 and 4 shangzheng and shencheng results show that the standard deviations of weighted methods are smaller showing wmrpe is able to ensure the results more robust besides wmpre can provide abundant dynamical properties of complex systems and demonstrate the intrinsic mechanism

brownian motion in cones we study the asymptotic behavior of brownian motion and its conditioned process in cones using an infinite series representation of its transition density a concise probabilistic interpretation of this series in terms of the skew product decomposition of brownian motion is derived and used to show properties of the transition density

the spectrum of a hermitian matrix sum abstract an announcement by lidskii b v lidskii functional anal appl 10 1982 76 77 russian 139 140 english claimed to establish the explicit description of the spectrum of a hermitian matrix sum in the form conjectured by horn a horn pacific j math 16 1962 225 241 but no supporting proof has been published this paper begins an analysis of the claim and is the first step towards bridging the distance between the elementary methods of horn s 1962 paper and the partial solution of the same problem using noncommutative harmonic analysis by dooley et al i dooley r repka n wildberger lin mult alg 36 1993 79 102 the methods of horn lidskii lead to combinatorial issues of independent interest

heat kernel expansion on a generalized cone it is shown that the short time expansion of the integrated heat kernel on a locally flat generalized cone c n as defined by cheeger consists of just the weyl volume term and the constant term this latter is explicitly evaluated when n is a lens space sd zm for odd d elliptic space sd z2 for all d and any of the three dimensional homogeneous space forms s3 agreement is found with the corresponding expansion on the orbifold version t4 z2 of the k3 surface and in fact with all td z2

an investigation of oil water two phase flow instability using multivariate multi scale weighted permutation entropy abstract the present study is devoted to the investigation of oil water two phase flow instability using multivariate multi scale weighted permutation entropy mwmpe four typical multivariate multi scale entropies namely mwmpe multivariate multi scale permutation entropy mmpe multivariate multi scale sample entropy mmse and multivariate multi scale approximate entropy mmae are applied in lorenz system with the addition of different kinds of noises the comparison results indicate that mwmpe presents the superiorities of being sensitive to the variation in scale showing a monotonous increasing trend as well as the best anti noise ability accordingly with the fluctuating signals from an eight electrode rotating electrical field conductance sensor used for the measurement of oil water flows we extract mwmpe for the whole experimental flow conditions in terms of which the effects of mixture velocity and water cut on oil water two phase flow instability are illuminated our research provides an effective method for uncovering the underlying evolution instability of the flow structures in oil water flows

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

interlacing inequalities matrix groups abstract this paper has two parts 1 interlacing inequalities in linear algebra eigenvalues singular values invariant polynomials attempts to construct a unifying theory 2 subgroups of the full linear group defined by polynomial identities

dispersive graded entropy on computing dynamical complexity abstract we propose a phase space based statistical disorder to investigate the dynamical complexity of chaotic models the statistical disorder is defined by introducing a grade function inversely maps the mean dispersion of the trajectories in the phase space we denote the associate entropy by the dispersive graded entropy dge numerical investigation shows that dge can quantify the dynamical complexity of discrete as well as continuous chaotic systems a comparative study is also made with the other phase space based entropy measures finally the proposed measure has been applied on three types of heart rate variability hrv signals the results support the clinical observations related to the dynamics of healthy and congestive hearts

uniformly regular families of commuting operators let a be an analytic family of commuting n n matrices with constant rank on an open connected set d then the range and the null space of ak become independent of in d for a certain power k n we obtain similar results for uniformly regular families of commuting operators on a banach space

weighted multiscale cumulative residual renyi permutation entropy of financial time series abstract in this paper based on cumulative residual entropy cre and renyi permutation entropy rpe multiscale cumulative residual renyi permutation entropy mcrrrpe and weighted multiscale cumulative residual renyi permutation entropy wmcrrpe are proposed as novelty measures to quantify the uncertainty of the nonlinear time series first the mcrrpe and wmcrrpe methods are performed on the synthetic data and the impact of the changes to parameters is discussed then the mcrrpe and wmcrrpe methods are applied to the closing prices of the us european and chinese stock markets we analyze the statistics and draw some conclusions from the comparison the stock markets can be divided into four categories 1 nasdaq and ftse100 2 s p500 dji and hsi 3 dax30 and cac40 and 4 sse and szse the standard deviation has a certain relationship with wmcrrpe compared with the mcrrpe method the wmcrrpe method can distinguish these financial stock markets effectively and better estimate the complexity of the time series containing amplitude information

generalized statistical complexity measures geometrical and analytical properties abstract we discuss bounds on the values adopted by the generalized statistical complexity measures m t martin et al phys lett a 311 2003 126 p w lamberti et al physica a 334 2004 119 introduced by lopez ruiz et al phys lett a 209 1995 321 and shiner et al phys rev e 59 1999 1459 several new theorems are proved and illustrated with reference to the celebrated logistic map

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

multiscale entropy analysis of complex physiologic time series there has been considerable interest in quantifying the complexity of physiologic time series such as heart rate however traditional algorithms indicate higher complexity for certain pathologic processes associated with random outputs than for healthy dynamics exhibiting long range correlations this paradox may be due to the fact that conventional algorithms fail to account for the multiple time scales inherent in healthy physiologic dynamics we introduce a method to calculate multiscale entropy mse for complex time series we find that mse robustly separates healthy and pathologic groups and consistently yields higher values for simulated long range correlated noise compared to uncorrelated noise

randomness and degrees of irregularity the fundamental question are sequential data random arises in myriad contexts often with severe data length constraints furthermore there is frequently a critical need to delineate nonrandom sequences in terms of closeness to randomness e g to evaluate the efficacy of therapy in medicine we address both these issues from a computable framework via a quantification of regularity apen approximate entropy defining maximal randomness for sequences of arbitrary length indicating the applicability to sequences as short as n 5 points an infinite sequence formulation of randomness is introduced that retains the operational and computable features of the finite case in the infinite sequence setting we indicate how the foundational definition of independence in probability theory and the definition of normality in number theory reduce to limit theorems without rates of convergence from which we utilize apen to address rates of convergence of a deficit from maximal randomness refining the aforementioned concepts in a computationally essential manner representative applications among many are indicated to assess i random number generation output ii well shuffled arrangements and iii the quality of bootstrap replicates

remark on exit times from cones in mathbb r n of brownian motion our purpose is to show how the asymptotics in corollary 1 3 of 2 can be obtained under much weaker hypotheses it turns out the problem essentially reduces to showing that if r s is a bessel process u 0 and 0 then p left mathop f limits 0 1 r s 2 ds leqq u right o t alpha as t we provide a simple proof of this fact