horizontal visibility graphs exact results for random time series networks this procedure allows us to apply methods of complex network theory for characterizing time series in this work we present the horizontal visibility algorithm a geometrically simpler and analytically solvable version of our former algorithm focusing on the mapping of random series series of independent identically distributed random variables after presenting some properties of the algorithm we present exact results on the topological properties of graphs associated with random series namely the degree distribution the clustering coefficient and the mean path length we show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series since any random series maps to a graph with an exponential degree distribution of the shape pk 1 32 3 k 2 independent of the probability distribution from which the series was generated accordingly visibility graphs with other pk are related to nonrandom series numerical simulations confirm the accuracy of the theorems for finite series in a second part we show that the method is able to distinguish chaotic series from independent and identically distributed i i d theory studying the following situations i noise free low dimensional chaotic series ii low dimensional noisy chaotic series even in the presence of large amounts of noise and iii high dimensional chaotic series coupled map lattice without needs for additional techniques such as surrogate data or noise reduction methods finally heuristic arguments are given to explain the topological properties of chaotic series and several sequences that are conjectured to be random are analyzed

lie properties of crossed products abstract let f g be a crossed product of a group g and the field f we study the lie properties of f g in order to obtain a characterization of those crossed products which are upper lower lie nilpotent and lie n m engel

horizontal visibility graphs from integer sequences the horizontal visibility graph hvg is a graph theoretical representation of a time series and builds a bridge between dynamical systems and graph theory in recent years this representation has been used to describe and theoretically compare different types of dynamics and has been applied to characterize empirical signals by extracting topological features from the associated hvgs which have shown to be informative on the class of dynamics among some other measures it has been shown that the degree distribution of these graphs is a very informative feature that encapsulates nontrivial information of the series s generative dynamics in particular the hvg associated to a bi infinite real valued series of independent and identically distributed random variables is a universal exponential law independent of the series marginal distribution most of the current applications have however only addressed real valued time series as no exact results are known for the topological properties of hvgs associated to integer valued series in this paper we explore this latter situation and address univariate time series where each variable can only take a finite number n of consecutive integer values we are able to construct an explicit formula for the parametric degree distribution which we prove to converge to the continuous case for large n and deviates otherwise a few applications are then considered

horizontal visibility graph transfer entropy hvg te a novel metric to characterize directed connectivity in large scale brain networks we propose a new measure horizontal visibility graph transfer entropy hvg te to estimate the direction of information flow between pairs of time series hvg te quantifies the transfer entropy between the degree sequences of horizontal visibility graphs derived from original time series twenty one rossler attractors unidirectionally coupled in the posterior to anterior direction were used to simulate 21 channel electroencephalography eeg brain networks and validate the performance of the hvg te we showed that the hvg te is robust to different levels of coupling strengths between the coupled rossler attractors a wide range of time delays different sample sizes the effects of noise and linear mixing and the choice of reference for eeg data we also applied hvg te to eeg data in 20 healthy controls and compared its performance to a recently introduces phase based te measure pte we found that compared with pte hvg te consistently detected stronger posterior to anterior information flow patterns in the alpha band 8 13hz eeg brain networks for three different references moreover in contrast to pte hvg te does not require an assumption on the periodicity of input signals therefore it can be more widely applicable even for non periodic signals this study shows that the hvg te is a directed connectivity measure to characterise the direction of information flow in large scale brain networks

new irreversibility measure and complexity analysis based on singular value decomposition abstract visibility graph algorithm is a powerful tool for visualization and analysis of complex dynamical system kullback leibler divergence kld based on the horizontal visibility graph algorithm has received largely extensive attention in the field of irreversibility analysis in this paper we will propose a new irreversibility measure built on the multi scale theory i e the kld based on singular value decomposition kld svd furthermore since matrix singular value can reflect the basic characteristics of the complex system we further bring forward the definition of shannon entropy with singular value decomposition se svd after considering that the shannon entropy has been well defined as a complexity measure in order to show the advantages of these two new measures in detecting the complexity of systems several simulation and real data experiments are chosen to examine the performance of them through these results we find that the average and variance of kld svd corresponding to each time series is less than that of the kld moreover kld svd s volatility is significantly weaker than kld s as there is a conceptual link between predictability and irreversibility we can argue that kld svd algorithm implies the higher predictability and lower confusion on the other hand se svd can do well in revealing the internal complexity of the different time series meanwhile it is useful to mention that se svd is robust against noise efficiently

noise reduction in chaotic time series data a survey of common methods this paper surveys some of the methods that have been suggested for reducing noise in time series data whose underlying dynamical behavior can be characterized as low dimensional chaos although the procedures differ in details all of them must solve three basic problems how to reconstruct an attractor from the data how to approximate the dynamics in various regions on the attractor and how to adjust the observations to satisfy better the approximations to the dynamics all current noise reduction methods have similar limitations but the basic problems are reasonably well understood the methods are an important tool in the experimentalist s repertoire for data analysis in our view they should be used more widely particularly in studies of attractor dimension lyapunov exponents prediction and control

a 5 engel associative algebra whose group of units is not 5 engel abstract let r be an associative ring with unity and let r and u r denote the associated lie ring with a b a b b a and the group of units of r respectively in 1983 gupta and levin proved that if r is a nilpotent lie ring of class c then u r is a nilpotent group of class at most c the aim of the present note is to show that in general a similar statement does not hold if r is n engel we construct an algebra r over a field of characteristic 2 3 such that the lie algebra r is 5 engel but the group u r is not

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

on lie solvable restricted enveloping algebras abstract in this note we study the lie derived lengths of a restricted enveloping algebra u l for a non abelian restricted lie algebra l over a field of positive characteristic p for p 2 we show that if the lie derived length of u l is minimal then u l is lie nilpotent moreover we investigate the case when the strong lie derived length of u l is minimal for odd p we establish a classification of lie centrally metabelian restricted enveloping algebras

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

time series characterization via horizontal visibility graph and information theory abstract complex networks theory have gained wider applicability since methods for transformation of time series to networks were proposed and successfully tested in the last few years horizontal visibility graph has become a popular method due to its simplicity and good results when applied to natural and artificially generated data in this work we explore different ways of extracting information from the network constructed from the horizontal visibility graph and evaluated by information theory quantifiers most works use the degree distribution of the network however we found alternative probability distributions more efficient than the degree distribution in characterizing dynamical systems in particular we find that when using distributions based on distances and amplitude values significant shorter time series are required we analyze fractional brownian motion time series and a paleoclimatic proxy record of enso from the pallcacocha lake to study dynamical changes during the holocene

the lie dimension subgroup conjecture abstract for every n 9 a group g n is constructed so that its n th th lie dimension subgroup contains properly its n th lower central subgroup settling negatively the lie dimension subgroup conjecture the question remains open for n 7 and 8

nonmatrix varieties and nil generated algebras whose units satisfy a group identity abstract letr denote the group of units of an associative algebrarover an infinite fieldf we prove that ifris unitarily generated by its nilpotent elements thenr satisfies a group identity precisely whenrsatisfies a nonmatrix polynomial identity as an application we examine the group algebrafgof a torsion groupgand the restricted enveloping algebrau l of ap nil restricted lie algebral giambruno sehgal and valenti recently proved that if the group of units fg satisfies a group identity thenfgsatisfies a polynomial identity thus confirming a conjecture of brian hartley we show that in fact fg satisfies a group identity if and only iffgsatisfies a nonmatrix polynomial identity in the case of restricted enveloping algebras we prove thatu l satisfies a group identity if and only ifu l satisfies the engel condition

presentations for unit groups of modular group algebras of groups of order 16 for a p group g and the field f of p elements let v denote the group of normalized units of the group algebra fg generators and relations are provided for v for each group g of order dividing 16 the presentations are sufficiently concise to permit transcription for machine calculation with v some applications are described a theoretical method for obtaining presentations for v is developed it is most effective when p 2 or when p 3 and the commutator subgroup g is of order 3 implementation involves extensive calculation in fg

volatility behavior of visibility graph emd financial time series from ising interacting system abstract a financial market dynamics model is developed and investigated by stochastic ising system where the ising model is the most popular ferromagnetic model in statistical physics systems applying two graph based analysis and multiscale entropy method we investigate and compare the statistical volatility behavior of return time series and the corresponding imf series derived from the empirical mode decomposition emd method and the real stock market indices are considered to be comparatively studied with the simulation data of the proposed model further we find that the degree distribution of visibility graph for the simulation series has the power law tails and the assortative network exhibits the mixing pattern property all these features are in agreement with the real market data the research confirms that the financial model established by the ising system is reasonable

visibility graph analysis on quarterly macroeconomic series of china based on complex network theory abstract the visibility graph approach and complex network theory provide a new insight into time series analysis the inheritance of the visibility graph from the original time series was further explored in the paper we found that degree distributions of visibility graphs extracted from pseudo brownian motion series obtained by the frequency domain algorithm exhibit exponential behaviors in which the exponential exponent is a binomial function of the hurst index inherited in the time series our simulations presented that the quantitative relations between the hurst indexes and the exponents of degree distribution function are different for different series and the visibility graph inherits some important features of the original time series further we convert some quarterly macroeconomic series including the growth rates of value added of three industry series and the growth rates of gross domestic product series of china to graphs by the visibility algorithm and explore the topological properties of graphs associated from the four macroeconomic series namely the degree distribution and correlations the clustering coefficient the average path length and community structure based on complex network analysis we find degree distributions of associated networks from the growth rates of value added of three industry series are almost exponential and the degree distributions of associated networks from the growth rates of gdp series are scale free we also discussed the assortativity and disassortativity of the four associated networks as they are related to the evolutionary process of the original macroeconomic series all the constructed networks have small world features the community structures of associated networks suggest dynamic changes of the original macroeconomic series we also detected the relationship among government policy changes community structures of associated networks and macroeconomic dynamics we find great influences of government policies in china on the changes of dynamics of gdp and the three industries adjustment the work in our paper provides a new way to understand the dynamics of economic development

topological properties and fractal analysis of a recurrence network constructed from fractional brownian motions many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks in this work we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional brownian motions fbms first our results indicate that the constructed recurrence networks have exponential degree distributions the average degree exponent increases first and then decreases with the increase of hurst index h of the associated fbms the relationship between h and can be represented by a cubic polynomial function we next focus on the motif rank distribution of recurrence networks so that we can better understand networks at the local structure level we find the interesting superfamily phenomenon i e the recurrence networks with the same motif rank pattern being grouped into two superfamilies last we numerically analyze the fractal and multifractal properties of recurrence networks we find that the average fractal dimension db of recurrence networks decreases with the hurst index h of the associated fbms and their dependence approximately satisfies the linear formula db 2 h which means that the fractal dimension of the associated recurrence network is close to that of the graph of the fbm moreover our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks and the multifractality of these networks becomes stronger at first and then weaker when the hurst index of the associated time series becomes larger from 0 4 to 0 95 in particular the recurrence network with the hurst index h 0 5 possesses the strongest multifractality in addition the dependence relationships of the average information dimension d 1 and the average correlation dimension d 2 on the hurst index h can also be fitted well with linear functions our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated fbm series

visibility graph analysis of economy policy uncertainty indices abstract uncertainty plays an important role in the global economy in this paper the economic policy uncertainty epu indices of the united states and china are selected as the proxy variable corresponding to the uncertainty of national economic policy by adopting the visibility graph algorithm the four economic policy uncertainty indices of the united states and china are mapped into complex networks and the topological properties of the corresponding networks are studied the hurst exponents of all the four indices are within 0 5 1 which implies that the economic policy uncertainty is persistent the degree distributions of the epu networks have power law tails and thus the networks are scale free the average clustering coefficients of the four epu networks are high and close to each other while these networks exhibit weak assortative mixing we also find that the epu network in united states based on daily data shows the small world feature since the average shortest path length increases logarithmically with the network size such that l n 0 626 ln n 0 405 our research highlights the possibility to study the epu from the view angle of complex networks

on universal lie nilpotent associative algebras abstract we study the quotient q i a of a free algebra a by the ideal m i a generated by the ith commutator of any elements in particular we completely describe such quotient for i 4 for i 3 this was done previously by feigin and shoikhet we also study properties of the ideals m i a e g when m i a m j a is contained in m i j 1 a by a result of gupta and levin it is always contained in m i j 2 a

characterization of spatiotemporal chaos from time series the analysis of time series generated by spatiotemporal chaotic systems is discussed we find that a grassberger procaccia algorithm with a suitable normalisation for the correction of systematic errors caused by the shape of the attractor is a reliable method for distinguishing between high dimensional chaos and noise we show that even a quantitative description of the attractor is possible by means of dimension densities the results obtained from the time series are in good agreement with values calculated directly from the generating equations of motion

group identities on the units of algebraic algebras with applications to restricted enveloping algebras abstract an algebra a is called a gi algebra if its group of units a satisfies a group identity we provide positive support for the following two open problems 1 does every algebraic gi algebra satisfy a polynomial identity 2 is every algebraically generated gi algebra locally finite

products of commutators in a lie nilpotent associative algebra abstract let f be a field and let f x be the free unital associative algebra over f freely generated by an infinite countable set x x 1 x 2 define a left normed commutator a 1 a 2 a n recursively by a 1 a 2 a 1 a 2 a 2 a 1 a 1 a n 1 a n a 1 a n 1 a n n 3 for n 2 let t n be the two sided ideal in f x generated by all commutators a 1 a 2 a n a i f x let f be a field of characteristic 0 in 2008 etingof kim and ma conjectured that t m t n t m n 1 if and only if m or n is odd in 2010 bapat and jordan confirmed the if direction of the conjecture if at least one of the numbers m n is odd then t m t n t m n 1 the aim of the present note is to confirm the only if direction of the conjecture we prove that if m 2 m and n 2 n are even then t m t n t m n 1 our result is valid over any field f

recurrence time analysis long term correlations and extreme events the recurrence times between extreme events have been the central point of statistical analyses in many different areas of science simultaneously the poincar e recurrence time has been extensively used to characterize nonlinear dynamical systems we compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series in particular we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function on the interval whereto the recurrences are observed and on the temporal correlations of time series in the case of long term correlations we verify the validity of the stretched exponential distribution which is uniquely defined by the exponent ensuremath gamma at the same time showing that it is restricted to the class of linear long term correlated processes simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different ensuremath gamma which shows a lack of invariance under the change of observables

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

the torsion subgroup of the additive group of a lie nilpotent associative ring of class 3 abstract let z x be the free unital associative ring freely generated by an infinite countable set x x 1 x 2 define a left normed commutator a 1 a 2 a n inductively by a b a b b a a 1 a 2 a n a 1 a n 1 a n n 3 for n 2 let t n be the two sided ideal in z x generated by all commutators a 1 a 2 a n a i z x let t 3 2 be the two sided ideal of the ring z x generated by all elements a 1 a 2 a 3 a 4 and a 1 a 2 a 3 a 4 a 5 a i z x it has been recently proved in 22 that the additive group of z x t 4 is a direct sum a b where a is a free abelian group isomorphic to the additive group of z x t 3 2 and b t 3 2 t 4 is an elementary abelian 3 group a basis of the free abelian summand a was described explicitly in 22 the aim of the present article is to find a basis of the elementary abelian 3 group b

topological properties of the limited penetrable horizontal visibility graph family the limited penetrable horizontal visibility graph algorithm was recently introduced to map time series in complex networks in this work we extend this algorithm to create a directed limited penetrable horizontal visibility graph and an image limited penetrable horizontal visibility graph we define two algorithms and provide theoretical results on the topological properties of these graphs associated with different types of real value series we perform several numerical simulations to check the accuracy of our theoretical results finally we present an application of the directed limited penetrable horizontal visibility graph to measure real value time series irreversibility and an application of the image limited penetrable horizontal visibility graph that discriminates noise from chaos we also propose a method to measure the systematic risk using the image limited penetrable horizontal visibility graph and the empirical results show the effectiveness of our proposed algorithms

time dependent limited penetrable visibility graph analysis of nonstationary time series abstract recent years have witnessed the development of visibility graph theory which allows us to analyze a time series from the perspective of complex network we in this paper develop a novel time dependent limited penetrable visibility graph tdlpvg two examples using nonstationary time series from rr intervals and gas liquid flows are provided to demonstrate the effectiveness of our approach the results of the first example suggest that our tdlpvg method allows characterizing the time varying behaviors and classifying heart states of healthy congestive heart failure and atrial fibrillation from rr interval time series for the second example we infer tdlpvgs from gas liquid flow signals and interestingly find that the deviation of node degree of tdlpvgs enables to effectively uncover the time varying dynamical flow behaviors of gas liquid slug and bubble flow patterns all these results render our tdlpvg method particularly powerful for characterizing the time varying features underlying realistic complex systems from time series

signatures of chaotic and stochastic dynamics uncovered with recurrence networks an old and important problem in the field of nonlinear time series analysis entails the distinction between chaotic and stochastic dynamics recently recurrence networks have been proposed as a t

lie structure of truncated symmetric poisson algebras abstract the paper naturally continues series of works on identical relations of group rings enveloping algebras and other related algebraic structures let l be a lie algebra over a field of characteristic p 0 consider its symmetric algebra s l n 0 u n u n 1 which is isomorphic to a polynomial ring it also has a structure of a poisson algebra where the lie product is traditionally denoted by this bracket naturally induces the structure of a poisson algebra on the ring s l s l x p x l which we call a truncated symmetric poisson algebra we study lie identical relations of s l namely we determine necessary and sufficient conditions for l under which s l is lie nilpotent strongly lie nilpotent solvable and strongly solvable where we assume that p 2 to specify the solvability we compute the strong lie nilpotency class of s l also we prove that the lie nilpotency class coincides with the strong lie nilpotency class in case p 3 shestakov proved that the symmetric algebra s l of an arbitrary lie algebra l satisfies the identity x y z 0 if and only if l is abelian we extend this result for the strong lie nilpotency and the strong solvability of s l we show that the solvability of s l and s l in case char k 2 is different from other characteristics namely we construct examples of such algebras which are solvable but not strongly solvable we use delta sets for lie algebras and the theory of identical relations of poisson algebras also we study filtrations in poisson algebras and prove results on products of terms of the lower central series for poisson algebras

lie centrally metabelian group rings let l r denote the associated lie ring of an associative ring r under the lie multiplication x y xy yx x y r then r is said to be lie centrally metabelian if the lie ring l r is centrally metabelian that is x1 x2 x3 x4 x5 0 for all x1 x2 x3 x4 x5 r let g be a group and k be a field such that the group ring kg is lie centrally metabelian if char k 0 then it is known that g is abelian in this paper we have obtained two results on lie centrally metabelian group rings when char k p 3 the first states that if char k p 3 then kg is lie centrally metabelian if and only if g is abelian the second result proves that if char k 3 and kg is lie centrally metabelian then the commutator subgroup g is a finite 3 engel 3 group of exponent at most 9 and consequently nilpotent of class at most 4 further it has been shown that if kg is lie solvable and char k p is sufficiently large then g is abelian at the end we obtain an interesting result which shows that 2 u r 1 2 l r r for any ring r where 2 u r and 2 l r denote the second derived term of the unit group u r and the associated lie ring l r of r respectively as a consequence the unit group u r is metabelian if the lie ring l r is metabelian

non classical law of the iterated logarithm behaviour for trimmed sums we study the law of the iterated logarithm for the partial sum of i i d random variables when the r n largest summands are excluded where r n o log logn this complements earlier work in which the case log logn o rn was considered a law of the iterated logarithm is again seen to prevail for a wide class of distributions but for reasons quite different from previously

pi algebras generated by nilpotent elements of bounded index abstract let r be an associative algebra over a field f of positive characteristic p we address the following problem if r is generated by nilpotent elements with bounded index under what conditions can we conclude that r itself is nil of bounded index we prove that whenever r satisfies the engel condition and p 0 or r is lie soluble and p 2 then the conclusion holds along the way we prove that if r satisfies the engel condition then there exists a positive integer n such that the map x x p n is additive the converse also holds when f is infinite or r is generated by nilpotent elements

statistics of return times a general framework and new applications in this paper we provide general estimates for the errors between the distribution of the first and more generally the k th return time suitably rescaled and the poisson law for measurable dynamical systems in the case that the system exhibits strong mixing properties these bounds are explicitly expressed in terms of the speed of mixing using these approximations the poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval

networks with time structure from time series we propose a method of constructing a network in which its time structure is directly incorporated based on a deterministic model from a time series to construct such a network we transform a linear model containing terms with different time delays into network topology the terms in the model are translated into temporal nodes of the network on each link connecting these nodes we assign a positive real number representing the strength of relationship or the distance between nodes specified by the parameters of the model the method is demonstrated by a known system and applied to two actual time series

time series analysis of s p 500 index a horizontal visibility graph approach the behavior of stock prices has been thoroughly studied throughout the last century and contradictory results have been reported in the corresponding literature in this paper a network theoretical approach is provided to investigate how crises affected the behavior of us stock prices we analyze high frequency data from s p500 via the horizontal visibility graph method and find that all major crises that took place worldwide in the last twenty years affected significantly the behavior of the price index nevertheless we observe that each of those crises impacted the index in a different way and magnitude interestingly our results suggest that the predictability of the price index series increases during the periods of crises