quantum variance of maass hecke cusp forms in this paper we study quantum variance for the modular surface x gamma backslash mathbb h where gamma sl 2 mathbb z is the full modular group we evaluate asymptotically the quantum variance which is introduced by s zelditch and describes the fluctuations of a quantum observable it is shown that the quantum variance is equal to the classical variance of the geodesic flow on s x the unit cotangent bundle of x but twisted by the central value of the maass hecke l functions

the freiheitssatz for poisson algebras abstract we prove the freiheitssatz for poisson algebras in characteristic zero we also give a new proof of the tameness of automorphisms for two generated free poisson algebras and show that an analogue of the commutator test theorem is equivalent to the two dimensional classical jacobian conjecture

a robust method for detecting interdependences application to intracranially recorded eeg we present a measure for characterizing statistical relationships between two time sequences in contrast to commonly used measures like cross correlations coherence and mutual information the proposed measure is non symmetric and provides information about the direction of interdependence it is closely related to recent attempts to detect generalized synchronization however we do not assume a strict functional relationship between the two time sequences and try to define the measure so as to be robust against noise and to detect also weak interdependences we apply our measure to intracranially recorded electroencephalograms of patients suffering from severe epilepsies

detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble a method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail mutual nonlinear prediction also provides information on the directionality of the coupling between systems furthermore the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony numerical examples studied include three classes of unidirectionally coupled systems systems with identical parameters nonidentical parameters and stochastic driving of a nonlinear system this technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool the interrelationships examined include single neuron unit firing the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex and intracellular measurements of integrated excitatory postsynaptic potentials epsp s dynamical interdependence perhaps generalized synchrony was identified in this neuronal network between simultaneous single unit firings between units and the population and between units and intracellular epsp s

nonlinear parametric model for granger causality of time series the notion of granger causality between two time series examines if the prediction of one series could be improved by incorporating information of the other in particular if the prediction error of the first time series is reduced by including measurements from the second time series then the second time series is said to have a causal influence on the first one we propose a radial basis function approach to nonlinear granger causality the proposed model is not constrained to be additive in variables from the two time series and can approximate any function of these variables still being suitable to evaluate causality usefulness of this measure of causality is shown in two applications in the first application a physiological one we consider time series of heart rate and blood pressure in congestive heart failure patients and patients affected by sepsis we find that sepsis patients unlike congestive heart failure patients show symmetric causal relationships between the two time series in the second application we consider the feedback loop in a model of excitatory and inhibitory neurons we find that in this system causality measures the combined influence of couplings and membrane time constants

synchronization likelihood an unbiased measure of generalized synchronization in multivariate data sets the study of complex systems consisting of many interacting subsystems requires the use of analytical tools which can detect statistical dependencies between time series recorded from these subsystems typical examples are the electroencephalogram eeg and magnetoencephalogram meg which may involve the simultaneous recording of 150 or more time series coherency which is often used to study such data is only sensitive to linear and symmetric interdependencies and cannot deal with non stationarity recently several algorithms based upon the concept of generalized synchronization have been introduced to overcome some of the limitations of coherency estimates e g physica d 134 1999 419 brain res 792 1998 24 however these methods are biased by the degrees of freedom of the interacting subsystems physica d 134 1999 419 physica d 148 2001 147 we propose a novel measure for generalized synchronization in multivariate data sets which avoids this bias and can deal with non stationary dynamics

sums of kloosterman sums in arithmetic progressions and the error term in the dispersion method we prove a bound for quintilinear sums of kloosterman sums with congruence conditions on the smooth summation variables this generalizes classical work of deshouillers and iwaniec and is key to obtaining power saving error terms in applications notably the dispersion method as a consequence assuming the riemann hypothesis for dirichlet l functions we prove a power saving error term in the titchmarsh divisor problem of estimating sum p leq x tau p 1 unconditionally we isolate the possible contribution of siegel zeroes showing it is always negative extending work of fouvry and tenenbaum we obtain power saving in the asymptotic formula for sum n leq x tau k n tau n 1 reproving a result announced by bykovski u i and vinogradov by a different method the gain in the exponent is shown to be independent of k if a generalized lindel of hypothesis is assumed

measuring synchronization in coupled model systems a comparison of different approaches the investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation for experimental data the underlying dynamics are usually not completely known therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis in this study we use three different coupled model systems to create a controlled setting for a comparison of six different measures of synchronization all measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications although certain dynamical features of a system under investigation e g power spectra dimension may render certain measures more suitable than others c 2006 elsevier b v all rights reserved

on moments of twisted l functions we study the average of the product of the central values of two l functions of modular forms f and g twisted by dirichlet characters to a large prime modulus q as our principal tools we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q and we use the theory of deligne and katz to prove new bounds on bilinear forms in kloosterman sums with power savings when both variables are near the square root of q when at least one of the forms f and g is non cuspidal we obtain an asymptotic formula for the mixed second moment of twisted l functions with a power saving error term in particular when both are non cuspidal this gives a significant improvement on m young s asymptotic evaluation of the fourth moment of dirichlet l functions in the general case the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in kloosterman sums

bivariate surrogate techniques necessity strengths and caveats the concept of surrogates allows testing results from time series analysis against specified null hypotheses in application to bivariate model dynamics we here compare different types of surrogates each designed to test against a different null hypothesis e g an underlying bivariate linear stochastic process two measures that aim at a characterization of interdependence between nonlinear deterministic dynamics were used as discriminating statistics we analyze eight different stochastic and deterministic models not only to demonstrate the power of the surrogates but also to reveal some pitfalls and limitations

the global extension problem crossed products and co flag non commutative poisson algebras let p be a poisson algebra e a vector space and e p an epimorphism of vector spaces with v ker the global extension problem asks for the classification of all poisson algebra structures that can be defined on e such that e p becomes a morphism of poisson algebras from a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component all such poisson algebra structures on e are classified by an explicitly constructed classifying set gph2 p v which is the coproduct of all non abelian cohomological objects ph2 p v v v which are the classifying sets for all extensions of p by v v v the second classical poisson cohomology group h2 p v appears as the most elementary piece among all components of gph2 p v several examples are provided in the case of metabelian poisson algebras or co flag poisson algebras over p the latter being poisson algebras q which admit a finite chain of epimorphisms of poisson algebras pn q npn 1 p1 1p0 p such that dim ker i 1 for all i 1 n

free poisson hopf algebras generated by coalgebras we construct the analogue of takeuchi s free hopf algebra in the setting of poisson hopf algebras more precisely we prove that there exists a free poisson hopf algebra on any coalgebra or equivalently that the forgetful functor from the category of poisson hopf algebras to the category of coalgebras has a left adjoint in particular we also prove that the category of poisson hopf algebras is a reflective subcategory of the category of poisson bialgebras along the way we describe coproducts and coequalizers in the category of poisson hopf algebras therefore showing that the latter category is cocomplete

estimates for rankin selberg l functions and quantum unique ergodicity abstract subconvex bounds in the weight aspect for rankin selberg l functions associated to two cusp forms are established these bounds are applied to prove the equidistribution of mass conjecture for cm forms

nonuniform state space reconstruction and coupling detection we investigate the state space reconstruction from multiple time series derived from continuous and discrete systems and propose a method for building embedding vectors progressively using information measure criteria regarding past current and future states the embedding scheme can be adapted for different purposes such as mixed modeling cross prediction and granger causality in particular we apply this method in order to detect and evaluate information transfer in coupled systems as a practical application we investigate in records of scalp epileptic eeg the information flow across brain areas

detection of weak directional coupling phase dynamics approach versus state space approach we compare two conceptually different approaches to the detection of weak directional couplings between two oscillatory systems from bivariate time series the first approach is based on the analysis of the systems phase dynamics whereas the other one tests for interdependencies in the reconstructed state spaces of the systems we analyze the sensitivity of both techniques to weak couplings in numerical experiments by considering couplings between almost identical as well as between significantly different nonlinear systems we study different degrees of phase diffusion test the robustness of the two techniques against observational noise and investigate the influence of the time series length our results show that none of the two approaches is generally superior to the other and we conclude that it is probably the combination of both techniques that would allow the most comprehensive and reliable characterization of coupled systems

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

universal enveloping algebras and universal derivations of poisson algebras abstract let k be an arbitrary field of characteristic 0 it is shown that for any n 1 the universal enveloping algebras of the poisson symplectic algebra p n k and the weyl algebra a n k are isomorphic and the canonical isomorphism between them easily leads to the moyal product a basis of the universal enveloping algebra p e of a free poisson algebra p k x 1 x n is constructed and it is proved that the left dependence of a finite number of elements of p e over p e is algorithmically recognizable we describe the poisson dependence of any two elements of a free poisson algebra in characteristic 0 in the language of universal derivatives the fox derivatives on free poisson algebras are defined and it is proved that an analogue of the jacobian conjecture for two generated free poisson algebras is equivalent to the two dimensional classical jacobian conjecture a new proof of the tameness of automorphisms of two generated free poisson algebras is also given

directionality of coupling from bivariate time series how to avoid false causalities and missed connections we discuss some problems encountered in inference of directionality of coupling or in the case of two interacting systems in inference of causality from bivariate time series we identify factors and influences that can lead to either decreased test sensitivity or false detections and propose ways to cope with them in order to perform tests with high sensitivity and a low rate of false positive results

trilinear forms with kloosterman fractions abstract we give new bounds for a m n m n a e a m n where m n and a are arbitrary coefficients improving upon a result of duke friedlander and iwaniec 7 we also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations

le probleme de tchebychev pour le douzieme polynome cyclotomique let p n denote the largest prime factor of the integer n and 12 n n 4 n 2 1 we prove that for x large enough we have p x x 1 c with c 10 26531

kloosterman sums for clifford algebras and a lower bound for the positive eigenvalues of the laplacian for congruence subgroups acting on hyperbolic spaces

brain behaviour and mathematics are we using the right approaches mathematics are used in biological sciences mostly as a quantifying tool for it is the science of numbers after all there is a long standing interest in the application of mathematical methods and concepts to neuroscience in attempts to decipher brain activity while there has been a very wide use of mathematical physical methodologies less effort has been made to formulate a comprehensive and integrative theory of brain function this review concentrates on recent developments uses and abuses of mathematical formalisms and techniques that are being applied in brain research particularly the current trend of using dynamical system theory to unravel the global collective dynamics of brain activity it is worth emphasising that the theoretician neuroscientist eager to apply mathematical analysis to neuronal recordings has to consider carefully some crucial anatomo physiological assumptions that may not be as accurate as the specific methods require on the other hand the experimentalist neuro physicist with an inclination to implement mathematical thoughts in brain science has to make an effort to comprehend the bases of the theoretical concepts that can be used as frameworks or as analysis methods of brain electrophysiological recordings and to critically inspect the accuracy of the interpretations of the results based on the neurophysiological ground it is hoped that this brief overview of anatomical and physiological presumptions and their relation to theoretical paradigms will help clarify some particular points of interest in current trends in brain science and may provoke further reflections on how certain or uncertain it is to conceptualise brain function based on these theoretical frameworks if the physiological and experimental constraints are not as accurate as the models prescribe

assessment of eeg synchronization based on state space analysis cortical computation involves the formation of cooperative neuronal assemblies characterized by synchronous oscillatory activity a traditional method for the identification of synchronous neuronal assemblies has been the coherence analysis of eeg signals here we suggest a new method called s estimator whereby cortical synchrony is defined from the embedding dimension in a state space we first validated the method on clusters of chaotic coupled oscillators and compared its performance to that of other methods for assessing synchronization then nine adult subjects were studied with high density eeg recordings while they viewed in the two hemifields hence with separate hemispheres identical sinusoidal gratings either arranged collinearly and moving together or orthogonally oriented and moving at 90 degrees the estimated synchronization increased with the collinear gratings over a cluster of occipital electrodes spanning both hemispheres whereas over temporo parietal regions of both hemispheres it decreased with the same stimulus and it increased with the orthogonal gratings separate calculations for different eeg frequencies showed that the occipital clusters involved synchronization in the beta band and the temporal clusters in the alpha band the gamma band appeared to be insensitive to stimulus diversity different stimulus configurations therefore appear to cause a complex rearrangement of synchronous neuronal assemblies distributed over the cortex in particular over the visual cortex

assessing causality from multivariate time series in this work we propose a general nonparametric test of causality for weakly dependent time series more precisely we study the problem of attribution i e the proper comparison of the relative influence that two or more external dynamics trigger on a given system of interest we illustrate the possible applications of the proposed methodology in very different fields like physiology and climate science

relative performance of mutual information estimation methods for quantifying the dependence among short and noisy data commonly used dependence measures such as linear correlation cross correlogram or kendall s tau cannot capture the complete dependence structure in data unless the structure is restricted to linear periodic or monotonic mutual information mi has been frequently utilized for capturing the complete dependence structure including nonlinear dependence recently several methods have been proposed for the mi estimation such as kernel density estimators kde k nearest neighbors knn edgeworth approximation of differential entropy and adaptive partitioning of the xy plane however outstanding gaps in the current literature have precluded the ability to effectively automate these methods which in turn have caused limited adoptions by the application communities this study attempts to address a key gap in the literature specifically the evaluation of the above methods to choose the best method particularly in terms of their robustness for short and noisy data based on comparisons with the theoretical mi estimates which can be computed analytically as well with linear correlation and kendall s tau here we consider smaller data sizes such as 50 100 and 1 000 where this study considers 50 and 100 data points as very short and 1 000 as short we consider a broader class of functions specifically linear quadratic periodicmore and chaotic contaminated with artificial noise with varying noise to signal ratios the case studies presented here are motivated by domain consideration in the earth sciences where the data are short and noisy our results indicate kde as the best choice for very short data at relatively high noise to signal levels whereas the performance of knn is the best for short data at relatively low noise levels as well as for short data consistently across noise levels in addition the optimal smoothing parameter of a gaussian kernel appears to be the best choice for kde while three nearest neighbors appear optimal for knn thus in situations where the approximate data sizes are known in advance and exploratory data analysis and or domain knowledge can be used to provide a priori insights on the noise to signal ratios the results in the paper point to a way forward for automating the process of mi estimation less

inferring the directionality of coupling with conditional mutual information uncovering the directionality of coupling is a significant step in understanding drive response relationships in complex systems in this paper we discuss a nonparametric method for detecting the directionality of coupling based on the estimation of information theoretic functionals we consider several different methods for estimating conditional mutual information the behavior of each estimator with respect to its free parameter is shown using a linear model where an analytical estimate of conditional mutual information is available numerical experiments in detecting coupling directionality are performed using chaotic oscillators where the influence of the phase extraction method and relative frequency ratio is investigated

intertwining the geodesic flow and the schrodinger group on hyperbolic surfaces we construct an explicit intertwining operator mathcal l between the schrodinger group e it frac triangle2 and the geodesic flow on certain hilbert spaces of symbols on the cotangent bundle t x of a compact hyperbolic surface x d we also define invariant eigendistributions of the geodesic flow ps j k nu j nu k patterson sullivan distributions out of pairs of triangle eigenfunctions generalizing the diagonal case j k treated in anantharaman and zelditch ann henri poincare 8 2 361 426 2007 the operator mathcal l maps ps j k nu j nu k to the wigner distribution w gamma j k studied in quantum chaos we define hilbert spaces mathcal h ps whose dual is spanned by ps j k nu j nu k resp mathcal h w whose dual is spanned by w gamma j k and show that mathcal l is a unitary isomorphism from mathcal h w to mathcal h ps

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

moments and distribution of central l values of quadratic twists of elliptic curves we show that if one can compute a little more than a particular moment for some family of l functions then one has upper bounds of the conjectured order of magnitude for all smaller positive real moments and a one sided central limit theorem holds we illustrate our method for the family of quadratic twists of an elliptic curve obtaining sharp upper bounds for all moments below the first we also establish a one sided central limit theorem supporting a conjecture of keating and snaith our work leads to a conjecture on the distribution of the order of the tate shafarevich group for rank zero quadratic twists of an elliptic curve and establishes the upper bound part of this conjecture assuming the birch swinnerton dyer conjecture

from granger causality to long term causality application to climatic data quantitative characterization of interaction between processes from time series is often required in different fields of natural science including geophysics and biophysics typically one estimates short term influences e g the widely used granger causality is defined via one step ahead predictions such an approach does not reveal how strongly the long term behavior of one process under study is affected by the others to overcome this problem we introduce the concept of long term causality which extends the concept of granger causality the long term causality is estimated from data via empirical modeling and analysis of model dynamics under different conditions apart from mathematical examples we apply both approaches to find out how strongly the global surface temperature gst is affected by variations in carbon dioxide atmospheric content solar activity and volcanic activity during the last 150 years influences of all the three factors on gst are detected with the granger causality however the long term causality shows that the rise in gst during the last decades can be explained only if the anthropogenic factor co 2 is taken into account in a model

automorphisms and derivations of free poisson algebras in two variables let p be a free poisson algebra in two variables over a field of characteristic zero we prove that the automorphisms of p are tame and that the locally nilpotent derivations of p are triangulable

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

poisson brackets and two generated subalgebras of rings of polynomials let a f x1 x2 xn be a ring of polynomials over a field f on the variables x1 x2 xn it is well known see for example 11 that the study of automorphisms of the algebra a is closely related with the description of its subalgebras by the theorem of p m cohn 4 a subalgebra of the algebra f x is free if and only if it is integrally closed the theorem of a zaks 13 says that the dedekind subalgebras of the algebra a are rings of polynomials in a single variable a nowicki and m nagata 8 proved that the kernel of any nontrivial derivation of the algebra f x y char f 0 is also a ring of polynomials in a single generator an original solution of the occurrence problem for the algebra a using the groebner basis was given by d shannon and m sweedler 9 however the method of the groebner basis does not give any information about the structure of concrete subalgebras recall that the solubility of the occurrence problem for rings of polynomials over fields of characteristic 0 was proved earlier by g noskov 7 the present paper is devoted to the investigation of the structure of twogenerated subalgebras of a in the sequel we always assume that f is an arbitrary field of characteristic 0 let us denote by f the highest homogeneous part of an element f a and by f1 f2 fk the subalgebra of a generated by the elements f1 f2 fk a definition 1 a pair of polynomials f1 f2 a is called reduced if they satisfy the following conditions 1 f1 f2 are algebraically dependent 2 f1 f2 are algebraically independent 3 f1 f2 f2 f1 recall that a pair f1 f2 with condition 3 is usually called reduced condition 1 means that we exclude the trivial case when f1 f2 are algebraically independent we do not consider the case when f1 f2 are algebraically dependent concerning this case recall the well known theorem of s s abhyankar and t t moh 1 which says that if f g f x and f g f x then f or f

on the rate of quantum ergodicity i upper bounds one problem in quantum ergodicity is to estimate the rate of decay of the sums s k lambda a frac 1 n lambda sum limits sqrt lambda j leqq lambda left a varphi j varphi j bar sigma a right k on a compact riemannian manifold m g with ergodic geodesic flow here j j are the spectral data of the of m g a is a 0 th order do bar sigma a is the liouville average of its principal symbol and n lambda j sqrt lambda j leqq lambda thats k a o 1 is proved in s z 1 cv 1 our purpose here is to show thats k a o log k 2 on a manifold of possibly variable negative curvature the main new ingredient is the central limit theorem for geodesic flows on such spaces r si

quantitative quantum ergodicity and the nodal domains of hecke maass cusp forms we prove a quantitative statement of the quantum ergodicity for hecke maass cusp forms on the modular surface as an application of our result along a density 1 subsequence of even hecke maass cusp forms we obtain a sharp lower bound for the l 2 norm of the restriction to a fixed compact geodesic segment of eta iy y 0 subset mathbb h we also obtain an upper bound of o epsilon left t phi 3 8 epsilon right for the l infty norm along a density 1 subsequence of hecke maass cusp forms for such forms this is an improvement over the upper bound of o epsilon left t phi 5 12 epsilon right given by iwaniec and sarnak in a recent work of ghosh reznikov and sarnak the authors proved for all even hecke maass forms that the number of nodal domains which intersect a geodesic segment of eta grows faster than t phi 1 12 epsilon for any epsilon 0 under the assumption that the lindelof hypothesis is true and that the geodesic segment is long enough upon removing a density zero subset of even hecke maass forms we prove without making any assumptions that the number of nodal domains grows faster than t phi 1 8 epsilon for any epsilon 0

detection of synchronization between chaotic signals an adaptive similarity based approach we present an adaptive similarity based approach to detect generalized synchronization gs with n m phase synchronization ps where n and m are integers and one of them is 1 this approach is based on the similarity index si and gaussian mixture model with the minimum description length criterion the clustering method which is shown to be superior to the closeness and connectivity of a continuous function is employed in this study to detect the existence of gs with n m ps we conducted a computer simulation and a finger lifting experiment to illustrate the effectiveness of the proposed method in the simulation of a rossler lorenz system our method outperformed the conventional si and gs with 2 1 ps within the coupled system was found in the experiment of self paced finger lifting movement cortico muscular gs with 1 2 and 1 3 ps was found between the surface electromyogram signals on the first dorsal interossei muscle and the magnetoencephalographic data in the motor area the gs with n m ps n or m 1 has been simultaneously resolved from both simulation and experiment the proposed approach thereby provides a promising means for advancing research into both nonlinear dynamics and brain science

polynomial and poisson dependence in free poisson algebras and free poisson fields abstract we prove that any two poisson dependent elements in a free poisson algebra and a free poisson field of characteristic zero are algebraically dependent thus answering positively a question from makar limanov and umirbaev 2007 8 we apply this result to give a new proof of the tameness of automorphisms for free poisson algebras of rank two see makar limanov and umirbaev 2011 9 makar limanov et al 2009 10

asymptotically stable phase synchronization revealed by autoregressive circle maps a specially designed of nonlinear time series analysis is introduced based on phases which are defined as polar angles in spaces spanned by a finite number of delayed coordinates a canonical choice of the polar axis and a related implicit estimation scheme for the potentially underlying autoregressive circle map next phase map guarantee the invertibility of reconstructed phase space trajectories to the original coordinates the resulting fourier approximated invertibility enforcing phase space map allows us to detect conditional asymptotic stability of coupled phases this comparatively general synchronization criterion unites two existing generalizations of the old concept and can successfully be applied e g to phases obtained from electrocardiogram and airflow recordings characterizing cardiorespiratory interaction

on the kuznetsov trace formula for pgl2 c abstract in this note using a representation theoretic method of cogdell and piatetski shapiro we prove the kuznetsov trace formula for an arbitrary discrete group in pgl 2 c that is cofinite but not cocompact an essential ingredient is a kernel formula recently proved by the author on bessel functions for pgl 2 c this approach avoids the difficult analysis in the existing method due to bruggeman and motohashi

statistics for mathematical properties of maps between time series embeddings we develop a set of statistics which are intended to characterize in terms of probabilities or confidence levels whether two data sets are related by a mapping with certain mathematical properties given these statistics we can ask how confident we can be that the mapping is continuous injective differentiable or has a differentiable inverse the intended use is for experimental or numerical situations in which multiple time series are generated and one wants to know what relation exists among them but the mapping between them is unknown or intractable examples of applications are testing filtered chaotic data for continuity and differentiability testing two data sets for synchronization in the most general sense testing one data set for determinism forward and backward in time and determining when transformations on two or three dimensional images are well behaved diffeomorphisms we test the statistics on several of these cases and show that they are useful for characterizing relations between data sets and for shedding light on phenomena which occur when data are transformed for example a dimension increase on filtering a chaotic data set

transition to weak generalized synchrony in chaotically driven flows we study regimes of strong and weak generalized synchronization in chaotically forced nonlinear flows the transition between these dynamical states can occur via a number of different routes and here we examine the onset of weak generalized synchrony through intermittency and blowout bifurcations the quantitative characterization of this dynamical transition is facilitated by measures that have been developed for the study of strange nonchaotic motion weak and strong generalized synchronous motion show contrasting sensitivity to parametric variation and have distinct distributions of finite time lyapunov exponents

comparison of six methods for the detection of causality in a bivariate time series in this comparative study six causality detection methods were compared namely the granger vector autoregressive test the extended granger test the kernel version of the granger test the conditional mutual information transfer entropy the evaluation of cross mappings between state spaces and an assessment of predictability improvement due to the use of mixed predictions seven test data sets were analyzed linear coupling of autoregressive models a unidirectional connection of two h enon systems a unidirectional connection of chaotic systems of r ossler and lorenz type and of two different r ossler systems an example of bidirectionally connected two species systems a fishery model as an example of two correlated observables without a causal relationship and an example of mediated causality we tested not only 20 phantom rule 0 16em 0ex 000 points long clean time series but also noisy and short variants of the data the standard and the extended granger tests worked only for the autoregressive models the remaining methods were more successful with the more complex test examples although they differed considerably in their capability to reveal the presence and the direction of coupling and to distinguish causality from mere correlation

testing for causality in reconstructed state spaces by an optimized mixed prediction method in this study a method of causality detection was designed to reveal coupling between dynamical systems represented by time series the method is based on the predictions in reconstructed state spaces the results of the proposed method were compared with outcomes of two other methods the granger var test of causality and the convergent cross mapping we used two types of test data the first test example is a unidirectional connection of chaotic systems of rossler and lorenz type the second one the fishery model is an example of two correlated observables without a causal relationship the results showed that the proposed method of optimized mixed prediction was able to reveal the presence and the direction of coupling and distinguish causality from mere correlation as well

construction of multivariate surrogate sets from nonlinear data using the wavelet transform abstract the use of surrogate data has become a crucial first step in the study of nonlinearity in time series data a widely used technique to construct surrogate data is to randomize the phases of the data in the fourier domain in this paper an alternative technique based on the resampling of wavelet coefficients is discussed this approach exploits between scale correlations that exist within nonlinear data but which are either absent or weak in stochastic data it proceeds by transforming the data into the wavelet domain and permuting the wavelet coefficients experimental and numerical time series data are used to demonstrate that the performance of the wavelet resampling technique is comparable to phase randomization in terms of the preservation of linear properties removal of nonlinear structure and computational demands however the wavelet technique may have specific and distinct advantages in the application to complex data sets such as numerical analysis of turbulence and experimental brain imaging data where wavelets give a more parsimonious representation of spatio temporal patterns than fourier modes it is shown that different techniques of resampling the data in the wavelet domain may optimize the construction of surrogate data according to the properties of the experimental time series and computational constraints

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

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

radial basis function approach to nonlinear granger causality of time series we consider an extension of granger causality to nonlinear bivariate time series in this frame if the prediction error of the first time series is reduced by including measurements from the second time series then the second time series is said to have a causal influence on the first one not all the nonlinear prediction schemes are suitable to evaluate causality indeed not all of them allow one to quantify how much knowledge of the other time series counts to improve prediction error we present an approach with bivariate time series modeled by a generalization of radial basis functions and show its application to a pair of unidirectionally coupled chaotic maps and to physiological examples

the spectral decomposition of theta 2 let theta be an elementary theta function such as the classical jacobi theta function we establish a spectral decomposition and surprisingly strong asymptotic formulas for langle theta 2 varphi rangle as varphi traverses a sequence of hecke translates of a nice enough fixed function the subtlety is that typically theta 2 notin l 2 applications to the subconvexity quantum variance and 4 norm problems we determine all pairs a f g a g h of generalized weighted quasi arithmetic means being square iterative roots of a f g a g h that is the equation a f g a g h circ a f g a g h a f g a g h is solved under three times differentiability of the functions f g h f g h as an application some special cases are presented are indicated

detecting generalized synchronization between chaotic signals a kernel based approach a unified framework for analysing generalized synchronization in coupled chaotic systems from data is proposed the key of the proposed approach is the use of the kernel methods recently developed in the field of machine learning several successful applications are presented which show the capability of the kernel based approach for detecting generalized synchronization and dynamical change of the coupling strength between two chaotic systems can be captured by the proposed approach it is also discussed how the kernel parameter is suitably chosen from data

the tame and the wild automorphisms of polynomial rings in three variables let c f x1 x2 xn be the polynomial ring in the variables x1 x2 xn over a field f and let autc be the group of automorphisms of c as an algebra over f an automorphism autc is called elementary if it has a form x1 xi 1 xi xi 1 xn 7 x1 xi 1 xi f xi 1 xn where 0 6 f f f x1 xi 1 xi 1 xn the subgroup of autc generated by all the elementary automorphisms is called the tame subgroup and the elements from this subgroup are called tame automorphisms of c non tame automorphisms of the algebra c are called wild it is well known 6 9 10 11 that the automorphisms of polynomial rings and free associative algebras in two variables are tame at present a few new proofs of these results have been found see 5 8 however in the case of three or more variables the similar question was open and known as the generation gap problem 2 3 or tame generators problem 8 the general belief was that the answer is negative and there were several candidate counterexamples see 5 8 12 7 19 the best known of them is the following automorphism aut f x y z constructed by nagata in 1972 see 12 x x x yz z y y 2 x yz x x yz z z z

dimension of interaction dynamics a method allowing one to distinguish interacting from noninteracting systems based on available time series is proposed and investigated some facts concerning generalized renyi dimensions that form the basis of our method are proved we show that one can find the dimension of the part of the attractor of the system connected with interaction between its parts we use our method to distinguish interacting from noninteracting systems on the examples of logistic and henon maps a classification of all possible interaction schemes is given

real zeros and size of rankin selberg l functions in the level aspect in this paper some asymptotic formulas are proved for the harmonic mollified second moment of a family of rankin selberg l functions one of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average a first consequence is a new subconvexity bound for rankin selberg l functions in the level aspect moreover infinitely many rankin selberg l functions having at most eight non trivial real zeros are produced and some new non trivial estimates for the analytic rank of the family studied are obtained

information directionality in coupled time series using transcripts in ordinal symbolic dynamics transcripts describe the algebraic relationship between ordinal patterns using the concept of transcript we exploit the mathematical structure of the group of permutations to derive properties and relations among information measures of the symbolic representations of time series these theoretical results are then applied for the assessment of coupling directionality in dynamical systems where suitable coupling directionality measures are introduced depending only on transcripts these measures improve the reliability of the information flow estimates and reduce to well established coupling directionality quantifiers when some general conditions are satisfied furthermore by generalizing the definition of transcript to ordinal patterns of different lengths several of the commonly used information directionality measures can be encompassed within the same framework

estimation of intensive quantities in spatio temporal systems from time series abstract we study multivariate time series generated by coupled map lattices exhibiting spatio temporal chaos and investigate to what extent we are able to estimate various intensive measures of the underlying system without explicit knowledge of the system dynamics using the rescaling and interleaving properties of the lyapunov spectrum of systems in a spatio temporally chaotic regime and paying careful attention to errors introduced by sub system boundary effects we develop algorithms that are capable of estimating the lyapunov spectrum from time series we analyse the performance of these and find that the choice of basis used to fit the dynamics is crucial when the local dynamics at a lattice site is well approximated by this basis we are able to accurately determine the full lyapunov spectrum however as the local dynamics moves away from the space spanned by this basis the performance of our algorithm deteriorates

granger causality and cross recurrence plots in rheochaos our stress relaxation measurements on wormlike micelles using a rheo sals rheology small angle light scattering apparatus allow simultaneous measurements of the stress and the scattered depolarized intensity the latter is sensitive to orientational ordering of the micelles to determine the presence of causal influences between the stress and the depolarized intensity time series we have used the technique of linear and nonlinear granger causality we find there exists a feedback mechanism between the two time series and that the orientational order has a stronger causal effect on the stress than vice versa we have also studied the phase space dynamics of the stress and the depolarized intensity time series using the recently developed technique of cross recurrence plots crps the presence of diagonal line structures in the crps unambiguously proves that the two time series share similar phase space dynamics

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom