on symmetric cauchy riemann manifolds the riemannian symmetric spaces play an important role in different branches of mathematics by definition a connected riemannian manifold m is called symmetric if to every a m there exists an involutory isometric diffeomorphism sa m m having a as isolated fixed point in m or equivalently if the differential dasa is the negative identity on the the tangent space ta tam of m at a in case such a transformation sa exists for a m it is uniquely determined and is the geodesic reflection of m about the point a as a consequence for every riemannian symmetric space m the group g gm generated by all symmetries sa a m is a lie group acting transitively on m in particular m can be identified with the homogeneous space g k for some compact subgroup k g using the elaborate theory of lie groups and lie algebras e cartan classified all riemannian symmetric spaces

information transfer between dynamical system components we present a rigorous formalism of information transfer for systems with dynamics fully known this follows from an accurate classification of the mechanisms for the entropy change of one component into a self evolution plus a transfer from the other component the formalism applies to both continuous flows and discrete maps the resulting transfer measure possesses a property of asymmetry and is qualitatively consistent with the classical measures it is further validated with the baker transformation and the henon map

information based detection of nonlinear granger causality in multivariate processes via a nonuniform embedding technique we present an approach framed in information theory to assess nonlinear causality between the subsystems of a whole stochastic or deterministic dynamical system the approach follows a sequential procedure for nonuniform embedding of multivariate time series whereby embedding vectors are built progressively on the basis of a minimization criterion applied to the entropy of the present state of the system conditioned to its past states a corrected conditional entropy estimator compensating for the biasing effect of single points in the quantized hyperspace is used to guarantee the existence of a minimum entropy rate at which to terminate the procedure the causal coupling is detected according to the granger notion of predictability improvement and is quantified in terms of information transfer we apply the approach to simulations of deterministic and stochastic systems showing its superiority over standard uniform embedding effects of quantization data length and noise contamination are investigated as practical applications we consider the assessment of cardiovascular regulatory mechanisms from the analysis of heart period arterial pressure and respiration time series and the investigation of the information flow across brain areas from multichannel scalp electroencephalographic recordings

a study of problems encountered in granger causality analysis from a neuroscience perspective granger causality methods were developed to analyze the flow of information between time series these methods have become more widely applied in neuroscience frequency domain causality measures such as those of geweke as well as multivariate methods have particular appeal in neuroscience due to the prevalence of oscillatory phenomena and highly multivariate experimental recordings despite its widespread application in many fields there are ongoing concerns regarding the applicability of granger causality methods in neuroscience when are these methods appropriate how reliably do they recover the system structure underlying the observed data what do frequency domain causality measures tell us about the functional properties of oscillatory neural systems in this paper we analyze fundamental properties of granger geweke gg causality both computational and conceptual specifically we show that i gg causality estimates can be either severely biased or of high variance both leading to spurious results ii even if estimated correctly gg causality estimates alone are not interpretable without examining the component behaviors of the system model and iii gg causality ignores critical components of a system s dynamics based on this analysis we find that the notion of causality quantified is incompatible with the objectives of many neuroscience investigations leading to highly counterintuitive and potentially misleading results through the analysis of these problems we provide important conceptual clarification of gg causality with implications for other related causality approaches and for the role of causality analyses in neuroscience as a whole

on the local solvability for top degree forms in hypo analytic structures l is the differential operator in the differential complex defined by the hypo analytic structure for most i differential forms of top degree m n the solution u should be an equivalence class of currents of bidegree m n 1 in the same paper it is also shown that the nonexistence of hypoanalytic functions h whose real parts have local extrema ensures the local solvability of 1 at least in the cases where either the tangent or the cotangent structure bundles have rank exactly equal to one and are real analytic vector bundles in the present paper we show that this remains true in the general case when the rank of the cotangent structure bundle is equal to 1 this is established by modifying the method in c h in this case the existence of hypo analytic functions whose real parts have local extrema is seen to be equivalent to the nonvanishing of the n 1 dimensional homology of the fibres of the structure see end of section 2 and cf t2 when it is the rank of the tangent structure bundle which is equal to 1 the result is derived in section 3 by reinterpreting moyer s proof of the necessity of condition nv i wish to thank paulo cordaro and jorge hounie for looking at the manuscript and pointing out errors and misprints needless to say if there are any left it is entirely my fault

instability of the cauchy kovalevskaya solution for a class of nonlinear systems we prove that in any c infty neighborhood of an analytic cauchy datum there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first order nonlinear systems we use a method initiated by g m etivier for elliptic systems based on the representation of solutions and on the fbi transform in our case the system can be hyperbolic at initial time but the characteristic roots leave the real line at positive times

some recent development in a concept of causality abstract the paper considers three separate but related topics i what is the relationship between causation and co integration if a pair of i 1 series are co integration there must be causation in at least one direction an implication is that some tests of causation based on different series may have missed one source of causation ii is there a need for a definition of instantaneous causation in a decision science it is argued that no such definition is required iii can causality tests be used for policy evaluation it is suggested that these tests are useful but that they should be evaluated with care

optimum cycle redundancy check codes with 16 bit redundancy optimal binary cyclic redundancy check codes with 16 parity bits crc 16 codes are presented and compared to those in existing standards for minimum distance undetected error probability on binary symmetric channels bscs and properness the codes in several cases are seen to be superior at block lengths of practical interest when they are used on low noise bscs the optimum minimum distance obtainable by some crc 16 codes is determined for all block lengths for several typical low noise bscs the minimum undetected error probability achievable with some crc 16 codes is given for all block lengths

kernel canonical correlation granger causality for multiple time series canonical correlation analysis as a multivariate statistical technique has been applied to multivariate granger causality analysis to infer information flow in complex systems it shows unique appeal and great superiority over the traditional vector autoregressive method due to the simplified procedure that detects causal interaction between multiple time series and the avoidance of potential model estimation problems however it is limited to the linear case here we extend the framework of canonical correlation to include the estimation of multivariate nonlinear granger causality for drawing inference about directed interaction its feasibility and effectiveness are verified on simulated data

exploration of synergistic and redundant information sharing in static and dynamical gaussian systems to fully characterize the information that two source variables carry about a third target variable one must decompose the total information into redundant unique and synergistic components i e obtain a partial information decomposition pid however shannon s theory of information does not provide formulas to fully determine these quantities several recent studies have begun addressing this some possible definitions for pid quantities have been proposed and some analyses have been carried out on systems composed of discrete variables here we present an in depth analysis of pids on gaussian systems both static and dynamical we show that for a broad class of gaussian systems previously proposed pid formulas imply that i redundancy reduces to the minimum information provided by either source variable and hence is independent of correlation between sources and ii synergy is the extra information contributed by the weaker source when the stronger source is known and can either increase or decrease with correlation between sources we find that gaussian systems frequently exhibit net synergy i e the information carried jointly by both sources is greater than the sum of information carried by each source individually drawing from several explicit examples we discuss the implications of these findings for measures of information transfer and information based measures of complexity both generally and within a neuroscience setting importantly by providing independent formulas for synergy and redundancy applicable to continuous time series data we provide an approach to characterizing and quantifying information sharing amongst complex system variables

normalizing the causality between time series recently a rigorous yet concise formula was derived to evaluate information flow and hence the causality in a quantitative sense between time series to assess the importance of a resulting causality it needs to be normalized the normalization is achieved through distinguishing a lyapunov exponent like one dimensional phase space stretching rate and a noise to signal ratio from the rate of information flow in the balance of the marginal entropy evolution of the flow recipient it is verified with autoregressive models and applied to a real financial analysis problem an unusually strong one way causality is identified from ibm international business machines corporation to ge general electric company in their early era revealing to us an old story which has almost faded into oblivion about seven dwarfs competing with a giant for the mainframe computer market

holomorphic extension and decomposition from a totally real manifold this paper is to develop an elementary cohomological approach for decomposing a function into boundary values of holomorphic functions and for discussing the corresponding microlocal analysis and hyperfunction theory

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

gevrey solvability and gevrey regularity in differential complexes associated to locally integrable structures in this work we study some properties of the differential complex associated to a locally integrable involutive structure acting on forms with gevrey coefficients among other results we prove that for such complexes gevrey solvability follows from smooth solvability under the sole assumption of a regularity condition as a consequence we obtain the proof of the gevrey solvability for a first order linear pde with real analytic coefficients satisfying the nirenberg treves condition p

uniqueness in certain first order nonlinear complex cauchy problems soit f t x une fonction holomorphe definie dans un voisinage ouvert complexe de t 0 x 0 0 0 r r n c c n on etudie l unicite des solutions c 2 a valeurs complexes u definies dans un voisinage de t 0 x 0 dans l espace reel du probleme de cauchy d ordre 1 non lineaire u t f t x u u x u t 0 x u 0 x avec u 0 x 0 0 u 0 x x 0 0

cr structures with group action and extendability of cr functions part i presents results on local embedding of cr structures we consider an abstract cr manifold whose structure is invariant under a transversal lie group action we show that such a manifold can always be locally embedded in complex space as a generic submanifold the proof is based on selection of canonical coordinates and repeated use of the newlander nirenberg theorem 13 when the lie group is abelian the embedding can be given a particularly simple form let l 1 be the codimension of our submanifold called m throughout the paper it is then convenient to denote by n i the dimension of the ambient complex space and by zl z w 1 w t the complex coordinates we shall systematically write

unified framework for information integration based on information geometry assessment of causal influences is a ubiquitous and important subject across diverse research fields drawn from consciousness studies integrated information is a measure that defines integration as the degree of causal influences among elements whereas pairwise causal influences between elements can be quantified with existing methods quantifying multiple influences among many elements poses two major mathematical difficulties first overestimation occurs due to interdependence among influences if each influence is separately quantified in a part based manner and then simply summed over second it is difficult to isolate causal influences while avoiding noncausal confounding influences to resolve these difficulties we propose a theoretical framework based on information geometry for the quantification of multiple causal influences with a holistic approach we derive a measure of integrated information which is geometrically interpreted as the divergence between the actual probability distribution of a system and an approximated probability distribution where causal influences among elements are statistically disconnected this framework provides intuitive geometric interpretations harmonizing various information theoretic measures in a unified manner including mutual information transfer entropy stochastic interaction and integrated information each of which is characterized by how causal influences are disconnected in addition to the mathematical assessment of consciousness our framework should help to analyze causal relationships in complex systems in a complete and hierarchical manner

the edge of the wedge theorem for systems of constant coefficient partial differential operators i on demontre des resultats sur l extendabilite holomorphe des fonctions holomorphes definies sur deux coins ou plus et pour lesquelles la somme des valeurs limites s annulent

on the probability of undetected error for linear block codes the problem of computing the probability of undetected error is considered for linear block codes used for error detection the recent literature is first reviewed and several results are extended it is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual using the dual code formulation the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes several bounds on the undetected error probability are then presented we conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained an efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results

time irreversibility of financial time series based on higher moments and multiscale kullback leibler divergence abstract irreversibility is an important property of time series in this paper we propose the higher moments and multiscale kullback leibler divergence to analyze time series irreversibility the higher moments kullback leibler divergence hmkld can amplify irreversibility and make the irreversibility variation more obvious therefore many time series whose irreversibility is hard to be found are also able to show the variations we employ simulated data and financial stock data to test and verify this method and find that hmkld of stock data is growing in the form of fluctuations as for multiscale kullback leibler divergence mkld it is very complex in the dynamic system so that exploring the law of simulation and stock system is difficult in conventional multiscale entropy method the coarse graining process is non overlapping however we apply a different coarse graining process and obtain a surprising discovery the result shows when the scales are 4 and 5 their entropy is nearly similar which demonstrates mkld is efficient to display characteristics of time series irreversibility

a generalization of borel s theorem and microlocal gevrey regularity in involutive structures in this paper we use borel s procedure to construct gevrey approximate solutions of an initial value problem for involutive systems of gevrey complex vector fields as an application we describe the gevrey wave front set of the boundary values of approximate solutions in wedges w of gevrey involutive structures m v we prove that the gevrey wave front set of the boundary value is contained in the polar of a certain cone t w contained in rv tx where x is a maximally real edge of w we also prove a partial converse

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

granger causality maps of diffusion processes granger causality is a statistical concept devised to reconstruct and quantify predictive information flow between stochastic processes although the general concept can be formulated model free it is often considered in the framework of linear stochastic processes here we show how local linear model descriptions can be employed to extend granger causality into the realm of nonlinear systems this novel treatment results in maps that resolve granger causality in regions of state space through examples we provide a proof of concept and illustrate the utility of these maps moreover by integration we convert the local granger causality into a global measure that yields a consistent picture for a global ornstein uhlenbeck process finally we recover invariance transformations known from the theory of autoregressive processes

existence of gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first order nonlinear pdes abstract given a g s involutive structure m v a gevrey submanifold x m which is maximally real and a gevrey function u 0 on x we construct a gevrey function u which extends u 0 and is a gevrey approximate solution for v we then use our construction to study gevrey micro local regularity of solutions u c 2 r n of a system of nonlinear pdes of the form f j x u u x 0 j 1 n where f j x 0 are gevrey functions of order s 1 and holomorphic in 0 c c n the functions f j satisfy an involutive condition and d f 1 d f n 0

on microlocal analyticity and smoothness of solutions of first order nonlinear pdes we study the microlocal analyticity and smoothness of solutions u of of the nonlinear pde ut f x t u ux under some assumptions on the repeated brackets of the linearized operator and its conjugate

sub bundles of the complexified tangent bundle we study embeddings of complex vector bundles especially line bundles in the complexification of the tangent bundle of a manifold the aim is to understand implications of properties of interest in partial differential equations

on the analyticity of solutions of first order nonlinear pde let x t e r m r and u e c 2 r m r we discuss local and microlocal analyticity for solutions u to the nonlinear equation u t f x t u u x here f x t 0 is complex valued and analytic in all arguments we also assume f to be holomorphic in 0 e c c m in particular we show that wf a u char l u where wf a denotes the analytic wave front set and char l u is the characteristic set of the linearized operator l u f j x t u u x x j

multi scale symbolic transfer entropy analysis of eeg abstract from both global and local perspectives we symbolize two kinds of eeg and analyze their dynamic and asymmetrical information using multi scale transfer entropy multi scale process with scale factor from 1 to 199 and step size of 2 is applied to eeg of healthy people and epileptic patients and then the permutation with embedding dimension of 3 and global approach are used to symbolize the sequences the forward and reverse symbol sequences are taken as the inputs of transfer entropy scale factor intervals of permutation and global way are 37 57 and 65 85 where the two kinds of eeg have satisfied entropy distinctions when scale factor is 67 transfer entropy of the healthy and epileptic subjects of permutation 0 1137 and 0 1028 have biggest difference and the corresponding values of the global symbolization is 0 0641 and 0 0601 which lies in the scale factor of 165 research results show that permutation which takes contribution of local information has better distinction and is more effectively applied to our multi scale transfer entropy analysis of eeg

an asymptotic formula for hypo analytic pseudodifferential operators an asymptotic expansion formula for hypo analytic pseudodifferential operators is proved and applications are given introduction in 2 we introduced hypo analytic pseudodifferential operators that are naturally associated with the hypo analytic structures of 1 in this paper we establish an asymptotic formula for these operators such an expansion is essential in several applications it allows us to define in a natural way the symbol of a hypo analytic pseudodifferential operator as well as the symbols of the adjoint transpose and composition of operators the paper is organized as follows in chapter i we discuss and develop the asymptotic formula chapter ii applies this formula to two results acknowledgment it is a pleasure to express my thanks to professor f treves for many stimulating discussions 1 asymptotic expansion 1 hypo analytic structures we will deal with structures which are a special case of the hypo analytic structures introduced by baouendi chang and treves in 1 we shall summarize the relevant concepts here let q be a c manifold of dimension m a hypo analytic structure of maximal dimension on q is the data of an open covering u of q and for each index al of m co functions z zm satisfying the following two conditions i dz dzm are linearly independent at each point of u ii if ua n u 7 0 there are open neighborhoods of z u un and of zfi u n uf and a holomorphic map fa of onto such that z fg o z on u n ua 8 on tflufi we will use the notation z z z m u cm a distribution h defined in an open neighborhood of a point po of q is hypo analytic at po received by the editors august 14 1989 1980 mathematics subject classilfication 1985 revision primary 35a20 secondary 35s99 1990 american mathematical society 0002 9947 90 1 00 25 per page

optimization of cyclic redundancy check codes with 24 and 32 parity bits the method developed by t fujiwara et al 1985 for efficiently computing the minimum distance of shortened hamming codes using the weight distribution of their dual codes is extended to treat arbitrary shortened cyclic codes using this method implemented on a high speed special purpose processor several classes of cyclic redundancy check crc codes with 24 and 32 parity bits are investigated the crc codes of each class are known to have the same minimum distance d sub min l in a certain range l of block lengths n and within each class that crc code has been determined the minimum distance of which exceeds d sub min l up to the largest block length the d sub min profiles of the resulting codes are presented and compared with the d sub min profiles of recent suggestions of p merkey and e c posner 1984 as well as with the d sub min profile of the widely used 32 parity bit standard code recommended in ieee 802

granger causality and transfer entropy are equivalent for gaussian variables granger causality is a statistical notion of causal influence based on prediction via vector autoregression developed originally in the field of econometrics it has since found application in a broader arena particularly in neuroscience more recently transfer entropy an information theoretic measure of time directed information transfer between jointly dependent processes has gained traction in a similarly wide field while it has been recognized that the two concepts must be related the exact relationship has until now not been formally described here we show that for gaussian variables granger causality and transfer entropy are entirely equivalent thus bridging autoregressive and information theoretic approaches to data driven causal inference

microglobal regularity and the global wavefront set in this paper we begin the study of regularity of partial differential equations in the space of global l q gevrey functions recently introduced in adwan et al j geom anal 27 3 1874 1913 2017 and hoepfner and raich indiana univ math j forthcoming and in a generalized and new function space called the space of global l q denjoy carleman functions we develop a wedge approach similar to bony s theorem bony in seminaire goulaouic schwartz 1976 1977 equations aux derivees partielles et analyse fonctionnelle exp no 3 centre math ecole polytech palaiseau 1977 and prove three main theorems the first establishes the existence of boundary values of continuous functions on a wedge next we borrow the fbi transform approach from hoepfner and raich forthcoming to define global wavefront sets and prove a relationship between the inclusion of a direction in the global wavefront set and the existence of boundary values of sums of weighted l p functions defined in wedges the final result is an application in which we prove a global version of a classical result namely the relationship between the global characteristic set of a partial differential operator p and the microglobal wavefront sets of u and pu

an f and m riesz theorem for planar vector fields we prove that solutions of the homogeneous equation lu 0 where l is a locally integrable vector field with smooth coefficients in two variables possess the f and m riesz property that is if omega is an open subset of the plane with smooth boundary u in c 1 omega satisfieslu 0 on omega has tempered growth at the boundary and its weak boundary value is a measure mu then mu is absolutely continuous with respect to lebesgue measure on the noncharacteristic portion of partial omega

expanding the transfer entropy to identify information circuits in complex systems we propose a formal expansion of the transfer entropy to put in evidence irreducible sets of variables which provide information for the future state of each assigned target multiplets characterized by a large contribution to the expansion are associated to the informational circuits present in the system with an informational character which can be associated to the sign of the contribution for the sake of computational complexity we adopt the assumption of gaussianity and use the corresponding exact formula for the conditional mutual information we report the application of the proposed methodology on two electroencephalography eeg data sets

approximate solutions and micro regularity in the denjoy carleman classes abstract we begin with a sequence m of positive real numbers and we consider the denjoy carleman class c m we show how to construct m approximate solutions for complex vector fields with c m coefficients we then use our construction to study micro local properties of boundary values of approximate solutions in general m involutive structures of codimension one where the approximate solution is defined in a wedge whose edge where the boundary value exists is a maximally real submanifold we also obtain a c m version of the edge of the wedge theorem

granger causality analysis with nonuniform sampling and its application to pulse coupled nonlinear dynamics the granger causality gc analysis is an effective approach to infer causal relations for time series however for data obtained by uniform sampling i e with an equal sampling time interval it is known that gc can yield unreliable causal inference due to aliasing if the sampling rate is not sufficiently high to solve this unreliability issue we consider the nonuniform sampling scheme as it can mitigate against aliasing by developing an unbiased estimation of power spectral density of nonuniformly sampled time series we establish a framework of spectrum based nonparametric gc analysis applying this framework to a general class of pulse coupled nonlinear networks and utilizing some particular spectral structure possessed by these nonlinear network data we demonstrate that for such nonlinear networks with nonuniformly sampled data reliable gc inference can be achieved at a low nonuniform mean sampling rate at which the traditional uniform sampling gc may lead to spurious causal inference

minimal sets in monotone and concave skew product semiflows ii two dimensional systems of differential equations abstract recurrent nonautonomous two dimensional systems of differential equations of ordinary finite delay and reaction diffusion types given by cooperative and concave vector fields define monotone and concave skew product semiflows whose dynamics is analyzed in this paper a complete description of all the minimal sets in a very interesting dynamical situation is provided and some criteria to determine its possible existence and the area on which they lie are given suitable examples prove the optimality of the results

granger causality for state space models granger causality has long been a prominent method for inferring causal interactions between stochastic variables for a broad range of complex physical systems however it has been recognized that a moving average ma component in the data presents a serious confound to granger causal analysis as routinely performed via autoregressive ar modeling we solve this problem by demonstrating that granger causality may be calculated simply and efficiently from the parameters of a state space ss model since ss models are equivalent to autoregressive moving average models granger causality estimated in this fashion is not degraded by the presence of a ma component this is of particular significance when the data has been filtered downsampled observed with noise or is a subprocess of a higher dimensional process since all of these operations commonplace in application domains as diverse as climate science econometrics and the neurosciences induce a ma component we show how granger causality conditional and unconditional in both time and frequency domains may be calculated directly from ss model parameters via solution of a discrete algebraic riccati equation numerical simulations demonstrate that granger causality estimators thus derived have greater statistical power and smaller bias than ar estimators we also discuss how the ss approach facilitates relaxation of the assumptions of linearity stationarity and homoscedasticity underlying current ar methods thus opening up potentially significant new areas of research in granger causal analysis

remarks on the integrability of first order complex pde abstract the classical method of characteristics for the integration of real first order pde is extended to complex equations via a symplectic structure on the complex cotangent bundle that is holomorphic along the fibres integrability is shown to be equivalent to a variant of darboux s theorem i e the existence of symplectic frames that incorporate the equations applications of the linear theory to the associated hamiltonian system yield integrability results for the nonlinear pde

causation entropy identifies indirect influences dominance of neighbors and anticipatory couplings abstract inference of causality is central in nonlinear time series analysis and science in general a popular approach to infer causality between two processes is to measure the information flow between them in terms of transfer entropy using dynamics of coupled oscillator networks we show that although transfer entropy can successfully detect information flow in two processes it often results in erroneous identification of network connections under the presence of indirect interactions dominance of neighbors or anticipatory couplings such effects are found to be profound for time dependent networks to overcome these limitations we develop a measure called causation entropy and show that its application can lead to reliable identification of true couplings

information flow within stochastic dynamical systems information flow or information transfer is an important concept in general physics and dynamical systems which has applications in a wide variety of scientific disciplines in this study we show that a rigorous formalism can be established in the context of a generic stochastic dynamical system an explicit formula has been obtained for the resulting transfer measure which possesses a property of transfer asymmetry and if the stochastic perturbation to the receiving component does not rely on the giving component has a form the same as that for the corresponding deterministic system this formula is further illustrated and validated with a two dimensional langevin equation a remarkable observation is that for two highly correlated time series there could be no information transfer from one certain series say x 2 to the other x 1 that is to say the evolution of x 1 may have nothing to do with x 2 even though x 1 and x 2 are highly correlated information flow analysis thus extends the traditional notion of correlation analysis and or mutual information analysis by providing a quantitative measure of causality between dynamical events

minimal sets in monotone and concave skew product semiflows i a general theory abstract the long term dynamics of a general monotone and concave skew product semiflow is analyzed paying special attention to the region delimited from below by the graph of a semicontinuous subequilibrium or by a minimal set admitting a flow extension different possibilities arise depending on the existence and number or absence of minimal sets strongly above the initially fixed one as well as on the coexistence or not of bounded and unbounded semiorbits on the region previous results are unified and extended and significative differences with the sublinear case are established some scenarios which are impossible in the autonomous or periodic cases may occur in this setting

on boundary properties of solutions of complex vector fields abstract this work presents results on the boundary properties of solutions of a complex planar smooth vector field l classical results in the h p theory of holomorphic functions of one variable are extended to the solutions of a class of nonelliptic complex vector fields

unraveling the cause effect relation between time series given two time series can one faithfully tell in a rigorous and quantitative way the cause and effect between them based on a recently rigorized physical notion namely information flow we solve an inverse problem and give this important and challenging question which is of interest in a wide variety of disciplines a positive answer here causality is measured by the time rate of information flowing from one series to the other the resulting formula is tight in form involving only commonly used statistics namely sample covariances an immediate corollary is that causation implies correlation but correlation does not imply causation it has been validated with touchstone linear and nonlinear series purportedly generated with one way causality that evades the traditional approaches it has also been applied successfully to the investigation of real world problems an example presented here is the cause and effect relation between the two climate modes el nino and the indian ocean dipole iod which have been linked to hazards in far flung regions of the globe in general the two modes are mutually causal but the causality is asymmetric el nino tends to stabilize iod while iod functions to make el nino more uncertain to el nino the information flowing from iod manifests itself as a propagation of uncertainty from the indian ocean

a multivariate faa di bruno formula with applications a multivariate faa di bruno formula for computing arbitrary partial derivatives of a function composition is presented it is shown by way of a general identity how such derivatives can also be expressed in the form of an infinite series applications to stochastic processes and multivariate cumulants are then delineated

information transfer in continuous processes we discuss a recently proposed quantity called transfer entropy which uses time series data to measure the amount of information transferred from one process to another in order to understand its foundation merits and limitations we review some aspects of information theoretic functionals while for symbol sequences these measures have an intuitive interpretation their application to continuous state processes and in particular their estimation from finite data sets is problematic for mutual information finite length scale estimates converge from below and can thus be used to reject the assumption that the observed processes are independent however mutual information does not provide any directional information conversely transfer entropy does resolve the directionality of information exchange but no similar monotonic convergence seems to hold thus only in the case of zero transfer entropy in one direction we can reliably infer an asymmetry of the information exchange

gevrey micro regularity for solutions to first order nonlinear pde abstract let x t r m r and u c 2 r m r we study the gevrey micro regularity of solutions u of the nonlinear equation u t f x t u u x where f x t 0 is a gevrey function of order s 1 and holomorphic in 0 we show that the gevrey wave front set of any c 2 solution u is contained in the characteristic set of the linearized operator l u t j 1 m f j x t u u x x j to achieve this we study the notion of gevrey approximate solutions a concept which we believe is of independent interest and could be applied to much more general situations

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

on the applicability of the lead lag ratio in causality assessment abstract within the large set of metrics that have been proposed to assess the presence of a causality relationship between time series the lead lag ratio llr has recently attracted increasing attention especially in the econophysics community it is based on the analysis of the asymmetry between the correlations obtained when one time series is shifted back or ahead in time with respect to the second one or in other words on the assessment of whether one time series is leading the other in spite of its popularity undoubtedly due to its conceptual and computational simplicity no empirical validation of the llr has hitherto been proposed we here take a first step towards this validation by presenting results corresponding to a large set of analyses on synthetic time series and comparing them with those yielded by the celebrated granger causality metric we show how in spite of behaving similarly to the granger causality in some specific situations the llr presents important drawbacks among which its unreliableness and the absence of a statistical significance test stand out that discourage its use in real world analyses

neural complexity a graph theoretic interpretation one of the central challenges facing modern neuroscience is to explain the ability of the nervous system to coherently integrate information across distinct functional modules in the absence of a central executive to this end tononi et al proc nat acad sci usa 91 5033 1994 proposed a measure of neural complexity that purports to capture this property based on mutual information between complementary subsets of a system neural complexity so defined is one of a family of information theoretic metrics developed to measure the balance between the segregation and integration of a system s dynamics one key question arising for such measures involves understanding how they are influenced by network topology sporns et al cereb cortex 10 127 2000 employed numerical models in order to determine the dependence of neural complexity on the topological features of a network however a complete picture has yet to be established while de lucia et al phys rev e 71 016114 2005 made the first attempts at an analytical account of this relationship their work utilized a formulation of neural complexity that we argue did not reflect the intuitions of the original work in this paper we start by describing weighted connection matrices formed by applying a random continuous weight distribution to binary adjacency matrices this allows us to derive an approximation for neural complexity in terms of the moments of the weight distribution and elementary graph motifs in particular we explicitly establish a dependency of neural complexity on cyclic graph motifs

the fbi transforms and their use in microlocal analysis abstract using a more general class of fbi transforms introduced by s berhanu and j hounie in 12 we completely characterize regularity and microregularity in denjoy carleman non quasi analytic classes which includes the gevrey classes and m christ version of the fbi transform defined in 22 as examples we also exhibit a result on microlocal regularity for solutions of first order partial differential equations in these classes that do not seem possible to prove using the classical fbi transform

calcul paradifferentiel precise et applications a des equations aux derivees partielles non semilineaires on demontre des theoremes de regularite pour une solution u suffisamment reguliere de l equation f x u u m 0 ou f est une fonction c de ses arguments dans le cas de l ordre 1 ou de la dimension 2

quasianalytic denjoy carleman classes and o minimality we show that the expansion of the real field generated by the functions of a quasianalytic denjoy carleman class is model complete and o minimal provided that the class satisfies certain closure conditions some of these structures do not admit analytic cell decomposition and they show that there is no largest o minimal expansion of the real field

content based multiple bitstream image transmission over noisy channels we propose a novel combined source and channel coding scheme for image transmission over noisy channels the main feature of the proposed scheme is a systematic decomposition of image sources so that unequal error protection can be applied according to not only bit error sensitivity but also visual content importance the wavelet transform is adopted to hierarchically decompose the image the association between the wavelet coefficients and what they represent spatially in the original image is fully exploited so that wavelet blocks are classified based on their corresponding image content the classification produces wavelet blocks in each class with similar content and statistics therefore enables high performance source compression using the set partitioning in hierarchical trees spiht algorithm to combat the channel noise an unequal error protection strategy with rate compatible punctured convolutional cyclic redundancy check rcpc crc codes is implemented based on the bit contribution to both peak signal to noise ratio psnr and visual quality at the receiving end a postprocessing method making use of the spiht decoding structure and the classification map is developed to restore the degradation due to the residual error after channel decoding experimental results show that the proposed scheme is indeed able to provide protection both for the bits that are more sensitive to errors and for the more important visual content under a noisy transmission environment in particular the reconstructed images illustrate consistently better visual quality than using the single bitstream based schemes

determination of the best crc codes with up to 10 bit redundancy all binary polynomials of degree up to 10 which are suitable to be used as generator polynomials of crc codes are classified and all the necessary data for the evaluation of the error control performance of the crc codes generated by the classified polynomials is calculated a procedure based on the computed data for choosing the best crc code is suggested