quantifying temporal correlations a test retest evaluation of functional connectivity in resting state fmri there have been many interpretations of functional connectivity and proposed measures of temporal correlations between bold signals across different brain areas these interpretations yield from many studies on functional connectivity using resting state fmri data that have emerged in recent years however not all of these studies used the same metrics for quantifying the temporal correlations between brain regions in this paper we use a public domain test retest resting state fmri data set to perform a systematic investigation of the stability of the metrics that are often used in resting state functional connectivity fc studies the fmri data set was collected across three different sessions the second session took place approximately eleven months after the first session and the third session was an hour after the second session the fc metrics composed of cross correlation partial cross correlation cross coherence and parameters based on an autoregressive model we discussed the strengths and weaknesses of each metric we performed roi level and full brain seed based voxelwise test retest analyses using each fc metric to assess its stability for both roi level and voxel level analyses we found that cross correlation yielded more stable measurements than the other metrics we discussed the consequences of this result on the utility of the fc metrics we observed that for negatively correlated rois their partial cross correlation is shrunk towards zero thus affecting the stability of their fc for the present data set we found greater stability in fc between the second and third sessions one hour between sessions compared to the first and second sessions approximately 11 months between sessions finally we report that some of the metrics showed a positive association between strength and stability in summary the results presented in this paper suggest important implications when choosing metrics for quantifying and assessing various types of functional connectivity for resting state fmri studies

congruences involving arithmetic progressions for weakly holomorphic modular forms abstract in this paper we give a classification of weights k such that there is a nonzero weakly holomorphic modular form f a n q n of weight k on 1 n having infinitely many congruences of the form a l n 0 mod l where l is a prime and is an integer in 0 1 l 1 these are similar to congruences for the partition function investigated by ramanujan furthermore we characterize linear combinations of shimura theta functions with odd characters in terms of these congruences as an application of our main theorem we consider a generalization of newman s conjecture for weakly holomorphic modular forms on 0 n with real dirichlet characters finally we use these results to study arithmetic properties of colored partitions and generalized frobenius partitions

blowup stability at optimal regularity for the critical wave equation abstract we establish strichartz estimates for the radial energy critical wave equation in 5 dimensions in similarity coordinates using these we prove the nonlinear asymptotic stability of the ode blowup in the energy space

short and long run causality measures theory and inference abstract the concept of causality introduced by wiener wiener n 1956 the theory of prediction in e f beckenback ed the theory of prediction mcgraw hill new york chapter 8 and granger granger c w j 1969 investigating causal relations by econometric models and cross spectral methods econometrica 37 424 459 is defined in terms of predictability one period ahead this concept can be generalized by considering causality at any given horizon h as well as tests for the corresponding non causality dufour j m renault e 1998 short run and long run causality in time series theory econometrica 66 1099 1125 dufour j m pelletier d renault e 2006 short run and long run causality in time series inference journal of econometrics 132 2 337 362 instead of tests for non causality at a given horizon we study the problem of measuring causality between two vector processes existing causality measures have been defined only for the horizon 1 and they fail to capture indirect causality we propose generalizations to any horizon h of the measures introduced by geweke geweke j 1982 measurement of linear dependence and feedback between multiple time series journal of the american statistical association 77 304 313 nonparametric and parametric measures of unidirectional causality and instantaneous effects are considered on noting that the causality measures typically involve complex functions of model parameters in var and varma models we propose a simple simulation based method to evaluate these measures for any varma model we also describe asymptotically valid nonparametric confidence intervals based on a bootstrap technique finally the proposed measures are applied to study causality relations at different horizons between macroeconomic monetary and financial variables in the us

temporal autocorrelation in univariate linear modeling of fmri data in functional magnetic resonance imaging statistical analysis there are problems with accounting for temporal autocorrelations when assessing change within voxels techniques to date have utilized temporal filtering strategies to either shape these autocorrelations or remove them shaping or coloring attempts to negate the effects of not accurately knowing the intrinsic autocorrelations by imposing known autocorrelation via temporal filtering removing the autocorrelation or prewhitening gives the best linear unbiased estimator assuming that the autocorrelation is accurately known for single event designs the efficiency of the estimator is considerably higher for prewhitening compared with coloring however it has been suggested that sufficiently accurate estimates of the autocorrelation are currently not available to give prewhitening acceptable bias to overcome this we consider different ways to estimate the autocorrelation for use in prewhitening after high pass filtering is performed a tukey taper set to smooth the spectral density more than would normally be used in spectral density estimation performs best importantly estimation is further improved by using nonlinear spatial filtering to smooth the estimated autocorrelation but only within tissue type using this approach when prewhitening reduced bias to close to zero at probability levels as low as 1 x 10 5

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

renormalization and blow up for charge one equivariant critical wave maps we prove the existence of equivariant finite time blow up solutions for the wave map problem from 2 1 s 2 of the form u t r q lambda t r mathcal r t r where u is the polar angle on the sphere q r 2 arctan r is the ground state harmonic map t t 1 and mathcal r t r is a radiative error with local energy going to zero as t 0 the number nu frac 1 2 can be prescribed arbitrarily this is accomplished by first renormalizing the blow up profile followed by a perturbative analysis

granger geweke causality estimation and interpretation abstract in a recent pnas article1 stokes and purdon performed numerical simulations to argue that granger geweke causality ggc estimation is severely biased or of high variance and ggc application to neuroscience is problematic because the ggc measure is independent of receiver dynamics here we use the same simulation examples to show that ggc measures when properly estimated either via the spectral factorization enabled nonparametric approach or the var model based parametric approach do not have the claimed bias and high variance problems further the receiver independence property of ggc does not present a problem for neuroscience applications when the nature and context of experimental measurements are taken into consideration ggc along with other spectral quantities yield neurophysiologically interpretable results

on the formation of singularities in the critical o 3 sigma model we study the phenomena of energy concentration for the critical o 3 sigma model also known as the wave map flow from 2 1 minkowski space into the sphere s 2 we establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities the construction and analysis are done in the context of the k equivariant symmetry reduction and we restrict to maps with homotopy class k 4 the concentration mechanism we uncover is essentially due to a resonant self focusing shrinking of a corresponding harmonic map we show that the phenomenon is generic e g in certain sobolev spaces in that it persists under small perturbations of initial data while the resulting blowup is bounded by a log modified self similar asymptotic

strichartz estimates in similarity coordinates and stable blowup for the critical wave equation we establish strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a time dependent self similar potential as an application we consider the critical wave equation and prove the asymptotic stability of the ode blowup profile in the energy space

construction of type ii blowup solutions for the 1 corotational energy supercritical wave maps abstract we consider the energy supercritical wave maps from r d into the d sphere s d with d 7 under an additional assumption of 1 corotational symmetry the problem reduces to the one dimensional semilinear wave equation t 2 u r 2 u d 1 r r u d 1 2 r 2 sin 2 u we construct for this equation a family of c solutions which blow up in finite time via concentration of the universal profile u r t q r t where q is the stationary solution of the equation and the speed is given by the quantized rates t c u t t l l n l d 1 2 the construction relies on two arguments the reduction of the problem to a finite dimensional one thanks to a robust universal energy method and modulation techniques developed by merle raphael and rodnianski 49 for the energy supercritical nonlinear schrodinger equation then we proceed by contradiction to solve the finite dimensional problem and conclude using the brouwer fixed point theorem

causality analysis of neural connectivity critical examination of existing methods and advances of new methods granger causality gc is one of the most popular measures to reveal causality influence of time series and has been widely applied in economics and neuroscience especially its counterpart in frequency domain spectral gc as well as other granger like causality measures have recently been applied to study causal interactions between brain areas in different frequency ranges during cognitive and perceptual tasks in this paper we show that 1 gc in time domain cannot correctly determine how strongly one time series influences the other when there is directional causality between two time series and 2 spectral gc and other granger like causality measures have inherent shortcomings and or limitations because of the use of the transfer function or its inverse matrix and partial information of the linear regression model on the other hand we propose two novel causality measures in time and frequency domains for the linear regression model called new causality and new spectral causality respectively which are more reasonable and understandable than gc or granger like measures especially from one simple example we point out that in time domain both new causality and gc adopt the concept of proportion but they are defined on two different equations where one equation for gc is only part of the other for new causality thus the new causality is a natural extension of gc and has a sound conceptual theoretical basis and gc is not the desired causal influence at all by several examples we confirm that new causality measures have distinct advantages over gc or granger like measures finally we conduct event related potential causality analysis for a subject with intracranial depth electrodes undergoing evaluation for epilepsy surgery and show that in the frequency domain all measures reveal significant directional event related causality but the result from new spectral causality is consistent with event related time frequency power spectrum activity the spectral gc as well as other granger like measures are shown to generate misleading results the proposed new causality measures may have wide potential applications in economics and neuroscience

functional connectivity studying nonlinear delayed interactions between bold signals correlation analysis has been widely used in the study of functional connectivity based on fmri data it assumes that the relevant information about the interactions of brain regions is reflected by a linear relationship between the values of two signals at the same time however this hypothesis has not been thoroughly investigated yet in this work we study in depth the information shared by bold signals of pairs of brain regions in particular we assess the amount of nonlinear and or nonsynchronous interactions present in data this is achieved by testing models reflecting linear synchronous interactions against more general models encompassing nonlinear nonsynchronous interactions many factors influencing measured bold signals are critical for the study of connectivity such as paradigm induced bold responses preprocessing motion artifacts and geometrical distortions interactions are also influenced by the proximity of brain regions the influence of all these factors is taken into account and the nature of the interactions is studied using various experimental conditions such that the conclusions reached are robust with respect to variation of these factors after defining nonlinear and or nonsynchronous interaction models in the framework of general linear models statistical tests are performed on different fmri data sets to infer the nature of the interactions finally a new connectivity metric is proposed which takes these inferences into account we find that bold signal interactions are statistically more significant when taking into account the history of the distant signal i e the signal from the interacting region than when using a model of linear instantaneous interaction moreover about 75 of the interactions are symmetric as assessed with the proposed connectivity metric the history dependent part of the coupling between brain regions can explain a high percentage of the variance in the data sets studied as these results are robust with respect to various confounding factors this work suggests that models used to study the functional connectivity between brain areas should in general take the bold signal history into account

measures of conditional linear dependence and feedback between time series abstract measures of linear dependence and feedback for two multiple time series conditional on a third are defined the measure of conditional linear dependence is the sum of linear feedback from the first to the second conditional on the third linear feedback from the second to the first conditional on the third and instantaneous linear feedback between the first and second series conditional on the third the measures are non negative and may be expressed in terms of measures of unconditional feedback between various combinations of the three series the measures of conditional linear feedback can be additively decomposed by frequency estimates of these measures are straightforward to compute and their distribution can be routinely approximated by bootstrap methods an empirical example involving real output money and interest rates is presented

threshold phenomenon for the quintic wave equation in three dimensions for the critical focusing wave equation square u u 5 rm on mathbb r 3 1 in the radial case we establish the role of the center stable manifold sigma constructed in krieger and schlag am j math 129 3 843 913 2007 near the ground state w 0 as a threshold between blowup and scattering to zero establishing a conjecture going back to numerical work by bizon et al nonlinearity 17 6 2187 2201 2004 the underlying topology is stronger than the energy norm

disclosing large scale directed functional connections in meg with the multivariate phase slope index abstract the phase slope index psi is a method to disclose the direction of frequency specific neural interactions from magnetoencephalographic meg time series a fundamental property of psi is that of vanishing for linear mixing of independent neural sources this property allows psi to cope with the artificial instantaneous connectivity among meg sensors or brain sources induced by the field spread nevertheless psi is limited by being a bivariate estimator of directionality as opposite to the multidimensional nature of brain activity as revealed by meg the purpose of this work is to provide a multivariate generalization of psi we termed this measure as the multivariate phase slope index mpsi in order to test the ability of mpsi in estimating the directionality and to compare the mpsi results to those obtained by bivariate psi approaches based on maximizing imaginary part of coherency and on canonical correlation analysis we used extensive simulations we proved that mpsi achieves the highest performance and that in a large number of simulated cases the bivariate methods as opposed to mpsi do not detect a statistically significant directionality finally we applied mpsi to assess seed based directed functional connectivity in the alpha band from resting state meg data of 61 subjects from the human connectome project the obtained results highlight a directed functional coupling in the alpha band between the primary visual cortex and several key regions of well known resting state networks e g dorsal attention network and fronto parietal network

construction of type ii blow up solutions for the energy critical wave equation in dimension 5 abstract we consider the semilinear wave equation with focusing energy critical nonlinearity in space dimension n 5 t t u u u 4 3 u with radial data it is known 7 that a solution u t u which blows up at t 0 in a neighborhood in the energy norm of the family of solitons w decomposes in the energy space as u t t u t w t u 0 u 1 o 1 where lim t 0 t t 0 and u 0 u 1 h 1 l 2 we construct a blow up solution of this type such that the asymptotic profile u 0 u 1 is any pair of sufficiently regular functions with u 0 0 0 for these solutions the concentration rate is t t 4 we also provide examples of solutions with concentration rate t t 1 for 8 related to the behavior of the asymptotic profile near the origin

renormalization and blow up for the critical yang mills problem we consider the yangs mills equations in 4 1 dimensions this is the energy critical case and we show that it admits a family of solutions which blow up in finite time they are obtained by the spherically symmetric ansatz in the so 4 gauge group and result by rescaling of the instanton solution the rescaling is done via a prescribed rate which in this case is a modification of the self similar rate by a power of vertical bar log t vertical bar the powers themselves take any value exceeding 3 2 and thus form a continuum of distinct rates leading to blow up the methods are related to the authors previous work on wave maps and the energy critical semi linear equation however in contrast to these equations the linearized yang mills operator around an instanton exhibits a zero energy eigenvalue rather than a resonance this turns out to have far reaching consequences amongst which are a completely different family of rates leading to blow up logarithmic rather than polynomial corrections to the self similar rate c 2009 elsevier inc all rights reserved

functional connectivity in resting state fmri is linear correlation sufficient functional connectivity fc analysis is a prominent approach to analyzing fmri data especially acquired under the resting state condition the commonly used linear correlation fc measure bears an implicit assumption of gaussianity of the dependence structure if only the marginals but not all the bivariate distributions are gaussian linear correlation consistently underestimates the strength of the dependence to assess the suitability of linear correlation and the general potential of nonlinear fc measures we present a framework for testing and estimating the deviation from gaussianity by means of comparing mutual information in the data and its gaussianized counterpart we apply this method to 24 sessions of human resting state fmri for each session matrix of connectivities between 90 anatomical parcel time series is computed using mutual information and compared to results from its multivariate gaussian surrogate that conserves the correlations but cancels any nonlinearity while the group level tests confirmed non gaussianity in the fc the quantitative assessment revealed that the portion of mutual information neglected by linear correlation is relatively minor on average only about 5 of the mutual information already captured by the linear correlation the marginality of the non gaussianity was confirmed in comparisons using clustering of the parcels the disagreement between clustering obtained from mutual information and linear correlation was attributable to random error we conclude that for this type of data practical relevance of nonlinear methods trying to improve over linear correlation might be limited by the fact that the data are indeed almost gaussian

concentration compactness and universal profiles for the non radial energy critical wave equation abstract in this paper we give an overview of the authors work on applications of the method of concentration compactness to global well posedness scattering blow up and universal profiles for the energy critical wave equation in the non radial setting new results and proofs are also given

analysis of fmri time series revisited again friston et al 1995 neuroimage 2 45 53 presented a method for detecting activations in fmr1 time series based on the general linear model and a heuristic analysis of the effective degrees of freedom in this communication we present corrected results that replace those of the previous paper and solve the same problem without recourse to heuristic arguments specifically we introduce a proper and unbiased estimator for the error terms and provide a more generally correct expression for the effective degrees of freedom the previous estimates of error variance were biased and in some instances could have led to a 10 20 overestimate of z values although the previous results are almost correct for the random regressors chosen for validation the present theoretical results are exact for any covariate or waveform we comment on some aspects of experimental design and data analysis in the light of the theoretical framework discussed here

asymptotic decomposition for semilinear wave and equivariant wave map equations in this paper we give a unified proof to the soliton resolution conjecture along a sequence of times for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations including the four dimensional radial yang mills equation without using outer energy type inequalities such inequalities have played a crucial role in previous works with similar results roughly speaking we prove that along a sequence of times t n to t the maximal time of existence the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space plus a free radiation term in the global case or a regular part in the finite time blow up case the main difficulty is that in general for instance for the radial four dimensional yang mills case and the radial six dimensional semilinear wave case we do not have a favorable outer energy inequality for the associated linear wave equations our main new input is the simultaneous use of two virial identities

analysis of economic growth fluctuations based on eemd and causal decomposition abstract in this paper we first apply ensemble empirical mode decomposition eemd to analyze the change rate of gdp time series for ten major countries each signal is decomposed into four modes with different scales and a residual trend the variance contribution and averaged period cycle for each mode is then calculated results show that economic growth fluctuations for most of the countries mainly fluctuate at a short period between 3 5 years then the causal decomposition proposed by yang et al is utilized to detect the mutual causation among different countries the novelty of this method is that the causal interaction is identified in instantaneous phase coherence at a specific time scale the strengths and directions of causal relationship for each mode among countries differ from each other which mirrors the fact that the world economic activities are fluctuated and changeable over varying periods however no visible causation for the fourth mode long term cycle among countries is characterized meaning that the development of economy over a long period for most countries depends on its own conditions

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

inelastic interaction of nearly equal solitons for the quartic gkdv equation this paper describes the interaction of two solitons with nearly equal speeds for the quartic gkdv equation partial tu partial x partial x 2u u 4 0 quad t x in mathbb r 0 1 we call soliton a solution of 0 1 of the form u t x qc x ct y0 where c 0 y0 and q c q c 4 cq c since 0 1 is not an integrable model the general question of the collision of two given solitons q c 1 x c 1 t q c 2 x c 2 t with c1 c2 is an open problem

l adic properties of the partition function abstract ramanujan s famous partition congruences modulo powers of 5 7 and 11 imply that certain sequences of partition generating functions tend l adically to 0 although these congruences have inspired research in many directions little is known about the l adic behavior of these sequences for primes l 13 using the classical theory of modular forms mod p as developed by serre in the 1970s we show that these sequences are governed by fractal behavior modulo any power of a prime l 5 these sequences of generating functions l adically converge to linear combinations of at most l 1 12 l 2 1 24 l many special q series for l 5 7 11 we have l 1 12 l 2 1 24 l 0 thereby giving a conceptual explanation of ramanujan s congruences we use the general result to reveal the theory of multiplicative partition congruences that atkin anticipated in the 1960s his results and observations are examples of systematic infinite families of congruences which exist for all powers of primes 13 l 31 since l 1 12 l 2 1 24 l 1 answering questions of mazur in appendix a we give a new general theorem which fits these results within the framework of overconvergent half integral weight p adic modular forms this result which is based on recent works by n ramsey is due to frank calegari

conditional correlation as a measure of mediated interactivity in fmri and meg eeg many measures have been proposed so far to extract brain functional interactivity from functional magnetic resonance imaging fmri and magnetoencephalography electroencephalography meg eeg data sets unfortunately none has been able to provide a relevant self contained and common definition of brain interaction in this paper we propose a first step in this direction we first introduce a common terminology together with a cross modal definition of interaction in this setting we investigate the commonalities shared by some measures of interaction proposed in the literature we show that temporal correlation nonlinear correlation mutual information generalized synchronization phase synchronization coherence and phase locking value plv actually measure the same quantity namely correlation when one is investigating linear interactions between independently and identically distributed gaussian variables we also demonstrate that these data driven measures can only partly account for the interaction patterns that can be expressed by the effective connectivity of structural equation modeling sem to bridge this gap we suggest the use of conditional correlation which is shown to be related to mediated interaction

nonlinear profile decomposition for the h 12 h 12 rd energy subcritical wave equation abstract we consider the nonlinear wave equation u t t u u u 4 d 1 0 in dimensions d 2 this equation is critical in h 1 2 h 1 2 r d and subcritical with respect to the energy we prove the nonlinear profile decomposition the proof must address the lack of compactness due to the lorentz symmetry

entropy and predictability of stock market returns abstract we examine the predictability of stock market returns by employing a new metric entropy measure of dependence with several desirable properties we compare our results with a number of traditional measures the metric entropy is capable of detecting nonlinear dependence within the returns series and is also capable of detecting nonlinear affinity between the returns and their predictions obtained from various models thereby serving as a measure of out of sample goodness of fit or model adequacy several models are investigated including the linear and neural network models as well as nonparametric and recursive unconditional mean models we find significant evidence of small nonlinear unconditional serial dependence within the returns series but fragile evidence of superior conditional predictability profit opportunity when using market switching versus buy and hold strategies

on the focusing critical semi linear wave equation the wave equation partial derivative tt psi delta psi psi 5 0 in r 3 is known to exhibit finite time blowup for data of negative energy furthermore it admits the special static solutions phi x a 3a 1 4 1 a x 2 1 2 for all a 0 which are linearly unstable we view these functions as a curve in the energy space h over dot 1 x l 2 we prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well defined long time asymptotic behavior as the sum of a bulk term plus a scattering term moreover this family forms a co dimension one manifold m of small diameter in a suitable topology loosely speaking m acts as a center stable manifold with the curve phi a as an attractor in m

kernel granger causality mapping effective connectivity on fmri data although it is accepted that linear granger causality can reveal effective connectivity in functional magnetic resonance imaging fmri the issue of detecting nonlinear connectivity has hitherto not been considered in this paper we address kernel granger causality kgc to describe effective connectivity in simulation studies and real fmri data of a motor imagery task based on the theory of reproducing kernel hilbert spaces kgc performs linear granger causality in the feature space of suitable kernel functions assuming an arbitrary degree of nonlinearity our results demonstrate that kgc captures effective couplings not revealed by the linear case in addition effective connectivity networks between the supplementary motor area sma as the seed and other brain areas are obtained from kgc

inelasticity of soliton collisions for the 5d energy critical wave equation for the focusing energy critical wave equation in 5d we construct a solution showing the inelastic nature of the collision of two solitons for any choice of sign speed scaling and translation parameters except the special case of two solitons of same scaling and opposite signs beyond its own interest as one of the first rigorous studies of the collision of solitons for a non integrable model the case of the quartic gkdv equation being partially treated in martel and merle ann math 174 2 757 857 2011 invent math 183 3 563 648 2011 int math res notices 2015 3 688 739 2015 this result can be seen as part of a wider program aiming at establishing the soliton resolution conjecture for the critical wave equation this conjecture has already been proved in the 3d radial case in duyckaerts et al camb j math 1 75 144 2013 and in the general case in 3 4 and 5d along a sequence of times in duyckaerts et al geom funct anal 27 4 798 862 2017 compared with the construction of an asymptotic two soliton in martel and merle arch ration mech anal 222 3 1113 1160 2016 the study of the nature of the collision requires a more refined approximate solution of the two soliton problem and a precise determination of its space asymptotics to prove inelasticity these asymptotics are combined with the method of channels of energy from duyckaerts et al camb j math 1 75 144 2013 kenig et al geom funct anal 24 610 647 2014

spatiotemporal nonlinearity in resting state fmri of the human brain in this work the spatiotemporal nonlinearity in resting state fmri data of the human brain was detected by nonlinear dynamics methods nine human subjects during resting state were imaged using single shot gradient echo planar imaging on a 1 5t scanner eigenvalue spectra for the covariance matrix correlation dimensions and spatiotemporal lyapunov exponents were calculated to detect the spatiotemporal nonlinearity in resting state fmri data by simulating adjusting and comparing the eigenvalue spectra of pure correlated noise with the corresponding real fmri data the intrinsic dimensionality was estimated the intrinsic dimensionality was used to extract the first few principal components from the real fmri data using principal component analysis which will preserve the correct phase dynamics while reducing both computational load and noise level of the data then the phase space was reconstructed using the time delay embedding method for their principal components and the correlation dimension was estimated by the grassberger procaccia algorithm of multiple variable series the spatiotemporal lyapunov exponents were calculated by using the method based on coupled map lattices through nonlinearity testing there are significant differences of correlation dimensions and spatiotemporal lyapunov exponents between fmri data and their surrogate data the fractal dimension and the positive spatiotemporal lyapunov exponents characterize the spatiotemporal nonlinear dynamics property of resting state fmri data therefore the results suggest that fluctuations presented in resting state may be an inherent model of basal neural activation of human brain cannot be fully attributed to noise

the instability of bourgain wang solutions for the l2 critical nls we consider the two dimensional l 2 critical nonlinear schr odinger equation i partial tu delta u uu 2 0 in their pioneering 1997 work bourgain and wang have constructed smooth solutions which blow up in finite time t nabla u t l 2 sim 1 over t t and which display some decoupling between the regular and the singular part of the solution at blow up time we prove that this dynamic is unstable more precisely we show that any such solution with small super critical l 2 mass lies on the boundary of both h 1 open sets of global solutions that scatter forward and backwards in time and solutions that blow up in finite time on the right in the log log regime exhibited in work by f merle and p rapha e l we moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in f merle s 1992 it comm pure appl math paper in the critical mass case

evaluating frequency wise directed connectivity of bold signals applying relative power contribution with the linear multivariate time series models in this article we propose a statistical method to evaluate directed interactions of functional magnetic resonance imaging fmri data the multivariate autoregressive mar model was combined with the relative power contribution rpc in this analysis the mar model was fitted to the data to specify the direction of connections and the rpc quantifies the strength of connections as the rpc is computed in the frequency domain we can evaluate the connectivity for each frequency component from this we can establish whether the specified connections represent low or high frequency connectivity which cannot be examined solely using the estimated mar coefficients we applied this analysis method to fmri data obtained during visual motion tasks confirming previous reports of bottom up connectivity around the frequency corresponding to the block experimental design furthermore we used the mar model with exogenous variables marx to extend our understanding of these data and to show how the input to v1 transfers to higher cortical areas

measurement of linear dependence and feedback between multiple time series abstract measures of linear dependence and feedback for multiple time series are defined the measure of linear dependence is the sum of the measure of linear feedback from the first series to the second linear feedback from the second to the first and instantaneous linear feedback the measures are nonnegative and zero only when feedback causality of the relevant type is absent the measures of linear feedback from one series to another can be additively decomposed by frequency a readily usable theory of inference for all of these measures and their decompositions is described the computations involved are modest

temporal aggregation of multivariate garch processes this paper derives results for the temporal aggregation of multivariate garch 1 1 processes in the general vector specification it is shown that the class of weak multivariate garch 1 1 processes is closed under temporal aggregation fourth moment characteristics turn out to be crucial for the low frequency dynamics for both stock and flow variables in some aspects the aggregation characteristics of multivariate garch processes are shown to be different from those of vector autoregressive moving average processes a numerical example illustrates some of the results

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

robust smoothness estimation in statistical parametric maps using standardized residuals from the general linear model the assessment of significant activations in functional imaging using voxel based methods often relies on results derived from the theory of gaussian random fields these results solve the multiple comparison problem and assume that the spatial correlation or smoothness of the data is known or can be estimated end results i e p values associated with local maxima clusters or sets of clusters critically depend on this assessment which should be as exact and as reliable as possible in some earlier implementations of statistical parametric mapping spm spm94 spm95 the smoothness was assessed on gaussianized t fields gt f that are not generally free of physiological signal this technique has two limitations first the estimation is not stable the variance of the estimator being far from negligible and second physiological signal in the gt f will bias the estimation in this paper we describe an estimation method that overcomes these drawbacks the new approach involves estimating the smoothness of standardized residual fields which approximates the smoothness of the component fields of the associated t field knowing the smoothness of these component fields is important because it allows one to compute corrected p values for statistical fields other than the t field or the gt f e g the f map and eschews bias due to deviation from the null hypothesis we validate the method on simulated data and demonstrate it using data from a functional mri study

detecting spatiotemporal nonlinear dynamics in resting state of human brain based on fmri datasets in this work a nonlinear dynamics method coupled map lattices was applied to functional magnetic resonance imaging fmri datasets to examine the spatiotemporal properties of resting state blood oxygen level dependent bold fluctuations spatiotemporal lyapunov exponent sple was calculated to study the deterministic nonlinearity in resting state human brain of nine subjects based on fmri datasets the results show that there is nonlinearity and determinism in resting state human brain furthermore the results demonstrate that there is a spatiotemporal chaos phenomenon in resting state brain and suggest that fluctuations of fmri data in resting state brain cannot be fully attributed to nuclear magnetic resonance noise at the same time the spatiotemporal chaos phenomenon suggests that the correlation between voxels varies with time and there is a dynamic functional connection or network in resting state human brain

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

profile decompositions for wave equations on hyperbolic space with applications the goal for this paper is twofold our first main objective is to develop bahouri gerard type profile decompositions for waves on hyperbolic space recently such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy with an eye towards further applications we develop this theory in a fairly general framework which includes the case of waves on hyperbolic space perturbed by a time independent potential our second objective is to use the profile decomposition to address a specific nonlinear problem namely the question of global well posedness and scattering for the defocusing energy critical semi linear wave equation on three dimensional hyperbolic space possibly perturbed by a repulsive time independent potential using the concentration compactness rigidity method introduced by kenig and merle we prove that all finite energy initial data lead to a global evolution that scatters to linear waves as t rightarrow pm infty this proof serves as a blueprint for the arguments in the work 45 by the authors where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane

finite time blow up for the fractional critical heat equation in rn abstract we consider the fractional critical heat equation u t s u u n 2 s n 2 s in r n 0 t u x 0 u 0 x in r n where 4 s n 6 s 0 s 1 for sufficiently small t 0 and given k distinct points q 1 q k we show that there exists an initial condition u 0 such that the solution u x t blows up at these k distinct points as t t more precisely the blow up profile around each concentration point takes the form of sharply scaled bubble and u t l r n t t n 2 s 6 s n as t t

nonparametric estimation and inference for conditional density based granger causality measures we propose a nonparametric estimation and inference for conditional density based granger causality measures that quantify linear and nonlinear granger causalities we first show how to write the causality measures in terms of copula densities thereafter we suggest consistent estimators for these measures based on a consistent nonparametric estimator of copula densities furthermore we establish the asymptotic normality of these nonparametric estimators and discuss the validity of a local smoothed bootstrap that we use in finite sample settings to compute a bootstrap bias corrected estimator and to perform statistical tests a monte carlo simulation study reveals that the bootstrap bias corrected estimator behaves well and the corresponding test has quite good finite sample size and power properties for a variety of typical data generating processes and different sample sizes finally two empirical applications are considered to illustrate the practical relevance of nonparametric causality measures

kernel granger causality and the analysis of dynamical networks we propose a method of analysis of dynamical networks based on a recent measure of granger causality between time series based on kernel methods the generalization of kernel granger causality to the multivariate case here presented shares the following features with the bivariate measures i the nonlinearity of the regression model can be controlled by choosing the kernel function and ii the problem of false causalities arising as the complexity of the model increases is addressed by a selection strategy of the eigenvectors of a reduced gram matrix whose range represents the additional features due to the second time series moreover there is no a priori assumption that the network must be a directed acyclic graph we apply the proposed approach to a network of chaotic maps and to a simulated genetic regulatory network it is shown that the underlying topology of the network can be reconstructed from time series of node s dynamics provided that a sufficient number of samples is available considering a linear dynamical network built by preferential attachment scheme we show that for limited data use of the bivariate granger causality is a better choice than methods using l1 minimization finally we consider real expression data from hela cells 94 genes and 48 time points the analysis of static correlations between genes reveals two modules corresponding to well known transcription factors granger analysis puts in evidence 19 causal relationships all involving genes related to tumor development

the effect of filtering on granger causality based multivariate causality measures in the past causality measures based on granger causality have been suggested for assessing directionality in neural signals in frequency domain analyses power or coherence of neural data it is common to preprocess the time series by filtering or decimating however in other fields it has been shown theoretically that filtering in combination with granger causality may lead to spurious or missed causalities we investigated whether this result translates to multivariate causality methods derived from granger causality with a a simulation study and b an application to magnetoencephalographic data to this end we performed extensive simulations of the effect of applying different filtering techniques and evaluated the performance of five different multivariate causality measures in combination with two numerical significance measures random permutation and leave one out method the analysis included three of the most widely used filters high pass low pass notch filter four different filter types butterworth chebyshev i and ii elliptic filter variation of filter order decimating and interpolation the simulation results suggest that preprocessing without a strong prior about the artifact to be removed disturbs the information content and time ordering of the data and leads to spurious and missed causalities only if apparent artifacts like a current or movement artifact are present filtering out the respective disturbance seems advisable while oversampling poses no problem decimation by a factor greater than the minimum time shift between the time series may lead to wrong inferences in general the multivariate causality measures are very sensitive to data preprocessing

on similarity in the evolution of semilinear wave and klein gordon equations numerical surveys the aim of this paper is threefold the first and also main purpose is to provide numerical evidence for the conjecture proposed by bizon et al dynamics near the threshold for blowup in the one dimensional focusing nonlinear klein gordon equation j math phys 52 103703 2011 10 1063 1 3645363 that the blowup evolution of spherically symmetric semilinear klein gordon equations is similar to the evolution of spherically symmetric semilinear wave equations i e the mass term can be neglected when the amplitude of a solution grows the second aim is to describe the relationship between different types of blowup for energy critical semilinear wave equations the third goal is to present numerical evidence for the fact that the special class of self similar profiles of semilinear wave equations found by kycia on self similar solutions of semilinear wave equations in higher space dimensions appl math comput 217 9451 9466 2011 10 1016 j amc 2011 04 039 play the same role in the evolution of se

the partition function and hecke operators abstract the theory of congruences for the partition function p n depends heavily on the properties of half integral weight hecke operators the subject has been complicated by the absence of closed formulas for the hecke images p z t l 2 where p z is the relevant modular generating function we obtain such formulas using euler s pentagonal number theorem and the denominator formula for the monster lie algebra as a corollary we obtain congruences for certain powers of ramanujan s delta function

source coding with feed forward rate distortion theorems and error exponents for a general source in this work we consider a source coding model with feed forward we analyze a system with a noiseless feed forward link where the decoder has knowledge of all previous source samples while reconstructing the present sample the rate distortion function for an arbitrary source with feed forward is derived in terms of directed information a variant of mutual information we further investigate the nature of the rate distortion function with feed forward for two common types of sources discrete memory less sources and gaussian sources we then characterize the error exponent for a general source with feed forward the results are then extended to feed forward with an arbitrary delay larger than the block length

inference on one way effect and evidence in japanese macroeconomic data this paper provides an approach to testing a variety of causal characteristics expressed in terms of the measures of one way effect for cointegrated vector time series for this purpose we propose wald tests and their computational algorithm by means of the measures of one way effect incorporating johansen s algorithm for the maximum likelihood estimates and the likelihood ratio tests using the wald test statistics the paper also provides a method of confidence set construction for the causal measures the approach proposed in the paper includes testing granger s non causality as an instance of its multiple applications as an illustration the paper presents a characterization of the causal structure of the recent japanese macroeconomy on the basis of the proposed method and the derived evidence

analysis of fmri time series revisited abstract this paper presents a general approach to the analysis of functional mri time series from one or more subjects the approach is predicated on an extension of the general linear model that allows for correlations between error terms due to physiological noise or correlations that ensue after temporal smoothing this extension uses the effective degrees of freedom associated with the error term the effective degrees of freedom are a simple function of the number of scans and the temporal autocorrelation function a specific form for the latter can be assumed if the data are smoothed in time to accentuate hemodynamic responses with a neural basis this assumption leads to an expedient implementation of a flexible statistical framework the importance of this small extension is that in contradistinction to our previous approach any parametric statistical analysis can be implemented we demonstrate this point using a multiple regression analysis that tests for effects of interest activations due to word generation while taking explicit account of some obvious confounds

functional connectivity and structural covariance between regions of interest can be measured more accurately using multivariate distance correlation studies of brain wide functional connectivity or structural covariance typically use measures like the pearson correlation coefficient applied to data that have been averaged across voxels within regions of interest rois however averaging across voxels may result in biased connectivity estimates when there is inhomogeneity within those rois e g sub regions that exhibit different patterns of functional connectivity or structural covariance here we propose a new measure based on distance correlation a test of multivariate dependence of high dimensional vectors which allows for both linear and non linear dependencies we used simulations to show how distance correlation out performs pearson correlation in the face of inhomogeneous rois to evaluate this new measure on real data we use resting state fmri scans and t1 structural scans from 2 sessions on each of 214 participants from the cambridge centre for ageing neuroscience cam can project pearson correlation and distance correlation showed similar average connectivity patterns for both functional connectivity and structural covariance nevertheless distance correlation was shown to be 1 more reliable across sessions 2 more similar across participants and 3 more robust to different sets of rois moreover we found that the similarity between functional connectivity and structural covariance estimates was higher for distance correlation compared to pearson correlation we also explored the relative effects of different preprocessing options and motion artefacts on functional connectivity because distance correlation is easy to implement and fast to compute it is a promising alternative to pearson correlations for investigating roi based brain wide connectivity patterns for functional as well as structural data

parity of the partition function abstract although much is known about the partition function little is known about its parity for the polynomials d x d x 2 1 24 where d 23 mod 24 we show that there are infinitely many m resp n for which p d m is even resp p d n is odd if there is at least one such m resp n we bound the first m and n if any in terms of the class number h d for prime d we show that there are indeed infinitely many even values to this end we construct new modular generating functions using generalized borcherds products and we employ galois representations and locally nilpotent hecke algebras

comparing alternative tests of causality in temporal systems abstract this paper discusses eight alternative tests of the absence of casual ordering all of which are asymptotically valid under the null hypothesis in the sense that their limiting size is known their behavior under alternatives is compared analytically using the concept of approximate slope and these results are supported by the outcomes of monte carlo experiments the implications of these comparisons for applied work are unambiguous wald variants of a test attributed to granger and a lagged dependent variable version of sim s test introduced in this paper are equivalent in all relevant respects and are preferred to the other tests discussed

center stable manifold of the ground state in the energy space for the critical wave equation we construct a center stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry as the dynamical threshold between scattering and blow up and also as a collection of solutions which stay close to the ground states up to energy slightly above the ground state this completes the 9 set classification of the global dynamics in our previous paper dcds 33 6 2013 we can also extend the manifold to arbitrary energy size by adding large radiation the manifold contains all the solutions scattering to the ground state solitons and also some of those blowing up in finite time by concentration of the ground states

does sunspot numbers cause global temperatures a reconsideration using non parametric causality tests abstract in a recent paper gupta et al 2015 analyzed whether sunspot numbers cause global temperatures based on monthly data covering the period 1880 1 2013 9 the authors find that standard time domain granger causality test fails to reject the null hypothesis that sunspot numbers do not cause global temperatures for both full and sub samples namely 1880 1 1936 2 1936 3 1986 11 and 1986 12 2013 9 identified based on tests of structural breaks however frequency domain causality test detects predictability for the full sample at short 2 2 6 months cycle lengths but not the sub samples but since full sample causality cannot be relied upon due to structural breaks gupta et al 2015 conclude that the evidence of causality running from sunspot numbers to global temperatures is weak and inconclusive given the importance of the issue of global warming our current paper aims to revisit this issue of whether sunspot numbers cause global temperatures using the same data set and sub samples used by gupta et al 2015 based on an nonparametric singular spectrum analysis ssa based causality test based on this test we however show that sunspot numbers have predictive ability for global temperatures for the three sub samples over and above the full sample thus generally speaking our non parametric ssa based causality test outperformed both time domain and frequency domain causality tests and highlighted that sunspot numbers have always been important in predicting global temperatures

validity of time reversal for testing granger causality inferring causal interactions from observed data is a challenging problem especially in the presence of measurement noise to alleviate the problem of spurious causality haufe 2013 proposed to contrast measures of information flow obtained on the original data against the same measures obtained on time reversed data they show that this procedure time reversed granger causality trgc robustly rejects causal interpretations on mixtures of independent signals while promising results have been achieved in simulations it was so far unknown whether time reversal leads to valid measures of information flow in the presence of true interaction here we prove that for linear finite order autoregressive processes with unidirectional information flow between two variables the application of time reversal for testing granger causality indeed leads to correct estimates of information flow and its directionality using simulations we further show that trgc is able to infer correct directionality with similar statistical power as the net granger causality between two variables while being much more robust to the presence of measurement noise

determination of the blow up rate for a critical semilinear wave equation in this paper we determine the blow up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance

estimation of the generalized prediction error variance of a multiple time series abstract for a multivariate stationary time series we propose a nonparametric estimator for its generalized prediction error variance using a multivariate analog of the szego kolmogorov formula replacing the integral by a sum and replacing the unknown spectral density matrix by a consistent estimator asymptotic normality of this estimator is established and its small sample behavior is assessed through simulation and application to two real data sets these examples show that the proposed method works reasonably well in comparison with parametric models in contrast to the univariate case where smoothing the periodogram is optional and generally not recommended in the multivariate case this smoothing is a necessity because the raw periodogram is a matrix of rank one to obtain a consistent estimator of a full rank spectral density matrix one must necessarily choose larger bandwidths for higher dimensional time series a bias correction factor is computed using the asymptotic properties of our propo

nonlinear estimation of the bold signal signal variations in functional magnetic resonance imaging experiments essentially reflect the vascular system response to increased demand for oxygen caused by neuronal activity termed the blood oxygenation level dependent bold effect the most comprehensive model to date of the bold signal is formulated as a mixed continuous discrete time system of nonlinear stochastic differential equations previous approaches to the analysis of this system have been based on linearised approximations of the dynamics which are limited in their ability to capture the inherent nonlinearities in the physiological system in this paper we present a nonlinear filtering method for simultaneous estimation of the hidden physiological states and the system parameters based on an iterative coordinate descent framework state estimates of the cerebral blood flow cerebral blood volume and deoxyhaemoglobin content are determined using a particle filter demonstrated via simulation to be accurate robust and efficient in comparison to linearisation based techniques the adaptive state and parameter estimation algorithm generates physiologically reasonable parameter estimates for experimental fmri data it is anticipated that signal processing techniques for modelling and estimation will become increasingly important in fmri analyses as limitations of linear and linearised modelling are reached

renormalization and blow up for wave maps from s 2 times mathbb r to s 2 we construct a one parameter family of finite time blow ups to the co rotational wave maps problem from s 2 x r to s 2 parameterized by nu is an element of 1 2 1 the longitudinal function u t alpha which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from r 2 to s 2 the domain of this harmonic map is identified with a neighborhood of the north pole in the domain s 2 via the exponential coordinates alpha theta in these coordinates u t alpha q lambda t alpha r t alpha where q r 2arctan r is the standard co rotational harmonic map to the sphere lambda t t 1 nu and r t alpha is the error with local energy going to zero as t 0 blow up will occur at t alpha 0 0 due to energy concentration and up to this point the solution will have regularity h1 nu

slow blow up solutions for the h 1 mathbb r 3 critical focusing semilinear wave equation given 1 2 and 0 arbitrary we prove the existence of energy solutions of ttu u u5 0 0 1 in r3 1 which blow up exactly at r t 0 as t 0 these solutions are radial and of the form u t 1 2w t r r t inside the cone r t where t t 1 w r 1 r2 3 1 2 is the stationary solution of 0 1 and is a radiation term with r t x t 2 t x t 2 x t 6 dx 0 t 0 outside of the light cone there is the energy bound r t u x t 2 ut x t 2 u x t 6 dx 0 the regularity of u increases with as in our accompanying article on wave maps 10 the argument is based on a renormalization method for the soliton profile w r

stable blow up dynamics for energy supercritical wave equations we study the semilinear wave equation displaystyle partial t 2 psi delta psi vert psi vert p 1 psi for p 3 with radial data in three spatial dimensions there exists an explicit solution which blows up at t t 0 given by displaystyle psi t t x c p t t frac 2 p 1 where c p is a suitable constant we prove that the blow up described by psi t is stable in the sense that there exists an open set in a topology strictly stronger than the energy of radial initial data that leads to a solution which converges to psi t as t to t in the backward lightcone of the blow up point t r t 0

testing causality between two vectors in multivariate autoregressive moving average models abstract in the analysis of economic time series a question often raised is whether a vector of variables causes another one in the sense of granger most of the literature on this topic is concerned with bivariate relationships or uses finite order autoregressive specifications the purpose of this article is to develop a causality analysis in the sense of granger for general vector autoregressive moving average arma models we give a definition of granger noncausality between vectors which is a natural and simple extension of the notion of granger noncausality between two variables in our context this definition is shown to be equivalent to a more complex definition proposed by tjostheim for the class of linear invertible processes we derive a necessary and sufficient condition for noncausality between two vectors of variables when the latter do not necessarily include all the variables considered in the analysis this result is then specialized to the class of stationary invertible arma process

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

congruence properties for a certain kind of partition functions in light of the modular equations of fifth and seventh order we derive some congruence properties for a certain kind of partition functions a n which satisfy n 0 a n qn q q k modm where k is a positive integer with 1 k 24 and m 2 3 in view of these properties we obtain many infinite families of congruences for c k n the number of generalized frobenius partitions of n with k colors and c k n the number of generalized frobenius partitions of n with k colors whose order is k under cyclic permutation of the k colors meanwhile we also apply the main theorems to some other kinds of partition functions

the decomposition and measurement of the interdependency between second order stationary processes for a given pair of multivariate stationary processes the process of one way effect is extracted from each of the processes each process is decomposed into two orthogonal processes namely into the process generated by the one way effect of the other process and the process orthogonal to it based on the decomposition three measures characterizing the interdependency of the pair of processes are introduced they are the measure of association the measure of one way effect and the measure of reciprocity each of the measures is defined as overall as well as frequencywise measure the paper shows that the measure of association is equal to the sum of the others it discusses the relationships of those measures to the ones proposed by gel fand yaglom and by geweke