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

a remark on liouville type theorems for the stationary navier stokes equations in three space dimensions abstract consider the 3d homogeneous stationary navier stokes equations in the whole space r 3 we deal with solutions vanishing at infinity in the class of the finite dirichlet integral by means of quantities having the same scaling property as the dirichlet integral we establish new a priori estimates as an application we prove the liouville theorem in the marginal case of scaling invariance

asymptotic behavior of solutions to elliptic and parabolic equations with unbounded coefficients of the second order in unbounded domains we study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain under the assumption that the solution belongs to l q with q in 2 infty we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients as a corollary we obtain the liouville type theorem in the case when the coefficients may grow at the spacial infinity we also study a corresponding parabolic problem in the n dimensional whole space and discuss the energy identity for solutions in l q as a corollary we show also the liouville type theorem for both forward and ancient solutions

multivariate granger causality and generalized variance granger causality analysis is a popular method for inference on directed interactions in complex systems of many variables a shortcoming of the standard framework for granger causality is that it only allows for examination of interactions between single univariate variables within a system perhaps conditioned on other variables however interactions do not necessarily take place between single variables but may occur among groups or ensembles of variables in this study we establish a principled framework for granger causality in the context of causal interactions among two or more multivariate sets of variables building on geweke s seminal 1982 work we offer additional justifications for one particular form of multivariate granger causality based on the generalized variances of residual errors taken together our results support a comprehensive and theoretically consistent extension of granger causality to the multivariate case treated individually they highlight several specific advantages of the generalized variance measure which we illustrate using applications in neuroscience as an example we further show how the measure can be used to define partial granger causality in the multivariate context and we also motivate reformulations of causal density and granger autonomy our results are directly applicable to experimental data and promise to reveal new types of functional relations in complex systems neural and otherwise

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

a liouville type theorem for axially symmetric d solutions to steady navier stokes equations abstract we study axially symmetric d solutions of three dimensional steady navier stokes equations we prove that if the velocity u decays like x 2 3 uniformly for z or the vorticity decays like x 5 3 uniformly for z then u vanishes here x denotes the distance to the axis

euler s equations and the maximum principle in this paper we use maximum principle in the far field region for the time dependent self similar euler equations to exclude discretely self similar blow up for the euler equations of the incompressible fluid flows our decay conditions near spatial infinity of the blow up profile are given explicitly in terms the coefficient in the equations we also deduce triviality of the discretely self similar solution to the magnetohydrodynamic system under suitable decay conditions near spatial infinity than the previous one applying similar argument directly to the euler equations we obtain a priori estimate of the vorticity in the far field region

an eulerian lagrangian approach for incompressible fluids local theory we study a formulation of the incompressible euler equations in terms of the inverse lagrangian map in this formulation the equations become a first order advective nonlinear system of partial differential equations

sensitivity results for a general class of generalized vector quasi equilibrium problems with set valued maps abstract in this paper we give sufficient conditions for the local openness local closedness properties and the lower upper semicontinuity properties of the solution sets of a general model which includes as special cases many generalized vector quasi equilibrium problems with set valued maps the obtained results generalize and improve several known results an application is also given for a model which can be interpreted as a system of generalized vector quasi equilibrium problems

liouville type theorems for the 3d stationary navier stokes mhd and hall mhd equations abstract in this paper we consider the liouville type problem of the three dimensional stationary incompressible navier stokes equations and establish an improved liouville type result only under the condition that the l p norm of a velocity u is finite moreover these results are extended to the mhd and hall mhd equations

finite energy of generalized suitable weak solutions to the navier stokes equations and liouville type theorems in two dimensional domains abstract introducing a new notion of generalized suitable weak solutions we first prove validity of the energy inequality for such a class of weak solutions to the navier stokes equations in the whole space r n although we need certain growth condition on the pressure we may treat the class even with infinite energy quantity except for the initial velocity we next handle the equation for vorticity in 2d unbounded domains under a certain condition on the asymptotic behavior at infinity we prove that the vorticity and its gradient of solutions are both globally square integrable as their applications loiuville type theorems are obtained

first order optimality conditions for elliptic mathematical programs with equilibrium constraints via variational analysis mathematical programs in which the constraint set is partially defined by the solutions of an elliptic variational inequality so called elliptic mpecs are formulated in reflexive banach spaces with the goal of deriving explicit first order optimality conditions amenable to the development of numerical procedures variational analytic concepts are both applied and further developed the paper is split into two main parts the first part concerns the derivation of conditions in which the lower level state constraints are assumed to be polyhedric sets this part is then completed by two examples the latter of which involves pointwise bilateral bounds on the gradient of the state the second part focuses on an important nonpolyhedric example namely when the lower level state constraints are presented by pointwise bounds on the euclidean norm of the gradient of the state a formula for the second order mosco epiderivative of the indicator function for this convex set is derived this result is then used to demonstrate the hadamard directional differentiability of the solution mapping of the variational inequality which then leads to the derivation of explicit strong stationarity conditions for this problem

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

a liouville theorem of navier stokes equations with two periodic variables abstract in this paper we study the liouville theorem of the stationary navier stokes equations in r 3 when the solution is periodic in two variables we can prove that actually the solution is trivial constant vector under the assumption that one component of the velocity vanishing at infinity has finite dirichlet integral and the other two components can have some growth with respect to the distance to the origin

sensitivity of solutions to variational inequalities on banach spaces we analyze the sensitivity of parameterized variational inequalities for convex polyhedric sets in reflexive banach spaces we compute a generalized derivative of the solution mapping where the formula for the derivative is given in terms of the solutions to an auxiliary variational inequality these results are distinguished from other work in this area by the fact that they do not depend on the uniqueness of the solutions to the variational inequalities to obtain our results we use second order epi derivatives to analyze the second order properties of polyhedric sets we apply our results to sensitivity analyses of stationary points and kkt pairs associated with constrained infinite dimensional optimization problems

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

liouville type theorems for the forced euler equations and the navier stokes equations in this paper we study the liouville type properties for solutions to the steady incompressible euler equations with forces in mathbb r n if we assume single signedness condition on the force then we can show that a c 1 mathbb r n solution v p with v 2 p in l frac q 2 mathbb r n q in frac 3n n 1 infty is trivial v 0 for the solution of the steady navier stokes equations satisfying v x to 0 as x to infty the condition int mathbb r 3 delta v frac 6 5 dx infty which is stronger than the important d condition int mathbb r 3 nabla v 2 dx infty but both having the same scaling property implies that v 0 in the appendix we reprove theorem 1 1 chae commun math phys 273 203 215 2007 using the self similar euler equations directly

decay rates of second order derivatives of axisymmetric d solutions to the stationary navier stokes equations abstract in this paper we are interested in the decay properties of the axisymmetric solutions to the stationary navier stokes equations in r 3 with finite dirichlet integral we establish some new a priori decay rates for the second order derivatives of the velocity the result is obtained by combining the weighted energy estimates the brezis galleout inequality and the scaling argument

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

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

semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems abstract in this paper we establish sufficient conditions for the solution set of parametric multivalued vector quasiequilibrium problems to be semicontinuous all kinds of semicontinuity are considered lower semicontinuity upper semicontinuity hausdorff upper semicontinuity and closedness moreover we investigate both the weak and strong solutions of quasiequilibrium problems

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

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

nonexistence of self similar singularities for the 3d incompressible euler equations we prove that there exists no self similar finite time blowing up solution to the 3d incompressible euler equations if the vorticity decays sufficiently fast near infinity in mathbb r 3 by a similar method we also show nonexistence of self similar blowing up solutions to the divergence free transport equation in mathbb r n this result has direct applications to the density dependent euler equations the boussinesq system and the quasi geostrophic equations for which we also show nonexistence of self similar blowing up solutions

unique continuation type theorem for the self similar euler equations abstract we prove a unique continuation type theorem for the self similar euler equations in r 3 assuming the time periodicity namely if a time periodic solution v y s of the time dependent self similar euler equations has the property that v 0 s 0 for all s 0 s 0 where s 0 is the time period and y 0 is a local extremal point of v y s near the origin then v y s 0 for all y s r 3 0 s 0 a similar result holds for more general system with arbitrary coefficients and also for the inviscid incompressible magnetohydrodynamic mhd system as a consequence we obtain new criteria for the absence of the discretely self similar singularities for the 3d euler equations and the inviscid 3d mhd

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

second order variational analysis in second order cone programming the paper conducts a second order variational analysis for an important class of nonpolyhedral conic programs generated by the so called second order lorentz ice cream cone mathcal q from one hand we prove that the indicator function of mathcal q is always twice epi differentiable and apply this result to characterizing the uniqueness of lagrange multipliers together with an error bound estimate in the general second order cone programming setting involving twice differentiable data on the other hand we precisely calculate the graphical derivative of the normal cone mapping to mathcal q under the metric subregularity constraint qualification and then give an application of the latter result to a complete characterization of isolated calmness for perturbed variational systems associated with second order cone programs the obtained results seem to be the first in the literature in these directions for nonpolyhedral problems without imposing any nondegeneracy assumptions

degree theory for generalized variational inequalities and applications in this paper a degree theory for finite dimensional generalized variational inequalities is built and employed to prove some results on solution existence and solution stability

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

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

on leray s self similar solutions of the navier stokes equations satisfying local energy estimates this paper proves that leray s self similar solutions of the three dimensional navier stokes equations must be trivial under very general assumptions for example if they satisfy local energy estimates

note on the liouville type problem for the stationary navier stokes equations in r3 abstract in this brief note we are concerned on the liouville type problem for the stationary navier stokes equations in r 3 we deduce an asymptotic formula for an integral involving the head pressure q 1 2 v 2 p and its derivative over domains enclosed by level surfaces of q this formula provides us with new sufficient condition for the triviality of solution to the navier stokes equations

remarks on liouville type theorems for the 3d steady axially symmetric navier stokes equations abstract in this note we investigate the 3d steady axially symmetric navier stokes equations and obtain liouville type theorems if the velocity or the vorticity satisfies some a priori decay assumptions which improve the recent result of seregin s in 15

on liouville type theorems for the steady navier stokes equations in r3 abstract in this paper we prove three different liouville type theorems for the steady navier stokes equations in r 3 in the first theorem we improve logarithmically the well known l 9 2 r 3 result in the second theorem we present a sufficient condition for the trivially of the solution v 0 in terms of the head pressure q 1 2 v 2 p the imposed integrability condition here has the same scaling property as the dirichlet integral in the last theorem we present fubini type condition which guarantee v 0

decay and vanishing of some axially symmetric d solutions of the navier stokes equations abstract we study axially symmetric d solutions of the 3 dimensional navier stokes equations the first result is an a priori decay estimate of the velocity for general domains the second is an a priori decay estimate of the vorticity in r 3 which improves the corresponding results in the literature in addition we prove a similar decay of full 3d solutions except for a small set of angles next we turn to d solutions which are periodic in the third variable and prove vanishing result under a reasonable condition as a corollary we prove that axially symmetric d solutions in the slab r 2 i with suitable boundary condition is 0 here i is any finite interval to the best of our knowledge this seems to be the first vanishing result on a 3 dimensional d solution without extra integral or decay or smallness assumption on the solution the tools used include brezis gallouet inequality dimension reduction scaling green s function bound and liouville theorems for navier stokes equations

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal

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