global l1 theory and regularity for the 3d nonlinear wigner poisson fokker planck system abstract a global existence uniqueness and regularity theorem is proved for the simplest markovian wigner poisson fokker planck model incorporating friction and dissipation mechanisms the proof relies on green function and energy estimates under mild formulation making essential use of the husimi function and the elliptic regularization of the fokker planck operator

on the quantum dissipative generator weak coupling approximation and stochastic approach for a quantum open system the so called schr dinger langevin picture has been revisited in a second order perturbation it is shown that a non markovian evolution for the stochastic state vector leads to a dissipative generator which has a kossakowski lindblad form in this context it is possible to analyse the completely positive condition the equivalence of this picture with the trace out technique in the weak coupling approximation has been proved

on the standard quantum brownian equation and an associated class of non autonomous master equations abstract it is shown that the standard quantum brownian equation qbe can violate positivity not only past the thermal correlation time but at arbitrarily long times at high system frequencies in an effort to improve the standard qbe exact operator solutions are provided for a class of non autonomous master equations these exact solutions are used to derive sufficient positivity conditions for the coefficients of the master equations

getting a stochastic process from a conservative lagrangian a first approach abstract the transition probability p v for a stochastic process generated by a conservative lagrangian l l 0 e v is obtained at first order from a perturbation series found using a path integral this p v corresponds to the transition probability for a random walk with a probability density given by the sum of a normal distribution and a perturbation which may be understood as the contribution of the interaction of the random walk with the external field it is also found that the moment generating function for p v can be expressed as the generating function of a normal distribution modified by a perturbation applications of these results to a linear potential a harmonic oscillator potential and an exponentially decaying potential are shown

path integral semiclassical and stochastic propagators for markovian open quantum systems we develop the path integral theory for master equations of general lindblad form positive semigroups describing markovian open quantum systems first the hamiltonian path integral expression for the propagator is derived which exhibits nicely the decoherence of pairs of phase space histories a very appealing picture arises in the semiclassical limit where the degree of decoherence is expressible in terms of a phase space decoherence distance functional for the important class of effective hamiltonians quadratic in the momenta we derive the lagrangian version of the path integral propagator we then evaluate the path integral approximately in a stationary phase approximation leading to a van vleck type propagator valid under semiclassical conditions we also derive the propagator for the soluble damped harmonic oscillator in closed form from path integrals finally connections to the active field of stochastic pure state descriptions of open quantum systems are established here in particular to linear quantum state diffusion

multiple positive radial solutions to some kirchhoff equations abstract in this paper we discuss the existence and multiplicity of positive radial solutions to some kirchhoff equations and elliptic equations by using the leggett williams three solutions theorem skillfully we obtain two positive solutions to kirchhoff equations under some appropriate conditions as a direct corollary we also obtain the same result to elliptic equations in order to facilitate the proof of main results we first establish a new simple three solutions theorems for the equation a b x x a x where a is a completely continuous operator a 0 b 0 and x k x for all x p where k is a nondecreasing nonnegative continuous function then apply it to prove the main results in addition we discover and prove a new inequality in the article in the end of the paper we present an example which makes the elliptic equation has infinite many positive radial solutions and give the approximate images of the nonlinear term

the quantum state diffusion model applied to open systems a model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation derived from a density operator equation such as the bloch equation and consistent with it the advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described the method is applied to the nonlinear absorber cascades of quantum transitions second harmonic generation and a measurement reduction process the model provides graphic illustrations of these processes with statistical fluctuations that mimic those of experiments the stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics the model is compared with the quantum jump model of dalibard 1992 carmichael and others which originated among experimenters looking for intuitive pictures and rules of computation

anomalous continuous time random walk spectral diffusion in coherent third order optical response recursive relations are derived for nonlinear optical response functions of a two level chromophore with stochastic frequency fluctuations described by a continuous time random walk stationary ensembles are constructed and signatures of anomalous relaxation in the photon echo signal are discussed for a two state jump modulation with a power law waiting time density function psi t approximately t alpha 1 stretched exponential decay of the photon echo signal is predicted for 0 alpha 1 and power law asymptotics for 1 alpha 2

a simple view of the brain through a frequency specific functional connectivity measure here we develop a measure of functional connectivity describing the degree of covariability between a brain region and the rest of the brain this measure is based on previous formulas for the mutual information mi between clusters of regions in the frequency domain under the current scenario the mi can be given as a simple monotonous function of the multiple coherence and it leads to an easy visual representation of connectivity patterns computationally efficient formulas adequate for short time series are presented and applied to functional magnetic resonance imaging fmri data measured in subjects n 34 performing a working memory task or being at rest while resting state coherence in high 0 17 0 25 hz and middle 0 08 0 17 hz frequency intervals is bilaterally salient in several limbic and temporal areas including the insula the amygdala and the primary auditory cortex low frequencies 0 08 hz have greatest connectivity in frontal structures results from the comparison between resting and n back conditions show enhanced low frequency coherence in many of the areas previously reported in standard fmri activation studies of working memory but task related reductions in high frequency connectivity are also found in regions of the default mode network finally potentially confounding effects of head movement and regional volume on mi are identified and addressed

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

quantum brownian motion and generalizations of hermite polynomials abstract we introduce new families of orthogonal polynomials h d n motivated by the non equilibrium evolution of a quantum brownian particle qbp the h d n s generalize non trivially the standard hermite polynomials employed for classical brownian motion we treat several models labelled by d for a non equilibrium qbp by means of the wigner function w in the presence of a heat bath at thermal equilibrium with and without ab initio friction for long times for a suitable class of initial conditions the non equilibrium wigner function w should approach in some sense the time independent equilibrium wigner function w e q d which describes the thermal equilibrium of the qbp with the heat bath and plays a central role w e q d is chosen to be the weight function which orthogonalizes the h d n s new results on w e q d and on the h d n s are reported we justify the key role of the h d n s as follows using the h d n s moments w e q d n and w n are introduced for w e q d and w respectively at equilibrium all moments w e q d n except the lowest one w e q d 0 vanish identically off equilibrium one expects that for long times for suitable initial conditions i all non equilibrium moments w n except the lowest moment w 0 will approach zero while ii the lowest non equilibrium moment w 0 will tend to w e q d 0 0 to complete the justification we outline how the approximate long time non equilibrium theories determined by w 0 for the different models d yield smoluchowski equations and irreversible evolutions of the qbp towards thermal equilibrium

the caldeira leggett quantum master equation in wigner phase space continued fraction solution and application to brownian motion in periodic potentials the continued fraction method of solving classical fokker planck equations has been adapted to tackle quantum master equations of the caldeira leggett type this was done taking advantage of the phase space wigner representation of the quantum density matrix the approach differs from those in which some continued fraction expression is found for a certain quantity in that the full solution of the master equation is obtained by continued fraction methods this allows us to study in detail the effects of the environment fluctuations and dissipation on several classes of nonlinear quantum systems we apply the method to the canonical problem of quantum brownian motion in periodic potentials both for cosine and ratchet potentials lacking inversion symmetry

quantum smoluchowski equation a systematic study the strong friction regime at low temperatures is analyzed systematically starting from the formally exact path integral expression for the reduced dynamics this quantum smoluchowski regime allows for a type of semiclassical treatment in the inverse friction strength so that higher order quantum corrections to the original quantum smoluchowski equation j ankerhold p pechukas and h grabert phys rev lett 87 086802 2001 j ankerhold and h grabert phys rev lett 101 119903 2008 can be derived drift and diffusion coefficients are determined by the equilibrium distribution in position and are directly related to the corresponding action of extremal paths and fluctuations around them it is shown that the inclusion of higher order corrections reproduces the quantum enhancement above crossover for the decay rate out of a metastable well exactly

construction of coarse grained order parameters in nonequilibrium systems we develop a renormalization group rg procedure that includes important system specific features the key ingredient is to systematize the coarse graining procedure that generates the rg flow the coarse graining technology comes from the control and operator theoretic model reduction the resulting generalized rg is a consistent generalization of the wilsonian rg we apply the procedure to a deterministic nonlinear wave equation nlwe with probabilistic initial conditions we derive the form of the projection operator from the dynamics of the nlwe and then use it to generate the rg flow for the distribution of initial conditions the probability density of the initial conditions is chosen to be a boltzmann weight that is quartic in the field variables in our calculation we find that in contrast to conventional implementations of the rg naive power counting breaks down we also show that the resulting rg equations are different from those derived from the conventional rg

the effect of interference of coloured additive and multiplicative white noises on escape rate in this paper we have calculated the escape rate from a meta stable state for coloured and correlated noise driven open systems based on the fokker planck description of the stochastic process we consider the effect of two correlation times due to the additive coloured noise and the correlation between additive coloured and multiplicative white noises the effect of the noise correlation strength on the rate has also been investigated

sample size and bentler and bonett s nonnormed fit index bentler and bonett s nonnormed fit index is a widely used measure of goodness of fit for the analysis of covariance structures this note shows that contrary to what has been claimed the nonnormed fit index is dependent on sample size specifically for a constant value of a fitting function the nonnormed index is inversely related to sample size a simple alternative fit measure is proposed that removes this dependency in addition it is shown that this new measure as well as the old nonnormed fit index can be applied to any fitting function that measures the deviation of the observed covariance matrix from the covariance matrix implied by the parameter estimates for a model

reduction of overdamped stochastic processes in the coherent state representation the adiabatic elimination of fast variables from overdamped stochastic processes by functional integration is demonstrated adiabatic elimination entails the evaluation of a reduced evolution operator from the full evolution equation for fokker planck processes the reduced evolution operator may be expressed as a ground state expectation and it is shown how this is represented as a coherent state path integral the elimination is then achieved by functionally integrating out all reference to the fast variables the end result is a decoupling of the full evolution equation into separate equations for the fast and slow variables the method is demonstrated for brownian motion and for a system with multiplicative colored noise

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

on sign changing solutions for nonlinear operator equations abstract in this paper the existence of sign changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory the main theorems are some new three solution theorems which are different from the famous amann s and leggett williams three solution theorems as well as the results in f li g han generalization for amann s and leggett williams three solution theorems and applications j math anal appl 298 2004 638 654 these three solutions are all nonzero one of them is positive another is negative and the third one is a sign changing solution furthermore the theoretical results are successfully applied to both integral and differential equations

decoherence of semiclassical wigner functions the lindblad master equation governs the general markovian evolution of a density operator for an open quantum system semiclassical wigner functions represent density operators in phase space in terms of chords on a classical manifold so that the amplitude and phase of each chord contribution is classically defined inserting such a wigner function into a phase space version of the master equation its explicit evolution is derived in the absence of dissipation there results a simple extension of the unitary evolution of the semiclassical wigner function which does not affect the phase of each chord contribution while dampening its amplitude exponentially projecting the wigner function on to an orthogonal position or momentum basis the dampening of long chords emerges as the exponential decay of off diagonal elements of the density matrix

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

comparing dynamic causal models this article describes the use of bayes factors for comparing dynamic causal models dcms dcms are used to make inferences about effective connectivity from functional magnetic resonance imaging fmri data these inferences however are contingent upon assumptions about model structure that is the connectivity pattern between the regions included in the model given the current lack of detailed knowledge on anatomical connectivity in the human brain there are often considerable degrees of freedom when defining the connectional structure of dcms in addition many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation and a formal procedure for directly comparing these competing hypotheses is highly desirable in this article we show how bayes factors can be used to guide choices about model structure both concerning the intrinsic connectivity pattern and the contextual modulation of individual connections the combined use of bayes factors and dcm thus allows one to evaluate competing scientific theories about the architecture of large scale neural networks and the neuronal interactions that mediate perception and cognition

a square root method for the density matrix and its applications to lindblad operators abstract the evolution of open systems subject to both hamiltonian and dissipative forces is studied by writing the nm element of the time t dependent density matrix in the form nm t 1 a 1 a n t m t the so called square root factors the t s are non square matrices and are averaged over a systems of the ensemble this square root description is exact evolution equations are then postulated for the t factors such as to reduce to the lindblad type evolution equations for the diagonal terms in the density matrix for the off diagonal terms they differ from the lindblad equations the square root factors t are not unique and the equations for the t s depend on the specific representation chosen two criteria can be suggested for fixing the choice of t s one is simplicity of the resulting equations and the other has to do with the reduction of the difference between the t formalism and the lindblad equations when the method is tested on cases which have been previously treated by other methods our results agree with them examples chosen are i molecular systems such that are either periodically driven near level degeneracies for which we calculate the decoherence occurring in multiple landau zener transitions or else when undergoing descent around conical intersections in the potential surfaces ii formal dissipative systems with lindblad type operators representing either a non markovian process or a two state system coupled to bosons attractive features of the present factorization method are complete positivity the no higher than linear increase of the implementation effort with the number of states involved and the introduction of randomness only at the start of the process

how good is good enough in path analysis of fmri data this paper is concerned with the problem of evaluating goodness of fit of a path analytic model to an interregional correlation matrix derived from functional magnetic resonance imaging fmri data we argue that model evaluation based on testing the null hypothesis that the correlation matrix predicted by the model equals the population correlation matrix is problematic because p values are conditional on asymptotic distributional results which may not be valid for fmri data acquired in less than 10 min as well as arbitrary specification of residual variances and effective degrees of freedom in each regional fmri time series we introduce an alternative approach based on an algorithm for automatic identification of the best fitting model that can be found to account for the data the algorithm starts from the null model in which all path coefficients are zero and iteratively unconstrains the coefficient which has the largest lagrangian multiplier at each step until a model is identified which has maximum goodness by a parsimonious fit index repeating this process after bootstrapping the data generates a confidence interval for goodness of fit of the best model if the goodness of the theoretically preferred model is within this confidence interval we can empirically say that the theoretical model could be the best model this relativistic and data based strategy for model evaluation is illustrated by analysis of functional mr images acquired from 20 normal volunteers during periodic performance for 5 min of a task demanding semantic decision and subvocal rehearsal a model including unidirectional connections from frontal to parietal cortex designed to represent sequential engagement of rehearsal and monitoring components of the articulatory loop is found to be irrefutable by hypothesis testing and within confidence limits for the best model that could be fitted to the data

stochastic quantum langevin equation the stochastic representation of quantum mechanics is used to give a purely stochastic treatment of the quantum langevin equation a general coupling involving positions and velocities of the particle and harmonic modes with arbitrary dependence on the particle s position is dealt with summing over the modes in the path integral and using stochastic integrals the author obtains explicitly the reduced measure that is the stochastic process representing the time evolution of the small system in the presence of noise contact is made with the operatorial representation and a relation is established between the correlations of the noise in both representations a general condition on the coupling constants is given for recovering time locality in the continuum limit and the effects of renormalisation are re analysed within this purely stochastic approach

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

propagator for narrow potential barriers the schrodinger equation in integral form is applied to the one dimensional scattering problem in the case of a narrow potential barrier since the kernel can be considered in a first approximation separable an explicit expression for the propagator is found by means of the complementary error function the problem of a particle confined in a half space and interacting with a narrow potential barrier is also considered and solved in an approximate way

multidimensional scaling of integrated neurocognitive function and schizophrenia as a disconnexion disorder multidimensional scaling mds is a multivariate statistical technique that can be used to define sub systems of functionally connected brain regions based on the analysis of functional magnetic resonance im aging fmri data here we introduce three way mul tidimensional scaling as a method for the analysis of a group of fmri data which yields both a generic inter regional configuration in low dimensional space and a measure of each individual s deviation from the ge neric configuration the distance between two generic interregional configurations obtained by mds of two groups of data can be minimized by generalized pro crustes analysis and the probability under the null hypothesis that the two groups are sampled from the same population of any residual group difference in interregional configurations can be assessed by a per mutation test these methods are developed and ap plied to activated fmri time series acquired from 19 patients with schizophrenia and 20 normal compari son subjects during the performance of a semantic categorization and subvocal rehearsal task the first three scaling dimensions are interpretable in terms of the major anatomical or functional subsystems of the activated system left right input processing other and subvocal output other we found no sig nificant global or local differences between groups in interregional configurations in this 3d space how ever there was significantly greater variability of interregional configurations within the group of patients with schizophrenia the implications for schizophrenia as a disconnexion disorder are

chemical langevin equation a path integral view of gillespie s derivation in 2000 gillespie rehabilitated the chemical langevin equation cle by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation cme in this work we construct an original path integral description of the cme and show how applying gillespie s two conditions to it directly leads to a path integral equivalent to the cle we compare this approach to the path integral equivalent of a large system size derivation and show that they are qualitatively different in particular both approaches involve converting many sums into many integrals and the difference between the two methods is essentially the difference between using the euler maclaurin formula and using riemann sums our results shed light on how path integrals can be used to conceptualize coarse graining biochemical systems and are readily generalizable

nonlinear schrodinger equation with a white noise potential phase space approach to spread and singularity we propose a phase space formulation for the nonlinear schr odinger equation with a white noise potential in order to shed light on two issues the rate of spread and the singularity formation in the average sense our main tools are the energy law and the variance identity the method is completely elementary for the problem of wave spread we show that the ensemble averaged dispersion in the critical or defocusing case follows the cubic in time law while in the supercritical and subcritical focusing cases the cubic law becomes an upper and lower bound respectively we have also found that in the critical and supercritical focusing cases the presence of a white noise random potential results in different conditions for singularity with positive probability from the homogeneous case but does not prevent singularity formation we show that in the supercritical focusing case the ensemble averaged self interaction energy and the momentum variance can exceed any fixed level in a finite time with positive probability c 2005 elsevier b v all rights reserved

phase space models for stochastic nonlinear parabolic waves wave spread and collapse we derive several kinetic equations to model the large scale low fresnel number behaviour of the generalized nonlinear schr odinger nls equation with a rapidly fluctuating random potential depending on the relative scale of fluctuation in the longitudinal and transverse directions we classify the kinetic equations into the longitudinal the transverse and the isotropic case the principal assumption of our derivation is that the rapid fluctuation in the linear potential does not give rise to rapid oscillations in the modulus of the wave amplitude for the longitudinal and the transverse cases we address two problems the rate of dispersion and the singularity formation using the nonlinear kinetic equations the main technique is the variance identities derived for the nonlinear kinetic equations for the problem of dispersion we show that in the longitudinal case the spread scales like time 3 2 whereas in the transverse case the spread is linearly proportional to time for the problem of singularity formation we show that the collapse conditions in the transverse case remain the same as those for the homogeneous nls equation with critical or supercritical self focusing nonlinearity whereas in the longitudinal case the small scale medium fluctuations tend to enhance the energy of the system and thus raise the energy barrier to wave collapse

six problems for causal inference from fmri neuroimaging e g fmri data are increasingly used to attempt to identify not only brain regions of interest rois that are especially active during perception cognition and action but also the qualitative causal relations among activity in these regions known as effective connectivity friston 1994 previous investigations and anatomical and physiological knowledge may somewhat constrain the possible hypotheses but there often remains a vast space of possible causal structures to find actual effective connectivity relations search methods must accommodate indirect measurements of nonlinear time series dependencies feedback multiple subjects possibly varying in identified regions of interest and unknown possible location dependent variations in bold response delays we describe combinations of procedures that under these conditions find feed forward sub structure characteristic of a group of subjects the method is illustrated with an empirical data set and confirmed with simulations of time series of non linear randomly generated effective connectivities with feedback subject to random differences of bold delays with regions of interest missing at random for some subjects measured with noise approximating the signal to noise ratio of the empirical data

existence of solutions to a class of nonlinear second order two point boundary value problems in this paper the existence and multiplicity results of solutions are obtained for the second order two point boundary value problem u t f t u t for all t 0 1 subject to u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively in argument quadratic root operator and its properties play an important role

two derivations of the master equation of quantum brownian motion central to many discussion of decoherence is a master equation for the reduced density matrix of a massive particle experiencing scattering from its surrounding environment such as that of joos and zeh such master equations enjoy a close relationship with spontaneous localization models like the grw model the aim of this paper is to present two derivations of the master equation the first derivation is a pedagogical model designed to illustrate the origins of the master equation as simply as possible focusing on physical principles and without the complications of s matrix theory this derivation may serve as a useful tutorial example for students attempting to learn this subject area the second is the opposite a very general derivation using non relativistic many body field theory it reduces to the equation of the type given by joos and zeh in the one particle sector but correcting certain numerical factors which have recently become significant in connection with experimental tests of decoherence this master equation also emphasizes the role of local number density as the preferred basis for decoherence in this model

relevant and irrelevant nonlinear schrodinger equations first we summarize the argument against deterministic nonlinear schrodinger equations we recall that any such equation activates quantum nonlocality in the sense that that information could be signalled in a finite time over arbitrarily large distances next we introduce a deterministic nonlinear schrodinger equation we justify it by showing that it is closest in a precise sense to the master equations for mixed states used to describe the evolution of open quantum systems we also illustrate some interesting properties of this equation finally we show that this equation can avoid the signalling problem if one adds noise to it in a precise way cases of both discrete and continuous noise are introduced explicitly and related to the density operator evolution the relevance for the classical limit of the obtained stochastic equations is illustrated on a classically chaotic kicked anharmonic oscillator

functional integral treatment of some quantum nondemolition systems in the scheme of a quantum nondemolition qnd measurement an observable is measured without perturbing its evolution in the context of studies of decoherence in quantum computing we examine the open quantum system of a two level atom or equivalently a spin 1 2 system in interaction with quantum reservoirs of either oscillators or spins under the qnd condition of the hamiltonian of the system commuting with the system reservoir interaction for completeness we also examine the well known non qnd spin bose problem for all these many body systems we use the methods of functional integration to work out the propagators the propagators for the qnd hamiltonians are shown to be analogous to the squeezing and rotation operators respectively for the two kinds of baths considered squeezing and rotation being both phase space area preserving canonical transformations this brings out an interesting connection between the energy preserving qnd hamiltonians and the homogeneous linear canonical transformations

a simple gauss newton procedure for covariance structure analysis with high level computer languages an implementation of the gauss newton algorithm for the analysis of covariance structures that is specifically adapted for high level computer languages is reviewed with this procedure one need only describe the structural form of the population covariance matrix and provide a sample covariance matrix and initial values for the parameters the gradient and approximate hessian which vary from model to model are computed numerically using this approach the entire method can be operationalized in a comparatively small program a large class of models can be estimated including many that utilize functional relationships among the parameters that are not possible in most available computer programs some examples are provided to illustrate how the algorithm can be used

effect of initial fmri data modeling on the connectivity reported between brain areas nearly all neuroimaging data analysis rests upon some form of variance partitioning conventional analyses with a general linear model glm partition the variance in the measured response variable into partitions described by a design matrix of explanatory variables this approach can also be adopted in the initial modeling of the data in studies using data led methods to summarize functional connectivity such as principle component analysis or studies of effective connectivity using for example structural equation modeling the point made in this technical note is that the partition of the original time series has to be precisely described to qualify the sources of variations that are taken into account for conventional analyses using the glm the partition investigated corresponds to the subspaces of the design matrix that are tested however in the analyses of functional and effective connectivity the particular subspaces considered are not always specified explicitly here we show that selecting different subspaces or variance partitions can have a profound effect both qualitatively and quantitatively on the sample covariances and the ensuing inferences about connectivity we will illustrate this using simulated data that include condition and block related effects and their interactions we will use these three subspaces to show how the correlation between two voxels depends on which sub partitions are examined we will also show how the partition of the design matrix influences the resulting correlation matrix observed when studying correlations between error terms we will finally demonstrate quantitatively the effect of the variance partitions considered on the correlations between two regions using a real fmri study of biological motion

symmetry of bilinear master equations for a quantum oscillator abstract we study the most general continuous transformation on the generators of bilinear master equations of a quantum oscillator we find that transformation operators that preserve the hermiticity of density operators and conserve the probability of reduced dynamics should be adjoint symmetric and they are not limited to the pure product of unitary operators in the bra and ket space but could be a mixture of them we need to include the more general transformation operators to explore the full symmetry of generic reduced dynamics we discuss how the operators are related to those considered in previous works and illustrate how they leave the reduced dynamics form invariant or map one into the other the positive semidefinite requirement on the density operator can be imposed to give a valid range of transformation parameters

zero point energies the uncertainty principle and positivity of the quantum brownian density operator high temperature and white noise approximations are frequently invoked when deriving the quantum brownian equation for an oscillator even if this white noise approximation is avoided it is shown that if the zero point energies of the environment are neglected as they often are the resultant equation will violate not only the basic tenet of quantum mechanics that requires the density operator to be positive but also the uncertainty principle when the zero point energies are included asymptotic results describing the evolution of the oscillator are obtained that preserve positivity and therefore the uncertainty principle

multiple sign changing solutions to the sturm liouville boundary value problem with resonance abstract in this paper we study the sturm liouville boundary value problem p x u q x u f x u x 0 1 subject to u 0 u 0 u 1 u 1 0 by constructing a new sobolev space h 1 0 1 we discuss the existence of multiple solutions especially the existence of multiple sign changing solutions to this problem when the nonlinear f is resonant both at 0 and by combining the methods of the morse theory the topological degree and the fixed point index we establish a multiple solutions theorem which guarantees that the problem has at least six nontrivial solutions if this problem has only finitely many solutions then of these solutions there are two positive solutions two negative solutions and two sign changing solutions

reaction rate theory weak to strong friction turnover in kramers fokker planck model abstract we present a unified treatment for both weak and strong friction including the turnover regime of kramers fokker planck model for activated rate processes no recourse to a specific microscopic model a la grabert et al or langevin dynamics a la graham will be made upon introducing some new theoretical concepts such as a constrained gaussian transformation cgt and a dynamical extension of phase space eps the analysis proceeds in terms of the unstable mode energy and is systematic in a small parameter we also introduce a nonlinear toy model for barrier recrossing for a simplified discussion of aspects like finite barrier corrections the boundary layer approximation and exact solution in the absence of potential conditions

entropic uncertainty bound for open pointer based simultaneous measurements of conjugate observables we discuss the information entropy for a general open pointer based simultaneous measurement and show how it is bound from below this entropic uncertainty bound is a direct consequence of the structure of the entropy and can be obtained from the formal solution of the measurement dynamics furthermore the structural properties of the entropy allow us to give an intuitive interpretation of the noisy influence of the pointers and the environmental heat bath on the measurement results

path integral for quantum operations in this paper we consider a phase space path integral for general time dependent quantum operations not necessarily unitary we obtain the path integral for a completely positive quantum operation satisfied lindblad equation quantum markovian master equation we consider the path integral for quantum operation with a simple infinitesimal generator

linearly coupled quantum oscillators with levy stable noise abstract the influence of an external noise on two linearly coupled quantum oscillators is investigated the noise is described by the probability measure corresponding to a symmetric levy stable distribution in calculations the path integral technique is used this paper is a continuation of investigations started by bose datta and feinsilver

non markovian expansion in quantum brownian motion abstract we consider the non markovian langevin evolution of a dissipative dynamical system in quantum mechanics in the path integral formalism after discussing the role of the frequency cutoff for the interaction of the system with the heat bath and the kernel and noise correlator that follow from the most common choices we derive an analytic expansion for the exact non markovian dissipation kernel and the corresponding colored noise in the general case that is consistent with the fluctuation dissipation theorem and incorporates systematically non local corrections we illustrate the modifications to results obtained using the traditional markovian langevin approach in the case of the exponential kernel and analyze the case of the non markovian brownian motion we present detailed results for the free and the quadratic cases which can be compared to exact solutions to test the convergence of the method and discuss potentials of a general nonlinear form

the wigner representation of classical mechanics quantization and classical limit abstract starting from the liouvillian formulation of classical physics it is possible by means of a fourier transform to introduce the wigner representation and to derive an operator structure to classical mechanics the importance of this new representation lies in the fact that it turns out to be the suitable route to establish a general method of quantization directly from the equations of motion without alluding to the existence of hamiltonian and lagrangian functions following this approach we quantize only the motion of a brownian particle with non linear friction in the markovian approximation the thermal bath may be quantum or classical thus when the bath is classically described we obtain a master equation which reduces to the caldeira leggett equation vide a o caldeira and a j leggett physica a 121 1983 587 for the linear friction case and when the reservoir is quantum we get an equation reducing to the one found by caldeira et al phys rev a 40 1989 3438 by neglecting the environmental influence we show that the system can be approximately described by equations of motion in terms of wave function such as the schrodinger langevin equation and equations of the caldirola kanai type finally to make the present study self consistent we evaluate the classical limit of these dynamical equations employing a new classical limiting method 0

connectivity analysis with structural equation modelling an example of the effects of voxel selection abstract structural equation modelling sem of neuroimaging data is commonly applied to a network of distributed brain regions we applied sem to an fmri dataset to identify condition specific effects in a simple experiment composed of visual stimulation and baseline conditions the visual network was composed of three well defined anatomical regions v1 v2 and v5 and three path connections v1 v2 v1 v5 and v2 v5 this network was used to test four hypotheses 1 whether the condition specific effects for all three connections vary according to the data selected for modelling 2 whether the summary measures that are often used are indeed appropriate 3 whether measures taken from the voxel timecourse can reliably predict the condition specific effects for each one of the three path connections and 4 whether all voxels within an anatomical region yield equivalent sem outcomes there was some variability in the significance of the condition specific effects across randomly selected voxels within regions however the sem outcome from the summary measures was comparable to the most frequent pattern of condition specific effects magnitude delay spread and goodness of fit measures taken from a gamma fit to the voxel time courses predicted reliably the significance of the sem condition specific effects for each connection this result enabled us to identify spatially coherent regions at the boundaries of v2 that displayed different condition specific effects from those seen in the majority of the voxels although the generality of these results awaits further investigation this example highlights a number of important issues for sem we have provided further evidence that the sem outcome does vary somewhat according to the voxels selected and that although the use of summary measures can give a generalised view of the connectivity pattern they could fail to capture functional differences within specialised areas

multiple solutions for nonlinear operator equations under the condition of two pairs of paralleled lower and upper solutions in this paper under the condition of two pairs of paralleled lower and upper solutions some new multiple solution theorems are obtained by using the fixed point index theory the theoretical results are applied to the nonlinear hammerstein integral equation

eigenvalues of the liouvillians of quantum master equation for a harmonic oscillator abstract the eigenvalues of the liouvillians of markovian master equation for a harmonic oscillator have a generic form the liouvillians considered are quadratic in the position coordinates or creation and annihilation operators as well as having positive renormalized frequencies we prove this by showing that a generic liouvillian of this form can be similarly related to the liouvillian of the kossakowski lindblad equation whose eigenvalues are already known the left and right eigenfunctions of the generic liouvillian also form a complete and biorthogonal set examples of similarly related right eigenfunctions are given

finite time thermodynamics of port hamiltonian systems abstract in this paper we identify a class of time varying port hamiltonian systems that is suitable for studying problems at the intersection of statistical mechanics and control of physical systems those port hamiltonian systems are able to modify their internal structure as well as their interconnection with the environment over time the framework allows us to prove the first and second laws of thermodynamics but also lets us apply results from optimal and stochastic control theory to physical systems in particular we show how to use linear control theory to optimally extract work from a single heat source over a finite time interval in the manner of maxwell s demon furthermore the optimal controller is a time varying port hamiltonian system which can be physically implemented as a variable linear capacitor and transformer we also use the theory to design a heat engine operating between two heat sources in finite time carnot like cycles of maximum power and we compare those two heat engines

integrable structure of conformal field theory ii q operator and ddv equation this paper is a direct continuation of 1 where we began the study of the integrable structures in conformal field theory we show here how to construct the operators bf q pm lambda which act in the highest weight virasoro module and commute for different values of the parameter these operators appear to be the cft analogs of the q matrix of baxter 2 in particular they satisfy baxter s famous t q equation we also show that under natural assumptions about analytic properties of the operators as the functions of the baxter s relation allows one to derive the nonlinear integral equations of destri de vega ddv 3 for the eigenvalues of the q operators we then use the ddv equation to obtain the asymptotic expansions of the q operators at large it is remarkable that unlike the expansions of the t operators of 1 the asymptotic series for q contains the dual nonlocal integrals of motion along with the local ones we also discuss an intriguing relation between the vacuum eigenvalues of the q operators and the stationary transport properties in the boundary sine gordon model on this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum hall system

erratum master equation in phase space applied to the quantum brownian motion in a tilted periodic potential quantum effects in the brownian motion in a tilted cosine potential are treated in the high temperature and weak bath particle coupling limit via the semiclassical master equation for the time evolution of the reduced wigner function in phase space the differential recurrence relation generated from the master equation by expanding the periodic wigner function expressed as a function of wave number k in fourier series via floquet s theorem is solved for bounded periodic initial conditions using matrix continued fractions yielding both the time independent and the time dependent bounded periodic solutions just as the classical problem the non periodic solution is determined from the recurrence relation by integrating w r t k over the first brillouin zone with the initial conditions generated by the non periodic wigner distribution corresponding to zero probability current the time independent periodic locked solution is used to estimate quantum effects in the dc current voltage characteristic of a josephson junction including the capacitance while the time dependent non periodic running solution is used to determine the dynamic structure factor and ultimately via integration over the first brillouin zone the lifetime of the zero voltage state the current voltage characteristics displayed as average velocity versus tilt reproduce in the high damping limit those from the semiclassical smoluchowski equation while the decrease in the lifetime of the zero voltage state due to the barrier lowering quantum effects appears to be very sensitive to the tilt

solutions of generic bilinear master equations for a quantum oscillator positive and factorized conditions on stationary states abstract we obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the gaussian form the well behavedness and positive semidefiniteness of the stationary states could be characterized by a three dimensional minkowski vector by requiring the stationary states to satisfy a factorized condition we obtain a generic class of master equations that includes the well known ones and their generalizations some of which are completely positive a further subset of the master equations with the gibbs states as stationary states is also obtained for master equations with not completely positive generators an analysis on the stationary states for a given initial state suggests conditions on the coefficients of the master equations that generate positive evolution

superoperator representation of nonlinear response unifying quantum field and mode coupling theories computing response functions by following the time evolution of superoperators in liouville space whose vectors are ordinary hilbert space operators offers an attractive alternative to the diagrammatic perturbative expansion of many body equilibrium and nonequilibrium green s functions the bookkeeping of time ordering is naturally maintained in real physical time allowing the formulation of wick s theorem for superoperators giving a factorization of higher order response functions in terms of two fundamental green s functions backward propagations and analytic continuations using artificial times keldysh loops and matsubara contours are avoided a generating functional for nonlinear response functions unifies quantum field theory and the classical mode coupling formalism of nonlinear hydrodynamics and may be used for semiclassical expansions classical response functions are obtained without the explicit computation of stability matrices

quantum moment hydrodynamics and the entropy principle this paper presents how a non commutative version of the entropy extremalization principle allows to construct new quantum hydrodynamic models our starting point is the moment method which consists in integrating the quantum liouville equation with respect to momentum p against a given vector of monomials of p like in the classical case the so obtained moment system is not closed inspired from levermore s procedure in the classical case 26 we propose to close the moment system by a quantum wigner distribution function which minimizes the entropy subject to the constraint that its moments are given in contrast to the classical case the quantum entropy is defined globally and not locally as the trace of an operator therefore the relation between the moments and the lagrange multipliers of the constrained entropy minimization problem becomes nonlocal and the resulting moment system involves nonlocal operators instead of purely local ones in the classical case in the present paper we discuss some practical aspects and consequences of this nonlocal feature

calderia leggett master equation and medium temperatures abstract the markovian approximation of quantum dissipation has been reconsidered in the model of caldeira and leggett physica a 121 1983 587 their high temperature master equation has been generalized to medium temperatures two additional damping terms in the master equation have been derived and shown to assure the lindblad form of the equation a transient term in the density matrix has been found and expressed in a simple form

generalization for amann s and leggett williams three solution theorems and applications abstract in this paper some new three solution theorems are obtained moreover the famous amann s and leggett williams three solution theorems in nonlinear functional analysis can be seen as their special cases namely these two famous theorems are united so they are both improved and that some new three solution theorems are applied to ordinary differential equations and some new existence results are obtained

filtering induces correlation in fmri resting state data abstract correlation based functional mri connectivity methods typically impose a temporal sample independence assumption on the data however the conventional use of temporal filtering to address the high noise content of fmri data may introduce sample dependence violation of the independence assumption has ramifications for the distribution of sample correlation which if unaccounted for may invalidate connectivity results to enable the use of temporal filtering for noise suppression while maintaining the integrity of connectivity results we derive the distribution of sample correlation between filtered timeseries as a function of the filter frequency response corrected distributions are also derived for statistical inference tests of sample correlation between filtered timeseries including fisher s z transformation and the student s t test crucially the proposed corrections are valid for any unknown true correlation and arbitrary filter specifications empirical simulations demonstrate the potential for temporal filtering to artificially induce connectivity by introducing sample dependence and verify the utility of the proposed corrections in mitigating this effect the importance of our corrections is exemplified in a resting state fmri connectivity analysis seed voxel correlation maps generated from filtered data using uncorrected test variates yield an unfeasible number of connections to the left primary motor cortex suggesting artificially induced connectivity while maps acquired from filtered data using corrected test variates exhibit bilateral connectivity in the primary motor cortex in conformance with expected results as seen in the literature

general lidstone problems multiplicity and symmetry of solutions abstract for the 2mth order lidstone boundary value problem y 2 m t f y t y t y 2 i t y 2 m 1 t t 0 1 y 2 i 0 y 2 i 1 0 0 i m 1 where 1 mf r m 0 thinsp is continuous growth conditions are imposed on f which yield the existence of at least three symmetric positive solutions this generalizes earlier papers which have applied avery s generalization of the leggett williams theorem to lidstone problems we then prove the analogous result for difference equations

quantum dynamical semigroups for diffusion models with hartree interaction we consider a class of evolution equations in lindblad form which model the dynamics of dissipative quantum mechanical systems with mean field interaction particularly this class includes the so called quantum fokker planck poisson model the existence and uniqueness of global in time mass preserving solutions is proved thus establishing the existence of a nonlinear conservative quantum dynamical semigroup the mathematical difficulties stem from combining an unbounded lindblad generator with the hartree nonlinearity

multiple positive solutions of nonlinear two point boundary value problems we improve the results obtained by erbe hu and wang in a recent paper we show that there exist at least two positive solutions of two point boundary value problems under conditions weaker than those used by erbe hu and wang

wigner distribution functions for complex dynamical systems the emergence of the wigner boltzmann equation the equation of motion for the reduced wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators the result is derived from the wigner function formulation of the feynman vernon influence functional theory it is shown how the true self energy for the equation of motion is connected with the influence functional for the path integral explicit expressions are derived in terms of the bare wigner propagator finally we show under which approximations the resulting equation of motion reduces to the wigner boltzmann equation

network discovery with dcm this paper is about inferring or discovering the functional architecture of distributed systems using dynamic causal modelling dcm we describe a scheme that recovers the dynamic bayesian dependency graph connections in a network using observed network activity this network discovery uses bayesian model selection to identify the sparsity structure absence of edges or connections in a graph that best explains observed time series the implicit adjacency matrix specifies the form of the network e g cyclic or acyclic and its graph theoretical attributes e g degree distribution the scheme is illustrated using functional magnetic resonance imaging fmri time series to discover functional brain networks crucially it can be applied to experimentally evoked responses activation studies or endogenous activity in task free resting state fmri studies unlike conventional approaches to network discovery dcm permits the analysis of directed and cyclic graphs furthermore it eschews implausible markovian assumptions about the serial independence of random fluctuations the scheme furnishes a network description of distributed activity in the brain that is optimal in the sense of having the greatest conditional probability relative to other networks the networks are characterised in terms of their connectivity or adjacency matrices and conditional distributions over the directed and reciprocal effective connectivity between connected nodes or regions we envisage that this approach will provide a useful complement to current analyses of functional connectivity for both activation and resting state studies

hamilton jacobi and fokker planck equations for the harmonic oscillator abstract using feynman s path integral formalism applied to stochastic classical processes we show the equivalence between the hamilton jacobi hj and fokker planck fp equations associated with an overdamped brownian harmonic oscillator in this case the langevin equation leads to a gaussian lagrangian function and then the path integration which defines the conditional probability density can be replaced by the extremal path due to this fact and following the classical dynamics formalism we prove the strict equivalence between the hj and fp equations we do this first for an ordinary brownian harmonic oscillator and then the proof is extended to an electrically charged brownian particle under the action of force fields magnetic field and additional time dependent force fields we observe that this extremal action principle allows us to derive in a straightforward way not only the hj differential equation but also its solution the extremal action