an algebraic construction of generalized coherent states for shape invariant potentials generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics we show that this generalized formalism is able to a supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system continuity of labelling resolution of unity temporal stability and action identity b reproduce results already known for shape invariant systems such as harmonic oscillator double anharmonic poschl teller and self similar potentials and c point to a formalism that provides a unified description of the different kind of coherent states for quantum systems

variational bayesian identification and prediction of stochastic nonlinear dynamic causal models in this paper we describe a general variational bayesian approach for approximate inference on nonlinear stochastic dynamic models this scheme extends established approximate inference on hidden states to cover i nonlinear evolution and observation functions ii unknown parameters and precision hyperparameters and iii model comparison and prediction under uncertainty model identification or inversion entails the estimation of the marginal likelihood or evidence of a model this difficult integration problem can be finessed by optimising a free energy bound on the evidence using results from variational calculus this yields a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables we derive such a variational bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models for both model identification and time series prediction the computational complexity of the scheme is comparable to that of an extended kalman filter which is critical when inverting high dimensional models or long time series using monte carlo simulations we assess the estimation efficiency of this variational bayesian approach using three stochastic variants of chaotic dynamic systems we also demonstrate the model comparison capabilities of the method its self consistency and its predictive power

irreducible representations of the lie superalgebras b 0 n with finite degenerated vacuum ii the problem of getting irreducible representations of lie superalgebras b 0 n n 1 2 is studied starting with a recently constructed family of linear representations in terms of differential operators on the space c n of cn valued c functions equivalent formulation via creation annihilation operators of a para bose system with n degrees of freedom is used and the domain d of any is shown to be a subset of c n containing a nonzero vacuum subspace by assuming its dimension finite the necessary conditions for existence of are derived the method is applied to the superalgebra b 0 1 and a one parameter family of nonequivalent irreducible representations in terms of unbounded linear operators on l2 r c2 is obtained each representation has a nondegenerated vacuum and for all z b 0 1 satisfying z z the operators z are essentially self adjoint

wigner quantum systems two particles interacting via a harmonic potential i two dimensional space a noncanonical quantum system consisting of two nonrelativistic particles interacting via a harmonic potential is considered the center of mass position and momentum operators obey the canonical commutation relations whereas the internal variables are assumed to be the odd generators of the lie superalgebra sl 1 2 this assumption implies a set of constraints in the phase space which are explicitly written in the paper all finite dimensional irreducible representations of sl 1 2 are considered particular attention is paid to the physical representations i e the representations corresponding to hermitian position and momentum operators the properties of the physical observables are investigated in particular the operators of the internal hamiltonian the relative distance the internal momentum and the orbital momentum commute with each other the spectrum of these operators is finite the distance between the constituents is preserved in time it can take no more than three different values for any non negative integer or half integer l there exists a representation where the orbital momentum is l in units of 2 the position of any one of the particles cannot be localized since the operators of the coordinates do not commute with each other the constituents are smeared with a certain probability within a finite surface which moves with a constant velocity together with the center of mass

the algebra of dual 1 hahn polynomials and the clebsch gordan problem of sl 1 2 the algebra h of the dual 1 hahn polynomials is derived and shown to arise in the clebsch gordan problem of sl 1 2 the dual 1 hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q 1 limit of the dual q hahn polynomials the hopf algebra sl 1 2 has four generators including an involution it is also a q 1 limit of the quantum algebra slq 2 and furthermore the dynamical algebra of the parabose oscillator the algebra h a two parameter generalization of u 2 with an involution as additional generator is first derived from the recurrence relation of the 1 hahn polynomials it is then shown that h can be realized in terms of the generators of two added sl 1 2 algebras so that the clebsch gordan coefficients of sl 1 2 are dual 1 hahn polynomials an irreducible representation of h involving five diagonal matrices and connected to the difference equation of the dual 1 hahn polynomials is constructed

analytic representations based on su 1 1 coherent states and their applications we consider two analytic representations of the su 1 1 lie group the representation in the unit disc based on the su 1 1 perelomov coherent states and the barut girardello representation based on the eigenstates of the su 1 1 lowering generator we show that these representations are related through a laplace transform a weak resolution of the identity in terms of the perelomov su 1 1 coherent states is presented which is valid even when the bargmann index k is smaller than various applications of these results in the context of the two photon realization of su 1 1 in quantum optics are also discussed

towards a statistical test for functional connectivity dynamics abstract sliding window correlation is an emerging method for mapping time resolved resting state functional connectivity to avoid mapping spurious connectivity fluctuations false positives leonardi and van de ville recently recommended choosing a window length exceeding the longest wavelength composing the bold signal usually assumed to be 100 s here we provide further statistical support for this rule of thumb however we demonstrate that non stationary fluctuations in functional connectivity can in theory be detected with much shorter window lengths e g 40 s while maintaining nominal control of false positives we find that statistical power is near maximal for window lengths chosen according to leonardi and van de ville s rule of thumb furthermore we lay some foundations for a parametric test to identify non stationary fluctuations in functional connectivity also noting limitations of the sinusoidal model upon which our work and the work of leonardi and van de ville is based most notably our analytical results pertain to covariances as does our statistical test whereas functional connectivity is more commonly measured using correlations

q deformed oscillator associated with the calogero model and its q coherent state we present the q deformed para bose oscillators associated with the two body calogero model the q deformed coherent state is also constructed and its resolution of unity is demonstrated

coherent states and their time dependence in fractional dimensions we construct representations of the lie algebra su 1 1 using representations of the momentum and position operators satisfying the r deformed heisenberg relations in which the fractional dimension d and angular momentum l appear as parameters the bargmann index k which characterizes representations of the positive discrete series of su 1 1 can take any positive value we construct coherent states in fractional dimensions in particular we extend the two well known analytic representations of coherent states for su 1 1 perelomov and barut girardello states from dimension one to any dimension d we generalize this construction to time dependent coherent states by means of the su 1 1 symmetries of the quantum time dependent harmonic oscillator in fractional dimensions we investigate the uncertainty relations of the momentum and position operators with respect to these coherent states and their dependence on the dimension

the barut girardello coherent states it is shown that the completeness problem of the sl 2 r coherent states proposed by barut and girardello leads to a moment problem not a mellin transform this moment problem which also appears in the theory of para bose oscillators has been solved following the sharma mehta mukunda sudarshan solution of the problem the matrix element of finite transformation in the coherent state basis is shown to satisfy a quasiorthogonality condition analogous to the orthogonality condition of the matrix element in the canonical basis finally the barut girardello hilbert space of entire analytic functions of growth 1 1 turns out to be only a subspace of bargmann s well known hilbert space of analytic functions this subspace which has been called the reduced bargmann space in a previous paper is an invariant subspace of sl 2 r with this identification the generators of the group in this realization turn out to be the well known boson operators of holman and biedenharn

the multiscale character of evoked cortical activity abstract both the architecture and the dynamics of the brain have characteristic features at different spatial scales however the existence nature and function of dynamical interdependencies between such scales have not been investigated we studied the multiscale properties of functional magnetic resonance imaging fmri data acquired while human subjects viewed a visual image traditional region of interest analysis of this data set revealed evoked activity in primary and extrastriate visual cortex wavelet transform in the spatial domain provides a multiscale representation of this evoked brain activity studying the correlation structure of this representation revealed strong and novel interdependencies in these data within and between different spatial scales we found that such correlations are stronger than those evident in the original data and comparable in magnitude to those obtained after gaussian smoothing however analysis of the data in the wavelet domain revealed additional structure such as positive correlations strong anti correlations and phase lagged interdependencies statistical significance of these effects was inferred through nonparametric bootstrap techniques we conclude that the spatial analysis of functional neuroimaging data in the wavelet domain provides novel information which may reflect complex spatiotemporal neuronal activity and information encoding it also affords a quantitative means of testing hierarchical and multiscale models of cortical activity

resolutions of the identity in terms of line integrals of su 1 1 coherent states resolutions of the identity in terms of contour integrals that involve coherent states and their complementary states are derived the complementary states are auxiliary states that help in the formulation of these resolutions of the identity since the coherent states are normalizable inside the unit disc and the complementary states are normalizable outside the unit disc enlargements of the hilbert space are considered which allow the construction of resolutions of the identity in terms of contour integrals in rings several examples of our formalism are presented

analytic and contour representations in the unit disk based on su 1 1 coherent states a contour representation in the unit disk based on su 1 1 coherent states is introduced the scalar product is given by a contour integral the regions of convergence of the functions representing ket and bra states are studied an analytic representation in the unit disk is also considered where the scalar product is represented by an integral over the unit disk with the lobachevsky measure various relations which connect these analytic functions with other phase space quantities are derived

representations and properties of generalized ar statistics coherent states and robertson uncertainty relations the generalization of ar statistics including bosonic and fermionic sectors is performed by means of the so called jacobson generators the corresponding fock spaces are constructed the bargmann representations are also considered for the bosonic ar statistics two inequivalent bargmann realizations are developed the first resp second realization induces in a natural way coherent states recognized as gazeau klauder resp klauder perelomov ones in the fermionic case the bargamnn realization leads to the klauder perelomov coherent states for each considered realization the inner product of two analytic functions is defined with respect to a measure explicitly computed the jacobson generators are realized as differential operators it is shown that the obtained coherent states minimize the robertson schr dinger uncertainty relation

generating functions for the mathfrak osp 1 2 clebsch gordan coefficients generating functions for clebsch gordan coefficients of are derived these coefficients are expressed as limits of the dual q hahn polynomials the generating functions are obtained using two different approaches respectively based on the coherent state representation and the position representation of

representation and properties of para bose oscillator operators ii coherent states and the minimum uncertainty states the energy position and momentum eigenstates of a para bose oscillator system were considered in paper i here we consider the bargmann or the analytic function description of the para bose system this brings in in a natural way the coherent states z defined as the eigenstates of the annihilation operator the transformation functions relating this description to the energy position and momentum eigenstates are explicitly obtained possible resolution of the identity operator using coherent states is examined a particular resolution contains two integrals one containing the diagonal basis z z and the other containing the pseudodiagonal basis z z we briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations the problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible dependence of this minimum value is considered

changes in dynamic functional connections with aging despite numerous studies on age related changes in static functional connections fcs the available literature on the changes in dynamic fcs with aging is lacking this study investigated the changes in dynamic fcs with aging based on resting state fmri data of 61 healthy adults aged 30 85 years the time resolved fcs among 160 pre defined regions of interest rois were first estimated using sliding window correlation based on the dynamic fc matrices we then analyzed the dynamic switches between different fc states using k means clustering and correlated age with the dwell time of each fc state across subjects the elderly were observed to spend more time in an fc state characterized by weak interactions throughout the brain and less time in an fc state characterized by strong interactions within the sensory motor network and the cognitive control network these results may reflect an overall weakening of connections in the elderly which support less efficient information transfer in them based on the dynamic fc matrices we also evaluated the variability and amplitude of fc time series which measure the relative to mean and absolute strength of fc fluctuations respectively and correlated age with the two measures across subjects relatively weak age vs variability correlations were observed but we did observe significant negative age vs amplitude correlations at both the global and regional level these results indicate that amplitude may be another effective metric for assessing fc fluctuations in addition to the widely used variability metric moreover the observed declines in the amplitude of fc fluctuations in the elderly may support the assumption that it should be the weakening of absolute interactions between brain regions rather than toggling between positive and negative correlations that causes the repeatedly reported widespread static fc decreases with aging overall the present results not only reflect an overall weakening of connections in the elderly but indicate the potential of dynamic fc analyses in studies of age related psychiatric and neurological disorders

wavelets and functional magnetic resonance imaging of the human brain the discrete wavelet transform dwt is widely used for multiresolution analysis and decorrelation or whitening of nonstationary time series and spatial processes wavelets are naturally appropriate for analysis of biological data such as functional magnetic resonance images of the human brain which often demonstrate scale invariant or fractal properties we provide a brief formal introduction to key properties of the dwt and review the growing literature on its application to fmri we focus on three applications in particular i wavelet coefficient resampling or wavestrapping of 1 d time series 2 to 3 d spatial maps and 4 d spatiotemporal processes ii wavelet based estimators for signal and noise parameters of time series regression models assuming the errors are fractional gaussian noise fgn and iii wavelet shrinkage in frequentist and bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps we conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fmri data

stereotypical modulations in dynamic functional connectivity explained by changes in bold variance abstract spontaneous activity measured in human subject under the absence of any task exhibits complex patterns of correlation that largely correspond to large scale functional topographies obtained with a wide variety of cognitive and perceptual tasks these resting state networks rsns fluctuate over time forming and dissolving on the scale of seconds to minutes while these fluctuations most prominently those of the default mode network have been linked to cognitive function it remains unclear whether they result from random noise or whether they index a nonstationary process which could be described as state switching in this study we use a sliding windows approach to relate temporal dynamics of rsns to global modulations in correlation and bold variance we compare empirical data phase randomized surrogate data and data simulated with a stationary model we find that rsn time courses exhibit a large amount of coactivation in all three cases and that the modulations in their activity are closely linked to global dynamics of the underlying bold signal we find that many properties of the observed fluctuations in fc and bold including their ranges and their correlations amongst each other are explained by fluctuations around the average fc structure however we also report some interesting characteristics that clearly support nonstationary features in the data in particular we find that the brain spends more time in the troughs of modulations than can be expected from stationary dynamics

test retest reliability of dynamic functional connectivity in resting state fmri abstract while static functional connectivity sfc of resting state fmri rfmri measures the average functional connectivity fc over the entire rfmri scan dynamic fc dfc captures the temporal variations of fc at shorter time windows although numerous studies have implemented dfc analyses only a few studies have investigated the reliability of dfc and this limits the biological interpretation of dfc here we used a large cohort n 820 of subjects and four rfmri scans from the human connectome project to systematically explore the relationship between sfc dfc and their test retest reliabilities through intra class correlation icc dfc icc was explored through the sliding window approach with three dfc statistics standard deviation alff and excursion excursion demonstrated the highest dfc icc and the highest age prediction accuracy dfc icc was generally higher at window sizes less than 40 s sfc and dfc were negatively correlated compared to sfc dfc was less reliable while sfc and sfc icc were positively correlated dfc and dfc icc were negatively correlated indicating that fc that was more dynamic was less reliable intra network fcs in the frontal parietal default mode sensorimotor and visual networks demonstrated high sfc and low dfc moreover iccs of both sfc and dfc in these regions were higher the above results were consistent across two brain atlases and independent component analysis based networks multiple window sizes and all three dfc statistics in summary dfc is less reliable than sfc and additional experiments are required to better understand the neurophysiological relevance of dfc

dem a variational treatment of dynamic systems this paper presents a variational treatment of dynamic models that furnishes time dependent conditional densities on the path or trajectory of a system s states and the time independent densities of its parameters these are obtained by maximising a variational action with respect to conditional densities under a fixed form assumption about their form the action or path integral of free energy represents a lower bound on the model s log evidence or marginal likelihood required for model selection and averaging this approach rests on formulating the optimisation dynamically in generalised coordinates of motion the resulting scheme can be used for online bayesian inversion of nonlinear dynamic causal models and is shown to outperform existing approaches such as kalman and particle filtering furthermore it provides for dual and triple inferences on a system s states parameters and hyperparameters using exactly the same principles we refer to this approach as dynamic expectation maximisation dem

quantum mechanical models in fractional dimensions we formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators p q satisfy the r deformed heisenberg relations which depend on an operator we find representations of p q in which the dimension d and angular momentum l appear as parameters related to the eigenvalues of we analyse the domain of p and find conditions which ensure that p is hermitian we investigate plane wave solutions and also free particle wavefunctions in fractional dimensions and show that as a consequence of wavefunction continuity l is quantized the representations of p q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d both algebraically and analytically we demonstrate that the formalism extends also to time dependent hamiltonians by solving the time dependent harmonic oscillator in any dimension d 0 using the method of lewis and riesenfeld

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

representations of the orthosymplectic lie superalgebra mathfrak osp 1 4 and paraboson coherent states we introduce and obtain two mode paraboson coherent states in appropriate subspaces these coherent states provide a decomposition of unity where the measure when expressed using the cat type states is positive definite bicoherent states where the mutually commuting lowering operators are diagonalized are also obtained matrix elements in the coherent state basis are calculated

exact solution of non adiabatic model hamiltonians in solid state physics and optics simple non adiabatic model hamiltonians are treated using the realisation of the vibrational modes on bargmann s hilbert space of analytic functions in this formulation the schrodinger equation is a system of linear first order differential equations the energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions the solutions are given in terms of neumann series the recurrence relations for the expansion coefficients have a simple structure under particular conditions for the interaction constant they allow for terminating neumann series isolated exact solutions in the general case the conditions for the eigenvalues are transcendental equations involving a continued fraction the continued fraction can be approximated to any desired degree of accuracy in a rapidly convergent process based on worpitzky s theorem and its relation to conformal mapping the eigenvalues are calculated a physical interpretation of the solutions which makes use of intuitive arguments is also given

a harmonic oscillator system with the generalized commutation relations the system of a one dimensional harmonic oscillator is discussed with the generalized commutation relations obtained by wigner the hamiltonian is shown to have self adjoint extensions the domain of self adjointness is explicitly specified in some cases the proof is carried out by the use of the lax milgram lemma a suitable rigged hilbert space is found for this system to reformulate the earlier arguments the main difficulty is that the momentum and hamiltonian operators contain the reflection r x x as well as singular terms