on optimum hamiltonians for state transformations for a prescribed pair of quantum states i and f we establish an elementary derivation of the optimum hamiltonian under constraints on its eigenvalues that generates the unitary transformation i f in the shortest duration the derivation is geometric in character and does not rely on variational calculus

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

nonlinear filtering and measure valued processes we construct a sequence of branching particle systems with time and space dependent branching mechanisms whose expectation converges to the solution of the zakai equation this gives an alternative numerical method to solve the filtering problem

almost invariant sets and invariant manifolds connecting probabilistic and geometric descriptions of coherent structures in flows abstract we study the transport and mixing properties of flows in a variety of settings connecting the classical geometrical approach via invariant manifolds with a probabilistic approach via transfer operators for non divergent fluid like flows we demonstrate that eigenvectors of numerical transfer operators efficiently decompose the domain into invariant regions for dissipative chaotic flows such a decomposition into invariant regions does not exist instead the transfer operator approach detects almost invariant sets we demonstrate numerically that the boundaries of these almost invariant regions are predominantly comprised of segments of co dimension 1 invariant manifolds for a mixing periodically driven fluid like flow we show that while sets bounded by stable and unstable manifolds are almost invariant the transfer operator approach can identify almost invariant sets with smaller mass leakage thus the transport mechanism of lobe dynamics need not correspond to minimal transport the transfer operator approach is purely probabilistic it directly determines those regions that minimally mix with their surroundings the almost invariant regions are identified via eigenvectors of a transfer operator and are ranked by the corresponding eigenvalues in the order of the sets invariance or leakiness while we demonstrate that the almost invariant sets are often bounded by segments of invariant manifolds without such a ranking it is not at all clear which intersections of invariant manifolds form the major barriers to mixing furthermore in some cases invariant manifolds do not bound sets of minimal leakage our transfer operator constructions are very simple and fast to implement they require a sample of short trajectories followed by eigenvector calculations of a sparse matrix

a state predictor for continuous time stochastic systems abstract this work investigates the state prediction problem for nonlinear stochastic differential systems affected by multiplicative state noise this problem is relevant in many state estimation frameworks such as filtering of continuous discrete systems i e stochastic differential systems with discrete measurements and time delay systems a very common heuristic to achieve the state prediction exploits the numerical integration of the deterministic nonlinear equation associated to the noise free system unfortunately these methods provide the exact solution only for linear systems instead here we provide the exact state prediction for nonlinear system in terms of the series expansion of the expected value of the state conditioned to the value in a previous time instant obtained according to the carleman embedding technique the truncation of the infinite series allows to compute the prediction at future times with an arbitrary approximation simulations support the effectiveness of the proposed state prediction algorithm in comparison to the aforementioned heuristic method

elementary derivation for passage times when a quantum system undergoes unitary evolution in accordance with a prescribed hamiltonian there is a class of states such that after the passage of a certain time is transformed into a state orthogonal to itself the shortest time for which this can occur for a given system is called the passage time we provide an elementary derivation of the passage time and demonstrate that the known lower bound due to fleming is typically attained except for special cases in which the energy spectra have particularly simple structures it is also shown using a geodesic argument that the passage times for these exceptional cases are necessarily larger than the fleming bound the analysis is extended to passage times for initially mixed states

geometric quantization and constrained systems the problem of obtaining the quantum theory of systems with first class constraints is discussed in the context of geometric quantization the precise structure needed on the constraint surface of the full phase space to obtain a polarization on the reduced phase space is displayed in a form that is particularly convenient for applications for unconstrained systems any polarization on the phase space leads to a mathematically consistent quantum description although not all of these descriptions may be viable from a physical standpoint it is pointed out that the situation is worse in the presence of constraints a general polarization on the full phase space need not lead to even a mathematically consistent quantum theory examples are given to illustrate the general constructions as well as the subtle difficulties

feedback particle filter the feedback particle filter introduced in this paper is a new approach to approximate nonlinear filtering motivated by techniques from mean field game theory the filter is defined by an ensemble of controlled stochastic systems the particles each particle evolves under feedback control based on its own state and features of the empirical distribution of the ensemble the feedback control law is obtained as the solution to an optimal control problem in which the optimization criterion is the kullback leibler divergence between the actual posterior and the common posterior of any particle the following conclusions are obtained for diffusions with continuous observations 1 the optimal control solution is exact the two posteriors match exactly provided they are initialized with identical priors 2 the optimal filter admits an innovation error based gain feedback structure 3 the optimal feedback gain is obtained via a solution of an euler lagrange boundary value problem the feedback gain equals the kalman gain in the linear gaussian case numerical algorithms are introduced and implemented in two general examples and a neuroscience application involving coupled oscillators in some cases it is found that the filter exhibits significantly lower variance when compared to the bootstrap particle filter

group averaging in the p q oscillator representation of sl 2 r we investigate refined algebraic quantization with group averaging in a finite dimensional constrained hamiltonian system that provides a simplified model of general relativity the classical theory has gauge group sl 2 r and a distinguished o p q observable algebra the gauge group of the quantum theory is the double cover of sl 2 r and its representation on the auxiliary hilbert space is isomorphic to the p q oscillator representation when p 2 q 2 and p q 0 mod 2 we obtain a physical hilbert space with a nontrivial representation of the o p q quantum observable algebra for p q 1 the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form

approximation of the zakai equation by the splitting up method the objective of this article is to apply an operator splitting method to the time integration of zakai equation using this approach one can decompose the numerical integration into a stochastic step and a deterministic one both of them much simpler to handle than the original problem a strong convergence theorem is given in the spirit of existing results for deterministic problems

modular invariance on the torus and abelian chern simons theory the implementation of modular invariance on the torus as a phase space at the quantum level is discussed in a group theoretical framework unlike the classical case at the quantum level some restrictions on the parameters of the theory should be imposed to ensure modular invariance two cases must be considered depending on the cohomology class of the symplectic form on the torus if it is of integer cohomology class n then full modular invariance is achieved at the quantum level only for those wave functions on the torus which are periodic if n is even or antiperiodic if n is odd if the symplectic form is of rational cohomology class n r a similar result holds the wave functions must be either periodic or antiperiodic on a torus r times larger in both directions depending on the parity of nr application of these results to the abelian chern simons theory is discussed

coherent sets for nonautonomous dynamical systems we describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems in the autonomous setting such objects are variously known as almost invariant sets metastable sets persistent patterns or strange eigenmodes and have proved to be important in a variety of applications in this current work we explain how to extend existing autonomous approaches to the nonautonomous setting we call the new time dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent the new methods are illustrated via detailed examples in both discrete and continuous time

refined algebraic quantization in the oscillator representation of sl 2 r we investigate refined algebraic quantization raq with group averaging in a constrained hamiltonian system with unreduced phase space t r4 and gauge group sl 2 r the reduced phase space m is connected and contains four mutually disconnected regular sectors with topology r s1 but these sectors are connected to each other through an exceptional set where m is not a manifold and where m has non hausdorff topology the raq physical hilbert space hphys decomposes as hphys hi where the four subspaces hi naturally correspond to the four regular sectors of m the raq observable algebra aobs represented on hphys contains natural subalgebras represented on each hi the group averaging takes place in the oscillator representation of sl 2 r on l2 r2 2 and ensuring convergence requires a subtle choice for the test state space the classical analog of this choice is to excise from m the exceptional set while nevertheless retaining information about the connections between the regular sectors a quantum

nonlinear filtering revisited a spectral approach the objective of this paper is to develop an approach to nonlinear filtering based on the cameron martin version of wiener chaos expansion this approach gives rise to a new numerical scheme for nonlinear filtering the main feature of this algorithm is that it allows one to separate the computations involving the observations from those dealing only with the system parameters and to shift the latter off line

time dependent hamiltonians with 100 evolution speed efficiency the evolution speed in projective hilbert space is considered for hermitian hamiltonians and for non hermitian nh ones based on the hilbert schmidt norm and the spectral norm of a hamiltonian resource related upper bounds on the evolution speed are constructed these bounds are valid also for nh hamiltonians and they are illustrated for an optical nh hamiltonian and for a nh pt symmetric matrix hamiltonian furthermore the concept of quantum speed efficiency is introduced as measure of the system resources directly spent on the motion in the projective hilbert space a recipe for the construction of time dependent hamiltonians which ensure 100 speed efficiency is given generally these efficient hamiltonians are nh but there is a hermitian efficient hamiltonian as well finally the extremal case of a nh non diagonalizable hamiltonian with vanishing energy difference is shown to produce a 100 efficient evolution with minimal resources consumption

group quantization of parametrized systems ii pasting hilbert spaces the method of group quantization described in the preceding paper j math phys 36 4612 1995 is extended so that it becomes applicable to some parametrized systems that do not admit a global transversal surface a simple completely solvable toy model is studied that admits a pair of maximal transversal surfaces intersecting all orbits the corresponding two quantum mechanics are constructed the similarity of the canonical group actions in the classical phase spaces on the one hand and in the quantum hilbert spaces on the other hand suggests how the two hilbert spaces are to be pasted together the resulting quantum theory is checked to be equivalent to that constructed directly by means of dirac s operator constraint method the complete system of partial hamiltonians for any of the two transversal surfaces is chosen and the quantum schrodinger or heisenberg pictures of time evolution are constructed

the quantum brachistochrone problem for non hermitian hamiltonians recently bender brody jones and meister found that in the quantum brachistochrone problem the passage time needed for the evolution of certain initial states into specified final states can be made arbitrarily small when the time evolution operator is taken to be non hermitian but pt symmetric here we demonstrate that such phenomena can also be obtained for non hermitian hamiltonians for which pt symmetry is completely broken i e dissipative systems we observe that the effect of a tunable passage time can be achieved by projecting between orthogonal eigenstates by means of a time evolution operator associated with a non hermitian hamiltonian it is not essential that this hamiltonian is pt symmetric

constraint rescaling in refined algebraic quantisation momentum constraint we investigate refined algebraic quantisation within a family of classically equivalent constrained hamiltonian systems that are related to each other by rescaling a momentum type constraint the quantum constraint is implemented by a rigging map that is motivated by group averaging but has a resolution finer than what can be peeled off from the formally divergent contributions to the averaging integral three cases emerge depending on the asymptotics of the rescaling function i quantisation is equivalent to that with identity scaling ii quantisation fails owing to nonexistence of self adjoint extensions of the constraint operator iii a quantisation ambiguity arises from the self adjoint extension of the constraint operator and the resolution of this purely quantum mechanical ambiguity determines the superselection structure of the physical hilbert space prospects of generalising the analysis to systems with several constraints are discussed

a bayesian tutorial for data assimilation data assimilation is the process by which observational data are fused with scientific information the bayesian paradigm provides a coherent probabilistic approach for combining information and thus is an appropriate framework for data assimilation viewing data assimilation as a problem in bayesian statistics is not new however the field of bayesian statistics is rapidly evolving and new approaches for model construction and sampling have been utilized recently in a wide variety of disciplines to combine information this article includes a brief introduction to bayesian methods paying particular attention to data assimilation we review linkages to optimal interpolation kriging kalman filtering smoothing and variational analysis discussion is provided concerning monte carlo methods for implementing bayesian analysis including importance sampling particle filtering ensemble kalman filtering and markov chain monte carlo sampling finally hierarchical bayesian modeling is reviewed we indicate how this approach can be used to incorporate significant physically based prior information into statistical models thereby accounting for uncertainty the approach is illustrated in a simplified advection diffusion model

approximation and limit results for nonlinear filters over an infinite time interval this paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval since the optimal filter can rarely be constructed one needs to compute with numerically feasible approximations the signal model can be a jump diffusion or just a process that is approximated by a jump diffusion the observation noise can be either white or of wide bandwidth the observations can be taken in either discrete or continuous time the cost of interest is the pathwise error per unit time over a long time interval it is shown under quite reasonable conditions on the approximating filter and the signal and noise processes that as time bandwidth process and filter approximation etc go to their limit in any way at all the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter were used analogous results are obtained with appropriate scaling if the observations are taken in discrete time and the sampling interval also goes to zero for these cases the approximating filter is a numerical approximation to the optimal filter for the presumed limit signal observation noise problem

non gaussian state space modeling of nonstationary time series abstract a non gaussian state space approach to the modeling of nonstationary time series is shown the model is expressed in state space form where the system noise and the observational noise are not necessarily gaussian recursive formulas of prediction filtering and smoothing for the state estimation and identification of the non gaussian state space model are given also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas significant merits of non gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples a typical application of this non gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes simple gaussian state space modeling is not adequate for this situation here the model with small system noise variance cannot detect jump whereas the one with large system noise variance yields unfavorable wiggle to work

a survey of numerical methods for nonlinear filtering problems the main goal of filtering is to obtain recursively in time good estimates of the state of a stochastic dynamical system based on noisy partial observations of the same in settings where the signal observation dynamics are significantly nonlinear or the noise intensities are high an extended kalman filter ekf which is essentially a first order approximation to an infinite dimensional problem can perform quite poorly it may require very frequent re initializations and in some situations may even diverge the theory of nonlinear filtering addresses these difficulties by considering the evolution of the conditional distribution of the state of the system given all the available observations in the space of probability measures we survey a variety of numerical schemes that have been developed in the literature for approximating the conditional distribution described by such stochastic evolution equations with a special emphasis on an important family of schemes known as the particle filters a numerical study is presented to illustrate that in settings where the signal observation dynamics are non linear a suitably chosen nonlinear scheme can drastically outperform the extended kalman filter

on the effectiveness of monte carlo for initial uncertainty forecasting in nonlinear dynamical systems abstract this paper considers the problem of initial uncertainty forecasting in deterministic nonlinear continuous time dynamical systems via particle ensembles the popular monte carlo method which while simple to implement faces fundamental issues in particular it is not clear how well the propagated particles continue to represent the true state uncertainty at future times this paper evaluates the performance of monte carlo forecasting by analyzing it in the context of markov chain monte carlo mcmc theory the propagated ensemble is viewed as the realization of a markov chain at each time instant generated by an associated instantaneous transition kernel it is shown that for a special class of nonlinear systems that have zero divergence the propagated kernel is in detailed balance with the true state probability density function this guarantees statistical consistency of the monte carlo ensemble with the truth at all times for such systems on the other hand no such guarantee is possible for systems with non zero divergence

a kushner stratonovich monte carlo filter applied to nonlinear dynamical system identification a monte carlo filter based on the idea of averaging over characteristics and fashioned after a particle based time discretized approximation to the kushner stratonovich ks nonlinear filtering equation is proposed a key aspect of the new filter is the gain like additive update designed to approximate the innovation integral in the ks equation and implemented through an annealing type iterative procedure which is aimed at rendering the innovation observation prediction mismatch for a given time step to a zero mean brownian increment corresponding to the measurement noise this may be contrasted with the weight based multiplicative updates in most particle filters that are known to precipitate the numerical problem of weight collapse within a finite ensemble setting a study to estimate the a priori error bounds in the proposed scheme is undertaken the numerical evidence presently gathered from the assessed performance of the proposed and a few other competing filters on a class of nonlinear dynamic system identification and target tracking problems is suggestive of the remarkably improved convergence and accuracy of the new filter c 2013 elsevier b v all rights reserved

stochastic sensitivity a computable lagrangian uncertainty measure for unsteady flows uncertainties in velocity data are often ignored when computing lagrangian particle trajectories of fluids modeling these as noise in the velocity field leads to a random deviation from each traje

modified particle filter methods for assimilating lagrangian data into a point vortex model the process of assimilating lagrangian particle trajectory data into fluid models can fail with a standard linear based method such as the kalman filter we implement a particle filtering approach that affords a nonlinear estimation and does not impose gaussianity on either the prior or the posterior distributions at the update step several schemes for reinitializing the particle filter specifically tailored to the lagrangian data assimilation problem are applied to a point vortex system a comparison with the extended kalman filter ekf for the same system demonstrates the effectiveness of particle filters for the assimilation of complex nonlinear lagrangian data

on the support of the ashtekar lewandowski measure we show that the ashtekar isham extension overline a g of the configuration space of yang mills theories a g is topologically and measure theoretically the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices

quantization of diffeomorphism invariant theories of connections with local degrees of freedom quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions this provides in particular a quantization of the husain kucha model the main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity in the riemannian case i e signature the approach appears to contain all the necessary ingredients already in the lorentzian case it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections

cubature kalman filtering for continuous discrete systems theory and simulations in this paper we extend the cubature kalman filter ckf to deal with nonlinear state space models of the continuous discrete kind to be consistent with the literature the resulting nonlinear filter is referred to as the continuous discrete cubature kalman filter cd ckf we use the ito taylor expansion of order 1 5 to transform the process equation modeled in the form of stochastic ordinary differential equations into a set of stochastic difference equations building on this transformation and assuming that all conditional densities are gaussian distributed the solution to the bayesian filter reduces to the problem of how to compute gaussian weighted integrals to numerically compute the integrals we use the third degree cubature rule for a reliable implementation of the cd ckf in a finite word length machine it is structurally modified to propagate the square roots of the covariance matrices the reliability and accuracy of the square root version of the cd ckf are tested in a case study that involves the use of a radar problem of practical significance the problem considered herein is challenging in the context of radar in two respects high dimensionality of the state and increasing degree of nonlinearity the results presented herein indicate that the cd ckf markedly outperforms existing continuous discrete filters

a combinatorial approach to diffeomorphism invariant quantum gauge theories quantum gauge theory in the connection representation uses functions of holonomies as configuration observables physical observables gauge and diffeomorphism invariant are represented in the hilbert space of physical states physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs then a family of graphs embedded in the space manifold satisfying certain properties induces a representation of the algebra of physical observables we construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes even though the two models are different at the kinematical level they provide unitarily equivalent representations of the algebra of physical observables in separable hilbert spaces of physical states their s knot basis is countable hence the combinator

the novel classes of finite dimensional filters with non maximal rank estimation algebra on state dimension four and rank of one ever since the technique of kalman bucy filter was popularised finding new classes of finite dimensional recursive filters has drawn much concern the idea of using estimation algebra to construct

algebras of semiclassical pseudodifferential operators associated with zoll type domains in cotangent bundles abstract we consider domains in cotangent bundles with the property that the null foliation of their boundary is fibrating and the leaves satisfy a bohr sommerfeld condition for example the unit disk bundle of a zoll metric given such a domain we construct an algebra of associated semiclassical pseudodifferential operators with singular symbols the schwartz kernels of the operators have frequency set contained in the union of the diagonal and the flow out of the null foliation of the boundary of the domain we develop a symbolic calculus prove the existence of projectors under a mild additional assumption whose range can be thought of as quantizing the domain give a symbolic proof of a szego limit theorem and study associated propagators

elementary solution to the time independent quantum navigation problem a quantum navigation problem concerns the identification of a time optimal hamiltonian that realizes a required quantum process or task under the influence of a prevailing background hamiltonian that cannot be manipulated when the task is to transform one quantum state into another finding the solution in closed form to the problem is nontrivial even in the case of time independent hamiltonians an elementary solution based on trigonometric analysis is found here when the hilbert space dimension is two difficulties arising from generalizations to higher dimensional systems are discussed

on galerkin approximations for the zakai equation with diffusive and point process observations this paper studies galerkin approximations applied to the zakai equation of stochastic filtering the basic idea of this approach is to project the infinite dimensional zakai equation onto some finite dimensional subspace generated by smooth basis functions this leads to a finite dimensional system of stochastic differential equations that can be solved numerically the contribution of the paper is twofold on the theoretical side existing convergence results are extended to filtering models with observations of point process or mixed type on the applied side various issues related to the numerical implementation of the method are considered in particular we propose working with a subspace that is constructed from a basis of hermite polynomials the paper closes with a numerical case study

sequential monte carlo samplers summary we propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space each distribution being known up to a normalizing constant these probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential monte carlo methods this methodology allows us to derive simple algorithms to make parallel markov chain monte carlo algorithms interact to perform global optimization and sequential bayesian estimation and to compute ratios of normalizing constants we illustrate these algorithms for various integration tasks arising in the context of bayesian inference

the inverse loop transform the loop transform in quantum gauge field theory can be recognized as the fourier transform or characteristic functional of a measure on the space of generalized connections modulo gauge transformations since this space is a compact hausdorff space conversely we know from the riesz markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular borel measure on the moduli space in the present article we show how one can compute the finite joint distributions of a given characteristic functional that is we derive the inverse loop transform

intelligent states for the anandan aharonov parameter based uncertainty relation we obtain the form of the intelligent states for the recent parameter based uncertainty relation of anandan and aharonov

quantization of non standard hamiltonian systems the quantization of classical theories that admit more than one hamiltonian description is considered this is done from a geometrical viewpoint both at the quantization level geometric quantization and at the level of the dynamics of the quantum theory a spin system is taken as an example in which all the steps can be completed it is shown that the geometry of the quantum theory imposes restrictions on the physically allowed non standard quantum theories

adaptive approximate bayesian computation sequential techniques can be adapted to the abc algorithm to enhance its efficiency for instance when sisson et al 2007 introduced the abc prc algorithm the goal was to improve upon existing abc mcmc algorithms marjoram et al 2003 while the abc prc method is based upon the theoretical developments of del moral et al 2006 the application to the setting of approximate bayesian computation induces a bias in the approximation to the posterior distribution of interest as we demonstrate in this paper via both theoretical reasoning and experimental results it is however possible to devise an alternative version based on genuine importance sampling arguments that we call abc pmc in connection with the population monte carlo method introduced in cappe et al 2004 furthermore the abc pmc algorithm is simpler than the abc prc algorithm in that it does not require any backward transition kernel and proposes an automatic scaling of the forward kernel in this paper we demonstrate the applicability of abc pmc and compare its performances with abc prc

an algebraic extension of dirac quantization examples an extension of the dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in dirac s original proposal these issues play an important role especially in the context of nonlinear diffeomorphism invariant theories such as general relativity recently an extension of the required type was proposed using algebraic quantization methods in this paper the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples the choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity however prior knowledge of general relativity is not assumed in the main discussion indeed the methods introduced and conclusions arrived at are applicable to any system with first class constraints in particular they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems

solution to the quantum zermelo navigation problem the solution to the problem of finding a time optimal control hamiltonian to generate a given unitary gate in an environment in which there exists an uncontrollable ambient hamiltonian e g a background field is obtained in the classical context finding the time optimal way to steer a ship in the presence of a background wind or current is known as the zermelo navigation problem whose solution can be obtained by working out geodesic curves on a space equipped with a randers metric the solution to the quantum zermelo problem which is shown here to take a remarkably simple form is likewise obtained by finding explicit solutions to the geodesic equations of motion associated with a randers metric on the space of unitary operators the result reveals that the optimal control in a sense goes along with the wind

generalised coherent states and group representations on hilbert spaces of analytic functions using the example for the group su 1 1 the author obtains the basis functions of a hilbert space of analytic functions from the perelomov 1972 definition of generalised coherent states the lie algebra in this space has the form of a holstein primakoff representation appropriate for su 1 1

constrained quantization on symplectic manifolds and quantum distribution functions a quantization scheme based on the extension of phase space with application of constrained quantization technique is considered the obtained method is similar to the geometric quantization for constrained systems the problem of scalar product on the reduced hilbert space is investigated and a possible solution to this problem is given generalization of the gupta bleuler like conditions is done by minimization of quadratic fluctuations of quantum constraints the scheme for the construction of generalized coherent states is considered and relation with berezin quantization is found the quantum distribution functions are introduced and their physical interpretation is discussed

a nonlinear filtering algorithm based on an approximation of the conditional distribution an effective form of the gaussian or moment approximation method for approximating optimal nonlinear filters with a diffusion signal process and discrete time observations is presented various computational simplifications reduce the dimensionality of the numerical integrations that need to be done this process combined with an iterative gaussian quadrature method makes the filter effective for real time use the advantages are illustrated by a model that captures the general flavour of modeling the highly uncertain behavior of a ship near obstacles such as a shore line into which it cannot go and must manoeuvre away in some unknown fashion the observations are of very poor quality yet the filter behaves well and is quite stable the procedure does not rely on linearization but attempts to compute the conditional moments directly by approximating the integrations used by the optimal filter

refined algebraic quantisation in a system with nonconstant gauge invariant structure functions in a previous work j louko and e martinez pascual constraint rescaling in refined algebraic quantisation momentum constraint j math phys 52 123504 2011 10 1063 1 3664336 refined algebraic quantisation raq within a family of classically equivalent constrained hamiltonian systems that are related to each other by rescaling one momentum type constraint was investigated in the present work the first steps to generalise this analysis to cases where more constraints occur are developed the system under consideration contains two momentum type constraints originally abelian where rescalings of these constraints by a non vanishing function of the coordinates are allowed these rescalings induce structure functions at the level of the gauge algebra providing a specific parametrised family of real valued scaling functions the implementation of the corresponding rescaled quantum momentum type constraints is performed using raq when the gauge algebra i remains abelian and ii undergoes int

multivariable feedback particle filter this paper presents the multivariable extension of the feedback particle filter fpf algorithm for the nonlinear filtering problem in continuous time the fpf is a control oriented approach to particle filtering the approach does not require importance sampling or resampling and offers significant variance improvements in particular the algorithm can be applied to systems that are not stable this paper describes new representations and algorithms for the fpf in the general multivariable nonlinear non gaussian setting theory surrounding the fpf is improved exactness of the fpf is established in the general setting as well as well posedness of the associated boundary value problem to obtain the filter gain a galerkin finite element algorithm is proposed for approximation of the gain its performance is illustrated in numerical experiments

a coherent structure approach for parameter estimation in lagrangian data assimilation abstract we introduce a data assimilation method to estimate model parameters with observations of passive tracers by directly assimilating lagrangian coherent structures our approach differs from the usual lagrangian data assimilation approach where parameters are estimated based on tracer trajectories we employ the approximate bayesian computation abc framework to avoid computing the likelihood function of the coherent structure which is usually unavailable we solve the abc by a sequential monte carlo smc method and use principal component analysis pca to identify the coherent patterns from tracer trajectory data our new method shows remarkably improved results compared to the bootstrap particle filter when the physical model exhibits chaotic advection

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

optimal filtering for a class of linear ito stochastic systems the dichotomic case this paper focuses on the problem of optimal h2 filtering for a class of continuous time stochastic systems without assuming their exponential stability in the mean square sense indeed the stability assumption is relaxed and we assume instead that the lyapunov operator associated to the dynamical stochastic system is exponentially dichotomic the optimal solution of the considered optimization problem is expressed in terms of the unique solution of a suitable algebraic lyapunov equation and the stabilizing solution of a certain algebraic riccati equation

detecting and locating near optimal almost invariant sets and cycles the behaviors of trajectories of nonlinear dynamical systems are notoriously hard to characterize and predict rather than characterizing dynamical behavior at the level of trajectories we consider following the evolution of sets there are often collections of sets that behave in a very predictable way in spite of the fact that individual trajectories are entirely unpredictable such special collections of sets are invisible to studies of long trajectories we describe a global set oriented method to detect and locate these large dynamical structures our approach is a marriage of new ideas in modern dynamical systems theory and the novel application of graph dissection algorithms

group theoretical quantization and the example of a phase space s1 r the group theoretical quantization scheme is reconsidered by means of elementary systems already the quantization of a particle on a circle shows that the standard procedure has to be supplemented by an additional condition on the admissibility of group actions a systematic strategy for finding admissible group actions for particular subbundles of cotangent spaces is developed two dimensional prototypes of which are t r and s s1 r interpreted as restrictions of t r and t s1 to positive coordinate and momentum respectively in this framework and under an additional natural condition an so 1 2 action on s results as the unique admissible group action furthermore for symplectic manifolds which are specific parts of phase spaces with known quantum theory a simple projection method of quantization is formulated for t r and s equivalent results to those of more established but more involved quantization schemes are obtained the approach may be of interest e g in attempts to quantize g

fleming s bound for the decay of mixed states fleming s inequality is generalized to the decay function of mixed states we show that for any symmetric hamiltonian h and for any density operator on a finite dimensional hilbert space with the orthogonal projection onto the range of the estimate holds for all t with h t 2 we show that equality either holds for all or it does not hold for a single t with 0 h t 2 all the density operators saturating the bound for all i e the mixed intelligent states are determined

the maximum speed of dynamical evolution abstract we discuss the problem of counting the maximum number of distinct states that an isolated physical system can pass through in a given period of time its maximum speed of dynamical evolution previous analyses have given bounds in terms of e the standard deviation of the energy of the system here we give a strict bound that depends only on e e0 the system s average energy minus its ground state energy we also discuss bounds on information processing rates implied by our bound on the speed of dynamical evolution for example adding 1 j of energy to a given computer can never increase its processing rate by more than about 3 1033 operations per second

group quantization of parametrized systems i time levels a method of quantizing parametrized systems is developed that is based on a kind of gauge invariant quantities the so called perennials a perennial must also be an integral of motion the problem of time in its particular form frozen time formalism global problem of time multiple choice problem is met as well as a related difficulty characteristic for this type of theory the paucity of perennials the present paper is an attempt to find some remedy in the ideas on forms of relativistic dynamics by dirac some aspects of dirac s theory are generalized to all finite dimensional first class parametrized systems the generalization is based on replacing the poincare group and the algebra of its generators as used by dirac by a canonical group of symmetries and by an algebra of elementary perennials a number of insights are gained the following are the main results first conditions are revealed under which the time evolution of the ordinary quantum mechanics or a generalization of it

engineering transient spectrum for a two level system interacting with two mode intelligent states abstract the problem of a single two level atom in its excited state placed in a bimodal cavity in which an arbitrary form of the intensity dependent atom field coupling is present is solved exactly expressions for the energy eigenvalues and eigenvectors are derived and an exact representation for the time dependent unitary evolution operator of the system is given by means of which we analyze the analytic form of the fluorescence spectrum using the transitions among the dressed states of the system the influence of parameter misfits on the quantum system is discussed an interesting relation between the fluorescence spectrum and the dynamical evolution is found when the initial field states are prepared in properly intelligent states by comparison with exact numerical results we demonstrate that features of the fluorescence spectrum are influenced significantly by the different kinds of the nonlinearity of the intensity dependent atom field coupling we provide numerous examples to illustrate this correspondence and provide evidence of its usefulness the present work considerably extends earlier results for two level systems

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

estimating long term behavior of flows without trajectory integration the infinitesimal generator approach the long term distributions of trajectories of a flow are described by invariant densities i e fixed points of an associated transfer operator in addition global slowly mixing structures such as almost invariant sets which partition phase space into regions that are almost dynamically disconnected can also be identified by certain eigenfunctions of this operator indeed these structures are often hard to obtain by brute force trajectory based analyses in a wide variety of applications transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system the computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories in this paper we propose to directly work with the infinitesimal generator instead of the operator completely avoiding trajectory integration we propose two different discretization schemes a cell based discretization and a spectral coll

approximation and limit results for nonlinear filters over an infinite time interval part ii random sampling algorithms the paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval since the optimal filter can rarely be constructed one needs to compute with numerically feasible approximations the signal model can be a jump diffusion reflected or not the observations can be taken either in discrete or continuous time the cost of interest is the pathwise error per unit time over a long time interval in a previous paper of the authors a budhiraja and h j kushner siam j control optim 37 1999 pp 1946 1979 it was shown under quite reasonable conditions on the approximating filter and on the signal and noise processes that as time bandwidth process and filter approximation etc go to their limit in any way at all the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter was used when suitable approximating filters cannot be readily constructed due to excessive computational requirements or to problems associated with a high signal dimension approximations based on random sampling methods or perhaps combinations of sampling and analytical methods become attractive and are the subject of a great deal of attention the work of the previous paper is extended to a wide class of such algorithms under quite broad conditions covering virtually all the cases considered to date it is shown that the pathwise average errors converge to the same limit that would be obtained if the optimal filter was used as time goes to infinity and the approximation parameter goes to its limit in any way at all all the extensions e g wide bandwidth observation or system driving noise in a budhiraja and h j kushner siam j control optim 37 1999 pp 1946 1979 hold for our random sampling algorithms as well