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

interpolation of precipitation under topographic influence at different time scales 1 in this paper new methodologies for interpolating rainfall data in individual time intervals ranging from a day to a year using gaussian copulas and unsymmetrical v copulas with a variety of treatments of altitude as an exogenous variable are described for shorter time aggregations zeros were treated as censored variables for each selected time step the marginal distributions of precipitation amounts were modeled using nonparametric density estimators while the spatial dependence structures were estimated using a maximum likelihood methodology the methodology was compared to other common geostatistical interpolators such as ordinary kriging and external drift kriging several measures of bias and error structure have been used to assess the efficacy of the methods in a range of comparative split sampling studies the data set chosen for the study comprises daily precipitation time series over 41 years in three regions in germany measured at more than 1200 locations over a large area 126 144 km2 among the many findings in the paper the ones that stand out are i correlation between precipitation and topography increases with the length of time interval and is significantly improved by directional smoothing of topography ii the copula methods are superior to kriging methods in terms of quality of interpolation bias and uncertainty estimation iii the treatment of zeros as censored variables improves interpolation quality for daily and pentad values monthly and annual data in this region present no dry periods iv the copula methods yield full conditional distributions of estimates at a target point in an interval improving substantially on the simple uncertainty estimates derived from kriging and v the gauss copula in particular performs best overall in terms of computational efficiency combined with useful error profiles in the interpolations and of all methods is the most realistic in its error estimates

pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

copula based geostatistical models for groundwater quality parameters 1 groundwater quality parameters exhibit considerable spatial variability geostatistical methods including the assessment of variograms are usually used to characterize this variability copulas offer an interesting opportunity to describe dependence structures for multivariate distributions bivariate empirical copulas can be used as an alternative to variograms and covariance functions for the description of the spatial variability rank correlations of these copulas express the strength of the dependence independently of the marginal distributions and thus offer an alternative to the variograms empirical copulas for four quality parameters chloride sulfate ph and nitrate obtained from a large scale groundwater quality measurement network in baden wurttemberg germany are calculated they indicate that the spatial dependence structure of the investigated parameters is not gaussian two theoretical copula based models are presented in this paper a gaussian and a non gaussian bootstrap based statistical tests using stochastic simulation of the multivariate distributions are used to investigate the appropriateness of the models according to the test results the gaussian copula is rejected for most of the parameters while the non gaussian alternative is not rejected in most cases

the structure of time optimal trajectories for single input systems in the plane the general real analytic case for arbitrary single input real analytic systems in the plane in which the control enters linearly we prove that the time optimal trajectories are finite concatenations of bang bang and singular arcs with local bounds on the number of switchings no nondegeneracy assumption is made other than real analyticity the analysis proceeds by applying our previous results on nondegenerate c infty systems to study the behavior of the trajectories on a neighborhood of every point except for the points in a discrete set of branch points the branch points are then handled by a combination of control theoretic arguments and the use of subanalytic set theory

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

optimal control in laser induced population transfer for two and three level quantum systems we apply the techniques of control theory and of sub riemannian geometry to laser induced population transfer in two and three level quantum systems the aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences sub riemannian geometry and singular riemannian geometry provide a natural framework for this minimization where the optimal control is expressed in terms of geodesics we first show that in two level systems the well known technique of pulse transfer in the rotating wave approximation emerges naturally from this minimization in three level systems driven by two resonant fields we also find the counterpart of the pulse transfer this geometrical picture also allows one to analyze the population transfer by adiabatic passage

time optimal synthesis for left invariant control systems on series so 3 consider the control system sigma given by dot x x f ug where x in so 3 u leq 1 and f g in so 3 define two perpendicular left invariant vector fields normalized so that f cos al and g sin al al in 0 pi 4 in this paper we provide an upper bound and a lower bound for n alpha the maximum number of switchings for time optimal trajectories of sigma more precisely we show that n s al leq n al leq n s al 4 where n s al is a suitable integer function of al such that n s al stackrel lower2pt hbox smash sim mbox tiny al to 0 pi 4 alpha the result is obtained by studying the time optimal synthesis of a projected control problem on rr p 2 where the projection is defined by an appropriate hopf fibration finally we study the projected control problem on the unit sphere s 2 it exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations

influence of soil structure and root water uptake strategy on unsaturated flow in heterogeneous media 1 we analyze the combined effects of the spatial variability of soil hydraulic properties and the water uptake by plant roots on unsaturated water flow for this analysis we use a simplified macroscopic root water uptake model which is usually applied only for homogeneous or layered soil and therefore we also determine whether it is applicable for multidimensional heterogeneous media analytical solutions for mean and variance of pressure head first order second moment approximations in layered media and numerical solutions of two dimensional 2 d autocorrelated multi gaussian and non multi gaussian parameter fields are analyzed for steady state and transient flow conditions for non gaussian topological features that have little influence on the mean and the variance of the pressure field if root water uptake is ignored we test whether the influence is significant if root water uptake is accounted for the results reveal that in field structures with large patches of coarse material local regions with pressure head values at the wilting point develop these are surrounded by wet material without a compensation mechanism for local stress the global transpiration demand is not met if local wilting occurs various compensation mechanisms are tested that depend respectively on the saturation the relative conductivity or a strategy where the deficit in the global uptake rate is equally distributed to unstressed locations the strategies lead to a global actual transpiration rate at the potential value and attenuate the formation of locally wilted areas wilted regions can however still occur and may be an artifact of the simplified model concept as root soil interactions are neglected therefore simplified macroscopic models for root water uptake should be used with caution in heterogeneous media

on the local structure of optimal trajectories in r 3 we analyze the structure of a control function u t corresponding to an optimal trajectory for the system dot q f q u g q in a three dimensional manifold near a point where some nondegeneracy conditions are satisfied the kind of optimality which is studied includes time optimality the control turns out to be the concatenation of some bang and some singular arcs studying the index of the second variation of the switching times the number of such arcs is bounded by four

multivariate copula analysis toolbox mvcat describing dependence and underlying uncertainty using a bayesian framework we present a newly developed multivariate copula analysis toolbox mvcat which includes a wide range of copula families with different levels of complexity mvcat employs a bayesian framework with a residual based gaussian likelihood function for inferring copula parameters and estimating the underlying uncertainties the contribution of this paper is threefold a providing a bayesian framework to approximate the predictive uncertainties of fitted copulas b introducing a hybrid evolution markov chain monte carlo mcmc approach designed for numerical estimation of the posterior distribution of copula parameters and c enabling the community to explore a wide range of copulas and evaluate them relative to the fitting uncertainties we show that the commonly used local optimization methods for copula parameter estimation often get trapped in local minima the proposed method however addresses this limitation and improves describing the dependence structure mvcat also enables evaluation of uncertainties relative to the length of record which is fundamental to a wide range of applications such as multivariate frequency analysis

information acquisition in auctions our aim in this paper is to study the incentives to acquire information we consider decision problems where the payoff has the single crossing property and signals are affiliated with the unknown parameter we introduce the notion of risk sensitiity and establish that the value of information is higher in decision problems that are more risk sensitive we apply this result to auctions we are able to show that a first price auction induces more information acquisition than a second price auction consider a decision maker choosing an action a to maximize the expected value of a payoff function u a the decision maker does not observe which he regards as a random variable v instead he observes the realization of a random variable x a signal which conveys information about w eassume that the decision maker can make x more informative at a cost making x more informative increases the expected payoff we analyze the returns to making x more informative we focus on problems where x and v are affiliated and u a has the weak

note on coherent states and adiabatic connections curvatures we give a possible generalization to the example in the paper of zanardi and rasetti phys lett a 264 94 1999 for this explicit forms of adiabatic connection curvature etc are given we also discuss the possibility of another generalization of their model

improved spatial prediction a combinatorial approach 1 this paper presents a combinatorial approach for improving spatial predictions first copulas are used to interpolate a spatially distributed point rainfall field to a uniform spatial grid it is observed that results vary substantially depending on the parameters chosen for interpolation leading to the hypothesis that it may be advantageous to estimate copula parameters locally or to combine local and global copula predictions it is found that by modifying the method of forecast combinations prediction errors in the spatial interpolation of rainfall can be reduced although this method of combining predictions is applied in the context of rainfall interpolation using local and global copula predictions it can be used on other spatial variables and interpolation methods

what model for entry in first price auctions a nonparametric approach abstract we develop a selective entry model for first price auctions that nests two polar models often estimated in the empirical literature on auctions levin and smith 1994 and samuelson 1985 the selective entry model features a pro competitive selection effect the selection effect is shown to be nonparametrically identifiable and a nonparametric test for its presence is proposed this test can be used to discriminate between the two polar models

bang bang control design for quantum state transfer based on hyperspherical coordinates and optimal time energy control we present a constructive control scheme for solving quantum state engineering problems based on the parameterization of the state vector in terms of complex hyperspherical coordinates unlike many control schemes based on factorization of unitary operators the scheme gives explicit expressions for all generalized euler angles in terms of the hyperspherical coordinates of the initial and final states the factorization when applicable has added benefits that phase rotations can be combined and performed concurrently the control procedure can be realized using simple bang bang or square wavefunction controls optimal time energy control is considered to find the optimal control amplitude the extension of the scheme to implement arbitrary unitary operators is also discussed

multivariate type g matern stochastic partial differential equation random fields for many applications with multivariate data random field models capturing departures from gaussianity within realizations are appropriate for this reason we formulate a new class of multivariate non gaussian models based on systems of stochastic partial differential equations with additive type g noise whose marginal covariance functions are of matern type we consider four increasingly flexible constructions of the noise where the first two are similar to existing copula based models in contrast with these the last two constructions can model non gaussian spatial data without replicates computationally efficient methods for likelihood based parameter estimation and probabilistic prediction are proposed and the flexibility of the models suggested is illustrated by numerical examples and two statistical applications less

quantum pontryagin principle under continuous measurements in this paper we develop the theory of the quantum pontryagin principle for continuous measurements and feedback the analysis is carried out under the assumption of compatible events in the output channel the plant is a quantum system which generally is in a mixed state coupled to a continuous measurement channel the pontryagin maximum principle is derived in both the schrodinger picture and heisenberg picture in particular in statistical moment coordinates to avoid solving stochastic equations we derive a linear quadratic gaussian scheme which is more suitable for control purposes finally we use the quantum harmonic oscillator as a concrete example to illustrate the performance of the controller

time optimal unitary operations in ising chains unequal couplings and fixed fidelity we analytically determine the minimal time and the optimal control laws required for the realization up to an assigned fidelity and with a fixed energy available of entangling quantum gates cnot between indirectly coupled qubits of a trilinear ising chain the control is coherent and open loop and it is represented by a local and continuous magnetic field acting on the intermediate qubit the time cost of this local quantum operation is not restricted to be zero when the matching with the target gate is perfect fidelity equal to 1 we provide exact solutions for the case of equal ising coupling for the more general case when some error is tolerated fidelity smaller than 1 we give perturbative solutions for unequal couplings comparison with previous numerical solutions for the minimal time to generate the same gates with the same ising hamiltonian but with instantaneous local controls shows that the latter are not time optimal

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

asymptotic normality of scrambled geometric net quadrature in a very recent work basu and owen found comput math 17 2017 467 496 propose the use of scrambled geometric nets in numerical integration when the domain is a product of s arbitrary spaces of dimension d having a certain partitioning constraint it was shown that for a class of smooth functions the integral estimate has variance o n 1 2 d log n s 1 for scrambled geometric nets compared to o n 1 for ordinary monte carlo the main idea of this paper is to expand on the work by loh ann statist 31 2003 1282 1324 to show that the scrambled geometric net estimate has an asymptotic normal distribution for certain smooth functions defined on products of suitable subsets of mathbb r d

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

streamflow prediction in ungauged catchments using copula based dissimilarity measures 1 there are many procedures in the available literature to perform prediction in ungauged basins commonly the euclidean metric is used as a proxy of the hydrologic dissimilarity here we propose a procedure to find a metric on the basis of dissimilarity measures that are estimated from pairwise empirical copula densities of runoff a metric is then defined in an transformed space of basin descriptors whose parameterization is obtained with a variance reducing technique a hydrologic model was run in an ungauged basin with sets of global parameters obtained from the k nearest neighboring donor basins using various metrics to take into account the uncertainty of its parameterization hydrologic model parameters were regionalized with a multiscale parameter regionalization technique whose transfer function parameters were found via calibration the streamflow in an ungauged basin was found as an ensemble streamflow prediction to account for the uncertainties of the transfer function parameters as well as those to define the metric this technique was applied in 38 german basins ranging in size from 70 to 4000 km2 for each basin a number of catchment descriptors and several climatic indices were quantified e g mean slope aspect shape factor mean elevation and mean monthly temperature in january among others daily streamflow time series correspond to the period from 1961 to 2000 simulated daily discharge was validated with a jackknife cross validation technique nash sutcliffe efficiencies obtained in this way ranged between 0 76 and 0 86 these results suggested that the proposed technique would produce reasonable results in ungauged basins

transforming low discrepancy sequences from a cube to a simplex sequences of points with a low discrepancy are the basic building blocks for quasi monte carlo methods traditionally these points are generated in a unit cube to develop point sets on a simplex we will transform the low discrepancy points from the unit cube to a simplex an advantage of this approach is that most of the known results on low discrepancy sequences can be re used after introducing several transformations their efficiency as well as their quality will be evaluated we present a koksma hlawka inequality which says that under certain conditions the order of convergence using the new point set is the same as that of the original set

time optimal cnot between indirectly coupled qubits in a linear ising chain we give analytical solutions for the time optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an ising hamiltonian interaction with a symmetric coupling j plus a local magnetic field acting on the intermediate qubit the energy available is fixed but we relax the standard assumption of instantaneous unitary operations acting on single qubits the time required for performing an entangling gate which is equivalent modulo local unitary operations to the cnot 1 3 between the indirectly coupled qubits 1 and 3 is i e faster than a previous estimate based on a similar hamiltonian and the assumption of local unitaries with zero time cost furthermore performing a simple walsh hadamard rotation in the hilbert space of qubit 3 shows that the time optimal synthesis of the cnot 1 3 which acts as the identity when the control qubit 1 is in the state 0 while if the control qubit is in the state 1 the target qubit 3 is flipped as also requires the same time t

transformations and hardy krause variation using a multivariable faa di bruno formula we give conditions on transformations tau 0 1 m to cal x where cal x is a closed and bounded subset of mathbb r d such that f circ tau is of bounded variation in the sense of hardy and krause for all f in c m cal x we give similar conditions for f circ tau to be smooth enough for scrambled net sampling to attain o n 3 2 epsilon accuracy some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition some other transformations due to fang and wang monogr statist appl probab 51 crc press boca raton fl 1993 satisfy the first but not the second condition we provide transformations for the simplex that makes f circ tau smooth enough to fully benefit from scrambled net sampling for all f in a class of generalized polynomials we also find sufficient conditions for conditional inversion in mathbb r 2 and for importance sampling to be of bounded variation in the sense of ha

regular synthesis and sufficiency conditions for optimality we propose a definition of regular synthesis that is more general than those suggested by other authors such as boltyanskii siam j control optim 4 1966 pp 326 361 and brunovsk math slovaca 28 1978 pp 81 100 and an even more general notion of regular presynthesis we give a complete proof of the corresponding sufficiency theorem a slightly weaker version of which had been stated in an earlier article with only a rough outline of the proof we illustrate the strength of our result by showing that the optimal synthesis for the famous fuller problem satisfies our hypotheses we also compare our concept of synthesis with the simpler notion of a family of solutions of the closed loop equation arising from an optimal feedback law and show by means of examples why the latter is inadequate and why the difficulty cannot be resolved by using other concepts of solution such as filippov solutions or the limits of sample and hold solutions recently proposed as feedback solutions by clarke et al ieee trans automat control 42 1997 pp 1394 1407 for equations with a non lipschitz and possibly discontinuous right hand side

quantum brachistochrone problem and the geometry of the state space in pseudo hermitian quantum mechanics a non hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining the inner product of physical hilbert state we study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space this allows for a careful treatment of the quantum brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using pt symmetric or other non hermitian hamiltonians than those given by hermitian hamiltonians

geostatistical interpolation using copulas 1 in many applications of geostatistical methods the dependence structure of the investigated parameter is described solely with the variogram or covariance functions which are susceptible to measurement anomalies and implies the assumption of gaussian dependence moreover the kriging variance respects only observation density data geometry and the variogram model to address these problems we borrow the idea from copulas to depict the dependence structure without the influence of the marginal distribution the methodology and basic hypotheses for application of copulas as geostatistical methods are discussed and the gaussian copula as well as a non gaussian copula are used in this paper copula parameters are estimated using a division of the observations into multipoint subsets and a subsequent maximization of the corresponding likelihood function the interpolation is carried out with two different copulas where the expected and median values are calculated from the copulas conditioned with the nearby observations the full conditional copulas provide the estimation distributions for the unobserved locations and can be used to define confidence intervals which depend on both the observation geometry and values observations of a large scale groundwater quality measurement network in baden wurttemberg are used to demonstrate the methodology five groundwater quality parameters chloride nitrate ph sulfate and dissolved oxygen are investigated all five parameters show non gaussian dependence the copula based interpolation results of the five parameters are compared to the results of conventional ordinary and indicator kriging different statistical measures including mean squared error relative differences and probability scores are used to compare cross validation and split sampling results of the interpolation methods the non gaussian copulas give better results than the geostatistical interpolations validation of the confidence intervals shows that they are more realistic than the estimation variances obtained by ordinary kriging

avoided level crossings diabolic points and branch points in the complex plane in an open double quantum dot we study the spectrum of an open double quantum dot as a function of different system parameters in order to receive information on the geometric phases of branch points in the complex plane bpcp we relate them to the geometrical phases of the diabolic points dps of the corresponding closed system the double dot consists of two single dots and a wire connecting them the two dots and the wire are represented by only a single state each the spectroscopic values follow from the eigenvalues and eigenfunctions of the hamiltonian describing the double dot system they are real when the system is closed and complex when the system is opened by attaching leads to it the discrete states as well as the narrow resonance states avoid crossing the dps are points within the avoided level crossing scenario of discrete states at the bpcp width bifurcation occurs here different riemann sheets evolve and the levels do not cross anymore the bpcp are physically meaningful the dps are unfolded into two bpcp with different chirality when the system is opened the geometric phase that arises by encircling the dp in the real plane is different from the phase that appears by encircling the bpcp this is found to be true even for a weakly opened system and the two bpcp into which the dp is unfolded

the structure of small time reachable sets in low dimensions this paper outlines a general method to determine the geometric structure of small time reachable sets for a single input control system with a bounded linear control the authors analysis relies on free nilpotent systems as a guide and hence their techniques only apply to nondegenerate situations the paper illustrates the effectiveness of the method in low dimensions among other results is given a precise description of the small time reachable set for a system dot x f x g x u u leqq 1 in dimension four under the generic assumption that the constant controls u equiv 1 and u equiv 1 are not singular as a corollary a local synthesis is obtained in dimension three for the time optimal control problem under the analogous generic condition

a globalized newton method for the accurate solution of a dipole quantum control problem a theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nanoelectronic devices and quantum computing this method is based on a reduced hessian krylov newton scheme applied to a norm preserving discrete model of a dipole quantum control problem the use of second order numerical methods for solving the control problem is justified proving the existence of optimal solutions and analyzing first and second order optimality conditions criteria for the discretization of the nonconvex optimization problem and for the formulation of the hessian are given to ensure accurate gradients and a symmetric hessian robustness of the newton approach is obtained using a globalization strategy with a robust linesearch procedure results of numerical experiments demonstrate that the newton approach presented in this paper is able to provide fast and accurate controls for high energy state transitions

classification of generic singularities for the planar time optimal synthesis this paper is concerned with control systems on the plane with control appearing linearly it is known that under generic conditions the problem of reaching points from the origin in minimum time admits a regular synthesis the minimum time function is piecewise smooth possibly nondifferentiable on a set that is a finite union of embedded submanifolds of dimension 1 or 0 called singularities the purpose of the present paper is to provide a classification of all types of singularities arising under generic conditions

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

exact pt symmetry is equivalent to hermiticity we show that a quantum system possessing an exact antilinear symmetry in particular pt symmetry is equivalent to a quantum system having a hermitian hamiltonian we construct the unitary operator relating an arbitrary non hermitian hamiltonian with exact pt symmetry to a hermitian hamiltonian we apply our general results to pt symmetry in finite dimensions and give the explicit form of the above mentioned unitary operator and hermitian hamiltonian in two dimensions our findings lead to the conjecture that non hermitian cpt symmetric field theories are equivalent to certain nonlocal hermitian field theories

the physics of non hermitian operators this special issue contains contributions presented at the workshop on the physics of non hermitian operators held at stellenbosch south africa in november 2005 a few additional articles are included from researchers who could not attend the meeting the workshop was the fourth meeting in a series on pseudo hermitian hamiltonians in quantum physics which commenced in prague in 2003 as a result of growing interest in non hermitian hamiltonians in general and pt symmetric quantum mechanics and quantum field theory in particular naturally hamiltonians of open systems are an obvious part of this topic but the focus here is rather on systems where unitary time evolution as such is not undermined this rather new and perhaps ambitious field thus explores a possible generalization of quantum mechanics which incorporates non hermitian hamiltonians into a consistent quantum mechanical framework new results of a physical and mathematical nature have brought about appreciable progress earlier work has been uncovered or even rediscovered and links between sub fields have been established this issue should provide a stimulus for initiating further advances in the field re focus on earlier perspectives and explore open and controversial questions while non hermitian operators have been encountered in theoretical physics since the advent of quantum mechanics their systematic mathematical investigation and their possible physical implications is a relatively new subject further explored in this issue we would like to thank the university of stellenbosch and the south african national research foundation for their financial support without which it would not have been possible to plan and organize the workshop organizational aspects and technical support were very ably taken care of by christine ruperti and tinus botha whom we would like to thank again finally we also appreciate and acknowledge the invitation from the editorial board of journal of physics a mathematical and general for the opportunity to publish this special issue and trust that this will indeed widen the interest and stimulate progress in the field

qmc designs optimal order quasi monte carlo integration schemes on the sphere we study equal weight numerical integration or quasi monte carlo qmc rules for functions in a sobolev space hs sd with smoothness parameter s d 2 defined over the unit sphere sd in rd 1 focusing on n point configurations that achieve optimal order qmc error bounds as is the case for efficient spherical designs we are led to introduce the concept of qmc designs these are sequences of n point configurations xn on s d such that the worst case error satisfies sup f h s f hs 1 1 n x xn f x sd f x d d x o n s d n with an implied constant that depends on the hs sd norm but is independent of n here d is the normalized surface measure on s d we provide methods for generation and numerical testing of qmc designs an essential tool is an expression for the worst case error in terms of a reproducing kernel for the space hs sd with s d 2 as a consequence of this and a recent result of bondarenko et al on the existence of spherical designs with appropriate number of points we show that minimizers of the n point energy for this kernel form a sequence of qmc designs for hs sd furthermore without appealing to the bondarenko et al result we prove that point sets that maximize the sum of suitable powers of the euclidean distance between pairs of points form a sequence of qmc designs for hs sd with s in the interval d 2 d 2 1 for such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order quasi monte carlo integration numerical experiments suggest that many familiar sequences of point sets on the sphere equal area points spiral points minimal coulomb or logarithmic energy points and fekete points are qmc designs for appropriate values of s for comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute qmc designs for any s d 2 if xn is a sequence of qmc designs for h s sd we prove that it is also a sequence of qmc designs for hs sd for all s d 2 s this leads to the question of determining the supremum of such s here called the qmc strength of the sequence for which we provide estimates based on computations for the aforementioned sequences received by the editor august 15 2012 and in revised form february 26 2013 2010 mathematics subject classification primary 65d30 65d32 secondary 11k38 41a55

isoholonomic problems and some applications we study the problem of finding the shortest loops with a given holonomy we show that the solutions are the trajectories of particles in yang mills potentials theorem 4 or equivalently the projections of kaluza klein geodesics theorem 2 applications to quantum mechanics berry s phase sect 3 and the optimal control of deformable bodies sect 6 are touched upon

sequential imputations and bayesian missing data problems abstract for missing data problems tanner and wong have described a data augmentation procedure that approximates the actual posterior distribution of the parameter vector by a mixture of complete data posteriors their method of constructing the complete data sets is closely related to the gibbs sampler both required iterations and similar to the em algorithm convergence can be slow we introduce in this article an alternative procedure that involves imputing the missing data sequentially and computing appropriate importance sampling weights in many applications this new procedure works very well without the need for iterations sensitivity analysis influence analysis and updating with new data can be performed cheaply bayesian prediction and model selection can also be incorporated examples taken from a wide range of applications are used for illustration

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 the curvature of monotone metrics and a conjecture concerning the kubo mori metric abstract the manifold of trace one positive complex n n matrices represents the space of faithful mixed states of a finite dimensional quantum system riemannian monotone metrics on these manifolds are in one to one correspondence with operator monotone functions these metrics generalize the classical fisher metric of statistical distinguishability of probability distributions to the quantum case and are of interest in quantum statistics and information theory thus it is natural to ask for curvature quantities of these riemannian metrics it is known that the classical fisher metric is of positive constant curvature the quantum case considered here is much more complicated most informations are available for the bures metric a special distinguished monotone metric partial curvature results are also known for the kubo mori metric it is the aim of this article to determine curvature quantities of an arbitrary riemannian monotone metric the resulting scalar curvature is explained in more detail for three examples in particular we consider an important conjecture of petz asserting that the scalar curvature of the kubo mori metric increases if one goes to more mixed states this assertion is shown up to a formal proof of the concavity of a certain function on r 3 however this concavity seems to be numerically evident and we give certain function plots supporting this concavity

on a mapping approach to investigating the bootstrap accuracy a simple mapping approach is proposed to study the bootstrap accuracy in a rather general setting it is demonstrated that the bootstrap accuracy can be obtained through this method for a broad class of statistics to which the commonly used edgeworth expansion approach may not be successfully applied we then consider some examples to illustrate how this approach may be used to find the bootstrap accuracy and show the advantage of the bootstrap approximation over the gaussian approximation for the multivariate kolmogorov smirnov statistic we show the error of bootstrap approximation is as small as that of the gaussian approximation for the multivariate kernel type density estimate we obtain an order of the bootstrap error which is smaller than the order of the error of the gaussian approximation given in rio 1994 we also consider an application of the bootstrap accuracy for empirical process to that for the copula process

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

robust incoherent control of qubit systems via switching and optimisation a robust incoherent quantum control scheme via projective measurements plus unitary transformations is proposed for driving a qubit system from an unknown initial mixed state to an arbitrary target pure state this scheme consists of two main steps projective measurement on the initial mixed state and optimal control between two pure states the first step projects the initial state into an eigenstate of the qubit system by projective measurement and guarantees that the proposed scheme is robust to different initial mixed states the second step finds a set of suitable optimal controls to drive the qubit system from the conditional eigenstate to the target pure state the connection between the two steps is accomplished by a switching strategy to accomplish the second step two approaches are presented in detail these approaches are time optimal transition with unbounded control and bang bang control with minimal switches the minimal time and minimal number of switches in these approaches can be calcu

exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation the isoholonomic problem in a homogeneous bundle is formulated and solved exactly the problem takes a form of a boundary value problem of a variational equation the solution is applied to the optimal control problem in holonomic quantum computer we provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a k dimensional unitary gate which operates on an n dimensional hilbert space with n 2k our construction is applied to several important unitary gates such as the hadamard gate the cnot gate and the two qubit discrete fourier transformation gate controllers for these gates are explicitly constructed

adiabatic control of the schrodinger equation via conical intersections of the eigenvalues in this paper we present a constructive method to control the bilinear schrodinger equation via two controls the method is based on adiabatic techniques and works if the spectrum of the hamiltonian admits eigenvalue intersections and if the latter are conical as it happens generically in this framework we are able to spread on several levels connected by conical intersections a state initially concentrated in a single energy level we provide sharp estimates on the dependence of the error with respect to the controllability time moreover we identify some special curves in the space of controls that improve the precision of the adiabatic approximation when passing through conical intersections with respect to classical adiabatic theory

dubins problem on surfaces ii nonpositive curvature let m be a complete connected two dimensional riemannian manifold with nonpositive gaussian curvature k we say that m satisfies the unrestricted complete controllability ucc property for the dubins problem if the following holds given any p 1 v 1 and p 2 v 2 in tm there exists for every eps 0 a curve gamma in m with geodesic curvature smaller than eps such that gamma connects p 1 to p 2 and for i 1 2 dot gamma is equal to v i at p i property ucc is equivalent to the complete controllability of a family of control systems of dubins type parameterized by eps it is well known that the poincare half plane does not verify property ucc in this paper we provide a complete characterization of the two dimensional nonpositively curved manifolds m with either uniformly negative or bounded curvature that satisfy property ucc more precisely if sup mk infty we show that ucc holds if and only if i m is of the first kind or ii the curvature satisfies a suitable integral decay condition at infinity

time optimal quantum evolution of mixed states we present a general formalism based on the variational principle for finding the time optimal quantum evolution of mixed states governed by a master equation when the hamiltonian and the lindblad operators are subject to certain constraints the problem may be reduced to solving first a fundamental equation which can be written down once the constraints are specified for the hamiltonian and then solving the constraints and the master equation for the lindblad and the density operators as an application of our formalism we study a simple one qubit model where the optimal lindblad operators can be simulated by a tunable coupling with an ancillary qubit

quasi monte carlo integration for twice differentiable functions over a triangle abstract we study quasi monte carlo integration for twice differentiable functions defined over a triangle we provide an explicit construction of infinite sequences of points including one by basu and owen 2015 as a special case which achieves the integration error of order n 1 log n 3 for any n 2 since a lower bound of order n 1 on the integration error holds for any linear quadrature rule the upper bound we obtain is best possible apart from the log n factor the major ingredient in our proof of the upper bound is the dyadic walsh analysis of twice differentiable functions over a triangle under a suitable recursive partitioning

the energy minimization problem for two level dissipative quantum systems in this article we study the energy minimization problem of dissipative two level quantum systems whose dynamics is governed by the kossakowski lindblad equations in the first part we classify the extremal curve solutions of the pontryagin maximum principle the optimality properties are analyzed using the concept of conjugate points and the hamilton jacobi bellman equation this analysis completed by numerical simulations based on adapted algorithms allows a computation of the optimal control law whose robustness with respect to the initial conditions and dissipative parameters is also detailed in the final section an application in nuclear magnetic resonance is presented

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

gaussian and non gaussian inverse modeling of groundwater flow using copulas and random mixing this paper presents a new copula based methodology for gaussian and non gaussian inverse modeling of groundwater flow the presented approach is embedded in a monte carlo framework and it is based on the concept of mixing spatial random fields where a spatial copula serves as spatial dependence function the target conditional spatial distribution of hydraulic transmissivities is obtained as a linear combination of unconditional spatial fields the corresponding weights of this linear combination are chosen such that the combined field has the prescribed spatial variability and honors all the observations of hydraulic transmissivities the constraints related to hydraulic head observations are nonlinear in order to fulfill these constraints a connected domain in the weight space inside which all linear constraints are fulfilled is identified this domain is defined analytically and includes an infinite number of conditional fields i e conditioned on the observed hydraulic transmissivities and the nonlinear constraints can be fulfilled via minimization of the deviation of the modeled and the observed hydraulic heads this procedure enables the simulation of a great number of solutions for the inverse problem allowing a reasonable quantification of the associated uncertainties the methodology can be used for fields with gaussian copula dependence and fields with specific non gaussian copula dependence further arbitrary marginal distributions can be considered

on automaton recognizability of abnormal extremals for a generic single input planar control system dot x f x u g x x in mathbb r 2 u in 1 1 f 0 0 we analyze the properties of abnormal extremals for the minimum time stabilization to the origin we prove that abnormal extremals are finite concatenations of bang arcs with switchings occurring on the set in which the vector fields f and g are collinear moreover all the generic singularities of one parametric family of extremal trajectories near to abnormal extremals are studied in particular we prove that all possible sequences of these singularities and hence all generic abnormal extremals can be classified by a set of words recognizable by an automaton

quantile based nonparametric inference for first price auctions abstract we propose a quantile based nonparametric approach to inference on the probability density function pdf of the private values in first price sealed bid auctions with independent private values our method of inference is based on a fully nonparametric kernel based estimator of the quantiles and pdf of observable bids our estimator attains the optimal rate of guerre et al 2000 and is also asymptotically normal with an appropriate choice of the bandwidth

regular synthesis for time optimal control of single input real analytic systems in the plane for arbitrary single input real analytic systems in the plane in which the control enters linearly we prove the existence of a regular synthesis for the optimal control problem in which it is desired to minimize the integral of a strictly positive real analytic lagrangian that does not depend on the control variable the analysis proceeds by applying our previous results on nondegenerate mathcal c infty systems as well as those on arbitrary real analytic ones to study the local structure of the time optimal trajectories the structure of the optimal trajectories for our problem is derived by reparametrization of time the existence of a synthesis is then proved by using subanalytic set theory

on average derivative quantile regression for fixed 0 1 the quantile regression function gives the th quantile x in the conditional distribution of a response variable y given the value x x of a vector of covariates it can be used to measure the effect of covariates not only in the center of a population but also in the upper and lower tails a functional that summarizes key features of the quantile specific relationship between x and y is the vector of weighted expected values of the vector of partial derivatives of the quantile function x in a nonparametric setting can be regarded as a vector of quantile specific nonparametric regression coefficients in survival analysis models e g cox s proportional hazard model proportional odds rate model accelerated failure time model and in monotone transformation models used in regression analysis gives the direction of the parameter vector in the parametric part of the model can also be used to estimate the direction of the parameter vector in semiparametric single index models popular in econometrics we show that under suitable regularity conditions the estimate of obtained by using the locally polynomial quantile estimate of chaudhuri 1991a is n 1 2 consistent and asymptotically normal with asymptotic variance equal to the variance of the influence function of the functional we discuss how the estimate of can be used for model diagnostics and in the construction of a link function estimate in general single index models

shortest paths synthesis for a car like robot this paper deals with the complete characterization of the shortest paths for a car like robot previous works have shown that the search for a shortest path may be limited to a simple family of trajectories our work completes this study by providing a way to select inside this family an optimal path to link any two configurations we combine the necessary conditions given by pontryagin s maximum principle with a geometric reasoning this approach enables us to complete the local information with a global analysis of different wave fronts we construct a partition of the configuration space in regions where the same kind of path is optimal to reach the origin in other words we determine a shortest path synthesis by providing at each point an optimal control law to steer the robot to the origin

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

exploring families of quantum controls for generating unitary transformations a numerical method is presented for exploring the intertwined roles of the control field structure and the final time t in determining the unitary evolution operator u t 0 for finite level quantum control systems the algorithm can a identify controls achieving a target unitary operator w at time t up to machine precision and b identify a continuous family of controls producing the same operator w over a continuous interval of final times the high degree of precision is obtained in part by exploiting the geometry of the unitary group in particular geodesics of the unitary group are followed both for tracking to a target transformation and for error management

efficient pairwise composite likelihood estimation for spatial clustered data spatial clustered data refer to high dimensional correlated measurements collected from units or subjects that are spatially clustered such data arise frequently from studies in social and health sciences we propose a unified modeling framework termed as geocopula to characterize both large scale variation and small scale variation for various data types including continuous data binary data and count data as special cases to overcome challenges in the estimation and inference for the model parameters we propose an efficient composite likelihood approach in that the estimation efficiency is resulted from a construction of over identified joint composite estimating equations consequently the statistical theory for the proposed estimation is developed by extending the classical theory of the generalized method of moments a clear advantage of the proposed estimation method is the computation feasibility we conduct several simulation studies to assess the performance of the proposed models and estimation methods for both gaussian and binary spatial clustered data results show a clear improvement on estimation efficiency over the conventional composite likelihood method an illustrative data example is included to motivate and demonstrate the proposed method

inference for first price auctions with guerre perrigne and vuong s estimator abstract we consider inference on the probability density of valuations in the first price sealed bid auctions model within the independent private value paradigm we show the asymptotic normality of the two step nonparametric estimator of guerre et al 2000 gpv and propose an easily implementable and consistent estimator of the asymptotic variance we prove the validity of the pointwise percentile bootstrap confidence intervals based on the gpv estimator lastly we use the intermediate gaussian approximation approach to construct bootstrap based asymptotically valid uniform confidence bands for the density of the valuations

factor copula models for replicated spatial data abstractwe propose a new copula model that can be used with replicated spatial data unlike the multivariate normal copula the proposed copula is based on the assumption that a common factor exists and affects the joint dependence of all measurements of the process moreover the proposed copula can model tail dependence and tail asymmetry the model is parameterized in terms of a covariance function that may be chosen from the many models proposed in the literature such as the matern model for some choice of common factors the joint copula density is given in closed form and therefore likelihood estimation is very fast in the general case one dimensional numerical integration is needed to calculate the likelihood but estimation is still reasonably fast even with large datasets we use simulation studies to show the wide range of dependence structures that can be generated by the proposed model with different choices of common factors we apply the proposed model to spatial temperature data and com

low discrepancy constructions in the triangle most quasi monte carlo research focuses on sampling from the unit cube many problems especially in computer graphics are defined via quadrature over the unit triangle quasi monte carlo methods for the triangle have been developed by pillards and cools j comput appl math 174 2005 pp 29 42 and by brandolini et al a koksma hlawka inequality for simplices in trends in harmonic analysis springer 2013 pp 33 46 this paper presents two quasi monte carlo constructions in the triangle with a vanishing discrepancy the first is a version of the van der corput sequence customized to the unit triangle it is an extensible digital construction that attains a discrepancy below 12 sqrt n the second construction rotates an integer lattice through an angle whose tangent is a quadratic irrational number it attains a discrepancy of o log n n which is the best possible rate previous work strongly indicated that such a discrepancy was possible but no constructions were available s

bayesian inference in econometric models using monte carlo integration methods for the systematic application of monte carlo integration with importance sampling to bayesian inference in econometric models are developed conditions under which the numerical approximation of a posterior moment converges almost surely to the true value as the number of monte carlo replications increases and the numerical accuracy of this approximation may be assessed reliably are set forth methods for the analytical verification of these conditions are discussed

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

characterization of the critical sets of quantum unitary control landscapes this work considers various families of quantum control landscapes i e objective functions for optimal control for obtaining target unitary transformations as the general solution of the controlled schrodinger equation we examine the critical point structure of the kinematic landscapes j f u u w a 2 and j p u a 4 tr aa w u 2 defined on the unitary group u h of a finite dimensional hilbert space h the parameter operator a e h is allowed to be completely arbitrary yielding an objective function that measures the difference in the actions of u and the target w on a subspace of state space namely the column space of a the analysis of this function includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima minima and saddles in addition we consider the question of whether these landscapes are morse bott functions on u h landscapes based on the intrinsic geodesic distance on u h and the projective unitary group pu h are also considered these results are then used to deduce properties of the critical set of the corresponding dynamical landscapes

a generic classification of time optimal planar stabilizing feedbacks consider the problem of stabilization at the origin in minimum time for a planar control system affine with respect to the control for a family of generic vector fields a topological equivalence relation on the corresponding time optimal feedback synthesis was introduced in a previous paper dynamics of continuous discrete and impulsive systems 3 1997 pp 335 371 the set of equivalence classes can be put in a one to one correspondence with a discrete family of graphs this provides a classification of the global structure of generic time optimal stabilizing feedbacks in the plane analogous to the classification of smooth dynamical systems developed by peixoto

multivariate hydrological frequency analysis using copulas 1 this article presents the modeling of multivariate extreme values using copulas our approach allows us to model the dependence structure independently of the marginal distributions which is not possible with standard classical methods the methodology has been applied on two different problems in hydrology the first application is concerned with the combined risk in the framework of frequency analysis four copulas have been tested on peak flows from the watershed of peribonka in quebec canada the second application relates to the joint modeling of peak flows and volumes three copulas have been applied to the watershed of the rimouski river in quebec canada this approach using copulas is promising since it allows us to take into account a wide range of correlation which can happen in hydrology

a multivariate faa di bruno formula with applications a multivariate faa di bruno formula for computing arbitrary partial derivatives of a function composition is presented it is shown by way of a general identity how such derivatives can also be expressed in the form of an infinite series applications to stochastic processes and multivariate cumulants are then delineated

identification and estimation of a discrete game of complete information we discuss the identification and estimation of discrete games of complete information following bresnahan and reiss 1990 1991 a discrete game is a generalization of a standard discrete choice model where utility depends on the actions of other players using recent algorithms to compute all of the nash equilibria to a game we propose simulation based estimators for static discrete games we demonstrate that the model is identified under weak functional form assumptions using exclusion restrictions and an identification at infinity approach monte carlo evidence demonstrates that the estimator can perform well in moderately sized samples as an application we study entry decisions by construction contractors to bid on highway projects in california we find that an equilibrium is more likely to be observed if it maximizes joint profits has a higher nash product uses mixed strategies and is not pareto dominated by another equilibrium

projective hilbert space structures at exceptional points a non hermitian complex symmetric 2 2 matrix toy model is used to study projective hilbert space structures in the vicinity of exceptional points eps the bi orthogonal eigenvectors of a diagonalizable matrix are puiseuxexpanded in terms of the root vectors at the ep it is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the ep limit can be resolved by projectively extending the original hilbert space the complementary normalization conditions correspond then to two different affine charts of this enlarged projective hilbert space geometric phase and phase jump behaviour are analysed and the usefulness of the phase rigidity as measure for the distance to ep configurations is demonstrated finally ep related aspects of pt symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated

geometrical phases for the g 4 2 grassmannian manifold we generalize the usual abelian berry phase generated for example in a system with two nondegenerate states to the case of a system with two doubly degenerate energy eigenspaces the parametric manifold describing the space of states of the first case is formally given by the g 2 1 grassmannian manifold while for the generalized system it is given by the g 4 2 one for the latter manifold which exhibits a much richer structure than its abelian counterpart we calculate the connection components the field strength and the associated geometrical phases that evolve nontrivially both of the degenerate eigenspaces a simple atomic model is proposed for their physical implementation

dynamic homotopy and landscape dynamical set topology in quantum control we examine the topology of the subset of controls taking a given initial state to a given final state in quantum control where state may mean a pure state an ensemble density matrix or a unitary propagator u 0 t the analysis consists in showing that the endpoint map acting on control space is a hurewicz fibration for a large class of affine control systems with vector controls exploiting the resulting fibration sequence and the long exact sequence of basepoint preserving homotopy classes of maps we show that the indicated subset of controls is homotopy equivalent to the loopspace of the state manifold this not only allows us to understand the connectedness of dynamical sets realized as preimages of subsets of the state space through this endpoint map but also provides a wealth of additional topological information about such subsets of control space

quasi monte carlo rules for numerical integration over the unit sphere mathbb s 2 we study numerical integration on the unit sphere mathbb s 2 subseteq mathbb r 3 using equal weight quadrature rules where the weights are such that constant functions are integrated exactly the quadrature points are constructed by lifting a 0 m 2 net given in the unit square 0 1 2 to the sphere mathbb s 2 by means of an area preserving map a similar approach has previously been suggested by cui and freeden siam j sci comput 18 2 595 609 1997 we prove three results the first one is that the construction is almost optimal with respect to discrepancies based on spherical rectangles further we prove that the point set is asymptotically uniformly distributed on mathbb s 2 and finally we prove an upper bound on the spherical cap l 2 discrepancy of order n 1 2 log n 1 2 where n denotes the number of points this improves upon the bound on the spherical cap l 2 discrepancy of the construction by lubotzky phillips and sarnak commun pure appl math 39 s suppl s149 s186 1986 by a factor of sqrt log n numerical results suggest that the 0 m 2 nets lifted to the sphere mathbb s 2 have spherical cap l 2 discrepancy converging with the optimal order of n 3 4