geometry of sets of quantum maps a generic positive map acting on a high dimensional system is not completely positive we investigate the set a of positive trace preserving maps acting on density matrices of size n and a sequence of its nested subsets the sets of maps which are b decomposable c completely positive and d extended by identity impose positive partial transpose and e are superpositive working with the hilbert schmidt euclidean measure we derive tight explicit two sided bounds for the volumes of all five sets a sample consequence is the fact that as n increases a generic positive map becomes not decomposable and a fortiori not completely positive due to the jamiolkowski isomorphism the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets such as states with positive partial transpose or separable states or supersets our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity high dimensional probability and geometry of banach spaces some of which are not standard

canonical coset parametrization and the bures metric of the three level quantum systems an explicit parametrization for the state space of an n level density matrix is given the parametrization is based on the canonical coset decomposition of unitary matrices we also compute explicitly the bures metric tensor over the state space of two and three level quantum systems

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

a parametrization of bipartite systems based on su 4 euler angles in this paper we give an explicit parametrization for all two qubit density matrices this is important for calculations involving entanglement and many other types of quantum information processing to accomplish this we present a generalized euler angle parametrization for su 4 and all possible two qubit density matrices the important group theoretical properties of such a description are then manifest we thus obtain the correct haar hurwitz measure and volume element for su 4 which follows from this parametrization in addition we study the role of this parametrization in the peres horodecki criteria for separability and its corresponding usefulness in calculating entangled two qubit states as represented through the parametrization

eeboost a general method for prediction and variable selection based on estimating equations the modern statistical literature is replete with methods for performing variable selection and prediction in standard regression problems however simple models may misspecify or fail to capture important aspects of the data generating process such as missingness correlation and over underdispersion in this article we describe eeboost a strategy for variable selection and prediction which can be applied in high dimensional settings where inference for low dimensional parameters would typically be based on estimating equations the method is simple flexible and easily implemented using existing software the eeboost algorithm is obtained as a modification of the standard boosting or functional gradient descent technique we show that eeboost is closely related to a class of l1 constrained projected likelihood ratio minimizations and therefore produces similar variable selection paths to penalized methods without the need to apply constrained optimization algorithms the flexibility of eeboost is

on reproducing kernels and quantization of states quantization of a mechanical system with the phase space a kahler manifold is studied it is shown that the calculation of the feynman path integral for such a system is equivalent to finding the reproducing kernel function the proposed approach is applied to a scalar massive conformal particle interacting with an external field which is described by deformation of a hermitian line bundle structure

hall normalization constants for the bures volumes of the n state quantum systems we report the results of certain integrations of quantum theoretic interest relying in this regard upon recently developed parametrizations of boya et al 1998 preprint quant ph 9810084 of the n n density matrices in terms of squared components of the unit n 1 sphere and the n n unitary matrices firstly we express the normalized volume elements of the bures minimal monotone metric for n 2 and 3 thereby obtaining bures prior probability distributions over the two and three state systems then as a first step in extending these results to n 3 we determine that the hall normalization constant cn for the marginal bures prior probablity distribution over the n 1 dimensional simplex of the n eigenvalues of the n n density matrices is for n 4 equal to 71 680 2 since we also find that c3 35 it follows that c4 is simply equal to 211c3 c2 itself is known to equal 2 the constant c5 is also found it too is associated with a remarkably simple decomposition involving the product of the eight consecutive prime numbers from 3 to 23 we also preliminarily investigate several cases n 5 with the use of quasi monte carlo integration we hope that the various analyses reported will prove useful in deriving a general formula which evidence suggests will involve the bernoulli numbers for the hall normalization constant for arbitrary n this would have diverse applications including quantum inference and universal quantum coding

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

weak galerkin method with r r 1 r 1 order finite elements for second order parabolic equations in this paper we propose a weak galerkin method with a stabilization term for a model problem of second order parabolic differential equations we establish both the continuous time and the discrete time weak galerkin finite element schemes which allow the use of totally discontinuous piecewise polynomial basis and the finite element partitions on shape regular polygons in addition we adapt the combination of polynomial spaces p r t 0 p r 1 e p r 1 t 2 that reduces the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation we show as well that the continuous time weak galerkin finite element method preserves the energy conservation law the optimal convergence order estimates in both discrete h1 and l2 norms are obtained we also present numerical experiments to illustrate the theoretical results

entropy as a function of geometric phase we give a closed form solution of von neumann entropy as a function of geometric phase modulated by visibility and average distinguishability in hilbert spaces of two and three dimensions we show that the same type of dependence also exists in higher dimensions albeit with other terms for non maximal mixing the results become more involved and generally depend also on the probability of the states we also outline a method for measuring both the entropy and the phase experimentally using a simple mach zehnder type interferometer which explains physically why the two concepts are related

quantum phases of a qutrit we consider various approaches to treat the phases of a qutrit although it is possible to represent qutrits in a convenient geometrical manner by resorting to a generalization of the poincare sphere we argue that the appropriate way of dealing with this problem is through phase operators associated with the algebra su 3 the rather unusual properties of these phases are caused by the small dimension of the system and are explored in detail we also examine the positive operator valued measures that can describe the qutrit phase properties

the diagonalization of cubic matrices this paper is devoted to analysing the problem of the diagonalization of cubic matrices we extend the familiar algebraic approach which is based on the cardano formulae we rewrite the complex roots of the associated resolvent secular equation in terms of transcendental functions and we derive the diagonalizing matrix

penalized estimating equations les modeles avec penalisation comme l estimateur ridge l estimateur de stein l estimateur en pont et le lasso ont ete proposes pour prendre en compte la collineariteen regression le lasso a ete applique au modele lineaire a la regression logistique au modele des hasards proportionnels de cox et aux reseaux de neurones cet article etudie le modele avec penalisation en pont pour une penalite j j applique aux equations d estimation et applique ce modele avec penalisation aux equations d estimation generalisees gee dans des etudes longitudinales le defaut de vraisemblance jointe dans les gee est surmonte par les equations d estimation penalisees pour lesquelles aucune vraisemblance jointe n est requise on donne les resultats asymptotiques pour l estimateur sons penalite on montre par des simulations et une application que les cee avec penalisation ameliorent potentiellement les resultats de l estimateur gee et presentent les memes proprietes que les modeles lineaires penalises

a priori probabilities of separable quantum states zyczkowski horodecki sanpera and lewenstein zhsl recently proposed a natural measure on the n dimensional quantum systems but expressed surprise when it led them to conclude that for n 2 2 disentangled separable systems are more probable 0 632 0 002 in nature than entangled ones we contend however that zhsl s rejected intuition has in fact a sound theoretical basis and that the a priori probability of disentangled 2 2 systems should more properly be viewed as considerably less than 0 5 we arrive at this conclusion in two quite distinct ways the first based on classical and the second quantum considerations both approaches however replace in whole or part the zhsl product measure by ones based on the volume elements of monotone metrics which in the classical case amounts to adopting the jeffreys prior of bayesian theory only the quantum theoretic analysis which yields the smallest probabilities of disentanglement uses the minimum number of parameters possible that is n2 1 as opposed to n2 n 1 although this over parametrization as recently indicated by byrd should be avoidable however despite substantial computation we are not able to obtain precise estimates of these probabilities and the need for additional possibly supercomputer analyses is indicated particularly so for higher dimensional quantum systems such as the 2 3 ones which we also study here

cyclic states berry phases and the schrodinger operator the evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval 0 t is considered it is shown that all the cyclic states of the system mod psi t ei phi mod psi 0 are determined by the eigenvalues in quasi energies and associated eigenvectors mod phi in t of the schrodinger operator sq t h t i delta delta t acting on some hilbert space the set of linearly independent cyclic quasi energy states possess some properties similar to the properties of the stationary states of a closed system the berry phases of the states associated with eigenvectors of the discrete spectrum of sq t which are single valued functions of omega 2 pi t are supplied by the partial derivatives of the corresponding eigenvalues quasi energies with respect to omega the approach developed is illustrated by several applications to time dependent systems the system under an adiabatic perturbation the forced harmonic oscillator and the two level system even in the case of a system with a time independent hamiltonian there exist nonstationary quasi energy states that have nontrivial berry phases beta 2 pi not an integer the criterion of existence of such states is formulated in terms of the energies of the system and the corresponding expression for the berry phases is obtained some examples of nonstationary cyclic states of closed systems including the coherent states of a force free oscillator and the wannier states of electrons in the parabolic band are considered

hamilton s turn and geometric phase for two level systems hamilton in the course of his studies on quaternions introduced an elegant geometric representation for the composition of su 2 elements in terms of turns on the unit sphere s2 the authors use these turns to study two level systems with particular reference to geometric phase the special roles played by piecewise geodesic circuits in the state space and evolution under constant hamiltonian are recognized

non cyclic geometric phases in a proposed two photon interferometric experiment an experimental arrangement is proposed which combines the idea of two photon interferometry with that of geometric phases a pair of photons generated by parametric down conversion is fed into a pair of mach zehnder interferometers in each of which there is phase shift produced by the helically wound optical fibre in the manner of tomita and chiao 1986 if the winding angle is a multiple of 2 pi there is a closed path in parameter space in accordance with the pioneering analysis of berry 1984 more general windings are also considered in which the path in parameter space is not closed and the geometric phase for such open paths is calculated in accordance with the gauge invariant prescription of aitchison and wanelik if the proposed experiment is realized it will provide a test for the noncyclic geometric phase for photons and for the set of prescriptions equivalent to that of aitchison and wanelik 1992 furthermore the geometric phase in this arrangement will be unequivocally quantum mechanical because of the non classical character of two photon interferometry

hilbert schmidt volume of the set of mixed quantum states we compute the volume of the convex n2 1 dimensional set n of density matrices of size n with respect to the hilbert schmidt measure the hyper area of the boundary of this set is also found and its ratio to the volume provides information about the structure of n similar investigations are also performed for the smaller set of all real density matrices as an intermediate step we analyse volumes of the unitary and orthogonal groups and of the flag manifolds

a generalized pancharatnam geometric phase formula for three level quantum systems we describe a recently developed generalization of the poincare sphere method to represent pure states of a three level quantum system in a convenient geometrical manner the construction depends on the properties of the group su 3 and its generators in the defining representation and uses geometrical objects and operations in an eight dimensional real euclidean space this construction is then used to develop a generalization of the well known pancharatnam geometric phase formula for evolution of a three level system along a geodesic triangle in state space

geometric phase and modulus relations for probability amplitudes as functions on complex parameter spaces we investigate general differential relations connecting the respective behaviors of the phase and modulus of probability amplitudes of the form f where f is a fixed state in hilbert space and is a variable state treated as a section of a u 1 bundle over a complex subspace of the corresponding ray space r cpn amplitude functions on such holomorphic line bundles while not strictly holomorphic nevertheless satisfy generalized cauchy riemann conditions involving the u 1 berry simon connection on the parameter space these conditions entail invertible relations between the gradients of the phase and modulus therefore allowing for the reconstruction of the phase from the modulus or vice versa and other conditions on the behavior of either polar component of the amplitude as a special case we consider amplitude functions valued on the space of pure states the ray space r cpn where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with t

imputation and variable selection in linear regression models with missing covariates across multiply imputed data sets variable selection methods such as stepwise regression and other criterion based strategies that include or exclude particular variables typically result in models with different selected predictors thus presenting a problem for combining the results from separate complete data analyses here drawing on a bayesian framework we propose two alternative strategies to address the problem of choosing among linear regression models when there are missing covariates one approach which we call impute then select its involves initially performing multiple imputation and then applying bayesian variable selection to the multiply imputed data sets a second strategy is to conduct bayesian variable selection and missing data imputation simultaneously within one gibbs sampling process which we call simultaneously impute and select sias the methods are implemented and evaluated using the bayesian procedure known as stochastic search variable selection for multivariate normal data sets but both strategies offer general frameworks within which different bayesian variable selection algorithms could be used for other types of data sets a study of mental health services utilization among children in foster care programs is used to illustrate the techniques simulation studies show that both its and sias outperform complete case analysis with stepwise variable selection and that sias slightly outperforms its

variable selection for cox s proportional hazards model and frailty model a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in fan and li 2001a it has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance such a property is called an oracle property the proposed procedures were illustrated in the context of linear regression robust linear regression and generalized linear models in this paper the nonconcave penalized likelihood approach is extended further to the cox proportional hazards model and the cox proportional hazards frailty model two commonly used semi parametric models in survival analysis as a result new variable selection procedures for these two commonly used models are proposed it is demonstrated how the rates of convergence depend on the regularization parameter in the penalty function further with a proper choice of the regularization parameter and the penalty function the proposed estimators possess an oracle property standard error formulae are derived and their accuracies are empirically tested simulation studies show that the proposed procedures are more stable in prediction and more effective in computation than the best subset variable selection and they reduce model complexity as effectively as the best subset variable selection compared with the lasso which is the penalized likelihood method with the l 1 penalty proposed by tibshirani the newly proposed approaches have better theoretic properties and finite sample performance

the lasso method for variable selection in the cox model summary i propose a new method for variable selection and shrinkage in cox s proportional hazards model my proposal minimizes the log partial likelihood subject to the sum of the absolute values of the parameters being bounded by a constant because of the nature of this constraint it shrinks coe cients and produces some coe cients that are exactly zero as a result it reduces the estimation variance while providing an interpretable final model the method is a variation of the lasso proposal of tibshirani designed for the linear regression context simulations indicate that the lasso can be more accurate than stepwise selection in this setting

on the landau zener formula for two level systems it is shown that the transition probability between eigenstates of a time dependent two level system is given by the fourier transform in the adiabatic parameter of a function determined by the hamiltonian sufficient and necessary conditions for the transition probability between levels to decrease exponentially in the adiabatic limit are established our method involves a paley wiener type theorem to study the fourier transform of functions belonging to a hardy class on a strip

boosting as a regularized path to a maximum margin classifier in this paper we study boosting methods from a new perspective we build on recent work by efron et al to show that boosting approximately and in some cases exactly minimizes its loss criterion with an l1 constraint on the coefficient vector this helps understand the success of boosting with early stopping as regularized fitting of the loss criterion for the two most commonly used criteria exponential and binomial log likelihood we further show that as the constraint is relaxed or equivalently as the boosting iterations proceed the solution converges in the separable case to an l1 optimal separating hyper plane we prove that this l1 optimal separating hyper plane has the property of maximizing the minimal l1 margin of the training data as defined in the boosting literature an interesting fundamental similarity between boosting and kernel support vector machines emerges as both can be described as methods for regularized optimization in high dimensional predictor space using a computational trick to make the calculation practical and converging to margin maximizing solutions while this statement describes svms exactly it applies to boosting only approximately

a new perspective on functional integration the core of this article is a general theorem with a large number of specializations given a manifold n and a finite number of one parameter groups of point transformations on n with generators y x 1 x d we obtain via functional integration over spaces of pointed paths on n paths with one fixed point a one parameter group of functional operators acting on tensor or spinor fields on n the generator of this group is a quadratic form in the lie derivatives lx in the x direction plus a term linear in ly the basic functional integral is over l2 1 paths x t n continuous paths with square integrable first derivative although the integrator is invariant under time translation the integral is powerful enough to be used for systems which are not time translation invariant seven nontrivial applications of the basic formula are given and the semiclassical expansion is computed the methods of proof are rigorous and combine albeverio ho egh krohn oscillatory integrals with elworthy s parametr

generalized euler angle parametrization for su n in a previous paper 1 an euler angle parametrization for su 4 was given here we present the derivation of a generalized euler angle parametrization for su n the formula for the calculation of the haar measure for su n as well as its relation to marinov s volume formula for su n 2 3 will also be derived as an example of this parametrization s usefulness the density matrix parametrization and invariant volume element for a qubit qutrit three qubit and two three state systems also known as two qutrit systems 4 will also be given

hamiltonian feynman path integrals via the chernoff formula the main aim of the present paper is using a chernoff theorem i e the chernoff formula to formulate and to prove some rigorous results on representations for solutions of schrodinger equations by the hamiltonian feynman path integrals feynman integrals over trajectories in the phase space the corresponding theorem is related to the original feynman approach to feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of nelson is related to the feynman approach to the feynman path integral over trajectories in the configuration space we also give a representation for solutions of some schrodinger equations by a series which represents an integral with respect to the complex poisson measure on trajectories in the phase space

coherent states for su 3 we define coherent states for su 3 using six bosonic creation and annihilation operators these coherent states are explicitly characterized by six complex numbers with constraints for the completely symmetric representations n 0 and 0 m only three of the bosonic operators are required for mixed representations n m all six operators are required the coherent states provide a resolution of identity satisfy the continuity property and possess a variety of group theoretic properties we introduce an explicit parametrization of the group su 3 and the corresponding integration measure finally we discuss the path integral formalism for a problem in which the hamiltonian is a function of su 3 operators at each site

penalized estimating functions and variable selection in semiparametric regression models we propose a general strategy for variable selection in semiparametric regression models by penalizing appropriate estimating functions important applications include semiparametric linear regression with censored responses and semiparametric regression with missing predictors unlike the existing penalized maximum likelihood estimators the proposed penalized estimating functions may not pertain to the derivatives of any objective functions and may be discrete in the regression coefficients we establish a general asymptotic theory for penalized estimating functions and present suitable numerical algorithms to implement the proposed estimators in addition we develop a resampling technique to estimate the variances of the estimated regression coefficients when the asymptotic variances cannot be evaluated directly simulation studies demonstrate that the proposed methods perform well in variable selection and variance estimation we illustrate our methods using data from the paul coverdell stroke registry

bloch sphere like construction of su 3 hamiltonians using unitary integration the bloch sphere is a familiar and useful geometrical picture of the time evolution of a single spin or a quantal two level system the analogous geometrical picture for three level systems is presented with several applications the relevant su 3 group and su 3 algebra are eight dimensional objects and are realized in our picture as two four dimensional manifolds that describe the time evolution operator the first called the base manifold is the counterpart of the s2 bloch sphere whereas the second called the fiber generalizes the single u 1 phase of a single spin now four dimensional it breaks down further into smaller objects depending on alternative representations that we discuss geometrical phases are also developed and presented for specific applications arbitrary time dependent couplings between three levels or between two spins qubits with su 3 hamiltonians can be conveniently handled through these geometrical objects

the extended bloch representation of quantum mechanics explaining superposition interference and entanglement an extended bloch representation of quantum mechanics was recently derived to offer a possible hidden measurements solution to the measurement problem in this article we use this representation to investigate the geometry of superposition and entangled states explaining interference effects and entanglement correlations in terms of the different orientations a state vector can take within the generalized bloch sphere we also introduce a tensorial determination of the generators of su n which we show to be particularly suitable for the description of multipartite systems from the viewpoint of the sub entities we then use it to show that non product states admit a general description where sub entities can remain in well defined states even when entangled this means that the completed version of quantum mechanics provided by the extended bloch representation where density operators are also considered to be representative of genuine states providing a complete description not only offers a pl

lie algebraic approach to fer s expansion for classical hamiltonian systems the so called fer s expansion is proposed as a solution for the time evolution operator of classical time dependent hamiltonian systems the quadratic hamiltonians treated as examples show that under very different regimes the second order approximation already gives extremely good results

meboost variable selection in the presence of measurement error we present a novel method for variable selection in regression models when covariates are measured with error the iterative algorithm we propose measurement error boosting meboost follows a path defined by estimating equations that correct for covariate measurement error we illustrate the use of meboost in practice by analyzing data from the box lunch study a clinical trial in nutrition where several variables are based on self report and hence measured with error where we are interested in performing model selection from a large data set to select variables that are related to the number of times a subject binge ate in the last 28 days furthermore we evaluated our method and compared its performance to the recently proposed convex conditioned lasso and to the naive lasso which does not correct for measurement error through a simulation study increasing the degree of measurement error increased prediction error and decreased the probability of accurate covariate selection but this loss of accuracy occurred to a lesser degree when using meboost through simulations we also make a case for the consistency of the model selected

volumes of conditioned bipartite state spaces we analyze the metric properties of conditioned quantum state spaces these spaces are the convex sets of density matrices that when partially traced over m degrees of freedom respectively yield the given n n density matrix for the case n 2 the volume of equipped with the hilbert schmidt measure can be conjectured to be a simple polynomial of the radius of in the bloch ball remarkably for we find numerically that the probability to find a separable state in is independent of except for pure for the same holds for the probability to find a state with a positive partial transpose in these results are proven analytically for the case of the family of 4 4 x states and thoroughly numerically investigated for the general case the important implications of these findings for the clarification of open problems in quantum theory are pointed out and discussed

differential geometry on su 3 with applications to three state systems the left and right invariant vector fields are calculated in an euler angle type parametrization for the group manifold of su 3 referred to here as euler coordinates the corresponding left and right invariant one forms are then calculated this enables the calculation of the invariant volume element or haar measure these are then used to describe the density matrix of a pure state and geometric phases for three state systems

berry s phase as the asymptotic limit of an exact evolution an example the authors exhibit for arbitrary time variations of the parameters of the generalised harmonic oscillator hamiltonian a canonical transformation which facilitates an exact analysis of quantal phase and classical angle formulae for berry s phase and hannay s angle are readily obtained by taking the adiabatic limit of the exact theory

realization of compact lie algebras in kahler manifolds the berezin quantization on a simply connected homogeneous kahler manifold which is considered as a phase space for a dynamical system enables a description of the quantal system in a finite dimensional hilbert space of holomorphic functions corresponding to generalized coherent states the lie algebra associated with the manifold symmetry group is given in terms of first order differential operators in the classical theory the lie algebra is represented by the momentum maps which are functions on the manifold and the lie product is the poisson bracket given by the kahler structure the kahler potentials are constructed for the manifolds related to all compact semi simple lie groups the complex coordinates are introduced by means of the borel method the kahler structure is obtained explicitly for any unitary group representation the cocycle functions for the lie algebra and the killing vector fields on the manifold are also obtained

boosting with the l2 loss this article investigates a computationally simple variant of boosting l2boost which is constructed from a functional gradient descent algorithm with the l2 loss function like other boosting algorithms l2boost uses many times in an iterative fashion a prechosen fitting method called the learner based on the explicit expression of refitting of residuals of l2boost the case with symmetric linear learners is studied in detail in both regression and classification in particular with the boosting iteration m working as the smoothing or regularization parameter a new exponential bias variance trade off is found with the variance complexity term increasing very slowly as m tends to infinity when the learner is a smoothing spline an optimal rate of convergence result holds for both regression and classification and the boosted smoothing spline even adapts to higher order unknown smoothness moreover a simple expansion of a smoothed 0 1 loss function is derived to reveal the importance of the d

exponentially small adiabatic invariant for the schrodinger equation we study an adiabatic invariant for the time dependent schrfdinger equation which gives the transition probability across a gap from time t to time t when the hamiltonian depends analytically on time and t oo t oo we give sufficient conditions so that this adiabatic invariant tends to zero exponentially fast in the adiabatic limit

double semiparametric method for missing covariates in cox regression models the problem of nuisance covariate model specification is considered in cox regression where the maximum semiparametric likelihood method is used to handle the missing covariates a component of the covariates is modeled nonparametrically to achieve robustness against covariate model misspecification and to reduce the number of possibly intractable integrations involved in the parametric modeling of the covariates the statistical properties of the proposed method are examined it is found that in some important situations the maximum semiparametric likelihood can be applied without making any additional parametric model assumptions on covariates the proposed method can yield a more efficient estimator than the nonparametric imputation methods and does not require specification of the missingness mechanism when compared with the inverse probability weighting method a real data example is analyzed to demonstrate use of the proposed method

on the hamiltonian reduction of geodesic motion on su 3 to su 3 su 2 the reduced hamiltonian system on t su 3 su 2 is derived from a riemannian geodesic motion on the su 3 group manifold parametrized by the generalized euler angles and endowed with a bi invariant metric our calculations show that the metric defined by the derived reduced hamiltonian flow on the orbit space su 3 su 2 s5 is not isometric or even geodesically equivalent to the standard riemannian metric on the five sphere s5 embedded into r6

quasiclassical path integral in coherent state manifolds quantization of classical dynamical systems with a poisson structure on homogeneous kahler manifolds is considered the quantization follows the method invented by berezin and represents the unitary transition operator as a quasiclassical path integral in the coherent state basis in case the coherent state manifold appears as a degenerate rank one co adjoint orbit of the symmetry group an explicit representation of the transition amplitude in terms of classical data can be derived for large values of the highest weight which corresponds to the quasiclassical approximation this representation is further shown to perfectly agree in contrast to some earlier approaches with the known exact results and may provide non trivial asymptotics of physical relevance

path integrals on homogeneous manifolds quantum kinematics are considered for two classes of systems i the coordinate space is a homogeneous riemannian manifold and the kinetic energy is the laplace beltrami operator ii the phase space is a homogeneous kahlerian manifold where the algebra of observables is the universal enveloping algebra of the symmetry group in both the cases the path integrals are derived from more fundamental principles the derivation enables a definite description of the classes of trajectories which constitute the domain of integration in the path integrals the specification of the classes of functions is important for consistency of the approach the standard action functional and the variational principle appear in the steepest descent method corresponding to the semiclassical approximation

model selection in estimating equations summary model selection is a necessary step in many practical regression analyses but for methods based on estimating equations such as the quasi likelihood and generalized estimating equation gee approaches there seem to be few well studied model selection techniques in this article we propose a new model selection criterion that minimizes the expected predictive bias epb of estimating equations a bootstrap smoothed cross validation bcv estimate of epb is presented and its performance is assessed via simulation for overdispersed generalized linear models for illustration the method is applied to a real data set taken from a study of the development of ewe embryos

berry s phase for compact lie groups the methods of kahler geometry are applied to generalize the results of berry obtained for su 2 namely the existence of a geometrical part in the adiabatic phase to any compact lie group we obtain explicit expressions for berry s geometric phases berry s connections and berry s curvatures in terms of parameters of the corresponding lie algebra valued hamiltonian it is demonstrated that the parameter space of the hamiltonian in the general theory is essentially a homogeneous kahler manifold the fundamental kahler potentials of this manifold completely determine berry s phase a general approach is exemplified by the lie algebra hamiltonians corresponding to su 2 and su 3 evolution groups

discrete time adiabatic theorems for quantum mechanical systems the theory of adiabatic asymptotics is adapted to systems with discrete time evolution the corresponding theorems about the approximation of physical time evolution by the adiabatic time evolution are shown to hold true in a discrete setting

approximations to the solution of cauchy problem for a linear evolution equation via the space shift operator second order equation example abstract we present a general method of solving the cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two one and zero the method is based on the chernoff approximation procedure applied to a specially constructed shift operator it is proven that approximations converge uniformly to the exact solution

unfolding a diabolic point a generalized crossing scenario the typical avoided crossings for hermitian quantum systems depending on parameters the diabolic crossing scenario are generalized to the non hermitian case e g for resonances two types of crossings appear for type i the real parts show an avoided and the imaginary parts a true crossing of the eigenenergies and for type ii the opposite is found a simple symmetric non hermitian two state matrix hamiltonian is analysed in detail the diabolic point bifurcates into two exceptional ones on exceptional lines where the matrices are defective the adiabatic transport of eigenvectors and eigenstates in parameter space is discussed in this generalized diabolic crossing scenario in particular the geometric berry phases for a cyclic variation of system parameters depending on the topology of the closed curves with respect to the exceptional lines

berry s phase hannay s angle and coherent states through the use of coherent states describing the action angle variables the authors establish a strict parallelism between berry s phase and hannay s angle in particular they define a geometrical transport for classical tori which leads to hannay s angle as in the case of berry s phase the transport can be associated with the minimization of a distance

homogeneous kahler manifolds paving the way towards new supersymmetric sigma models homogeneous kahler manifolds give rise to a broad class of supersymmetric sigma models containing as a rather special subclass the more familiar supersymmetric sigma models based on hermitian symmetric spaces in this article all homogeneous kahler manifolds with semisimple symmetry groupg are constructed and are classified in terms of dynkin diagrams explicit expressions for the complex structure and the kahler structure are given in terms of the lie algebra g ofg it is shown that for compactg one can always find an einstein kahler structure which is unique up to a constant multiple and for which the kahler potential takes a simple form

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

a geometrical approach to non adiabatic transitions in quantum theory applications to nmr over barrier reflection and parametric excitation of quantum oscillator this paper deals with non adiabatic processes i e processes excluded by the adiabatic theorem from the geometrical group theoretical point of view an approximate formula for the probabilities of the non adiabatic transitions is derived in the adiabatic regime for the case when the parameter dependent hamiltonian represents a smooth curve in the lie algebra and the quantal dynamics is determined by the corresponding lie group evolution operator we treat the spin precession in a time dependent magnetic field and the over barrier reflection problem in a uniform way using the first order dynamical equations on su 2 and su 1 1 group manifolds respectively a comparison with analytic solutions for simple solvable models is provided

cox regression with incomplete covariate measurements abstract this article provides a general solution to the problem of missing covariate data under the cox regression model the estimating function for the vector of regression parameters is an approximation to the partial likelihood score function with full covariate measurements and reduces to the pseudolikelihood score function of self and prentice in the special setting of case cohort designs the resulting parameter estimator is consistent and asymptotically normal with a covariance matrix for which a simple and consistent estimator is provided extensive simulation studies show that the large sample approximations are adequate for practical use the proposed approach tends to be more efficient than the complete case analysis especially for large cohorts with infrequent failures for case cohort designs the new methodology offers a variance covariance estimator that is much easier to calculate than the existing ones and allows multiple subcohort augmentations to improve efficiency real data taken f

solving nonlocal initial boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the adomian decomposition method in this paper we present a new approach to solve nonlocal initial boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations subject to initial and nonlocal boundary conditions of integral type we first transform the given nonlocal initial boundary value problems of integral type for the linear and nonlinear parabolic and hyperbolic partial differential equations into local dirichlet initial boundary value problems and then use a relatively new modified adomian decomposition method adm furthermore we investigate the fourier adomian method which also does not require any a priori assumptions on the solution for the solution of nonlocal initial boundary value problems combined with our new approach several examples are presented to demonstrate the efficiency of the adm

performance of generalized estimating equations in practical situations moment methods for analyzing repeated binary responses have been proposed by liang and zeger 1986 biometrika 73 13 22 and extended by prentice 1988 biometrics 44 1033 1048 in their generalized estimating equations gee both liang and zeger 1986 and prentice 1988 estimate the parameters associated with the expected value of an individual s vector of binary responses as well as the correlations between pairs of binary responses in this paper we discuss one step estimators i e estimators obtained from one step of the generalized estimating equations and compare their performance to that of the fully iterated estimators in small samples in simulations we find the performance of the one step estimator to be qualitatively similar to that of the fully iterated estimator when the sample size is small and the association between binary responses is high we recommend using the one step estimator to circumvent convergence problems associated with the fully iterated gee algorithm furthermore we find the gee methods to be more efficient than ordinary logistic regression with variance correction for estimating the effect of a time varying covariate

feynman formulae and phase space feynman path integrals for tau quantization of some levy khintchine type hamilton functions evolution semigroups generated by pseudo differential operators are considered these operators are obtained by different parameterized by a number procedures of quantization from a certain class of functions or symbols defined on the phase space this class contains hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the levy khintchine formula the considered semigroups are represented as limits of n fold iterated integrals when n tends to infinity such representations are called feynman formulae some of these representations are constructed with the help of another pseudo differential operator obtained by the same procedure of quantization such representations are called hamiltonian feynman formulae some representations are based on integral operators with elementary kernels these are called lagrangian feynman formulae langrangian feynman formulae provide approximations of evolution semigroups suitable for direct computatio

a projected likelihood function for semiparametric models summary in a sequence of papers mcleish 1984 hutton nelson 1986 and godambe heyde 1987 have motivated the use of the quasi score function as an estimating function for semiparametric models by showing that it is the projection of the true score into a class of unbiased estimating functions thus the quasi score approximates the score within this class of functions in this paper we introduce a class of inference functions for independent observations that contain analogues of likelihood ratios under semiparametric assumptions this class of functions which contains the semiparametric estimating functions as a subspace is built through the use of tensor products for models with independent observations we consider the projection of the likelihood ratios into this class of inference functions it is shown that this projected likelihood is first order equivalent to the quasi likelihood

solution giving formula to cauchy problem for multidimensional parabolic equation with variable coefficients we present a general method of solving the cauchy problem for multidimensional parabolic diffusion type equation with variable coefficients which depend on spatial variable but do not change over time we assume the existence of the c0 semigroup this is a standard assumption in the evolution equations theory which guarantees the existence of the solution and then find the representation based on the family of translation operators of the solution in terms of coefficients of the equation and initial condition it is proved that if the coefficients of the equation are bounded infinitely smooth and satisfy some other conditions then there exists a solution giving c0 semigroup of contraction operators we also represent the solution as a feynman formula i e as a limit of a multiple integral with multiplicity tending to infinity with generalized functions appearing in the integral kernel we present a general method of solving the cauchy problem for multidimensional parabolic diffusion type equation with variable coefficients which depend on spatial variable but do not change over time we assume the existence of the c0 semigroup this is a standard assumption in the evolution equations theory which guarantees the existence of the solution and then find the representation based on the family of translation operators of the solution in terms of coefficients of the equation and initial condition it is proved that if the coefficients of the equation are bounded infinitely smooth and satisfy some other conditions then there exists a solution giving c0 semigroup of contraction operators we also represent the solution as a feynman formula i e as a limit of a multiple integral with multiplicity tending to infinity with generalized functions appearing in the integral kernel

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

an analysis of a second order time accurate scheme for a finite volume method for parabolic equations on general nonconforming multidimensional spatial meshes the present work is an extension of our previous work bradji and fuhrmann 2011 4 which dealt with error analysis of a finite volume scheme of first order both in time and space for parabolic equations on general nonconforming multidimensional spatial meshes introduced recently in eymard et al 2010 12 we aim in this paper to get a higher order time accurate scheme for a finite volume method for parabolic equations using the same class of spatial generic meshes stated above we derive a finite volume scheme approximating the heat equation as a model for parabolic equations in which the discretization in time is performed using the crank nicolson method we derive an a priori estimate for the discrete problem and we prove that the error estimate of the finite volume scheme is of order two in time and it is of optimal order in space the error estimate is analysed in several norms which allow us to derive approximations for the exact solution and its first derivatives both spatial and temporal whose the convergence order is two in time and it is optimal in space we prove in particular when the discrete flux is calculated using a stabilized discrete gradient that the convergence order is k 2 h d where h d resp k is the mesh size of the spatial resp time discretization these estimates are valid under the regularity assumption u c 3 0 t c 2 w for the exact solution u the proof of these error estimates is based essentially on a new a priori estimate for the discrete problem and a comparison between the finite volume approximate solution and an auxiliary finite volume approximation

l1 regularization path algorithm for generalized linear models summary we introduce a path following algorithm for l1 regularized generalized linear models the l1 regularization procedure is useful especially because it in effect selects variables according to the amount of penalization on the l1 norm of the coefficients in a manner that is less greedy than forward selection backward deletion the generalized linear model path algorithm efficiently computes solutions along the entire regularization path by using the predictor corrector method of convex optimization selecting the step length of the regularization parameter is critical in controlling the overall accuracy of the paths we suggest intuitive and flexible strategies for choosing appropriate values we demonstrate the implementation with several simulated and real data sets

strict quantization of coadjoint orbits a strict quantization of a compact symplectic manifold s on a subset i r containing 0 as an accumulation point is defined as a continuous field of c algebras a i with a0 c0 s and a set of continuous cross sections q f f c s for which q0 f f here q f q f for all i whereas for 0 one requires that i q f q g q f g in norm we discuss general conditions which guarantee that a deformation quantization in a more physical sense leads to one in the above sense using ideas of berezin lieb simon and others we construct a strict quantization of an arbitrary integral coadjoint orbit o of a compact connected lie group g associated to a highest weight here i 0 1 n so that 1 k k n and a1 k is defined as the c algebra of all matrices on the finite dimensional hilbert space vk carrying the irreducible representation uk g with highest weight k the quantization maps q1 k f are constructed from coherent states in vk and have the special feature of being positive

quasi feynman formulas a method of obtaining the evolution operator for the schrodinger equation abstract for a densely defined self adjoint operator h in hilbert space f the operator exp i t h is the evolution operator for the schrodinger equation i t h i e if 0 x 0 x then t x exp i t h 0 x for x q the space f here is the space of wave functions defined on an abstract space q the configuration space of a quantum system and h is the hamiltonian of the system in this paper the operator exp i t h for all real values of t is expressed in terms of the family of self adjoint bounded operators s t t 0 which is chernoff tangent to the operator h one can take s t exp t h or use other simple families s that are listed in the paper the main theorem is proven on the level of semigroups of bounded operators in f so it can be used in a wider context due to its generality two examples of application are provided

on the quantization of the modified poschl teller potential it is shown that the connection one form on the prequantization line bundle that generates the correct energy spectrum for the modified poschl teller potential consists of the canonical one form on the corresponding cotangent bundle together with an adjustment term that arises from the berry s connection

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

on using the cox proportional hazards model with missing covariates summary we propose two methods for handling missing covariates in using the cox proportional hazards model the maximum partial likelihood estimator based only on study subjects having complete covariates does not utilise all available information also it is biased when the probability of missingness depends on the failure or censoring time our suggestion is to impute the conditional expectation of the statistic involving missing covariates given the available information the proposed method provides a consistent regression parameter estimator when the probability of missingness depends on the failure or censoring time as well as on the observed covariates also the proposed estimator is more efficient than the estimator suggested previously by lin ying 1993 when data are missing completely at random

geometric amplitude adiabatic invariants quantization and strong stability of hamiltonian systems considered is a linear set of ordinary differential equations with a matrix depending on a set of adiabatically varying parameters asymptotic solutions have been constructed as has been shown an important characteristic determining the qualitative portrait of the system is the real part of berry s complex geometric phase which we call the geometric amplitude for systems with purely imaginary eigenvalues the equivalence has been proven in the adiabatic approximation of system sets without geometric amplitude hamiltonian systems quantizable systems and strongly stable systems classification of the systems without geometric amplitude is given with respect to the kind of matrix of the initial system

fluctuation path integrals in the generalized coherent states representation fluctuations around classical solutions are evaluated by the path integral method based on the generalized coherent states a simple expression for the semiclassical propagator is derived and applied to the su 2 character formula it is also pointed out that several versions of the coherent states path integral can be considered owing to the nonorthogonal character of the coherent states and they are compared by applying them to the su 2 character formula

greedy function approximation a gradient boosting machine function estimation approximation is viewed from the perspective of numerical optimization in function space rather than parameter space a connection is made between stagewise additive expansions and steepest descent minimization a general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion specific algorithms are presented for least squares least absolute deviation and huber m loss functions for regression and multiclass logistic likelihood for classification special enhancements are derived for the particular case where the individual additive components are regression trees and tools for interpreting such treeboost models are presented gradient boosting of regression trees produces competitive highly robust interpretable procedures for both regression and classification especially appropriate for mining less than clean data connections between this approach and the boosting methods of freund and shapire and friedman hastie and tibshirani 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

reachability and steering of rolling polyhedra a case study in discrete nonholonomy rolling a ball on a plane is a standard example of nonholonomy reported in many textbooks and the problem is also well understood for any smooth deformation of the surfaces for nonsmoothly deformed surfaces however much less is known although it may seem intuitive that nonholonomy is conserved think e g to polyhedral approximations of smooth surfaces current definitions of nonholonomy are inherently referred to systems described by ordinary differential equations and are thus inapplicable to such systems in this paper we study the set of positions and orientations that a polyhedral part can reach by rolling on a plane through sequences of adjacent faces we provide a description of such reachable set discuss conditions under which the set is dense or discrete or has a compound structure and provide a method for steering the system to a desired reachable configuration robustly with respect to model uncertainties based on ideas and concepts encountered in this case study and in some other examples we provide we turn back to the most general aspects of the problem and investigate the possible generalization of the notion of kinematic nonholonomy to nonsmooth discrete and hybrid dynamical systems to capture the essence of phenomena commonly regarded as nonholonomic at least two irreducible concepts are to be defined of internal and external nonholonomy which may coexist in the same system these definitions are instantiated by examples

unitary irreducible representations of sl 2 c in discrete and continuous su 1 1 bases we derive the matrix elements of generators of unitary irreducible representations of sl 2 c with respect to basis states arising from a decomposition into irreducible representations of su 1 1 this is done with regard to a discrete basis diagonalized by j3 and a continuous basis diagonalized by k1 and for both the discrete and continuous series of su 1 1 for completeness we also treat the more conventional su 2 decomposition as a fifth case the derivation proceeds in a functional differential framework and exploits the fact that state functions and differential operators have a similar structure in all five cases the states are defined explicitly and related to su 1 1 and su 2 matrix elements

an explicit computation of the bures metric over the space of n dimensional density matrices the aim of this paper is to provide a method for explicit computation of the bures metric over the space of n level quantum system based on the coset parametrization of density matrices

su 3 revisited the d matrices for all states of the two fundamental representations and octet are given in the euler angle parametrization of su 3 the raising and lowering operators are given in terms of linear combinations of the left invariant vector fields of the group manifold in this parametrization using these differential operators the highest weight state of an arbitrary irreducible representation is found and a description of the calculation of clebsch gordon coefficients is given

augmented inverse probability weighted estimator for cox missing covariate regression summary this article investigates an augmented inverse selection probability weighted estimator for cox regression parameter estimation when covariate variables are incomplete this estimator extends the horvitz and thompson 1952 journal of the american statistical association 47 663 685 weighted estimator this estimator is doubly robust because it is consistent as long as either the selection probability model or the joint distribution of covariates is correctly specified the augmentation term of the estimating equation depends on the baseline cumulative hazard and on a conditional distribution that can be implemented by using an em type algorithm this method is compared with some previously proposed estimators via simulation studies the method is applied to a real example

full asymptotic expansion of transition probabilities in the adiabatic limit the authors consider a two level quantum mechanical system driven by an analytic time dependent hamiltonian of the form h in t in the adiabatic limit in or 0

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures