chow s theorem and universal holonomic quantum computation a theorem from control theory relating the lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to holonomic quantum computation conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated with a non abelian gauge connection when applied to the optical holonomic computer these conditions determine that the holonomy group of the two qubit interaction model contains su 2 su 2 in particular a universal two qubit logic gate is attainable for this model

a geometric setting for internal motions of the quantum three body system quantum mechanics for internal motions of the three body system is set up on the basis of the complex vector bundle theory the three body system is called a triatomic molecule in the born oppenheimer approximation the internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator this bundle is equipped with a linear connection which is a natural consequence of a geometric interpretation of the so called eckart condition the coupling of the internal motion with the rotation is understood naturally in terms of this connection the internal hamiltonian operator is obtained which includes the internal motion rotation coupling and a centrifugal potential the complex vector bundle for the triatomic molecule proves to be a trivial bundle though the geometric setting for the internal motion is independent of whether the bundle is trivial or not

adaptive estimation of eigensubspace in a recent work we recast the problem of estimating the minimum eigenvector eigenvector corresponding to the minimum eigenvalue of a symmetric positive definite matrix into a neural network framework we now extend this work using an inflation technique to estimate all or some of the orthogonal eigenvectors of the given matrix based on these results we form a cost function for the finite data case and derive a newton based adaptive algorithm the inflation technique leads to a highly modular and parallel structure for implementation the computational requirement of the algorithm is o n sup 2 n being the size of the covariance matrix we also present a rigorous convergence analysis of this adaptive algorithm the algorithm is locally convergent and the undesired stationary points are unstable computer simulation results are provided to compare its performance with that of two adaptive subspace estimation methods proposed by yang and kaveh 1988 and an improved version of one of them for stationary and nonstationary signal scenarios the results show that the proposed approach performs identically to one of them and is significantly superior to the remaining two

geometric phase geometric distance and length of the curve in quantum evolution the geometric phase and the geometric distance function are intimately related via length of the curve a concept the author introduces for any parametric evolution of the quantum system he offers an interpretation of the nonadiabatic berry phase as the integral over a difference between the differentials of two geometric quantities which enables him to say that the geometric phase is just half the integral of the contracted length of the curve that the system traverses during a cyclic excursion

orthogonal eigensubspace estimation using neural networks we present a neural network nn approach for simultaneously estimating all or some of the orthogonal eigenvectors of a symmetric nonindefinite matrix corresponding to its repeated minimum in magnitude eigenvalue this problem has its origin in the constrained minimization framework and has extensive applications in signal processing we recast this problem into the nn framework by constructing an appropriate energy function which the nn minimizes the nn is of feedback type with the neurons having sigmoidal activation function the proposed approach is analyzed to characterize the nature of the minimizers of the energy function the main result is that the matrix w is a minimizer of the energy function if and only if the columns of w are the orthogonal eigenvectors with a given norm corresponding to the smallest eigenvalue of the given matrix further all minimizers are global minimizers bounds on the integration time step that is required to numerically solve the system of differential equations which define the dynamics of the nn have also been derived results of computer simulations are presented to support our analysis

linear dynamical systems coherent state manifolds flows and matrix riccati equation the classical motion and the quantum evolution determined by linear hamiltonians on coherent state manifolds of compact and noncompact hermitian symmetric space structure are considered a matrix riccati equation with opposite signs of the quadratic terms in the compact and noncompact cases determines the classical motion and the quantum evolution the geometric meaning of equations of motion as flow on a complex grassmann manifold and his noncompact dual is emphasized possibilities of generalizing the results to flag manifolds are pointed out

optimal control for holonomic and nonholonomic mechanical systems with symmetry and lagrangian reduction in this paper we establish necessary conditions for optimal control using the ideas of lagrangian reduction in the sense of reduction under a symmetry group the techniques developed here are designed for lagrangian mechanical control systems with symmetry the benefit of such an approach is that it makes use of the special structure of the system especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems lagrangian reduction can do in one step what one can alternatively do by applying the pontryagin maximum principle followed by an application of poisson reduction the idea of using lagrangian reduction in the sense of symmetry reduction was also obtained by bloch and crouch proc 33rd cdc ieee 1994 pp 2584 2590 in a somewhat different context and the general idea is closely related to those in montgomery comm math phys 128 1990 pp 565 592 and vershik and gershkovich dynamical systems vii v arnold and s p novikov eds springer verlag 1994 here we develop this idea further and apply it to some known examples such as optimal control on lie groups and principal bundles such as the ball and plate problem and reorientation examples with zero angular momentum such as the satellite with moveable masses however one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of bloch krishnaprasad marsden and murray arch rational mech anal 1996 to appear the snakeboard is used to illustrate the method

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

control aspects of holonomic quantum computation a unifying framework for the control of quantum systems with non abelian holonomy is presented it is shown that from a control theoretic point of view holonomic quantum computation can be treated as a control system evolving on a principal fiber bundle an extension of methods developed for these classical systems may be applied to quantum holonomic systems to obtain insight into the control properties of such systems and to construct control algorithms for two established examples of the computing paradigm

a dual purpose principal and minor subspace gradient flow the dual purpose principal and minor subspace gradient flow can be used to track principal subspace ps and if altered simply by the sign it can also serve as a minor subspace ms trackor this is of practical significance in the implementations of algorithms in this paper a unified information criterion is proposed and a dual purpose principal and minor subspace gradient flow is derived based on the information criterion in this dual purpose gradient flow the weight matrix length is self stabilizing i e moving towards unit length in each learning step the energy function associated with the dual purpose gradient flow for tracking ps and ms is given and it exhibits a unique global minimum attained if and only if its state matrices span the ps or ms of the autocorrelation matrix of a vector data stream the other stationary points of its energy function are unstable saddle points the proposed dual purpose gradient flow can efficiently track an orthonormal basis of the ps or ms which is illustrated through simulation experiments

on the inflation method in adaptive noise subspace estimator it is shown that yang and kaveh s 1987 1988 inflation method is not necessary to accomplish an adaptive noise subspace estimator without the inflation method an alternative noise subspace estimator involving less computation and simpler parallel implementation is developed we also prove that if the initial noise subspace basis consists of a set of mutually orthonormal vectors the proposed estimator has a better performance than that with the inflation method both in the mean sense and in the mean square sense computer simulations are provided to substantiate the superiority of the estimator without the inflation method

energy function for the one unit oja algorithm the one unit oja algorithm plays a very important role in the study of principal component analysis neural networks in this paper we propose an energy function whose steepest descent direction i e negative gradient direction is the same as the average evolution direction of the one unit oja algorithm and the energy function has two global minimal points corresponding to the two converged points of the one unit oja algorithm and it has no other local minimal points

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

neural network learning algorithms for tracking minor subspace in high dimensional data stream a novel random gradient based algorithm is developed for online tracking the minor component mc associated with the smallest eigenvalue of the autocorrelation matrix of the input vector sequence the five available learning algorithms for tracking one mc are extended to those for tracking multiple mcs or the minor subspace ms in order to overcome the dynamical divergence properties of some available random gradient based algorithms we propose a modification of the oja type algorithms called ojam which can work satisfactorily the averaging differential equation and the energy function associated with the ojam are given it is shown that the averaging differential equation will globally asymptotically converge to an invariance set the corresponding energy or lyapunov functions exhibit a unique global minimum attained if and only if its state matrices span the ms of the autocorrelation matrix of a vector data stream the other stationary points are saddle unstable points the globally convergence of ojam is also studied the ojam provides an efficient online learning for tracking the ms it can track an orthonormal basis of the ms while the other five available algorithms cannot track any orthonormal basis of the ms the performances of the relative algorithms are shown via computer simulations

development and analysis of a neural network approach to pisarenko s harmonic retrieval method pisarenko s harmonic retrieval phr method is perhaps the first eigenstructure based spectral estimation technique the basic step in this method is the computation of eigenvector corresponding to the minimum eigenvalue of the autocorrelation matrix of the underlying data the authors recast a known constrained minimization formulation for obtaining this eigenvector into the neural network nn framework using the penalty function approach they develop an appropriate energy function for the nn this nn is of feedback type with the neurons having sigmoidal activation function analysis of the proposed approach shows that the required eigenvector is a minimizer with a given norm of this energy function further all its minimizers are global minimizers bounds on the integration time step that is required to numerically solve the system of nonlinear differential equations which define the network dynamics have been derived results of computer simulations are presented to support their analysis

adaptive eigensubspace algorithms for direction or frequency estimation and tracking the authors present an adaptive estimator of the complete noise or signal subspace of a sample covariance matrix as well as the estimator s practical implementations the general formulation of the proposed estimator results from an asymptotic argument which shows the signal or noise subspace computation to be equivalent to a constrained gradient search procedure a highly parallel algorithm denoted the inflation method is introduced for the estimation of the noise subspace the simulation results of these adaptive estimators show that the adaptive subspace algorithms perform substantially better than p a thompson s 1980 adaptive version of v f pisarenko s technique 1973 in estimating frequencies or directions of arrival doa of plane waves for tracking nonstationary parameters the simulation results also show that the adaptive subspace algorithms are better than direct eigendecomposition methods for which computational complexity is much higher than the adaptive versions

a globally convergent mc algorithm with an adaptive learning rate this brief deals with the problem of minor component analysis mca artificial neural networks can be exploited to achieve the task of mca recent research works show that convergence of neural networks based mca algorithms can be guaranteed if the learning rates are less than certain thresholds however the computation of these thresholds needs information about the eigenvalues of the autocorrelation matrix of data set which is unavailable in online extraction of minor component from input data stream in this correspondence we introduce an adaptive learning rate into the ojan mca algorithm such that its convergence condition does not depend on any unobtainable information and can be easily satisfied in practical applications

a minor subspace analysis algorithm this paper proposes a learning algorithm which extracts adaptively the minor subspace spanned by the eigenvectors corresponding to the smallest eigenvalues of the autocorrelation matrix of an input signal we show both analytically and by simulation results that the weight vectors provided by the proposed algorithm are guaranteed to converge to the minor subspace of the input signal for wider applications we also present the complex valued version of the proposed minor subspace analysis algorithm

pontryagin maximum principle on almost lie algebroids the fundamental theorem of the theory of optimal control the pontryagin maximum principle pmp is extended to the setting of almost lie al algebroids geometrical objects generalizing lie algebroids this formulation of the pmp yields in particular a scheme comprising reductions of optimal control problems similar to the reduction for the rigid body in analytical mechanics on the other hand in the presented approach the reduced and unreduced pmps are parts of the same universal formalism the framework is based on a very general concept of homotopy of measurable paths and the geometry of al algebroids

least squares type algorithm for adaptive implementation of pisarenko s harmonic retrieval method pisarenko s harmonic retrieval method involves determining the minimum eigenvalue and the corresponding eigenvector of the covariance matrix of the observed random process recently thompson 9 suggested a constrained gradient search procedure for obtaining an adaptive version of pisarenko s method and his simulations have verified that the frequency estimates provided by his procedure were unbiased however the main cost of this technique was that the initial convergence rate could be very slow for certain poor initial conditions restating the constrained minimization as an unconstrained nonlinear problem we derived an alternative gauss newton type recursive algorithm which also used the second derivative matrix or hessian this algorithm may also be viewed as an approximate least squares algorithm simulations have been performed to compare this algorithm to a slight variant of thompson s original algorithm the most important conclusions are that the least squares type algorithm has faster convergence in the beginning while its convergence rate close to the true parameters depends on the signal to noise ratio of the input signal the approximate least squares algorithm resolves the sinusoids much faster than the gradient version

explicitly solvable non linear optimal control problems the method of collective hamiltonians is applied to the study of optimal control problems on manifolds this approach enables us to single out exactly solvable situations various examples are considered necessary results from hamiltonian formalism are given including an industrial technique for the construction of completely integrable systems

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

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

a unified self stabilizing neural network algorithm for principal and minor components extraction recently many unified learning algorithms have been developed for principal component analysis and minor component analysis these unified algorithms can be used to extract principal components and if altered simply by the sign can also serve as a minor component extractor this is of practical significance in the implementations of algorithms this paper proposes a unified self stabilizing neural network learning algorithm for principal and minor components extraction and studies the stability of the proposed unified algorithm via the fixed point analysis method the proposed unified self stabilizing algorithm for principal and minor components extraction is extended for tracking the principal subspace ps and minor subspace ms the averaging differential equation and the energy function associated with the unified algorithm for tracking ps and ms are given it is shown that the averaging differential equation will globally asymptotically converge to an invariance set and the corresponding energy function exhibit a unique global minimum attained if and only if its state matrices span the ps or ms of the autocorrelation matrix of a vector data stream it is concluded that the proposed unified algorithm for tracking ps and ms can efficiently track an orthonormal basis of the ps or ms simulations are carried out to further illustrate the theoretical results achieved

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

global convergence of oja s subspace algorithm for principal component extraction oja s principal subspace algorithm is a well known and powerful technique for learning and tracking principal information in time series a thorough investigation of the convergence property of oja s algorithm is undertaken in this paper the asymptotic convergence rates of the algorithm is discovered the dependence of the algorithm on its initial weight matrix and the singularity of the data covariance matrix is comprehensively addressed

lagrangian submanifolds and dynamics on lie algebroids in some previous papers a geometric description of lagrangian mechanics on lie algebroids has been developed in this topical review we give a hamiltonian description of mechanics on lie algebroids in addition we introduce the notion of a lagrangian submanifold of a symplectic lie algebroid and we prove that the lagrangian hamiltonian dynamics on lie algebroids may be described in terms of lagrangian submanifolds of symplectic lie algebroids the lagrangian hamiltonian formalism on lie algebroids permits us to deal with lagrangian hamiltonian functions not defined necessarily on tangent cotangent bundles thus we may apply our results to the projection of lagrangian hamiltonian functions which are invariant under the action of a symmetry lie group as a consequence we obtain that lagrange poincare hamilton poincare equations are the euler lagrange hamilton equations associated with the corresponding atiyah algebroid moreover we prove that lagrange poincare hamilton poincare equations are the local equations defining certain lagrangian submanifolds of symplectic atiyah algebroids

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

oscillating about coplanarity in the 4 body problem for the newtonian 4 body problem in space we prove that any zero angular momentum bounded solution suffers infinitely many coplanar instants that is times at which all 4 bodies lie in the same plane this result generalizes a known result for collinear instants syzygies in the zero angular momentum planar 3 body problem and extends to the d 1 body problem in d space the proof begins by identifying the translation reduced configuration space with real d times d matrices the degeneration locus set of coplanar configurations when d 3 with the set of matrices having determinant zero and the mass metric with the frobenius standard euclidean norm let s denote the signed distance from a matrix to the hypersurface of matrices with determinant zero the proof hinges on establishing a harmonic oscillator type ode for s along solutions bounds on inter body distances then yield an explicit lower bound omega for the frequency of this oscillator guaranteeing a degeneration within every time interval of length pi omega the non negativity of the curvature of oriented shape space the quotient of the translation reduced configuration space by the rotation group plays a crucial role in the proof

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

a generalized information criterion for generalized minor component extraction generalized minor component analysis gmca is an essential technique in data classification and signal processing in this paper we propose an information criterion for gmca and derive a fast gmca algorithm for extracting the first generalized minor component gmc by using quasi newton method to this information criterion in order to extract multiple gmcs through the weighed matrix method this information criterion is extended into a weighted one which has a unique global maximum attained if and only if its state matrices converge to the gmcs of the matrix pencil composed of the autocorrelation matrices of two stochastic processes a gradient algorithm is also derived based on this weighted information criterion theoretical analysis shows that the gradient algorithm has self stabilizing property and does not need the normalization operation required in other algorithms the global convergence analysis of the proposed algorithm is accomplished through the lyapunov method numerical simulations and real application are carried out to further demonstrate the accuracy and speed advantages of the proposed algorithms

the mechanics and control for multi particle systems this paper deals with the mechanics and control for multi particle systems from a geometric point of view the centre of mass system is viewed as a principal fibre bundle with structure group so 3 the base space of which is called the internal or shape space a natural connection and a natural riemannian metric are both defined on the centre of mass system the equations of motion for the multi particle system are derived in the lagrangian formalism adapted to the bundle structure and then reduced with the conserved total angular momentum in contrast with this the control problem is studied with non holonomic constraints i e with the vanishing total angular momentum and equations of motion are determined for an optimally controlled multi particle system the resultant equations derived in each of the mechanical and control systems are to be compared

symmetry reduction of optimal control systems and principal connections this paper explores the role of symmetries and reduction in nonlinear control and optimal control systems the focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry in particular we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems affine and kinematic optimal control systems are of particular interest we explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems

performance analysis of adaptive eigenanalysis algorithms we present a rigorous analysis of several popular forms of short memory adaptive eigenanalysis algorithms using a stochastic averaging method a first order analysis shows that the algorithms do not have any equilibrium points despite published claims to the contrary through averaging analysis we show that they hover around an appropriate eigenvector a second order analysis is also given without the gaussian noise assumption and our results greatly outperform an earlier approximation in the literature the second order analysis has been of much interest in the offline study but in the dynamic adaptive case is uncommon

rotational search methods for adaptive pisarenko harmonic retrieval two adaptation algorithms for adaptive pisarenko harmonic retrieval are described they are derived by considering the associated minimum eigenvalue problem as an optimization problem which seeks the minimum of a quadratic cost function given a hyperspherical constraint an iterative search procedure is used in which each search path is constrained to lie on the unit hypersphere computational complexity per iteration is approximately one third that of previous adaptive phr algorithms simulations reveal that at low snr the trial eigenvector can converge to the true minimum eigenvector of the sample covariance matrix long before this matrix is a good estimate of the true covariance matrix

adaptive multiple minor directions extraction in parallel using a pca neural network a principal component analysis pca neural network is developed for online extraction of the multiple minor directions of an input signal the neural network can extract the multiple minor directions in parallel by computing the principal directions of the transformed input signal so that the stability speed problem of directly computing the minor directions can be avoided to a certain extent on the other hand the learning algorithms for updating the net weights use constant learning rates this overcomes the shortcoming of the learning rates approaching zero in addition the proposed algorithms are globally convergent so that it is very simple to choose the initial values of the learning parameters this paper presents the convergence analysis of the proposed algorithms by studying the corresponding deterministic discrete time ddt equations rigorous mathematical proof is given to prove the global convergence the theoretical results are further confirmed via simulations

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

adaptive minor component extraction with modular structure an information criterion for adaptively estimating multiple minor eigencomponents of a covariance matrix is proposed it is proved that the proposed criterion has a unique global minimum at the minor subspace and that all other equilibrium points are saddle points based on the gradient search approach of the proposed information criterion an adaptive algorithm called adaptive minor component extraction amex is developed the proposed algorithm automatically performs the multiple minor component extraction in parallel without the inflation procedure similar to the adaptive lattice filter structure the amex algorithm also has the flexibility wherein increasing the number of the desired minor component does not affect the previously extracted minor components the amex algorithm has a highly modular structure and the various modules operate completely in parallel without any delay simulation results are given to demonstrate the effectiveness of the amex algorithm for both the minor component analysis mca and the minor subspace analysis msa

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

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

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

holonomy reduced dynamics of triatomic molecules whereas it is easy to reduce the translational symmetry of a molecular system using e g jacobi coordinates the situation is much more involved for rotational symmetry in this paper we address the latter problem using holonomy reduction we suggest that the configuration space may be considered as the reduced holonomy bundle with a connection induced by the mechanical connection using the fact that for the special case of the three body problem the holonomy group is so 2 as opposed to so 3 like in systems with more than three bodies we obtain a holonomy reduced configuration space of topology the dynamics then takes place on the cotangent bundle over the holonomy reduced configuration space on this phase space there is an s1 symmetry action coming from the conserved reduced angular momentum which can be reduced using the standard symplectic reduction method using a theorem by arnold it follows that the resulting symmetry reduced phase space is again a natural mechanical phase space i e a cotangent bundle this is different from what is obtained from the usual approach where symplectic reduction is used from the outset this difference is discussed in some detail and a connection between the reduced dynamics of a triatomic molecule and the motion of a charged particle in a magnetic field is established

fast subspace tracking and neural network learning by a novel information criterion we introduce a novel information criterion nic for searching for the optimum weights of a two layer linear neural network nn the nic exhibits a single global maximum attained if and only if the weights span the desired principal subspace of a covariance matrix the other stationary points of the nic are unstable saddle points we develop an adaptive algorithm based on the nic for estimating and tracking the principal subspace of a vector sequence the nic algorithm provides a fast on line learning of the optimum weights for the two layer linear nn we establish the connections between the nic algorithm and the conventional mean square error mse based algorithms such as oja s algorithm oja 1989 lmser past apex and gha the nic algorithm has several key advantages such as faster convergence which is illustrated through analysis and simulation