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 branch and cut decomposition algorithm for solving chance constrained mathematical programs with finite support we present a new approach for exactly solving chance constrained mathematical programs having discrete distributions with finite support and random polyhedral constraints such problems have been notoriously difficult to solve due to nonconvexity of the feasible region and most available methods are only able to find provably good solutions in certain very special cases our approach uses both decomposition to enable processing subproblems corresponding to one possible outcome at a time and integer programming techniques to combine the results of these subproblems to yield strong valid inequalities computational results on a chance constrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixed integer programming formulation and a simple decomposition approach that does not use strong valid inequalities we also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels as would be done when solving for the efficient frontier of risk and cost

a new concept of geometric phase in parameter space coupling as a parameter we have obtained the geometric phase of the tripartite spin 1 2 system with spin spin coupling and discussed the influence of this coupling on the geometric phase we have discovered the coupling independent geometric phase and explained it based on this work we propose the concept of quasimagnetic field and assimilate the coupling into the parameter space by this theory the relation between the geometric phase and the solid angle in parameter space still holds for a system with spin spin coupling if the concept of the parameter is properly revised in the way we have elucidated

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

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

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

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

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

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

a comparative study of the hannay s angles associated with a damped harmonic oscillator and a generalized harmonic oscillator in this paper we show that a damped harmonic oscillator the simplest dissipative physical system is canonically equivalent to a generalized harmonic oscillator a conservative system even for time dependent parameters as a consequence the hannay s angles and the adiabatic invariants of the two systems appear to be the same modulo this equivalence this raises the question of whether this analogy can be extended to other nonlinear dissipative systems

quantum response at finite fields and breakdown of chern numbers we show that the response to an electric field in models of the integral quantum hall effect is analytic in the field and has isolated essential singularity at zero field we also study the breakdown of chern numbers associated with the response of floquet states we argue and give evidence that the breakdown of chern numbers in floquet states is a discontinuous transition at zero field this follows from an observation of independent interest that chern numbers for finite dimensional floquet operators are generically zero these results rule out the possibility that the breakdown of the hall conductance is a phase transition at finite fields for a large class of models

regularization methods for optimization problems with probabilistic constraints we analyze nonlinear stochastic optimization problems with probabilistic constraints on nonlinear inequalities with random right hand sides we develop two numerical methods with regularization for their numerical solution the methods are based on first order optimality conditions and successive inner approximations of the feasible set by progressive generation of p efficient points the algorithms yield an optimal solution for problems involving concave probability distributions for arbitrary distributions the algorithms solve the convex hull problem and provide upper and lower bounds for the optimal value and nearly optimal solutions the methods are compared numerically to two cutting plane methods

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

wave functions for a duffin kemmer petiau particle in a time dependent potential we present the solution of the duffin kemmer petiau equation of spin 0 in a time dependent linear scalar field the treatment is based on the use of the lewis jr riensfield invariant method

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

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

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

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

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

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

single commodity stochastic network design under probabilistic constraint with discrete random variables single commodity networks are considered where demands at the nodes are random the problem is to find minimum cost optimal built in capacities at the nodes and arcs subject to the constraint that all demands should be met on a prescribed probability level reliability constraint and some deterministic constraints should be satisfied the reliability constraint is formulated in terms of the gale hoffman feasibility inequalities but their number is reduced by elimination technique the concept of a p efficient point is used in a smart way to convert and then relax the problem into an lp the p efficient points are simultaneously generated with the solution of the lp the joint distribution of the demands is used to obtain the p efficient points for all stochastic inequalities that were not eliminated and the solution of a multiple choice knapsack problem is used to generate p efficient points the model can be applied to planning in interconnected power systems flood control networks design of shelter

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

large deviations for the one dimensional edwards model in this article we prove a large deviation principle for the empirical drift of a one dimensional brownian motion with self repellence called the edwards model our results extend earlier work in which a law of large numbers and a central limit theorem were derived in the edwards model a path of length t receives a penalty e ht w hereht is the selfintersection local time of the path and 0 is a parameter called the strength of self repellence we identify the rate function in the large deviation principle for the endpoint of the path as 2 3 i 1 3 with i given in terms of the principal eigenvalues of a one parameter family of sturm liouville operators we show that there exist numbers 0 b b such that 1 i is linearly decreasing on 0 b 2 i is real analytic and strictly convex on b 3 i is continuously differentiable at b and 4 i has a unique zero at b the latter fact identifies b as the asymptotic drift of the endpoint the critical drift b is associated with a crossover in the optimal strategy of the path for b b the path assumes local drift b during the full time t while for 0 b b it assumes local drift b during time b b 2b t and local drift b during the remaining time b b 2b t thus in the second regime the path makes an overshoot of size b b 2 t so as to reduce its intersection local time

concavity and efficient points of discrete distributions in probabilistic programming we consider stochastic programming problems with probabilistic constraints involving integer valued random variables the concept of a p efficient point of a probability distribution is used to derive various equivalent problem formulations next we introduce the concept of r concave discrete probability distributions and analyse its relevance for problems under consideration these notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables the results are illustrated with numerical examples

convergence analysis for mathematical programs with distributionally robust chance constraint convergence analysis for optimization problems with chance constraints concerns impact of variation of probability measure in the chance constraints on the optimal value and the optimal solutions and research on this topic has been well documented in the literature of stochastic programming in this paper we extend such analysis to optimization problems with distributionally robust chance constraints where the true probability distribution is unknown but it is possible to construct an ambiguity set of probability distributions and the chance constraint is based on the most conservative selection of probability distribution from the ambiguity set the convergence analysis focuses on impact of the variation of the ambiguity set on the optimal value and the optimal solutions we start by deriving general convergence results under abstract conditions such as continuity of the robust probability function and uniform convergence of the robust probability functions and followed with detailed analysis of these

central limit theorem for the edwards model the edwards model in one dimension is a transformed path measure for standard brownian motion discouraging self intersections we prove a central limit theorem for the endpoint of the path extending a law of large numbers proved by westwater the scaled variance is characterized in terms of the largest eigenvalue of a one parameter family of differential operators introduced and analyzed by van der hofstad and den hollander interestingly the scaled variance turns out to be independent of the strength of self repellence and to be strictly smaller than one the value for free brownian motion

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

large deviations and exponential decay for the magnetization in a gaussian random field we consider a continuous model for transverse magnetization of spins diffusing in a homogeneous gaussian random longitudinal field lambda v x x in bbb r d where lambda is the coupling constant giving the intensity of the random field in this setting the transverse magnetization is given by the formula m t bbb e exp lambda 2 int 0 t int 0 t k b r b s ds dr where b t t ge 0 is the standard process of brownian motion and k x is the covariance function of the original random field v x we use large deviation techniques to show that the limit s lambda lim t rightarrow infty frac 1 t ln m t exists we also determine the small lambda behavior of the rate s lambda and show that it is indeed decaying as conjectured in the physics literature

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

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

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

solutions of a toda equations based on noncompact group su 1 1 and infinite dimensional grassmann manifolds special solutions of a toda equations are constructed based on noncompact group su 1 1 these solutions are derived from the holomorphic construction of infinite dimensional nonlinear grassmann models it is also shown that the solutions give a family of solutions of self dual einstein equations in a certain continuum limit

the constants in the central limit theorem for the one dimensional edwards model the edwards model in one dimension is a transformed path measure for one dimensional brownian motion discouraging self intersections we study the constants appearing in the central limit theorem clt for the endpoint of the path which represent the mean and the variance and the exponential rate of the normalizing constant the same constants appear in the weak interaction limit of the one dimensional domb joyce model the domb joyce model is the discrete analogue of the edwards model based on simple random walk where each self intersection of the random walk path recieves a penalty e 2 we prove that the variance is strictly smaller than 1 which shows that the weak interaction limits of the variances in both clts are singular the proofs are based on bounds for the eigenvalues of a certain one parameter family of sturm liouville differential operators obtained by using monotonicity of the zeros of the eigen functions in combination with computer plots

moyal products a new perspective on quasi hermitian quantum mechanics the rationale for introducing non hermitian hamiltonians and other observables is reviewed and open issues identified we present a new approach based on moyal products to compute the metric for quasi hermitian systems this approach is not only an efficient method of computation but also suggests a new perspective on quasi hermitian quantum mechanics which invites further exploration in particular we present some first results which link the berry connection and curvature to non perturbative properties and the metric

weak interaction limits for one dimensional random polymers in this paper we present a new and flexible method to show that in one dimension various self repellent random walks converge to self repellent brownian motion in the limit of weak interaction after appropriate space time scaling our method is based on cutting the path into pieces of an appropriately scaled length controlling the interaction between the different pieces and applying an invariance principle to the single pieces in this way we show that the self repellent random walk large deviation rate function for the empirical drift of the path converges to the self repellent brownian motion large deviation rate function after appropriate scaling with the interaction parameters the method is considerably simpler than the approach followed in our earlier work which was based on functional analytic arguments applied to variational representations and only worked in a very limited number of situations

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

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

nonadiabatic holonomy operators in classical and quantum completely integrable systems given a completely integrable system we associate to any connection on a fiber bundle in invariant tori over a parameter manifold the classical and quantum holonomy operator generalized berry s phase factor without any adiabatic approximation

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

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

an integer programming approach for linear programs with probabilistic constraints linear programs with joint probabilistic constraints pclp are difficult to solve because the feasible region is not convex we consider a special case of pclp in which only the right hand side is random and this random vector has a finite distribution we give a mixed integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint we obtain two strengthened formulations as a byproduct of this analysis we obtain new results for the previously studied mixing set subject to an additional knapsack inequality we present computational results which indicate that by using our strengthened formulations instances that are considerably larger than have been considered before can be solved to optimality

scaling for a random polymer letq n be the law of then step random walk on d obtained by weighting simple random walk with a factore for every self intersection domb joyce model of soft polymers it was proved by greven and den hollander 1993 that ind 1 and for every 0 there exist 0 1 and such that under the lawq n asn begin array l i theta beta is the lim it empirical speed of the random walk ii mu beta is the limit empirical distribution of the local times end array a representation was given for and in terms of a largest eigenvalue problem for a certain family of x matrices in the present paper we use this representation to prove the following scaling result as 0 begin array l i beta textstyle 1 over 3 theta beta to b ii beta textstyle 1 over 3 mu beta left left lceil cdot beta textstyle 1 over 3 right rceil right to l 1 eta cdot end array the limitsb 0 and are identified in terms of a sturm liouville problem which turns out to have several interesting properties the techniques that are used in the proof are functional analytic and revolve around the notion of epi convergence of functionals onl 2 our scaling result shows that the speed of soft polymers ind 1 is not right differentiable at 0 which precludes expansion techniques that have been used successfully ind 5 hara and slade 1992a b in simulations the scaling limit is seen for 10 2

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

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

eventual convexity of probability constraints with elliptical distributions probability constraints are often employed to intuitively define safety of given decisions in optimization problems they simply express that a given system of inequalities depending on a decision vector and a random vector is satisfied with high enough probability it is known that even if this system is convex in the decision vector the associated probability constraint is not convex in general in this paper we show that some degree of convexity is still preserved for the large class of elliptical random vectors encompassing for example gaussian or student random vectors more precisely our main result establishes that under mild assumptions eventual convexity holds i e the probability constraint is convex when the safety level is large enough we also provide tools to compute a concrete convexity certificate from nominal problem data our results are illustrated on several examples including the situation of polyhedral systems with random technology matrices and arbitrary covariance structure

spinors and scalars on riemann surfaces we study magnetic dirac and schrodinger operators for general two dimensional manifolds we find relations between them and use these as tools for calculating ground state degeneracies of various electronic systems with and without spin we find examples which have a degenerate ground state although there are no symmetry operators commuting with the schrodinger operator the connection between the two operators is related to a berry phase

on edwards model for polymer chains ii the self consistent potential we obtain an existence and uniqueness theorem for the self consistent potential in edwards model for polymer chains and confirm the asymptotic analysis proposed by him on the basis of wkb arguments

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

product integral formalism and non abelian stokes theorem we make use of the properties of product integrals to obtain a surface product integral representation for the wilson loop operator the result can be interpreted as the non abelian version of stokes theorem

nonanticipative duality relaxations and formulations for chance constrained stochastic programs we propose two new lagrangian dual problems for chance constrained stochastic programs based on relaxing nonanticipativity constraints we compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed integer programming mip formulation for a given dual solution the associated lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem we also derive two new primal mip formulations and demonstrate that for chance constrained linear programs the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds we propose a new heuristic method and two new exact algorithms based on these duals and formulations the first exact algorithm applies to chance constrained binary programs and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems the second exact method is a branch and cut algorithm for solving either of the primal formulations our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently and the gaps between the best dual bounds and the heuristic solutions are small

berry s transport and minimization of averaged distances in this paper we consider the metric approach to the berry s transport geometrical part of the adiabatic evolution of any pure or mixed state for possibly degenerate hamiltonians we emphasize that explicit formulae for the transport of such states need the introduction of an averaging procedure in analogy with the classical case this quantum averaging involves multiplying each energy eigenstate by a different phase and then integrating over these phases we show in particular how the transport of non stationary pure states rays and density matrices arise from the minimization of hilbert fubini study and bures averaged distances respectively

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

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

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

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

mathematical programming approaches for generating p efficient points abstract probabilistically constrained problems in which the random variables are finitely distributed are non convex in general and hard to solve the p efficiency concept has been widely used to develop efficient methods to solve such problems those methods require the generation of p efficient points pleps and use an enumeration scheme to identify pleps in this paper we consider a random vector characterized by a finite set of scenarios and generate pleps by solving a mixed integer programming mip problem we solve this computationally challenging mip problem with a new mathematical programming framework it involves solving a series of increasingly tighter outer approximations and employs as algorithmic techniques a bundle preprocessing method strengthening valid inequalities and a fixing strategy the method is exact resp heuristic and ensures the generation of pleps resp quasi pleps if the fixing strategy is not resp is employed and it can be used to generate multiple pleps to the best of our knowledge generating a set of pleps using an optimization based approach and developing effective methods for the application of the p efficiency concept to the random variables described by a finite set of scenarios are novel we present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques

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

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

universal integrals of motion and universal invariants of quantum systems universal quantum integrals of motion are introduced and their relation with the universal quantum invariants is established the invariants concerned are certain combinations of the second and higher order moments variances of quantum mechanical operators which are preserved in time independently of the concrete form of the coefficients of the schr dinger equation provided the hamiltonian is either a generic quadratic form of the coordinate and momenta operators or a linear combination of generators of some finite dimensional algebra in particular any semisimple lie algebra using the phase space representation of quantum mechanics in terms of the wigner function the relations between the quantum invariants and the classical universal integral invariants by poincar and cartan are elucidated examples of the universal invariant solutions of the schr dinger equation i e self consistent eigenstates of the universal integrals of motion are given applications to the physics of optical and particle beams 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

gradient formulae for nonlinear probabilistic constraints with gaussian and gaussian like distributions probabilistic constraints represent a major model of stochastic optimization a possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods to do so one has to provide sufficiently precise approximations for values and gradients of probability functions for linear probabilistic constraints under gaussian distribution this can be done successfully by analytically reducing these values and gradients to values of gaussian distribution functions and computing the latter for instance by genz s code for nonlinear models one may fall back on the spherical radial decomposition of gaussian random vectors and apply for instance deak s sampling scheme for the uniform distribution on the sphere in order to compute values of corresponding probability functions the present paper demonstrates how the same sampling scheme can be used to simultaneously compute gradients of these probability functions more precisely we prove a formula repre

geometric quantization of time dependent completely integrable hamiltonian systems a time dependent completely integrable hamiltonian system is quantized with respect to time dependent action angle variables near an instantly compact regular invariant manifold its hamiltonian depends only on action variables and has a time independent countable energy spectrum

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

a central limit theorem for a one dimensional polymer measure let s n n n 0 be a random walk on the integers having bounded steps the self repellent resp self avoiding walk is a sequence of transformed path measures which discourage resp forbid self intersections this is used as a model for polymers previously we proved a law of large numbers that is we showed the convergence of s n n toward a positive number under the polymer measure the present paper proves a classical central limit theorem for the self repellent and self avoiding walks that is we prove the asymptotic normality of s n n n for large n the proof refines and continues results and techniques developed previously

geometric quantization of mechanical systems with time dependent parameters quantum systems with adiabatic classical parameters are widely studied e g in the modern holonomic quantum computation we here provide complete geometric quantization of a hamiltonian system with time dependent parameters without the adiabatic assumption a hamiltonian of such a system is affine in the temporal derivative of parameter functions this leads to the geometric berry factor phenomena

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

a survey of one dimensional random polymers in the last decade there has been an enormous progress in the mathematical understanding of one dimensional polymer measures which are path measures that suppress self intersections we are currently in the situation that many interesting questions have either been answered or that essential new ideas are needed in this survey paper we discuss the most relevant results open questions and heuristics

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

unitary transformation approach to the exact solution for a class of time dependent nonlinear hamiltonian systems by performing unitary transformations on the time dependent schrodinger equation the exact solution for a class of nonlinear hamiltonian systems is obtained and it is shown that these time dependent problems are related to the associated time independent problems in addition the evolution operator is derived nonadiabatic berry s phase is calculated on the basis of the exact solution the theory is applied to some illustrative examples

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

self avoiding walk in five or more dimensions i the critical behaviour we use the lace expansion to study the standard self avoiding walk in thed dimensional hypercubic lattice ford 5 we prove that the numbercn ofn step self avoiding walks satisfiescn a n where is the connective constant i e 1 and that the mean square displacement is asymptotically linear in the number of steps i e v 1 2 a bound is obtained forcn x the number ofn step self avoiding walks ending atx the correlation length is shown to diverge asymptotically like z 1 2 the critical two point function is shown to decay at least as fast as x 2 and its fourier transform is shown to be asymptotic to a multiple ofk 2 ask 0 i e 0 we also prove that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion the infinite self avoiding walk is constructed in this paper we prove these results assuming convergence of the lace expansion the convergence of the lace expansion is proved in a companion paper