correspondence between berry s phase and lewis s phase for quadratic hamiltonians the authors show the existence of a relation between berry s phase and lewis s phase for the case of quadratic hamiltonians

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

geometrical phase in the cyclic evolution of non hermitian systems the authors derive by a biorthonormal state approach the analogy of berry s phase factor for open non conservative systems for both adiabatic and non adiabatic evolution in the latter case a non unitary evolution operator method is exploited an application is given to the optical supermode propagation in the free electron laser

the appearance of a geometric type instability in dynamic systems with adiabatically varying parameters dynamic systems which are described by homogeneous ordinary differential equations with slowly variable coefficients are considered the eigenvalues of these equations correspond to stable states of the system the exponential growth that has been discovered within the wkb approximation corresponds to a new type of instability it results in a physically reversible continuous energy transfer to the system due to adiabatic variation of two or more parameters a general theory is presented and simple mathematical and physical examples are given a particular case of the effect where the amplitude of the solutions does not change is the increase of the so called geometric berry hannay phase

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

berry s phase and the planar three body problem berry s angular 2 form in seen as a correction to the symplectic structure in a separation of variables type scheme where the variables are canonically non separable this view is applied to the planar three body problem where the rotational and vibrational motions are not separable it is shown that the corrected symplectic structure gives the correct quantisation

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

a 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

hannay berry phase and the restricted three vortex problem abstract asymptotic expansions for the hannay berry phase are computed for a class of hamiltonian systems in perturbed action angle form with slowly varying parameters where the parameters recur on a slow time scale a multi scale perturbation method is used which in principle can be carried to any desired order of accuracy as a specific application we use the method to examine the restricted three vortex problem a completely integrable configuration of two co rotating point vortices a distance d apart and a tracer particle rotating around one of the vortices with distance r in the asymptotic limit r d e we compute the particle trajectory through o e and find that the hannay berry phase is given by 2 3 r 0 d o 1 d 2 where r 0 is the conserved action

the degenerate parametric oscillator and ince s equation we construct green s function for the quantum degenerate parametric oscillator in the coordinate representation in terms of standard solutions of ince s equation in a framework of a general approach to variable quadratic hamiltonians exact time dependent wavefunctions and their connections with dynamical invariants and su 1 1 group are also discussed an extension to the degenerate parametric oscillator with time dependent amplitude and phase is also mentioned

on the inefficiency of the adaptive design for monitoring clinical trials summary adaptive designs which allow the sample size to be modified based on sequentially computed observed treatment differences have been advocated recently for monitoring clinical trials although such methods have a great deal of appeal on the surface we show that such methods are inefficient and that one can improve uniformly on such adaptive designs using standard group sequential tests based on the sequentially computed likelihood ratio test statistic

coherent states for the quantum mechanics on a compact manifold we review our recent observations concerning quantum mechanics on such compact manifolds as a circle torus and sphere

the bargmann transform and canonical transformations this paper concerns a relationship between the kernel of the bargmann transform and the corresponding canonical transformation we study this fact for a bargmann transform introduced by thomas and wassell j math phys 36 5480 5505 1995 when the configuration space is the two sphere s2 and for a bargmann transform that we introduce for the three sphere s3 it is shown that the kernel of the bargmann transform is a power series in a function which is a generating function of the corresponding canonical transformation a classical analog of the bargmann transform we show in each case that our canonical transformation is a composition of two other canonical transformations involving the complex null quadric in c3 or c4 we also describe quantizations of those two other canonical transformations by dealing with spaces of holomorphic functions on the aforementioned null quadrics some of these quantizations have been studied by bargmann and todorov j math phys 18 1141 1148 1977 and the other qu

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

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

non abelian geometric phase for general three dimensional quantum systems adiabatic u 2 geometric phases are studied for arbitrary quantum systems with a three dimensional hilbert space necessary and sufficient conditions for the occurrence of the non abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous hamiltonian the parameter space of such systems which has the structure of is explicitly constructed the results of this article are applicable for arbitrary multipole interaction hamiltonians and their linear combinations for spin j 1 systems in particular it is shown that the nuclear quadrupole hamiltonian does actually lead to non abelian geometric phases for j 1 this system being bosonic is time reversal invariant therefore it cannot support abelian adiabatic geometrical phases

berry s phase and wavefunctions for time dependent hamilton systems an alternative derivation of berry s phase 1985 to that given by morales 1988 is given for non autonomous hamiltonian systems which admit an energy like first integral the generality of the problems treated is greater

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

berry s phase and hannay s angle from quantum canonical transformations using an action angle coherent states formalism introduced by the authors in a preceding paper it is shown that berry s phase and hannay s angle can both be derived from the same quantum unitary transformation their relationship is easily established in this framework

a factorization property of the berry phase it is shown that for systems with hamiltonians expressible in terms of the generators of a lie group the berry phase appears in the form of a product of a representation dependent term and a purely geometrical factor

fuzzy spheres from inequivalent coherent states quantizations the existence of a family of coherent states cs solving the identity in a hilbert space allows under certain conditions to quantize functions defined on the measure space of cs parameters the application of this procedure to the 2 sphere provides a family of inequivalent cs quantizations based on the spin spherical harmonics the cs quantization from usual spherical harmonics appears to give a trivial issue for the cartesian coordinates we compare these cs quantizations to the usual madore construction of the fuzzy sphere due to these differences our procedure yields new types of fuzzy spheres moreover the general applicability of cs quantization suggests similar constructions of fuzzy versions of a large variety of sets

berry phase in arbitrary dimensions we study the properties of the berry phase in quantum systems where momentum and position operators satisfy the r deformed heisenberg relations and the noninteger dimension d and angular momentum appear as parameters we focus on the dynamics of the generalized time dependent harmonic oscillator which satisfies the schr dinger wave mechanics formulated in arbitrary dimensions we also examine quantum systems which can be expressed as linear combination of generators with time dependent coefficients in arbitrary dimensions we show the explicit dependence of the berry phase on d and the parity of quantum systems which satisfy the r deformed heisenberg relations using an explicit relation of the fubini study metric in fractional dimensions we highlight a geometrical interpretation of d and of systems which are quantized in arbitrary dimensions

increasing the sample size when the unblinded interim result is promising increasing the sample size based on unblinded interim result may inflate the type i error rate and appropriate statistical adjustments may be needed to control the type i error rate at the nominal level we briefly review the existing approaches which allow early stopping due to futility or change the test statistic by using different weights or adjust the critical value for final test or enforce rules for sample size recalculation the implication of early stopping due to futility and a simple modification to the weighted z statistic approach are discussed in this paper we show that increasing the sample size when the unblinded interim result is promising will not inflate the type i error rate and therefore no statistical adjustment is necessary the unblinded interim result is considered promising if the conditional power is greater than 50 per cent or equivalently the sample size increment needed to achieve a desired power does not exceed an upper bound the actual sample size increment may be determined by important factors such as budget size of the eligible patient population and competition in the market the 50 per cent conditional power approach is extended to a group sequential trial with one interim analysis where a decision may be made at the interim analysis to stop the trial early due to a convincing treatment benefit or to increase the sample size if the interim result is not as good as expected the type i error rate will not be inflated if the sample size may be increased only when the conditional power is greater than 50 per cent if there are two or more interim analyses in a group sequential trial our simulation study shows that the type i error rate is also well controlled

invariance of the generalised berry phase under unitary transformations application to the time dependent generalised harmonic oscillator a generalisation of the berry phase which reduces to the usual berry phase in the adiabatic and cyclic case is derived in a gauge invariant way this generalised berry phase unlike the berry phase is invariant under unitary transformations a time dependent generalised harmonic oscillator with a cross term in the hamiltonian is taken as an example the hamiltonian of this system is not in general the energy an energy the time derivative of which is the power is obtained from the equation of motion when the system is quantised the generalised berry phase is zero and remains invariant under unitary transformations on the other hand if the energy is chosen to be the hamiltonian a non zero generalised berry phase is obtained but the power is given incorrectly in this case the total phase which includes the dynamical phase and the generalised berry phase is equal in the adiabatic limit to the correct total phase

adaptive and nonadaptive group sequential tests summary methods have been proposed for redesigning a clinical trial at an interim stage in order to increase power in order to preserve the type i error rate methods for unplanned design change have to be defined in terms of nonsufficient statistics and this calls into question their efficiency and the credibility of conclusions reached we evaluate schemes for adaptive redesign extending the theoretical arguments for use of sufficient statistics of tsiatis mehta 2003 and assessing the possible benefits of preplanned adaptive designs by numerical computation of optimal tests these optimal adaptive designs are concrete examples of optimal sequentially planned sequential tests proposed by schmitz 1993 we conclude that the flexibility of unplanned adaptive designs comes at a price and we recommend that the appropriate power for a study should be determined as thoroughly as possible at the outset then standard error spending tests possibly with unevenly spaced analyses provide efficient designs but it is still possible to fall back on flexible methods for redesign should study objective change unexpectedly once the trial is under way

on geometric phases and dynamical invariants a formalism is developed for calculating berry phases for nonadiabatic time periodic quantum systems when a dynamical invariant is known it is found that when the invariant is periodic and has a nondegenerate spectrum this method allows a convenient way to obtain generalized berry phases and the proper cyclic initial states the method is applied to the generalized harmonic oscillator and the two level system where the invariant operators are explicitly constructed formulae for the conventional berry s phases are readily obtained by taking the adiabatic limit of the exact results

analysis of berry s phase by the evolution operator method berry 1984 discovered that an eigenstate undergoing an adiabatic evolution in the parameter space will acquire a topological phase aharonov and anandan 1987 on the other hand showed that an eigenstate transporting round a closed circuit in the projective hilbert space suffices to generate the topological phase the authors employ the evolution operator method to study the propagation of an eigenstate they show that berry s phase can be represented as a closed path integral in the hilbert space and is independent of the choice for the base mod n t the treatments of berry and aharonov and anandan are shown to correspond to two different choices of the base therefore their two approaches are unified they have acquired a more general viewpoint on the origin of berry s phase

semiclassical sum rules and generalized coherent states various semiclassical sum rules are obtained for matrix elements of smooth observables using semiclassical estimates of time evolved coherent states together with stationary phase theorems

can one measure hannay angles it has been overlooked that with the exception of especially favourable circumstances the answer seems to be negative for single frequency systems however an averaging procedure is shown to allow for an experimental investigation of the hannay angles

adaptive group sequential designs for clinical trials combining the advantages of adaptive and of classical group sequential approaches a general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing this allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs the concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans

evaluation of experiments with adaptive interim analyses summary a general method for statistical testing in experiments with an adaptive interim analysis is proposed the method is based on the observed error probabilities from the disjoint subsamples before and after the interim analysis formally an intersection of individual null hypotheses is tested by combining the twop values into a global test statistic stopping rules for fisher s product criterion in terms of critical limits for the p value in the first subsample are introduced including early stopping in the case of missing effects the control of qualitative treatment stage interactions is considered a generalization to three stages is outlined the loss of power when using the product criterion instead of the optimal classical test on the whole sample is calculated for the test of the mean of a normal distribution depending on increasing proportions of the first subsample in relation to the total sample size an upper bound on the loss of power due to early stopping is derived a general example is presented and rules for assessing the sample size in the second stage of the trial are given the problems of interpretation and precautions to be taken for applications are discussed finally the sources of bias for estimation in such designs are described

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

group sequential tests for delayed responses with discussion summary group sequential methods are used routinely to monitor clinical trials and to provide early stopping when there is evidence of a treatment effect a lack of an effect or concerns about patient safety in many studies the response of clinical interest is measured some time after the start of treatment and there are subjects at each interim analysis who have been treated but are yet to respond we formulate a new form of group sequential test which gives a proper treatment of these pipeline subjects these tests can be applied even when the continued accrual of data after the decision to stop the trial is unexpected we illustrate our methods through a series of examples we define error spending versions of these new designs which handle unpredictable group sizes and provide an information monitoring framework that can accommodate nuisance parameters such as an unknown response variance by studying optimal versions of our new designs we show how the benefits of lower expected sample size that are normally achieved by a group sequential test are reduced when there is a delay in response the loss of efficiency for larger delays can be ameliorated by incorporating data on a correlated short term end point fitting a joint model for the two end points but still making inferences on the original longer term end point we derive p values and confidence intervals on termination of our new tests

generalized berry phase for the most general time dependent damped harmonic oscillator the most general time dependence hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum it describes in general a harmonic oscillator with a time dependent mass a time dependent friction or antifriction constant and a time dependent spring constant acted upon by a time dependent force the energy whose time derivative is the power is in general different from the hamiltonian the generalized berry phase for a given energy eigenstate has a state dependent part which vanishes if there is no damping and an arbitrary state independent part if the hamiltonian is identified as the energy both the energy eigenvalues and the generalized berry phase are different in the adiabatic limit both approaches give the same total phase which is the sum of the dynamical and berry phases

existence of the hannay angle for single frequency systems the hannay angle was introduced by hannay 1985 as a means of measuring an anholonomy effect in classical mechanics closely corresponding to berry s phase in quantum mechanics such classical adiabatic angle shifts the hannay angles arise when an integrable classical hamiltonian involves time dependent parameters which undergo a closed adiabatic excursion in this paper two proofs of an averaging type of theorem for a single frequency dynamical system are given as a consequence one can establish the existence of the hannay angle for a class of smooth classical hamiltonian systems with one degree of freedom moreover a review of the rotated rotator illustrates the usefulness of the averaging theorem

classical non adiabatic angles if a family of tori in phase space is driven by a time dependent hamiltonian flow in such a way as to return after some time to the original family there generally results a shift in the angle variables one realisation of this process is in the cyclic adiabatic change of a classical hamiltonian and the angle change has previously been shown to separate naturally into a dynamical part and a geometrical part here the same geometrical angle change is extracted when the return is achieved non adiabatically and the dynamical remainder calculated two examples are given the precession of a spin and the rotation of phase space ellipses

the bargmann representation for the quantum mechanics on a sphere the bargmann representation is constructed corresponding to the coherent states for a particle on a sphere introduced in kowalski and rembielinski j phys a 33 6035 2000 the connection is discussed between the introduced formalism and the standard approach based on the hilbert space of square integrable functions on a sphere s2

generalization of berry s geometric phase equivalence of the hamiltonian nature quantizability and strong stability of linear oscillatory systems and conservation of adiabatic invariants a linear set of ordinary differential equations with a matrix depending on a set of adiabatically varying parameters is considered its asymptotic solutions are constructed to an arbitrary accuracy in the adiabaticity parameter e by extending the phase space of the system not only with the space of parameters like in the theory of berry s phase but also with the space of their derivatives it proves possible to represent the phase of the solution as a sum of the dynamic phase and a generalized geometric phase determined by a contour integral in the space of parameters and derivatives hence it is possible to obtain the asymptotic results to any degree of accuracy in e while earlier they were obtainable only in the first order adiabatic approximation namely for linear oscillatory adiabatic systems the quantizability parametric stability and the hamiltonian nature of the system are equivalent properties as a consequence one can obtain the well known result regarding conservation with accuracy

modification of sample size in group sequential clinical trials in group sequential clinical trials sample size reestimation can be a complicated issue when it allows for change of sample size to be influenced by an observed sample path our simulation studies show that increasing sample size based on an interim estimate of the treatment difference can substantially inflate the probability of type i error in most practical situations a new group sequential test procedure is developed by modifying the weights used in the traditional repeated significance two sample mean test the new test has the type i error probability preserved at the target level and can provide a substantial gain in power with the increase of sample size generalization of the new procedure is discussed

on the removability of berry s phase the authors present a thorough analysis of the removability of the berry phase they show that for any system they can by properly restricting the parameter space turn the geometrical phase into an extra shift of the dynamical one however this extra shift in the dynamical phase retains its truly geometrical character the removability is thus only apparent and the geometrical phase can still be measured in interference experiments as indeed has already been done several times

generalized geometric phase of a classical oscillator the equation of a linear oscillator with adiabatically varying eigenfrequency et e 1 is the adiabaticity parameter is considered the asymptotic solutions to the equation have been obtained to terms of order e3 it is shown that imaginary terms of order e2 form a generalized geometric phase determined by the geometry of the system s contour in the plane the real terms of orders e and e3 as predicted bliokh k yu 2002 j math phys 43 5624 do not form geometric amplitudes but are responsible for local relationships between the solution amplitudes and the parameters that is for adiabatic invariants

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

adaptive sample size modification in clinical trials start small then ask for more we consider sample size re estimation in a clinical trial in particular when there is a significant delay before the measurement of patient response mehta and pocock have proposed methods in which sample size is increased when interim results fall in a promising zone where it is deemed worthwhile to increase conditional power by adding more subjects our analysis reveals potential pitfalls in applying this approach mehta and pocock use results of chen demets and lan to identify when increasing sample size but applying a conventional level significance test at the end of the trial does not inflate the type i error rate we have found the greatest gains in power per additional observation are liable to lie outside the region defined by this method mehta and pocock increase sample size to achieve a particular conditional power calculated under the current estimate of treatment effect this leads to high increases in sample size for a small range of interim outcomes whereas we have found it more efficient to make moderate increases in sample size over a wider range of cases if the aforementioned pitfalls are avoided we believe the broad framework proposed by mehta and pocock is valuable for clinical trial design working in this framework we propose sample size rules that apply explicitly the principle of adding observations when they are most beneficial the resulting trial designs are closely related to efficient group sequential tests for a delayed response proposed by hampson and jennison copyright 2015 john wiley sons ltd

the connection whose holonomy is the classical adiabatic angles of hannay and berry and its generalization to the non integrable case we show how averaging defines an ehresmann connection whose holonomy is the classical adiabatic angles which hannay defined for families of completely integrable systems the averaging formula we obtain for the connection only requires that the family of hamiltonians has a continuous symmetry group this allows us to extend the notion of the hannay angles to families of non integrable systems with symmetry we state three geometric axioms satisfied by the connection these axioms uniquely determine the connection thus enabling us to find new formulas for the connection and its curvature two examples are given

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

geometric quantization and the generalized segal bargmann transform for lie groups of compact type let k be a connected lie group of compact type and let t k be its cotangent bundle this paper considers geometric quantization of t k first using the vertical polarization and then using a natural kahler polarization obtained by identifying t k with the complexified group k the first main result is that the hilbert space obtained by using the kahler polarization is naturally identifiable with the generalized segal bargmann space introduced by the author from a different point of view namely that of heat kernels the second main result is that the pairing map of geometric quantization coincides with the generalized segal bargmann transform introduced by the author this means that the pairing map in this case is a constant multiple of a unitary map for both results it is essential that the half form correction be included when using the kahler polarization

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

on wave action and phase in the non canonical hamiltonian formulation the long timeevolution of disturbances to slowlyvarying solutions of partial differential equations is subject to the adiabatic invariance of the wave action generally this approximate conservati

the geometric phase for chaotic unitary families we consider the geometric phase for a family of quantum classical hamiltonians in which the effect of changing parameters is simply to induce unitary canonical transformations in this case the classical limit of the geometric phase is easily obtained even when the classical motion is chaotic the results agree with those previously obtained for general chaotic families but may be expressed in a simpler form not involving time integrals of correlation functions it is also straightforward to establish some results which are problematic in the general case for example the form of periodic orbit corrections and the closeness of the classical 2 form if the parameters are regarded as dynamical variables evolving slowly so as to maintain adiabaticity they are subject to geometric magnetism but not in contrast to the general case deterministic friction and born oppenheimer forces examples including families of translated and rotated systems 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

maslov s complex germ method and berry s phase in the adiabatic approximation the connection of the beny phase with the quasi classical trajectory coherent slates of the schrirdinger type equation with the arbitrary scalar pseudo differential operator and the dirac equation in the external penodic electromagnetic field is studied

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

adiabatic invariance in the standard map adiabatic invariance of the action is investigated in the standard map under slow changes of the stochasticity parameter k a fixed action representation of the rotational tori is developed perturbatively in k and is connected with the usual kam representation at fixed winding number the notion of adiabatic invariance to a given order in k k 1 is then introduced it involves an approximation to exact dynamics by essentially a power series expansion in both k and a slowness parameter adiabatic invariance is explicitly verified to second order and the dependence of the nonadiabaticity on the form of the switching function and the slowness of the change is investigated the case of adiabatic switching to larger values of k k 1 is approached phenomenologically taking into account the problem of separatrix crossing it is shown that this crossing leads in general to nonadiabatic effects in the limit of infinitely slow change these reflect the finite widths or intrinsic action uncertainties associated with the main island chains crossed the latter are determined from the farey tree in the range of variation of the winding number an estimate of the critical slowness parameter corresponding to the onset of the intrinsic nonadiabatic effects is derived it involves typical time scales associated with the main island chains crossed

berry phases for the nonlocal gross pitaevskii equation with a quadratic potential a countable set of asymptotic space localized solutions is constructed by the complex germ method in the adiabatic approximation for the nonstationary gross pitaevskii equation with nonlocal nonlinearity and a quadratic potential the asymptotic parameter is 1 t where t 1 is the adiabatic evolution time a generalization of the berry phase of the linear schrodinger equation is formulated for the gross pitaevskii equation for the solutions constructed the berry phases are found in explicit form

near optimal nonholonomic motion planning for a system of coupled rigid bodies how does a falling cat change her orientation in midair without violating angular momentum constraint this has become an interesting problem to both control engineers and roboticists in this paper we address this problem together with a constructive solution first we show that a falling cat problem is equivalent to the constructive nonlinear controllability problem thus the same principle and algorithm used by a falling cat can be used for space robotic applications such as reorientation of a satellite using rotors and attitude control of a space structure using internal motion and other robotic tasks such as dextrous manipulation with multifingered robotic hands and nonholonomic motion planning for mobile robots then using ideas from ritz approximation theory we develop a simple algorithm for motion planning of a falling cat finally we test the algorithm through simulation on two widely accepted models of a falling cat it is interesting to note that one set of simulation results closely resembles the real trajectories employed by a falling cat

parallel transport along a space curve and related phases the authors investigate geometrical properties of a space curve and of its spherical images they show that a space curve is characterised by two phase like quantities and comment on the relation of these quantities to the berry phase

coherent states on spheres we describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d dimensional sphere sd the coherent states are labeled by points in the associated phase space t sd these coherent states are not of perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators we describe as well the segal bargmann representation for the system the associated unitary segal bargmann transform and a natural inversion formula although many of these results are in principle special cases of the results of hall and stenzel we give here a substantially different description based on ideas of thiemann and of kowalski and rembielinski all of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space we focus on the sphere case in order to carry out the calculations in a self contained and explicit way

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

the geometric phase for chaotic systems the geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two form through a surface in the system s parameter space we obtain the classical limit of this two form in a form applicable to systems whose classical dynamics is chaotic for integrable systems the expression is equivalent to the hannay two form we discuss various properties of the classical two form derive semiclassical corrections to it associated with classical periodic orbits and consider implications for the semiclassical density of degeneracies

the segal bargmann transform for noncompact symmetric spaces of the complex type we consider the generalized segal bargmann transform defined in terms of the heat operator for a noncompact symmetric space of the complex type for radial functions we show that the segal bargmann transform is a unitary map onto a certain l 2 space of meromorphic functions for general functions we give an inversion formula for the segal bargmann transform involving integration against an unwrapped version of the heat kernel for the dual compact symmetric space both results involve delicate cancellations of singularities

efficient group sequential designs when there are several effect sizes under consideration we consider the construction of efficient group sequential designs where the goal is a low expected sample size not only at the null hypothesis and the alternative taken to be the minimal clinically meaningful effect size but also at more optimistic anticipated effect sizes pre specified type i error rate and power requirements can be achieved both by standard group sequential tests and by more recently proposed adaptive procedures we investigate four nested classes of designs a group sequential tests with equal group sizes and stopping boundaries determined by a monomial error spending function the rho family b as a but the initial group size is allowed to be different from the others c group sequential tests with arbitrary group sizes and arbitrary boundaries fixed in advance d adaptive tests as c but at each analysis future group sizes and critical values are updated depending on the current value of the test statistic by examining the performance of optimal procedures within each class we conclude that class b provides simple and efficient designs with efficiency close to that of the more complex designs of classes c and d we provide tables and figures illustrating the performances of optimal designs within each class and defining the optimal procedures of classes a and b

are flexible designs sound flexible designs allow large modifications of a design during an experiment in particular the sample size can be modified in response to interim data or external information a standard flexible methodology combines such design modifications with a weighted test which guarantees the type i error level however this inference violates basic inference principles in an example with independent n mu 1 observations the test rejects the null hypothesis of mu or 0 while the average of the observations is negative we conclude that flexible design in its most general form with the corresponding weighted test is not valid several possible modifications of the flexible design methodology are discussed with a focus on alternative hypothesis tests

on the classical limit of berry s phase integrable systems berry s phase is given by integration of a characteristic two form we consider integrable systems defined by weyl quantized classical hamiltonians it is shown that the limit of i times this tow form is the curvature of the classical connection whose holonomy is the hannay angels a result of this type was derived by berry b2

designed extension of studies based on conditional power we propose a flexible method of extending a study based on conditional power the possibility for extension when the p value at the planned end is small but not statistically significant is built in to the design of the study the significance of the treatment difference at the planned end is used to determine the number of additional observations needed and the critical value necessary for use after accruing those additional observations it may therefore be thought of as a two stage procedure even though the observed treatment difference at stage 1 is used to make decisions the type i error rate is protected

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis

non abelian geometric phase floquet theory and periodic dynamical invariants for a periodic hamiltonian periodic dynamical invariants may be used to obtain non degenerate cyclic states this observation is generalized to the degenerate cyclic states and the relation between the periodic dynamical invariants and the floquet decompositions of the time evolution operator is elucidated in particular a necessary condition for the occurrence of cyclic non adiabatic non abelian geometrical phase is derived degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field