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

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

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

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

on the time fractional navier stokes equations abstract this paper is concerned with the navier stokes equations with time fractional derivative of order 0 1 this type of equations can be used to simulate anomalous diffusion in fractal media we establish the existence and uniqueness of local and global mild solutions in h q meanwhile we also give local mild solutions in j q moreover we prove the existence and regularity of classical solutions for such equations in j q

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

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

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

notes on the space time decay rate of the stokes flows in the half space in this paper a stokes equations in the half space r n n 2 has been considered we derive a rapid decay rate of the stokes flow in space and time when the initial data decreases fast enough and satisfies some additional condition initial data decreasing too slowly to be x h l1 r n are also considered

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

decay results of solutions to the incompressible navier stokes flows in a half space abstract in a half space we consider the asymptotic behavior of the strong solution for the non stationary navier stokes equations in particular the decay rates of the second order derivatives of the navier stokes flows in l r r n n 2 with 1 r are derived by using l q l r estimates and a clever analysis on the fractional powers of the stokes operator in addition we prove that the strong solution and its first and second derivatives decay in time more rapidly than observed in general if the initial datum lies in a suitable weighted space

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

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

small cancellation theory over burnside groups abstract we develop a version of small cancellation theory in the variety of burnside groups more precisely we show that there exists a critical exponent n 0 such that for every odd integer n n 0 the well known classical c 1 6 small cancellation theory as well as its graphical generalization and its version for free products produce examples of infinite n periodic groups our result gives a powerful tool for producing uncountable collections of examples of n periodic groups with prescribed properties it can be applied without any prior knowledge in the subject of n periodic groups as applications we show the undecidability of markov properties in classes of n periodic groups we produce n periodic groups whose cayley graph contains an embedded expander graphs and we give an n periodic version of the rips construction we also obtain simpler proofs of some known results like the existence of uncountably many finitely generated n periodic groups and the sq universality in the class of n periodic groups of free burnside groups

genericity the arzhantseva ol shanskii method and the isomorphism problem for one relator groups we show that the isomorphism problem is solvable in at most exponential time for a class of one relator groups which is exponentially generic in the sense of ol shanskii this is obtained by applying the arzhantseva ol shanskii graph minimization method to prove the general result that for fixed integers m 2 and n 1 there is an exponentially generic class of non free m generator n relator groups with the property that there is only one nielsen equivalence class of m tuples which generate a non free subgroup in particular every m generated subgroup in such a generic group g is either free or is equal to g itself and such groups are thus co hopfian these results are obtained by elementary methods without using the deep results of sela about co hopficity and the isomorphism problem for torsion free hyperbolic groups

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

extinction decay and blow up for keller segel systems of fast diffusion type abstract we consider the quasi linear keller segel system of singular type where the principal part u m represents a fast diffusion like 0 m 1 we first construct a global weak solution with small initial data in the scaling invariant norm l n q m 2 for all dimensions n 2 and all exponents q 2 as for the large initial data we show that there exists a blow up solution in the case of n 2 in the second part the decay property in l r with 1 r for 1 2 n m 1 with the mass conservation is shown on the other hand in the case of 0 m 1 2 n the extinction phenomenon of solution is proved it is clarified that the case of m 1 2 n exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across 1 2 n for the borderline case of m 1 2 n our solution decays in l r exponentially as t

asymptotic behavior for the stokes flow and navier stokes equations in half spaces abstract using the solution formula in ukai 1987 27 for the stokes equations we find asymptotic profiles of solutions in l 1 r n n 2 for the stokes flow and non stationary navier stokes equations since the projection operator p l 1 r n l 1 r n is unbounded we use a decomposition for p u u to overcome the difficulty and prove that the decay rate for the first derivatives of the strong solution u of the navier stokes system in l 1 r n is controlled by t 1 2 1 t n 2 2 for any t 0

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

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

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

pointwise decay estimate of navier stokes flows in the half space with slowly decreasing initial value abstract in this paper we study the spatial and temporal decay estimate of the navier stokes flow in the half space corresponding to auniformly but slowly decreasing initial velocity we show the local in time solvability of the navier stokes equations with u x t c 0 1 x t and u x t c 0 t 1 2 1 x t when 1 x t e t a h l r n 0 and t 1 2 1 x t e t a h l r n 0 0 n asymptotically it holds that u x t e t a h x o t x 0 n as x the solution exists globally in time for 1 when the data is small enough and it holds that u x t e t a h x o x t 1 n as x t

kazhdan groups with infinite outer automorphism group for each countable group q we produce a short exact sequence 1 to n to g to q to 1 where g is f g and has a graphical frac16 presentation and n is f g and satisfies property t as a consequence we produce a group n with property t such that out n is infinite using the tools developed we are also able to produce examples of nonhopfian and non cohopfian groups with property t one of our main tools is the use of random groups to achieve certain properties

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

on l 1 summability and asymptotic profiles for smooth solutions to navier stokes equations in a 3d exterior domain the exterior nonstationary problem is studied for the 3d navier stokes equations the l1 summability is proved for smooth solutions which correspond to initial data satisfying certain symmetry and moment conditions the result is then applied to show that such solutions decay in time more rapidly than observed in general furthermore an asymptotic expansion is deduced and a lower bound estimate is given for the rates of decay in time

rips segev torsion free groups without the unique product property abstract we generalize the graphical small cancellation theory of gromov to a graphical small cancellation theory over the free product we extend gromov s small cancellation theorem to the free product we explain and generalize rips segev s construction of torsion free groups without the unique product property by viewing these groups as given by graphical small cancellation presentations over the free product our graphical small cancellation theorem then provides first examples of gromov hyperbolic groups without the unique product property we construct uncountably many non isomorphic torsion free groups without the unique product property we show that the presentations of generalized rips segev groups are not generic among finite presentations of groups

a new test for asphericity and diagrammatic reducibility of group presentations we present a new test for studying asphericity and diagrammatic reducibility of group presentations our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails we use this criterion to generalize results of j howie and s m gersten on asphericity of lots and of adian presentations and derive new results on solvability of equations over groups we also use our methods to investigate a conjecture of s v ivanov related to kaplansky s problem on zero divisors we strengthen ivanov s result for locally indicable groups and prove a weak version of the conjecture

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

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

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

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

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

global existence and decay properties for a degenerate keller segel model with a power factor in drift term abstract the following degenerate parabolic system modelling chemotaxis is considered ks u t u m u q 1 v x r n t 0 v t v v u x r n t 0 u x 0 u 0 x v x 0 v 0 x x r n where m 1 q 2 0 or 1 and n 1 the aim of this paper is to prove the existence of a time global weak solution u v of ks with the l 0 l r n bound such a global bound is obtained in the case of i m q 2 n for large initial data and ii 1 m q 2 n for small initial data in the case of ii the decay properties of the solution u v are also discussed

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

decay rates for the incompressible navier stokes flows in 3d exterior domains abstract the exterior nonstationary problem is studied for the 3d navier stokes equations for which the associated total net force to the boundary may not vanish the time decay properties of the strong solution including the first and second derivatives are shown in l q and weighted spaces in particular the relation of weighted l 1 summability for smooth solutions is discussed in details between the time decay and the total net force exerted by the fluid to the body the conclusions in this article improve and extend results in h bae b jin asymptotic behavior for the navier stokes equations in 2d exterior domains j funct anal 240 2006 508 529 h bae b jin temporal and spatial decay rates of navier stokes solutions in exterior domains bull korean math soc 44 2007 547 567 and in c he t miyakawa on l 1 summability and asymptotic profiles for smooth solutions to navier stokes equations in a 3d exterior domain math z 245 2003 387 417 c he t miyakawa on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain j differential equations 246 2009 2355 2386 respectively

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

weighted decay properties for the incompressible stokes flow and navier stokes equations in a half space abstract using the stokes solution formula and l q l r estimates of the stokes operator semigroup we establish the weighted decay properties for the stokes flow and navier stokes equations including their spatial derivatives in half spaces in addition the unboundedness of the projection operator p l r n l r n is overcome by employing a decomposition for the nonlinear term and l asymptotic behavior for the second derivatives of navier stokes flows in half spaces is given

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

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