bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

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

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

eigenvalues outside the bulk of inhomogeneous erdős renyi random graphs in this article an inhomogeneous erd s renyi random graph on 1 ldots n is considered where an edge is placed between vertices i and j with probability varepsilon n f i n j n for i le j the choice being made independently for each pair the integral operator i f associated with the bounded function f is assumed to be symmetric non negative definite and of finite rank k we study the edge of the spectrum of the adjacency matrix of such an inhomogeneous erd s renyi random graph under the assumption that n varepsilon n rightarrow infty sufficiently fast although the bulk of the spectrum of the adjacency matrix scaled by sqrt n varepsilon n is compactly supported the kth largest eigenvalue goes to infinity it turns out that the largest eigenvalue after appropriate scaling and centering converges to a gaussian law if the largest eigenvalue of i f has multiplicity 1 if i f has k distinct non zero eigenvalues then the joint distribution of the k largest eigenvalues converge jointly to a multivariate gaussian law the first order behaviour of the eigenvectors is derived as a byproduct of the above results the results complement the homogeneous case derived by 18

existence and bifurcation behavior of positive solutions for a class of kirchhoff type problems abstract we study the existence of positive solutions for the nonlinear kirchhoff type equation a b u 2 d x u u u 3 with dirichlet boundary condition where r are two real parameters we provide a description of a two dimensional set in the plane which correspond to the existence of positive solutions for the above kirchhoff type equation

extremal eigenvalues and eigenvectors of deformed wigner matrices we consider random matrices of the form h w lambda v lambda in mathbb r where w is a real symmetric or complex hermitian wigner matrix of size n and v is a real bounded diagonal random matrix of size n with i i d entries that are independent of w we assume subexponential decay of the distribution of the matrix entries of w and we choose lambda sim 1 so that the eigenvalues of w and lambda v are typically of the same order further we assume that the density of the entries of v is supported on a single interval and is convex near the edges of its support in this paper we prove that there is lambda in mathbb r such that the largest eigenvalues of h are in the limit of large n determined by the order statistics of v for lambda lambda in particular the largest eigenvalue of h has a weibull distribution in the limit n rightarrow infty if lambda lambda moreover for n sufficiently large we show that the eigenvectors associated to the largest eigenvalues are partially localized for lambda lambda while they are completely delocalized for lambda lambda similar results hold for the lowest eigenvalues

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

stationary kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity this paper deals with the existence and the asymptotic behavior of non negative solutions for a class of stationary kirchhoff problems driven by a fractional integro differential operator lk and involving a critical nonlinearity in particular we consider the problem m u 2 lku f x u u 2s 2uin u 0in rn where rn is a bounded domain 2s is the critical exponent of the fractional sobolev space hs rn the function f is a subcritical term and is a positive parameter the main feature as well as the main difficulty of the analysis is the fact that the kirchhoff function m could be zero at zero that is the problem is degenerate the adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

theory of levy matrices we investigate the statistical properties of the spectrum of large symmetrical matrices with each element mathit h mathit i mathit j chosen according to a broad distribution ensuremath rho h decaying for large h as mathit h mathrm ensuremath 1 mathrm ensuremath mathrm ensuremath mu for ensuremath mu g2 mathit h 2 is finite and the well known gaussian orthogonal ensemble goe results are recovered when ensuremath mu 2 the semicircular law is replaced by a density which extends over the whole energy axis furthermore while all states are extended in the case of goe matrices we show numerically and analytically that two mobility edges appear separating extended from localized states with an intermediate mixed phase in between the unusual nature of these localized states is discussed

the cauchy problem of coupled elliptic sine gordon equations with noise analysis of a general kernel based regularization and reliable tools of computing abstract developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering in this paper we explore the regularization method for the coupled elliptic sine gordon equations along with cauchy data the system of equations originates from the static case of the coupled hyperbolic sine gordon equations modeling the coupled josephson junctions in superconductivity and so far it addresses the josephson junctions in general the cauchy problem is not well posed and herein the hadamard instability occurs drastically generalizing the kernel based regularization method we propose a stable approximate solution confirmed by the error estimate this solution strongly converges to the exact solution in l 2 norm the main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation we employ the proposed techniques that successfully approximated the highly oscillatory integral and apply the picard like iteration to organize an efficient and reliable tool of computations the results are viewed as the improvement as well as the generalization of many previous works the paper is also accompanied by a numerical example that demonstrates the potential of this idea

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

geometric tempus for a class of ergodic classical systems using the symplectic formalism of classical mechanics a smooth holonomy effect is identified in a special class of classical ergodic hamiltonian systems generalized canonical families that are cycled adiabatically it is a smooth shift in a time resembling observable tempus that is canonically conjugated to the phase space volume contained in the energy shell the curvature 2 form of the adiabatic connection obtained is closely related to hannay s and the robbins berry 2 form the measurability of the tempus is discussed

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

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

a modified integral equation method of the nonlinear elliptic equation with globally and locally lipschitz source the paper is devoted to investigating a cauchy problem for nonlinear elliptic pdes in the abstract hilbert space the problem is hardly solved by computation since it is severely ill posed in the sense of hadamard we shall use a modified integral equation method to regularize the nonlinear problem with globally and locally lipschitz source terms convergence estimates are established under priori assumptions on exact solution a numerical test is provided to illustrate that the proposed method is feasible and effective these results extend some earlier works on a cauchy problem for elliptic equations

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

asymptotic stability for anisotropic kirchhoff systems we study the question of asymptotic stability as time tends to infinity of solutions of dissipative anisotropic kirchhoff systems involving the p x laplacian operator governed by time dependent nonlinear damping forces and strongly nonlinear power like variable potential energies this problem had been considered earlier for potential energies which arise from restoring forces whereas here we allow also the effect of amplifying forces global asymptotic stability can then no longer be expected and should be replaced by local stability the results are further extended to the more delicate problem involving higher order damping terms

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

eigenvector localization for random band matrices with power law band width it is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors provided the band width w raised to a power remains smaller than the matrix size n for a gaussian band ensemble with matrix elements given by i i d centered gaussians within a band of width w the estimate 8 holds

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

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

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

wigner random matrices with non symmetrically distributed entries we show that the spectral radius of an n n random symmetric matrix with i i d bounded centered but non symmetrically distributed entries is bounded from above by 2 o n 6 11 e where 2 is the variance of the matrix entries and e is an arbitrary small positive number our bound improves the earlier results by z furedi and j komlos 1981 and van vu 2005

approximate solution of the backward problem for kirchhoff s model of parabolic type with discrete random noise abstract in this paper we consider the final boundary value problem for non local kirchhoff s model of parabolic type with discrete random noise we first discuss the instability of solutions then we present the regularized solution by the trigonometric method in non parametric regression associated with the truncated expansion method in addition under a prior assumption on the exact solution the convergence rate is obtained in the last section the effectiveness and suitability of our results are justified via numerical experiments

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

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

on the law of addition of random matrices normalized eigenvalue counting measure of the sum of two hermitian or real symmetric matrices a n and b n rotated independently with respect to each other by the random unitary or orthogonal haar distributed matrix u n i e a n u n b n u n is studied in the limit of large matrix order n convergence in probability to a limiting nonrandom measure is established a functional equation for the stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of a n and b n is obtained and studied

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

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

on cauchy problem for nonlinear fractional differential equation with random discrete data abstract this paper is concerned with finding the solution u x t of the cauchy problem for nonlinear fractional elliptic equation with perturbed input data this study shows that our forward problem is severely ill posed in sense of hadamard for this ill posed problem the trigonometric of non parametric regression associated with the truncation method is applied to construct a regularized solution under prior assumptions for the exact solution the convergence rate is obtained in both l2 and h q for q 0 norm moreover the numerical example is also investigated to justify our results

on universality of local edge regime for the deformed gaussian unitary ensemble we consider the deformed gaussian ensemble h n h n 0 m n in which h n 0 is a hermitian matrix possibly random and m n is the gaussian unitary random matrix gue independent of h n 0 assuming that the normalized counting measure of h n 0 converges weakly in probability if random to a non random measure n 0 with a bounded support and assuming some conditions on the convergence rate we prove the universality of the local eigenvalue statistics near the edge of the limiting spectrum of h n

from gumbel to tracy widom the tracy widom distribution that has been much studied in recent years can be thought of as an extreme value distribution we discuss interpolation between the classical extreme value distribution exp exp x the gumbel distribution and the tracy widom distribution there is a family of determinantal processes whose edge behaviour interpolates between a poisson process with density exp x and the airy kernel point process this process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by moshe neuberger and shapiro we also consider the deformed gue ensemble m m 0 sqrt 2s v with m 0 diagonal with independent elements and v from gue here we do not see a transition from tracy widom to gumbel but rather a transition from tracy widom to gaussian

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

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

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

on a nonlocal equation arising in population dynamics we study a one dimensional nonlocal variant of fisher s equation describing the spatial spread of a mutant in a given population and its generalization to the so called monostable nonlinearity the dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modelled by a convolution operator we prove that as in the classical local problem there exist travelling wave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity our proofs rely on an appropriate version of the maximum principle qualitative properties of solutions and approximation schemes leading to singular limits

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

on a backward problem for the kirchhoff s model of parabolic type abstract we study for the first time the backward problem for nonlocal nonlinear boundary value problem of kirchhoff s model of parabolic type first we show that the problem is severely ill posed in the sense of hadamard we propose two methods the fourier truncation method for stabilizing the problem with homogeneous source and the quasi reversibility method for regularizing the problem with nonlinear source under some priori assumptions on the exact solution we establish some stability estimates in the h 0 1 norm

a critical kirchhoff type problem involving a nonlocal operator abstract in this paper we show the existence of non negative solutions for a kirchhoff type problem driven by a nonlocal integrodifferential operator that is m u z 2 l k u f x u u 2 2 u in u 0 in r n where l k is an integrodifferential operator with kernel k is a bounded subset of r n m and f are continuous functions z is a functional norm and 2 is a fractional sobolev exponent

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

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

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

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

stochastic burgers equation with fractional derivative driven by multiplicative noise abstract this article is devoted to the study of the existence and uniqueness of mild solution to time and space fractional stochastic burgers equation perturbed by multiplicative white noise the required results are obtained by stochastic analysis techniques fractional calculus and semigroup theory moreover we prove the regularity properties of mild solution to this generalized burgers equation in the framework of bochner spaces

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

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

existence uniqueness and regularity for a class of semilinear stochastic volterra equations with multiplicative noise abstract we consider a class of semilinear volterra type stochastic evolution equation driven by multiplicative gaussian noise the memory kernel not necessarily analytic is such that the deterministic linear equation exhibits a parabolic character under appropriate lipschitz type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean p holder continuous with values in an appropriate sobolev space depending on the kernel and the data in particular we obtain pathwise space time sobolev holder regularity of the solution together with a maximal type bound on the spatial sobolev norm as one of the main technical tools we establish a smoothing property of the derivative of the deterministic evolution operator family

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

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

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

localization and delocalization of eigenvectors for heavy tailed random matrices consider an n times n hermitian random matrix with above the diagonal independent entries with alpha stable symmetric distribution and 0 2 we give vanishing bounds on the l p norm of the eigenvectors normalized to have unit l 2 norm on the contrary when 0 alpha 2 3 we prove that these eigenvectors are localized

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

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