high order asymptotic expansions for robust tests summary this paper provides high order asymptotic expansions for the m ratio and the signedroot m ratio robust test statistics which allow one to compute accurate approximations to significance or critical values using the edgeworth approximation the bartlett correction the variance correction or the saddlepoint approximation specific results are obtained for the linear regression model with the huber m estimator a monte carlo study illustrates the numerical accuracy of these approximations with respect to the usual first order approximations

the gauss bonnet chern theorem a probabilistic perspective we prove that the euler form of a metric connection on a real oriented vector bundle e over a compact oriented manifold m can be identified as a current with the expectation of the random current defined by the zero locus of a certain random section of the bundle we also explain how to reconstruct probabilistically the metric and the connection on e from the statistics of random sections of e contents

time dependent quantum tunnelling via crossover processes it is shown that the propagator k xt mod x 0 for a particle in a potential field is derivable from the classical path starting from x and reaching x at time t considering a wavepacket as the particle initial state lying mainly on one side of a barrier the propagator is used for obtaining the wavefunction on the other side tunnelling is then discussed in terms of the evolving wavefunction it is argued that while the particle s expectation energy is taken to be lower than the barrier s height the transformation by which the wavefunction is produced entails energetically crossover flights as required by the dynamics for the classical paths involved in the propagation process the parabolic repeller exemplifies the formalism and exact results for the probability and current densities are given a computation involving ballistic tunnelling shows that the current density rises to a saturation value proportional to the particles injection rate

a quantitative central limit theorem for the excursion area of random spherical harmonics over subdomains of s2 in recent years considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere spherical harmonics in this paper we extend those results to proper subsets of the sphere s2 i e spherical caps focussing in particular on the excursion area precisely we show that the asymptotic behaviour of the excursion area is dominated by the so called second order chaos component and we exploit this result to establish a quantitative central limit theorem in the high energy limit these results generalize analogous findings for the full sphere their proofs however require more sophisticated techniques in particular a careful analysis of some independent interest for smooth approximations of the indicator function for spherical cap subsets

the barrier transmission problem treated by the amplitude phase method and expressed in terms of an invariant of the ermakov lewis type transmission and reflection of a quantal particle by a single hump potential barrier are analysed by means of an amplitude phase decomposition of the wavefunction on both sides of the barrier the amplitude phase analysis of the wavefunction provides a particular invariant of the ermakov lewis type which originates in the matching process the transmission and reflection coefficients turn out to be simple functions of this invariant numerical calculations of the invariant for an eckart epstein potential barrier provide very accurate results

gibbs entropy invariants of motion abstract invariants of motion are constructed departing from the gibbs entropy the method we applied also allows us to find the subspace of a real vector space to which the system is confined to during its temporal evolution we also find that the uncertainty principle is itself an invariant of motion for harmonic oscillator like hamiltonians

a connection between quantum hilbert space and classical phase space operators we develop a general classicalization procedure that links hilbert space and phase space operators using weyl s operator then we transform the time dependent schrodinger equation into a phase space picture using free parameters they include position q and momentum p we expand the phase space hamiltonian in an taylor series and fix parameters with the condition that coefficients of 0 i1 q and i1 p vanish this condition results in generalized hamilton equations and a natural link between classical and quantum dynamics while the quantum motion equation remains exact in this picture the schrodinger equation reduces in the classical limit to a generalized liouville equation for the quantum mechanical system state we modify glauber s coherent states with a suitable phase factor s q p t and use them to obtain phase space representations of quantum dynamics and quantum mechanical quantities

on nonlinear functionals of random spherical eigenfunctions we prove central limit theorems and stein like bounds for the asymptotic behaviour of nonlinear functionals of spherical gaussian eigenfunctions our investigation combines asymptotic analysis of higher order moments for legendre polynomials and in addition recent results on malliavin calculus and total variation bounds for gaussian subordinated fields we discuss applications to geometric functionals like the defect and invariant statistics e g polyspectra of isotropic spherical random fields both of these have relevance for applications especially in an astrophysical environment

the evolution operator technique in solving the schrodinger equation and its application to disentangling exponential operators and solving the problem of a mass varying harmonic oscillator the authors present a method for finding the evolution operator for the schrodinger equation for the hamiltonian expressible as h t a1 t j4 a2 t j0 a3 t j where j j0 and j are the su 2 group generators such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary as a demonstration of our general approach they solved the problem of a harmonic oscillator with a varying mass

exact time dependence of solutions to the time dependent schrodinger equation solutions of the schrodinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time independent operators using time dependent unitary transformations exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time dependent hamiltonians in d dimensions including non hermitean symmetric hamiltonians hamiltonians are constructed using time dependent unitary spatial transformations comprising dilatations translations and rotations and solutions are found in several forms as eigenfunctions of a quadratic invariant as coherent state eigenfunctions of boson operators as plane wave solutions from which the general solution is obtained as an integral transform by means of the fourier transform and as distributional solutions for which the initial wavefunction is the dirac function for the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d 1 including barut girardello and perelomov coherent states i e vector coherent states which are shown to be related to eigenfunctions of the quadratic invariant by the transformation this transformation which leaves the ermakov equation invariant implements su 1 1 transformations on linear dynamical invariants coherent states are derived also for the time dependent linear potential exact solutions are found for hamiltonians with electromagnetic interactions in which the time dependent magnetic and electric fields are not necessarily spatially uniform as an example it is shown how to find exact solutions of the time dependent schrodinger equation for the dirac magnetic monopole in the presence of time dependent magnetic and electric fields of a specified form

representations of one dimensional hamiltonians in terms of their invariants a general formalism for representing the hamiltonian of a system with one degree of freedom in terms of its invariants is developed those hamiltonians h q p t are derived for which any particular function i q p t is an invariant for each of those hamiltonians a function canonically conjugate to i q p t is derived which is also an invariant of h q p t the formalism is also presented for the case in which i q p t is expressed as a function of two canonically conjugate functions the formalism is illustrated by applying it to the case of a particle moving in a time dependent potential some earlier results are recovered and an invariant is found for a new potential lines for further study are outlined that may be fruitful for finding more examples of integrable systems

variance of the volume of random real algebraic submanifolds let mathcal x be a complex projective manifold of dimension n defined over the reals and let m denote its real locus we study the vanishing locus z s d in m of a random real holomorphic section s d of mathcal e otimes mathcal l d where mathcal l to mathcal x is an ample line bundle and mathcal e to mathcal x is a rank r hermitian bundle when r in 1 dots n 1 we obtain an asymptotic of order d r frac n 2 as d goes to infinity for the variance of the linear statistics associated to z s d including its volume given an open set u subset m we show that the probability that z s d does not intersect u is a o of d frac n 2 when d goes to infinity when n geq 3 we also prove almost sure convergence for the linear statistics associated to a random sequence of sections of increasing degree our framework contains the case of random real algebraic submanifolds of mathbb rp n obtained as the common zero set of r independent kostlan shub smale polynomials

on the number of nodal domains of random spherical harmonics let n f be a number of nodal domains of a random gaussian spherical harmonic f of degree n we prove that as n grows to infinity the mean of n f n 2 tends to a positive constant a and that n f n 2 exponentially concentrates around a this result is consistent with predictions made by bogomolny and schmit using a percolation like model for nodal domains of random gaussian plane waves

on the bogomolny schmit conjecture bogomolny and schmit proposed that the critical edge percolation on the square lattice is a good model for the nodal domains of a random plane wave based on this they made a conjecture about the number of nodal domains recent computer experiments showed that the mean number of clusters per vertex and the mean number of nodal domains per unit area are very close but different since the original argument was mostly supported by numerics it was believed that the percolation model is wrong in this paper we give some numerical evidence in favour of the percolation model

discretisation schemes for level sets of planar gaussian fields smooth random gaussian functions play an important role in mathematical physics a main example being the random plane wave model conjectured by berry to give a universal description of high energy eigenfunctions of the laplacian on generic compact manifolds our work is motivated by questions about the geometry of such random functions in particular relating to the structure of their nodal and level sets we study four discretisation schemes that extract information about level sets of planar gaussian fields each scheme recovers information up to a different level of precision and each requires a maximum mesh size in order to be valid with high probability the first two schemes are generalisations and enhancements of similar schemes that have appeared in the literature beffara and gayet in publ math ihes 2017 https doi org 10 1007 s10240 017 0093 0 mischaikow and wanner in ann appl probab 17 980 1018 2007 these give complete topological information about the level sets on either a local or global scale as an application we improve the results in beffara and gayet 2017 on russo seymour welsh estimates for the nodal set of positively correlated planar gaussian fields the third and fourth schemes are to the best of our knowledge completely new the third scheme is specific to the nodal set of the random plane wave and provides global topological information about the nodal set up to visible ambiguities the fourth scheme gives a way to approximate the mean number of excursion domains of planar gaussian fields

volume distribution of nodal domains of random band limited functions we study the volume distribution of nodal domains of families of naturally arising gaussian random fields on generic manifolds namely random band limited functions it is found that in the high energy limit a typical instance obeys a deterministic universal law independent of the manifold some of the basic qualitative properties of this law such as its support monotonicity and continuity of the cumulative probability function are established

generalized hamiltonian structures for ermakov systems we construct poisson structures for ermakov systems using the ermakov invariant as the hamiltonian two classes of poisson structures are obtained one of them is degenerate in which case we derive the casimir functions in some situations the existence of casimir functions can give rise to superintegrable ermakov systems finally we characterize the cases where linearization of the equations of motion is possible

the sojourn time of the inverted harmonic oscillator on the noncommutative plane the sojourn time of the gaussian wavepacket that is stationed at the center of the inverted harmonic oscillator is investigated on the noncommutative plane in detail in ordinary commutative space quantum mechanics the sojourn time of the gaussian wavepacket is always a monotonically decreasing function of the curvature parameter of the potential however in this paper we find that the spatial noncommutativity makes the sojourn time a concave function of with a minimum at an inflection point 0 furthermore if is larger than a certain critical value the sojourn time will become infinity thus the ordinary intuitive physical picture about the relation between the sojourn time and the shape of the inverted oscillator potential is changed when the spatial noncommutativity is considered

a second invariant for one degree of freedom time dependent hamiltonians given a first invariant an explicit formula for a second invariant of a one degree of freedom time dependent hamiltonian is derived in terms of the hamiltonian and an assumed first invariant if the first invariant is expressed as a function of two canonical functions a transformation to an autonomous hamiltonian system is possible

coherent states and their time dependence in fractional dimensions we construct representations of the lie algebra su 1 1 using representations of the momentum and position operators satisfying the r deformed heisenberg relations in which the fractional dimension d and angular momentum l appear as parameters the bargmann index k which characterizes representations of the positive discrete series of su 1 1 can take any positive value we construct coherent states in fractional dimensions in particular we extend the two well known analytic representations of coherent states for su 1 1 perelomov and barut girardello states from dimension one to any dimension d we generalize this construction to time dependent coherent states by means of the su 1 1 symmetries of the quantum time dependent harmonic oscillator in fractional dimensions we investigate the uncertainty relations of the momentum and position operators with respect to these coherent states and their dependence on the dimension

nodal length fluctuations for arithmetic random waves using the spectral multiplicities of the standard torus we endow the laplace eigenspaces with gaussian probability measures this induces a notion of random gaussian laplace eigenfunctions on the torus arithmetic random waves we study the distribution of the nodal length of random eigenfunctions for large eigenvalues and our primary result is that the asymptotics for the variance is nonuniversal our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy

time evolution of a time dependent inverted harmonic oscillator in arbitrary dimensions the time evolution of a time dependent inverted harmonic oscillator tdiho of arbitrary dimensions is investigated using the algebraic method we obtain the exact orthogonal basis of solutions of a tdiho in which the dimensionality d and angular momentum l appear as parameters and also discuss its properties with the wavepacket as the initial state the general expressions of sojourn time of a tdiho are given the method is also applied to the inverted caldirola kanai harmonic oscillator the results show that the sojourn time appears as an increasing function of the dissipation parameter in arbitrary dimensions but a decreasing function of the dimensionality

ermakov systems of arbitrary order and dimension structure and linearization ermakov systems of arbitrary order and dimension are constructed these inherit an underlying linear structure based on that recently established for the classical ermakov system as an application alignment of a 2 1 dimensional ermakov and integrable ernst system is shown to produce a novel integrable hybrid of a 2 1 dimensional sinh gordon system and of a conventional ermakov system

non oscillating solutions to uncoupled ermakov systems and the semi classical limit the amplitude phase formulation of the schrodinger equation is investigated within the context of uncoupled ermakov systems whereby the amplitude function is given by the auxiliary non linear equation the classical limit of the amplitude and phase functions is analysed by setting up a semi classical ermakov system in this limit it is shown that classical quantities such as the classical probability amplitude and the reduced action are obtained only when the semi classical amplitude and the accumulated phase are non oscillating functions respectively of the space and energy variables conversely among the infinitely many arbitrary exact quantum amplitude and phase functions corresponding to a given wavefunction only the non oscillating ones yield classical quantities in the limit 0

the minimum numbers of nodal domains of spherical grushin harmonics abstract in this paper nodal lines and nodal domains of homogeneous grushin harmonic polynomials of degree k in three variables are studied it is proved that for the cases k 4 m m n there exist spherical grushin harmonics f of degree k such that the nodal curves of f divide the unit gauge sphere s into two parts and for the case k 4 m m n there is no a spherical grushin harmonic f of degree k such that the nodal curve of f divides s into two parts this provides a different picture comparing with that of classical spherical harmonics

analytic approximation to the harmonic oscillator equation with a sub period time dependent parameter analytic solutions are presented for the time dependent harmonic oscillator equation as well as for the corresponding amplitude ermakov and phase equations the solutions are adequate approximations for an initially constant amplitude and a time dependent parameter that varies continuously in a time much shorter than the characteristic period the amplitude and phase representation of the solution is shown not to be unique the relationship between the phase independent amplitudes is derived from the orthogonal functions invariant

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

gaussian random measures generated by berry s nodal sets we consider vectors of random variables obtained by restricting the length of the nodal set of berry s random wave model to a finite collection of possibly overlapping smooth compact subsets of mathbb r 2 our main result shows that as the energy diverges to infinity and after an adequate normalisation these random elements converge in distribution to a gaussian vector whose covariance structure reproduces that of a homogeneous independently scattered random measure a by product of our analysis is that when restricted to rectangles the dominant chaotic projection of the nodal length field weakly converges to a standard wiener sheet in the banach space of real valued continuous mappings over a fixed compact set an analogous study is performed for complex valued random waves in which case the nodal set is a locally finite collection of random points

on the number of excursion sets of planar gaussian fields the nazarov sodin constant describes the average number of nodal set components of smooth gaussian fields on large scales we generalise this to a functional describing the corresponding number of level set components for arbitrary levels using results from morse theory we express this functional as an integral over the level densities of different types of critical points and as a result deduce the absolute continuity of the functional as the level varies we further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields including the important special case of the random plane wave

on probability measures arising from lattice points on circles a circle centered at the origin and with radius chosen so that it has non empty intersection with the integer lattice mathbb z 2 gives rise to a probability measure on the unit circle in a natural way such measures and their weak limits are said to be attainable from lattice points on circles we investigate the set of attainable measures and show that it contains all extreme points in the sense of convex geometry of the set of all probability measures that are invariant under some natural symmetries further the set of attainable measures is closed under convolution yet there exist symmetric probability measures that are not attainable to show this we study the geometry of projections onto a finite number of fourier coefficients and find that the set of attainable measures has many singularities with a fractal structure this complicated structure in some sense arises from prime powers singularities do not occur for circles of radius sqrt n if n is square free

statistics of nodal lines and points in chaotic quantum billiards perimeter corrections fluctuations curvature for real time reversal symmetric quantum billiards the mean length l of nodal line is calculated for the nth mode n 1 with wavenumber k using a gaussian random wave model adapted locally to satisfy dirichlet or neumann boundary conditions the leading term is of order k i e n and the first perimeter correction dominated by an unanticipated long range boundary effect is of order log k i e log n with the same sign negative for both boundary conditions the leading order state to state fluctuations l are of order log k for the curvature of nodal lines and 2 are of order k but 3 and higher moments diverge for complex e g aharonov bohm billiards the mean number n of nodal points phase singularities in the mode has a leading term of order k2 i e n the perimeter correction again a long range effect is of order klog k i e nlog n and positive notwithstanding nodal depletion near the boundary and the fluctuations n are of order k log k generalizations of the results for mixed robin boundary conditions are stated

an amplitude phase approach to calculating regge pole positions and residues new amplitude phase formulae for regge pole positions and residues are derived the derivation makes use of certain invariants of the ermakov lewis type the formulas allow calculation to be made on the real r axis with an additional flexibility to optimize its numerical aspects

exercises in exact quantization the formalism of exact 1d quantization is reviewed in detail and applied to the spectral study of three concrete schrodinger hamiltonians d2 dq2 v q on the half line q 0 with a dirichlet or neumann condition at q 0 emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters we first discuss the homogeneous potential v q qn as n versus its solvable n limit an infinite square well useful distinctions are established between regular and singular behaviours of spectral quantities various identities among the square well spectral functions are unravelled as limits of finite n properties the second model is the quartic anharmonic oscillator the zero energy spectral determinants det d2 dq2 q4 vq2 are explicitly analysed in detail revealing many special values algebraic identities between taylor coefficients and functional equations of a quartic type coupled to asymptotic v properties of airy type the third study addresses the potentials v q qn vqn 2 1 of even degree their zero energy spectral determinants prove computable in closed form and the generalized eigenvalue problems with v as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case corresponding to n 2 these results partly reflect the presence of quasi exactly solvable potentials in the family above

a note on generalization of the lewis invariant and the ermakov systems in this paper we examine several possible generalizations of the well known lewis invariant and the ermakov systems the analysis relies on the application of the theorem of emmy noether the problem of constructing lewis type invariants is reduced to a problem of solving a single partial differential equation the so called potential equation within this approach the role of the auxiliary equation is clarified by an analysis of linear systems generalization of the ermakov type procedure for derivation of the lewis invariant is also discussed moreover different forms of general solution of the problem will be provided which extend the range of possible applications of the analysis

solution of the schrodinger equation for time dependent 1d harmonic oscillators using the orthogonal functions invariant an extension of the classical orthogonal functions invariant to the quantum domain is presented this invariant is expressed in terms of the hamiltonian unitary transformations which involve the auxiliary function of this quantum invariant are used to solve the time dependent schrodinger equation for a harmonic oscillator with time dependent parameter the solution thus obtained is in agreement with the results derived using other methods which invoke the lewis invariant in their procedures

quantitative quantum ergodicity and the nodal domains of hecke maass cusp forms we prove a quantitative statement of the quantum ergodicity for hecke maass cusp forms on the modular surface as an application of our result along a density 1 subsequence of even hecke maass cusp forms we obtain a sharp lower bound for the l 2 norm of the restriction to a fixed compact geodesic segment of eta iy y 0 subset mathbb h we also obtain an upper bound of o epsilon left t phi 3 8 epsilon right for the l infty norm along a density 1 subsequence of hecke maass cusp forms for such forms this is an improvement over the upper bound of o epsilon left t phi 5 12 epsilon right given by iwaniec and sarnak in a recent work of ghosh reznikov and sarnak the authors proved for all even hecke maass forms that the number of nodal domains which intersect a geodesic segment of eta grows faster than t phi 1 12 epsilon for any epsilon 0 under the assumption that the lindelof hypothesis is true and that the geodesic segment is long enough upon removing a density zero subset of even hecke maass forms we prove without making any assumptions that the number of nodal domains grows faster than t phi 1 8 epsilon for any epsilon 0

local universality for zeros and critical points of monochromatic random waves this paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves phi lambda of frequency lambda on a compact smooth riemannian manifold m g as lambda rightarrow infty we prove global variance estimates for the measures of integration over the zeros and critical points of phi lambda these global estimates hold for a wide class of manifolds for example when m g has no conjugate points and rely on new local variance estimates on zeros and critical points of phi lambda in balls of radius approx lambda 1 around a fixed point our local results hold under conditions about the structure of geodesics that are generic in the space of all metrics on m

non recursive functions knots with thick ropes and self clenching thick hyperspheres we introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n dimensional sphere for n 5 sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem this recursion theoretic approach is applied in this paper to a class of functionals on the space of c1 1 smooth hypersurfaces diffeomorphic to sn in rn 1 where n is any fixed number 5 the simplest of these functionals kv is defined by the formula kv n vol n 1 n r n where r n denotes the radius of injectivity of the normal exponential map for n rn l we prove the existence of an infinite set of distinct locally minimal values of kv on the space of c1 1 smooth topological hyperspheres in rn 1 for any n 5 the functional kv naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of thick circles and more generally thick spheres into euclidean spaces we introduce the notion of knot with thick rope types the theory of knot with thick rope types turns out to be quite different from the classical knot theory because of the following result there exists an infinite set of non trivial knot with thick rope types in codimension one for every dimension greater than or equal to five

arithmetic zeros and nodal domains on the sphere we obtain lower bounds for the number of nodal domains of hecke eigenfunctions on the sphere assuming the generalized lindelof hypothesis we prove that the number of nodal domains of any hecke eigenfunction grows with the eigenvalue of the laplacian by a very different method we show unconditionally that the average number of nodal domains of degree l hecke eigenfunctions grows significantly faster than the uniform growth obtained under lindelof

effects of a transient barrier on wavepacket traversal an analytical treatment of a propagating wavepacket incident on a transient barrier reveals an effect in which for a particular time interval the time varying transmission probability exceeds that for the free propagation of the wavepacket we show that this effect can be interpreted semiclassically this effect is quantified and it is shown that its magnitude is in one to one correspondence with the strength of the barrier a feature that has the potential to be used in a scheme for key generation it is found that the speed with which the information about the barrier perturbation propagates across the wavepacket can exceed the group velocity of the wavepacket an application to the speed up of entanglement generation is also considered

hamilton jacobi theory and the quantum action variable a quantum action variable is defined in the context of a quantum hamilton jacobi theory the action variable can be used to find the exact bound state energy levels of a quantum system without solving an equation of motion for the system wave functions

weak measurements of trajectories in quantum systems classical bohmian and sum over paths weak values obtained from weak measurements attempt to describe the properties of a quantum system as it evolves from an initial to a final state without practically altering this evolution trajectories can be defined from weak measurements of the position or inferred from weak values of the momentum operator the former can be connected to the asymptotic form of the feynman propagator and the latter to bohmian trajectories employing a time dependent oscillator as a model this work analyzes to what extent weak measurements can shed light on the underlying dynamics of a quantum system expressed in terms of trajectories in particular by comparing the two approaches

hole probability for nodal sets of the cut off gaussian free field abstract let g be a closed connected surface equipped with a riemannian metric let n n n and n n n be the increasing sequence of eigenvalues and the sequence of corresponding l 2 normalized eigenfunctions of the laplacian on for each l 0 we consider l 0 n l n n n where the n are i i d centered gaussians with variance 1 as l l converges a s to the gaussian free field on in the sense of distributions we first compute the asymptotic behavior of the covariance function for this family of fields as l we then use this result to obtain the asymptotics of the probability that l is positive on a given open proper subset with smooth boundary in doing so we also prove the concentration of the supremum of l around 1 2 ln l

a symbolic algorithm for studying adjustments to the profile likelihood summary a symbolic algorithm for the derivation of the asymptotic expansion of the maximum likelihood estimate was first introduced by andrews stafford 1993 this algorithm is quite general and can be modified to give asymptotic expansions for the maximum likelihood estimate based on either the conditional modified or adjusted profile likelihoods expansions are obtained quickly with no chance for human clerical error as a result these three adjustments can be easily studied and some new statistical results are obtained bartlett corrections of each adjusted likelihood and asymptotic expansions for the first four moments of the maximum likelihood estimate based on the conditional and adjusted profile likelihoods are derived

on the nodal count statistics for separable systems in any dimension we consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in an arbitrary dimension we give an explicit expression for the limiting distribution of normalized nodal counts and analyse some of its universal properties our results are illustrated by the detailed discussion of simple examples and numerical nodal count distributions

note on invariants for uncoupled ermakov systems several invariants for an ermakov system consisting of a parametric oscillator and a milne oscillator in one dimension are derived this system appears in amplitude phase analysis of parametric oscillator solutions first and second order invariants for such a system are derived using the wronskian relations of a fundamental pair of parametric oscillator solutions in this way wronskian invariants of the parametric oscillator system alone imply ermakov lewis type invariants for the combined ermakov system as well as invariants for the milne oscillator alone

stein malliavin approximations for nonlinear functionals of random eigenfunctions on s d abstract we investigate stein malliavin approximations for nonlinear functionals of geometric interest for random eigenfunctions on the unit d dimensional sphere s d d 2 all our results are established in the high energy limit i e as the corresponding eigenvalues diverge in particular we prove a quantitative central limit theorem for the excursion volume of gaussian eigenfunctions this goal is achieved by means of several results of independent interest concerning the asymptotic analysis for the variance of moments of gaussian eigenfunctions the rates of convergence in various probability metrics for hermite subordinated processes and quantitative central limit theorems for arbitrary polynomials of finite order or general square integrable nonlinear transforms some related issues were already considered in the literature for the 2 dimensional case s 2 our results are new or improve the existing bounds even in these special circumstances proofs are based on the asymptotic analysis of moments of all order for gegenbauer polynomials and make extensive use of the recent literature on so called fourth moment theorems by nourdin and peccati

quantum tunnelling of a damped and driven inverted harmonic oscillator using the evolution operator method we derive the exact propagator of the generalized parametric oscillator in its more general form this result is exploited to obtain the exact wavefunction of a damped and driven inverted harmonic oscillator of the caldirola kanai type taking a gaussian wavepacket as the initial state we discuss the tunnelling process of such a system the probability density and the persistence probability are evaluated the expression for the sojourn time is derived for a small external force and is the sum of two terms whose explicit forms are obtained in the case of an extended wavepacket the first term is an increasing function of the dissipation parameter gamma whereas the second one is strictly due to the presence of the driving force

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

exact invariants for a class of time dependent nonlinear hamiltonian systems a method of generalizing a class of invariants for a time dependent linear oscillator is developed for the motion of a mass point in one dimension with a general time dependent nonlinear potential formulas are derived for the allowable time dependent potentials and for the corresponding invariants the method by which these conclusions are reached is interesting theoretically and is explained in detail

on the linearization of the generalized ermakov systems a linearization procedure is proposed for ermakov systems with frequency depending on dynamic variables the procedure applies to a wide class of generalized ermakov systems which are shown to be linearizable in a manner similar to that applicable to usual ermakov systems the kepler ermakov systems belong to this category but other more generic systems are also included

tools for the symbolic computation of asymptotic expansions summary this paper describes a collection of procedures for the systematic computation of asymptotic expansions that are common in statistical theory and practice expansions of functions of sums of independent and identically distributed random variables the procedures permit the expansion of maximum likelihood estimates the associated deviance or drop in likelihood and more general functions of random variables with distributions involving one or more parameters the procedures are illustrated with examples involving general and specific laws much of statistical theory and practice is based on asymptotic expansions many programs are available to assist in the numerical evaluation of such expansions but there is a need for computational tools to assist in their derivation and symbolic evaluation heller 1991 shows how symbolic calculation may be used in a wide variety of statistical problems kendall 1988 1990 gives procedures for the symbolic computation of expressions in the analysis of the diffusion of euclidean shape silverman and young 1987 use computer algebra to evaluate criteria on which the decision to smooth a bootstrap distribution is based young and daniels 1990 apply symbolic computation to evaluate expressions in the assessment of bootstrap bias venables 1985 utilizes symbolic computation heavily to obtain expansions of maximum marginal likelihood estimates most notably fisher s a statistic barndorff nielsen and blasild 1986 describe procedures for the numerical calculation of bartlett factors in cases where cumulants of the likelihood function may be specified most of these references involve the evaluation of complicated formulae in particular cases and not the derivation of the formulae themselves here we give general procedures for both the derivation of formulae and their evaluation in specific cases the derivation of asymptotic expansions is typically a simple but laborious task consider for example the calculation of the expectation of the likelihood ratio test statistic for a one parameter family to order 1 n this may be accomplished in general

a note on the calculation of bartlett adjustments summary we describe for the numerical calculation of bartlett adjustments a method which may be of use when the cumulants of the log likelihood derivatives are easy to determine in one parametrization while the hypotheses to be tested are all linear in some other parametrization the method relies on the cartesian tensorial nature of those cumulants and should be particularly convenient in connection with statistical packages with structure similar to glim

quantum tunnelling effect for the inverted caldirola kanai hamiltonian in order to study the behaviour of the inverted caldirola kanai hamiltonian in the quantum tunnelling effect the authors consider a wavepacket as the initial state and they calculate exactly the probability density they also obtain the transmission and reflection probability the expectation value of the particle s energy and the sojourn time which appears to be an increasing function of the dissipation parameter gamma a similar result is also obtained by considering a normalized plane wave as the initial state

generalization of the amplitude phase s matrix formula for coupled scattering states the amplitude phase method is generalized to coupled schr dinger scattering states with a common angular momentum quantum number a pair of exponential type amplitude phase solutions u j r exp ij r for each channel is obtained containing a common complex scalar phase function j r and two column vector amplitudes u j r the amplitude functions satisfy certain nonlinear generalized milne equations and the scalar product of the two amplitudes determines the derivative of the common phase function fundamental amplitude phase matrix solutions that are proportional to jost like schr dinger matrix solutions are constructed it is shown how a generalized amplitude phase s matrix formula can be derived from wronskian relations involving the two amplitude phase matrix solutions and a regular matrix solution

variation of the nazarov sodin constant for random plane waves and arithmetic random waves abstract this is a manuscript containing the full proofs of results announced in 10 together with some recent updates we prove that the nazarov sodin constant which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random gaussian field genuinely depends on the field we then infer the same for arithmetic random waves i e random toral laplace eigenfunctions

scattering s matrix derived from invariants of the ermakov lewis type an amplitude phase formula for the s matrix due to a central potential is derived the derivation makes use of invariants of the ermakov lewis type

nodal domains of a non separable problem the right angled isosceles triangle we study the nodal set of eigenfunctions of the laplace operator on the right angled isosceles triangle a local analysis of the nodal pattern provides an algorithm for computing the number n of nodal domains for any eigenfunction in addition an exact recursive formula for the number of nodal domains is found to reproduce all existing data eventually we use the recursion formula to analyse a large sequence of nodal counts statistically our analysis shows that the distribution of nodal counts for this triangular shape has a much richer structure than the known cases of regular separable shapes or completely irregular shapes furthermore we demonstrate that the nodal count sequence contains information about the periodic orbits of the corresponding classical ray dynamics

a conditional saddlepoint approximation for testing problems a saddlepoint approximation is provided for the distribution function of one m statistic conditional on another m statistic many interesting statistics based on dependent quantities e g spacings multinomial frequencies rank differences can be expressed in terms of independent identically distributed random variables conditioned on their sum so that this conditional saddlepoint approximation yields accurate approximations for the distribution of such statistics this saddlepoint approximation can also be used in conditional testing where nuisance parameters are eliminated by conditioning on sufficient statistics

on parameter transformations and interval estimation summary parameterizations which reduce the asymptotic bias and skewness of various pivotal quantities arising in large sample theory are discussed for models depending on an unknown scalar parameter transformation formulae by which such parameterizations can be obtained are derived and these formulae extend those for one dimensional curved exponential families given by hougaard 1982 to assess the accuracy of normal approximations to the distributions of the pivots their second order properties are considered and comparisons with the signed square root of the likelihood ratio statistic are drawn

nodal statistics on quantum graphs it has been suggested that the distribution of the suitably normalized number of zeros of laplacian eigenfunctions contains information about the geometry of the underlying domain we study this distribution more precisely the distribution of the nodal surplus for laplacian eigenfunctions of a metric graph the existence of the distribution is established along with its symmetry one consequence of the symmetry is that the graph s first betti number can be recovered as twice the average nodal surplus of its eigenfunctions furthermore for graphs with disjoint cycles it is proven that the distribution has a universal form it is binomial over the allowed range of values of the surplus to prove the latter result we introduce the notion of a local nodal surplus and study its symmetry and dependence properties establishing that the local nodal surpluses of disjoint cycles behave like independent bernoulli variables

the bootstrap to smooth or not to smooth summary the bootstrap and smoothed bootstrap are considered as alternative methods of estimating properties of unknown distributions such as the sampling error of parameter estimates criteria are developed for determining whether it is advantageous to use the smoothed bootstrap rather than the standard bootstrap key steps in the argument leading to these criteria include the study of the estimation of linear functionals of distributions and the approximation of general functionals by linear functionals consideration of an example the estimation of the standard error in the variance stabilized sample correlation coefficient elucidates previously published simulation results and also illustrates the use of computer algebraic manipulation as a useful technique in asymptotic statistics finally the various approximations used are vindicated by a simulation study

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m