quasilinearization method and its verification on exactly solvable models in quantum mechanics the proof of the convergence of the quasilinearization method of bellman and kalaba whose origin lies in the theory of linear programming is extended to large and infinite domains and to singular functionals in order to enable the application of the method to physical problems this powerful method approximates solution of nonlinear differential equations by treating the nonlinear terms as a perturbation about the linear ones and is not based unlike perturbation theories on existence of some kind of small parameter the general properties of the method particularly its uniform and quadratic convergence which often also is monotonic are analyzed and verified on exactly solvable models in quantum mechanics namely application of the method to scattering length calculations in the variable phase method shows that each approximation of the method sums many orders of the perturbation theory and that the method reproduces properly the singular structure of the exact solutions the method provides final and reasonable answers for infinite values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist

slopes of eigencurves over boundary disks let p be a prime number we study the slopes of u p eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra we give a sharp lower bound of the corresponding newton polygon the computation happens over a definite quaternion algebra by jacquet langlands correspondence it generalizes a prior work of jacobs slopes of compact hecke operators thesis university of london imperial college 2004 who treated the case of p 3 with a particular level in case when the modular forms have a finite character of conductor highly divisible by p we improve the lower bound to show that the slopes of u p eigenvalues grow roughly like arithmetic progressions as the weight k increases this is the first very positive evidence for buzzard kilford s conjecture on the behavior of the eigencurve near the boundary of the weight space that is proved for arbitrary p and general level we give the exact formula of a fraction of the slope sequence

jwkb method and quasilinearisation the eikonal solution of the radial schrodinger equation for spherically symmetric potentials is treated by the quasilinearisation technique the first iteration contains all the terms of the full expansion of the eikonal series obtained by froman and froman 1965 but with different coefficients the strength of the centrifugal potential appropriately modified in each order of approximation is obtained stipulating the proper behaviour of the wavefunction at the origin it leads to an iterative map which converges very fast due to quadratic convergence of quasilinearisation the energy eigenvalues for the coulomb potential are obtained by this technique

improved generalized quasilinearization method and rapid convergence for reaction diffusion equations abstract in the method of quasilinearization or the method of generalized quasilinearization we assume that k f t x u u k exists where f t x u is the forcing function and obtain kth order of convergence in this paper we assume a weaker condition namely k 1 f t x u u k 1 exists and lipschitzian in u and develop generalized quasilinearization method to reaction diffusion equations the iterates will be different depending on k 1 f t x u u k 1 is nondecreasing or nonincreasing in u and k being even or odd finally we prove that the sequences generated by the generalized quasilinearization method converge to the unique solution of the nonlinear reaction diffusion equation and the convergence is of order k

numerical investigation of generalized quasilinearization method for reaction diffusion systems in we have developed generalized quasilinearization method for reaction diffusion systems when the forcing functions are the sum of convex and concave functions the solutions of the corresponding linear systems converge monotonically uniformly and quadratically to the unique solution of the nonlinear problem as a byproduct of our result we have discussed the reaction diffusion system when the forcing function satisfies mixed quasimonotone property in this paper we have established the application of the theoretical results developed in with numerical examples we also demonstrate the consistency of the finite different scheme and discuss the stability and convergence of the scheme for the examples considered here

an efficient multi step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes an efficient multi step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes

triple product p adic l functions for balanced weights we construct p adic triple product l functions that interpolate square roots of central critical l values in the balanced region thus our construction complements that of harris and tilouine there are four central critical regions for the triple product l functions and two opposite settings according to the sign of the functional equation in the first case three of these regions are of interpolation having positive sign they are called the unbalanced regions and one gets three p adic l functions one for each region of interpolation this is the harris tilouine setting in the other setting there is only one region of interpolation called the balanced region we produce the corresponding p adic l function our triple product p adic l function arises as p adic period integrals interpolating normalizations of the local archimedean period integrals the latter encode information about classical representation theoretic branching laws the main step in our construction of p adic period integrals is showing that these branching laws vary in a p adic analytic fashion this relies crucially on the ash stevens theory of highest weight representations over affinoid algebras

p adic families of siegel modular cuspforms let p be an odd prime and g 2 an integer we prove that a nite slope siegel cuspidal eigenform of genus g can be p adically deformed over the g dimensional weight space the proof of this theorem relies on the construction of a family of sheaves of locally analytic overconvergent modular forms contents

a sensitivity analysis for least squares ill posed problems using the haar basis this paper applies the wavelet theory to provide some information about the stability and resolution limit of inverse problems a sensitivity analysis for one dimensional least squares ill posed problems is done by using the haar basis three mappings from the parameter space l 2 r into the observation space l 2 r are considered the sensitivity in the direction of a haar function of width h is o h gamma which approximates the lower bound of sensitivity of the mapping where gamma 0 implying that the sensitivity in the haar basis directions has a rapid decay when h to 0 this gives a quantitative relation between the sensitivity of the mapping and the parameter resolution scale in this way the haar basis decomposes the parameters to be estimated into parts with different sensitivity order providing an easy method to identify the suitability of multiresolution algorithms for inverse problems and to choose a convenient discretization size h

hida families and rational points on elliptic curves let e be an elliptic curve over q of conductor n mp having a prime p n of multiplicative reduction because e is modular it corresponds to a normalised weight two eigenform on 0 n whose q expansion is denoted f n anq letx hom z p z p z p 1 z zp which contains z as a dense subset by associating to k z the character x xk 2 denote by a u the ring of cp valued p adic analytic functions on a compact open subset u of x hida s theory associates to f a neighborhood u of 2 x which can be assumed for simplicity to be contained in the residue class of 2 modulo p 1 and a formal q expansion

on the coefficients of the characteristic series of the u operator abstract a conceptual proof is given of the fact that the coefficients of the characteristic series of the u operator acting on families of overconvegent modular forms lie in the iwasawa algebra

an efficient iterative method for computing deflections of bernoulli euler von karman beams on a nonlinear elastic foundation abstract an efficient iterative method is developed for the static analysis of large deflections of an infinite beam with variable cross section resting on a nonlinear foundation a pseudo spring constant is added and explicit matrix operators are introduced to perform differentiation through green s function the nonlinearity of the problem is handled with quasilinearization to compute the solution of the quasilinear differential equation with prescribed accuracy a new discretization method for solving quasilinear differential equations involving up to the 4th order derivative is used the discretization method is based on relating discretizations of up to the fourth order derivative of the solution with a discretization of the solution by using a suitable green function numerical experiments show that the error incurred by the discretization can be made small for the two first derivatives and that the method proposed in the paper converges fast and has good accuracy

quasilinearisation and three dimensional oscillators the anharmonic symmetric oscillator is a frequently used model for physical systems the recently developed quasilinearisation technique is applied to this model as well as to the simpler one without the quartic term in the potential

the eigencurve over the boundary of weight space author s liu ruochuan wan daqing xiao liang abstract we prove that the eigencurve associated to a definite quaternion algebra over qq satisfies the following properties as conjectured by coleman mazur and buzzard kilford a over the boundary annuli of weight space the eigencurve is a disjoint union of countably infinitely many connected components each finite and flat over the weight annuli b the u p slopes of points on each fixed connected component are proportional to the p adic valuations of the parameter on weight space and c the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference in particular as a point moves towards the boundary on an irreducible connected component of the eigencurve the slope converges to zero

representation reduction and solution space contraction in quasi exactly solvable systems in quasi exactly solvable problems partial analytic solutions energy spectrum and associated wavefunctions are obtained if some potential parameters are assigned specific values we introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters to obtain a larger solution space one varies the energy over a discrete set the spectrum by simply changing the value of a given integer a unified treatment that includes the standard as well as the new class of quasi exactly solvable problems is presented and a few examples are given the solution space is spanned by discrete square integrable basis functions in which the matrix representation of the hamiltonian is tridiagonal consequently the wave equation gives a three term recursion relation for the expansion coefficients of the wavefunction imposing quasi exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component the finite is real and exactly solvable whereas the infinite is complex and associated with zero norm states consequently the whole physical space contracts to a finite dimensional subspace with normalizable states

une correspondance de jacquet langlands p adique we extend the jacquet langlands correspondence between the hecke modules of usual and quaternionic modular forms to overconvergent p adic forms of finite slope we show that this correspondence respects p adic families and is induced by an isomorphism between some associated eigencurves

nonlinearity scale and sensitivity for parameter estimation problems we study model nonlinearity and sensitivity for parameter estimation problems previous work has revealed a correlation between high nonlinearity low sensitivity and short scale for an ode model in this paper we investigate whether this correlation is valid for a larger class of model functions we set forth a proposition that says in essence that the correlation holds when the forward model output is an integral solutions to odes and pdes can be viewed as integral models if the proposition is true it may explain an apparent conflict between earlier works on uncertainty analysis for parameter estimation problems it could also have impact on the choice of solution algorithm for such problems the validity of the proposition is assessed by studying nonlinearity and sensitivity for fairly general nonlinear models and corresponding integrals of these models theory is developed and its predictions are tested through numerical examples the focus is on general effects for the model classes integrated nonintegrated i e effects that do not depend on any specific choice of model function within each class

unified qlm for regular and arbitrary singular potentials a parameterless numerical implementation of the quasilinearization method qlm is constructed and tested to 21 25 digits precision to give quadratically convergent energies e of the klauder effect exhibiting spiked harmonic oscillator with the r or exp r type spikes in a riccati reformulation of the schrodinger equation the radial solution is uniformly quadratic convergent to the same precision as e except in the small minorization interval where the self correcting property of qlm assures geometric convergence like in the picard algorithm to about 12 16 digits sufficient not to affect the convergence of e confirming what is expected on physical grounds it was shown before that for regular potentials immediate onset of quadratic convergence is guaranteed by the initial iteration of the wkb form and that for quadratic convergence of e for power type spikes it suffices to augment this by a nonlinear integration point distribution and by minorization of negative solution values the form of the riccati equation used allows the minorization function to easily be formally defined over the entire interval without the need for a cutoff radius of application and dependence on its scale factor is plateau like and negligible

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

higher even order convergence and coupled solutions for second order boundary value problems on time scales in this paper for a second order boundary value problem on time scales a method of generalized quasilinearization under coupled upper and lower solutions is discussed an attempt for the method is to establish sufficient conditions for generating monotone iterative schemes whose elements converge rapidly to the unique solution of the given problem furthermore the convergence is of order k k 2 which is even finally two examples are provided to illustrate our results

overconvergent modular forms and the fontaine mazur conjecture we prove a conjecture of fontaine and mazur on modularity of representations of g which are potentially semi stable at p for representations coming from finite slope overconvergent eigenforms we also give an application of our technique to a question of gouvea on overconvergence of certain modular forms if is a suitable representation of g s on a two dimensional vector space over a finite field product of m and the s a finite set of primes we construct a rigid analytic subspace of the universal deformation space of which is defined by a purely representation theoretic condition and contains the eigencurve of coleman mazur conjecturally it is equal to this curve

a multiresolution method for distributed parameter estimation a multiresolution method for distributed parameter estimation or inverse problems is studied numerically the identification of the coefficient of an elliptic equation in one dimension is considered as our model problem first multiscale bases are used to analyze the degree of ill posedness of the inverse problem second based on some numerical results it is shown that the method of scale by scale multiresolution yields robust and fast convergence finally it is shown how the method gives a natural regularization approach which is complementary to tikhonov s regularization

p adic banach spaces and families of modular forms let p be a prime cp the completion of an algebraic closure of the p adic numbers qp and k a nite extension of qp contained in cp let v be the valuation on cp such that v p 1 and let be the absolute value on cp such that x p v x for x cp suppose n is a positive integer prime to p let x1 np denote the modular curve over k which represents elliptic curves with 1 np structure and let up be the hecke operator on modular forms on x1 np which takes a form with q expansion n anq n to the modular form with q expansion n anpq n a modular form f is said to have slope q if there is a polynomial r t over cp such that r up f 0 and such that the newton polygon of r t has only one side and its slope is

analytic calculation of energies and wave functions of the quartic and pure quartic oscillators ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the schrodinger equation into a nonlinear riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method qlm in the qlm the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions the qlm is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter our explicit analytic results are then compared with exact numerical and also with wkb solutions and it is found that our ground state wave functions using a range of small to large coupling constants yield a precision of between 0 1 and 1 percent and are more accurate than wkb solutions by two to three orders of magnitude in addition our qlm wave functions are devoid of unphysical turning point singularities and thus allow one to make analytic

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

completed cohomology of shimura curves and a p adic jacquet langlands correspondence we study indefinite quaternion algebras over totally real fields f and give an example of a cohomological construction of p adic jacquet langlands functoriality using completed cohomology we also study the tame levels of p adic automorphic forms on these quaternion algebras and give an analogue of mazur s level lowering principle

on the semi simplicity of the u_p operator on modular forms for and positive integers let denote the vector space of cuspidal modular forms of level and weight this vector space is equipped with the usual hecke operators if we need to consider several levels or weights at the same time we will denote this by or if is a prime number dividing our is also known under the name one of our main results can be stated very easily if and does not divide then the operator is semi simple we can prove the same result for weight under the assumption that certain crystalline frobenius elements are semi simple milne has shown in 11 0 2 that this semi simplicity is implied by tate s conjecture claiming that for 1 projective and smooth over a finite field of characteristic and 23 54 687 9 ab c1d fehgjilknmo equals the order of pq c1r sb at 2 ulmer proved in 17 that is semi simple for 5 t and u not dividing under the assumption of the birch swinnerton dyer conjecture for elliptic curves over function fields in characteristic his method is quite different from ours assuming that is not semi simple he really shows that the birch swinnerton dyer conjecture does not hold for an explicitly given elliptic curve the structure of our proof is as follows using the theory of newforms the problem is shown to be equivalent to the problem of showing that for a normalized newform v of weight prime to level and character w the polynomial x uzy x w8 ba c has no double root this polynomial happens to be the characteristic polynomial of the frobenius element at in the two dimensional galois representations associated to v it is also the characteristic polynomial of the crystalline frobenius asociated to v we show that this crystalline frobenius cannot be a scalar in sections 2 and 3 we prove the results concerning these frobenius elements for d and e f respectively section 2 is quite elementary whereas in section 3 we use a lot of the machinary for comparing adic etale and crystalline cohomology in section 4 we give some applications the ramanujan inequality is a strict inequality in certain cases certain hecke algebras are reduced hence have non zero discriminant section 5 gives some results due to abbes and ullmo concerning the discriminants of certain hecke algebras g partially supported by the institut universitaire de france

the analysis of an ill posed problem using multi scale resolution and second order adjoint techniques abstract a wavelet regularization approach is presented for dealing with an ill posed problem of adjoint parameter estimation applied to estimating inflow parameters from down flow data in an inverse convection case applied to the two dimensional parabolized navier stokes equations the wavelet method provides a decomposition into two subspaces by identifying both a well posed as well as an ill posed subspace the scale of which is determined by finding the minimal eigenvalues of the hessian of a cost functional measuring the lack of fit between model prediction and observed parameters the control space is transformed into a wavelet space the hessian of the cost is obtained either by a discrete differentiation of the gradients of the cost derived from the first order adjoint or by using the full second order adjoint the minimum eigenvalues of the hessian are obtained either by employing a shifted iteration method x zou i m navon f x le dimet tellus 44a 4 1992 273 or by using the rayleigh quotient the numerical results obtained show the usefulness and applicability of this algorithm if the hessian minimal eigenvalue is greater or equal to the square of the data error dispersion in which case the problem can be considered as well posed i e regularized if the regularization fails i e the minimal hessian eigenvalue is less than the square of the data error dispersion of the problem the following wavelet scale should be neglected followed by another algorithm iteration the use of wavelets also allowed computational efficiency due to reduction of the control dimension obtained by neglecting the small scale wavelet coefficients

beyond wkb quantum corrections to hamilton jacobi theory in this paper we develop quantum mechanics of quasi one dimensional systems upon the framework of the quantum mechanical hamilton jacobi theory we will show that the schrodinger point of view and the hamilton jacobi point of view are fully equivalent in their description of physical systems but differ in their descriptive manner as a main result of this a wavefunction in hamilton jacobi theory can be decomposed into travelling waves in any point in space not only asymptotically using the quasi linearization technique we derive quantum correction functions in every order of the quantum correction functions will remove the turning point singularity that plagues the wkb series expansion already in zeroth order and thus provide an extremely good approximation to the full solution of the schrodinger equation in the language of quantum action it is also possible to elegantly solve the connection problem without asymptotic approximations the use of quantum action further allows us to derive an equation by which the maslov index is directly calculable without any approximations stationary quantum trajectories will also be considered and thoroughly discussed

accurate solution of the thomasfermi equation using the fractional order of rational chebyshev functions in this paper the nonlinear singular thomasfermi differential equation for neutral atoms is solved using the fractional order of rational chebyshev orthogonal functions frcs of the first kind ftn t l on a semi infinite domain where l is an arbitrary numerical parameter first using the quasilinearization method the equation be converted into a sequence of linear ordinary differential equations ldes and then these ldes are solved using the frcs collocation method using 300 collocation points we have obtained a very good approximation solution and the value of the initial slope y 0 1 5880710226113753127186845094239501095 highly accurate to 37 decimal places

anharmonic oscillators and generalized squeezed states we present a new method to study the eigenvalues and eigenfunctions of anharmonic oscillators it involves a new class of bogoliubov transformations and leads to the introduction of k photon coherent states we consider the hamiltonians for the simple harmonic and anharmonic oscillators as the two generators of a lie algebra whose other generators may be found exactly or up to any desired order of the perturbation parameter involved an element of this lie group turning out to be the multi photon operator transforms the anharmonic hamiltonian to the harmonic one thus facilitating the calculation of the eigenvalues and eigenfunctions of the former the transformation of the ordinary annihilation and creation operators leads to generalized ones corresponding to generalized oscillation modes and also helps us out to introduce multi photon coherent states we specifically consider four photon coherent states in detail and study time dependent position and momentum uncertainties in these states