lifschitz tails for random schrodinger operator in bernoulli distributed potentials this paper presents an elementary proof of lifschitz tail behavior for random discrete schrodinger operators with a bernoulli distributed potential the proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value this is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the jump were infinite

on the significance of absolutely continuous invariant measures abstract let be a mapping from an interval i into itself for a large class of such maps there exist ergodic measures invariant under which are absolutely continuous with respect to lebesgue measure and others which are continuous but not absolutely continuous the aim of this paper is to deal with the question which of these ergodic measures best describes the real dynamics of to do this we shall model the dynamics of by a markov chain which reflects the perturbations inherent in experimental work or the truncation error in computations when is expanding it is the absolutely continuous invariant measure that acts as a global attractor this explains the observed effect that the absolutely continuous invariant measure is the one that appears in experimental and numerical work

vibration analysis by discrete singular convolution this paper explores the utility of a discrete singular convolution algorithm for vibration analysis a number of different realizations of singular convolution kernels are selected to illustrate the present algorithm vibration analysis of strings rods beams diatomic molecules membranes waveguides and thin plates are utilized to test numerical accuracy and speed of convergence of the present approach numerical experiments indicate that the discrete singular convolution is a simple and reliable algorithm for vibration analysis

point wise error estimate of a conservative difference scheme for the fractional schrodinger equation in this paper a new conservative difference scheme is proposed for solving the nonlinear space fractional schrodinger equation the riesz space fractional derivative is approximated by the second order accurate weighted and shifted grunwald difference operator building on a careful analysis of this operator the conservation properties including the mass and energy are analyzed and some norm equivalences are established then the numerical solution is shown to be bounded in various norms uniformly with respect to the discretization parameters and optimal order bounds on the global error of the scheme in the l 2 norm semi h 2 norm and l norm are derived the uniform bounds on the numerical solutions as well as the error bounds hold unconditionally in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed numerical tests are performed to support our theoretical results

fractional equivalent lagrangian densities for a fractional higher order equation in this communication we show that the equivalent lagrangian densities elds of a fractional higher order nonlinear schrodinger equation with stable soliton like solutions can be related in a hitherto unknown way this new relationship is described in terms of a new fractional operator that includes both left and right sided fractional derivatives using this operator it is possible to generate new elds that contain different fractional parts in addition to the already known elds which only differ by a sum of first order partial derivatives of two arbitrary functions

on the nonlocality of the fractional schrodinger equation a number of papers over the past eight years have claimed to solve the fractional schrodinger equation for systems ranging from the one dimensional infinite square well to the coulomb potential to one dimensional scattering with a rectangular barrier however some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal preventing the fractional schrodinger equation from being solved in the usual piecewise fashion we focus on the one dimensional infinite square well and show that the purported ground state which is based on a piecewise approach is definitely not a solution of the fractional schrodinger equation for the general fractional parameter on a more positive note we present a solution to the fractional schrodinger equation for the one dimensional harmonic oscillator with 1

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

enlargement of obstacles for the simple random walk we consider a continuous time simple random walk moving among obstacles which are sites resp bonds of the lattice z d we derive in this context a version of the technique of enlargement of obstacles developed by sznitman in the brownian case this method gives controls on exponential moments of certain death times as well as good lower bounds for certain principal eigenvalues we give an application to recover an asymptotic result of donsker and varadhan on the number of sites visited by the random walk and another application to the number of bonds visited by the random walk

multiplicity and concentration results for fractional schrodinger system with steep potential wells abstract this paper is concerned with the fractional coupled schrodinger system by using the nehari manifold and fibering map we obtain the multiplicity and concentration of solutions for the given problem with steep potential wells where some new estimates will be established in particular although there exist concave convex nonlinearities in the coupled system it is not necessary to assume that the corresponding lebesgue norms of the weight functions of the convex terms need to be small enough

collocation method for fractional quantum mechanics we show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional laplacian using a collocation approach based on little sinc functions which discretizes the schrodinger equation on a uniform grid the different boundary conditions are naturally implemented using sets of functions with the appropriate behavior good convergence properties are observed a comparison with results based on a wentzel kramers brillouin analysis is performed

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

why computers like lebesgue measure let 0 1 0 1 be a transformation which has an absolutely continuous invariant measure let be a realistic deterministic model for we prove that if has long computer trajectories either periodic or non periodic then these computer trajectories have histograms which approach the density of for a large class of piecewise linear transformations we prove the existence of long periodic trajectories

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

vibrations of non homogeneous rectangular membranes abstract an exact solution for the title problem is obtained for the case where the two subdomains of the non homogeneous membrane are of rectangular shape it is shown that the fundamental frequency coefficient obtained by means a of the classical kantorovish method and b the optimized kantorovich method are in very good agreement with the result predicted by the exact method

randomly chosen chaotic maps can give rise to nearly ordered behavior abstract parrondo s paradox j m r parrondo g p harmer d abbott new paradoxical games based on brownian ratchets phys rev lett 85 2000 5226 5229 see also o e percus j k percus can two wrongs make a right coin tossing games and parrondo s paradox math intelligencer 24 3 2002 68 72 states that two losing gambling games when combined one after the other either deterministically or randomly can result in a winning game that is a losing game followed by a losing game a winning game inspired by this paradox a recent study j almeida d peralta salas m romera can two chaotic systems give rise to order physica d 200 2005 124 132 asked an analogous question in discrete time dynamical system can two chaotic systems give rise to order namely can they be combined into another dynamical system which does not behave chaotically numerical evidence is provided in j almeida d peralta salas m romera can two chaotic systems give rise to order physica d 200 2005 124 132 that two chaotic quadratic maps when composed with each other create a new dynamical system which has a stable period orbit the question of what happens in the case of random composition of maps is posed in j almeida d peralta salas m romera can two chaotic systems give rise to order physica d 200 2005 124 132 but left unanswered in this note we present an example of a dynamical system where at each iteration a map is chosen in a probabilistic manner from a collection of chaotic maps the resulting random map is proved to have an infinite absolutely continuous invariant measure acim with spikes at two points from this we show that the dynamics behaves in a nearly ordered manner when the foregoing maps are applied one after the other deterministically as in o e percus j k percus can two wrongs make a right coin tossing games and parrondo s paradox math intelligencer 24 3 2002 68 72 the resulting composed map has a periodic orbit which is stable

existence of solutions for perturbed fractional p laplacian equations abstract the purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p laplacian operator as follows p s u v x u p 2 u a x u r 2 u b x u q 2 u in r n where is a real parameter p s is the fractional p laplacian operator with 0 s 1 p p r min q p s and v a b r n 0 are three positive weights using variational methods we obtain nonexistence and multiplicity results for the above mentioned equations depending on and according to the integrability properties of the ratio a q p b r p our results extend the previous work of autuori and pucci 2013 5 to the fractional p laplacian setting furthermore we weaken one of the conditions used in their paper hence the results of this paper are new even in the fractional laplacian case

solving fractional schrodinger type spectral problems cauchy oscillator and cauchy well this paper is a direct offspring of the work of garbaczewski and stephanovich levy flights and nonlocal quantum dynamics j math phys 54 072103 2013 where basic tenets of the nonlocally induced random and quantum dynamics were analyzed a number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so called fractional quantum mechanics spectral problems presently we give a decisive computer assisted proof for an exemplary finite and ultimately infinite cauchy well problem that spectral solutions proposed so far were plainly wrong as a constructive input we provide an explicit spectral solution of the finite cauchy well the infinite well emerges as a limiting case in a sequence of deepening finite wells the employed numerical methodology algorithm based on the strang splitting method has been tested for an exemplary cauchy oscillator problem whose analytic solution is available an impact of the inherent spatial nonlocality of motion generators upon computer assisted outcomes potentially defective in view of various cutoffs i e detailed eigenvalues and shapes of eigenfunctions has been analyzed

some solutions to the space fractional schrodinger equation using momentum representation method the space fractional schrodinger equation with linear potential delta function potential and coulomb potential is studied under momentum representation using fourier transformation by use of mellin transform and its inverse transform we obtain the energy levels and wave functions expressed in h function for a particle in linear potential field the wave function expressed also by the h function and the unique energy level of the bound state for the particle of even parity state in delta function potential well which is proved to have no action on the particle of odd parity state is also obtained the integral form of the wave functions for a particle in coulomb potential field is shown and the corresponding energy levels which have been discussed in laskin s paper phys rev e 66 056108 2002 are proved to satisfy an equality of infinite limit of the h function all of these results contain the ones of the standard quantum mechanics as their special cases

dynamic parrondo s paradox abstract this paper is focused on studying parrondo s paradox in non linear dynamics specifically how the periodic combination of the individual maps f and g can give rise to chaos or order we construct dynamical systems exhibiting the paradox for several notions of chaos derived from topological dynamics the effect of altering the order of the combination i e considering f g or g f is analyzed as well as the robustness of the parrondo effect under small perturbations conditions for avoiding parrondian dynamics are also obtained placing special emphasis on the notion of chaos given by turbulence

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

constructive approximations to densities invariant under nonexpanding transformations let 0 1 0 1 be a piecewise monotonic nonexpanding map which has an invariant densityg and is topologically conjugate to a piecewise monotonic expanding map where the conjugacy is absolutely continuous an effective computable method is presented for approximatingg

ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential in this paper we show the that the ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential is controlled asymptotically as the system size n goes to infinity by the random variable l n the length the longest consecutive sequence of sites on the lattice with potential equal to zero specifically we will show that for almost every realization of the potential the ground state energy behaves asymptotically as frac pi 2 ell n 1 2 in the sense that the ratio of the quantities goes to one

crank nicolson difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative in this paper the crank nicolson cn difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative is studied the existence of this difference solution is proved by the brouwer fixed point theorem the stability and convergence of the cn scheme are discussed in the l 2 norm when the fractional order is two all those results are in accord with the difference scheme developed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also presented

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

on the number of invariant measures for higher dimensional chaotic transformations lets be a bounded region inr n and letp s l i 1 m be a partition ofs into a finite number of closed subsets having piecewisec2 boundaries of finite n 1 dimensional measure let s s be piecewisec2 onp and expanding in the sense that there exists 0 1 such that for anyi 1 2 m dt i 1 wheredt i 1 is the derivative matrix oft i 1 and is the euclidean matrix norm we prove that for some classes of such mappings for example jabtonski transformations or convexity preserving transformations the number of crossing points constitutes a bound for the number of ergodic absolutely continuous invariant measures we give examples showing that in general the simple bound of one dimensional dynamics cannot be generalized to higher dimensions in fact we show that it is possible to construct piecewise expandingc2 transformations on a fixed partition with a finite number of elements but which have an arbitrarily large number of ergodic absolutely continuous invariant measures

collocation on uniform grids in this paper we derive four sets of sinc like functions defined on a finite interval and obeying different boundary conditions the functions in each set are orthogonal and their nodes are uniformly distributed on the interval we have applied each set to solve a large class of eigenvalue equations with different boundary conditions both on finite intervals and on the real line showing that precise numerical results can be obtained efficiently and rapidly a comparison with results available in the literature is also performed

a linearly implicit conservative difference scheme for the space fractional coupled nonlinear schrodinger equations abstract in this paper a linearly implicit conservative difference scheme for the coupled nonlinear schrodinger equations with space fractional derivative is proposed this scheme conserves the mass and energy in the discrete level and only needs to solve a linear system at each step the existence and uniqueness of the difference solution are proved the stability and convergence of the scheme are discussed and it is shown to be convergent of order o 2 h 2 in the discrete l 2 norm with the time step and mesh size h when the fractional order is two all those results are in accord with the difference scheme proposed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also reported

higher dimensional point transformations and asymptotic measures for cellular automata abstract the evolution of a cellular automata is modelled by a higher dimensional point transformation on configuration space the transformation reflects the interaction of neighbours and has a probabilistic interpretation for certain point transformations the asymptotic measure of sets of configurations can be computed by using computer orbits

collapsing effects in numerical simulation of a class of chaotic dynamical systems and random mappings with a single attracting centre computer simulations of dynamical systems contain discretizations where finite machine arithmetic replaces continuum state spaces in some circumstances complicated theoretical behaviour has a tendency to collapse to trivial and degenerate behaviour as a result of discretizations characteristics of such collapsing effects often seem to depend on the corresponding discretization only in a random way we describe a procedure to construct a stochastic process with similar statistical characteristics results of computer modelling are discussed

absolutely continuous invariant measures for random maps with position dependent probabilities abstract a random map is discrete time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process usually the map k is chosen from a finite collection of maps with constant probability p k in this note we allow the p k s to be functions of position in this case the random map cannot be considered to be a skew product the main result provides a sufficient condition for the existence of an absolutely continuous invariant measure for position dependent random maps on 0 1 geometrical and topological properties of sets of absolutely continuous invariant measures attainable by means of position dependent random maps are studied theoretically and numerically

mass conservative fourier spectral methods for solving the fractional nonlinear schrodinger equation we propose three fourier spectral methods i e the split step fourier spectral ssfs the crank nicolson fourier spectral cnfs and the relaxation fourier spectral refs methods for solving the fractional nonlinear schrodinger nls equation all of them are mass conservative and time reversible and they have the spectral order accuracy in space and the second order accuracy in time in addition the cnfs and refs methods are energy conservative the performance of these methods in simulating the plane wave and soliton dynamics is discussed the ssfs method preserves the dispersion relation and thus it is more accurate for studying the long time behaviors of the plane wave solutions furthermore our numerical simulations suggest that the ssfs method is better in solving the defocusing nls but the cnfs and refs methods are more effective for the focusing nls

effects of phase space discretization on the long time behavior of dynamical systems abstract we investigate the effects that roundoff errors have on discrete time dynamical systems computer generated orbits terminate in loops whose lengths scale with the machine precision in a universal way we present arguments to explain this fact

collapsing of chaos in one dimensional maps in their numerical investigation of the family of one dimensional maps f x d 1 2jxj where 2 diamond et al p diamond et al physica d 86 1999 559 571 have observed the surprising numerical phenomenon that a large fraction of initial conditions chosen at random eventually wind up at 1 a repelling fixed point this is a numerical artifact because the continuous maps are chaotic and almost every true trajectory can be shown to be dense in 1 1 the goal of this paper is to extend and resolve this obvious contradiction we model the numerical simulation with a randomly selected map while they used 27 bit precision in computing f we prove for our model that this numerical artifact persists for an arbitrary high numerical prevision the fraction of initial points eventually winding up at 1 remains bounded away from 0 for every numerical precision 2000 elsevier science b v all rights reserved

comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates this paper presents a comprehensive comparison study between the discrete singular convolution dsc and the well known global method of generalized differential quadrature gdq for vibration analysis so as to enhance the understanding of the dsc algorithm the dsc method is implemented through lagrange s delta sequence kernel dsc lk which utilizes local lagrange polynomials to calculate weighting coefficients whereas the gdq requires global ones moreover it is shown that the treatments of boundary conditions and the use of grid systems are different in the two methods comparison study is carried out on 21 rectangular plates of different combinations of simply supported s clamped c and transversely supported with nonuniform elastic rotational restraint e edges and five rectangular plates of mixed supporting edges some of which with a range of aspect ratios and rotational spring coefficients all the results of the dsc lk agree very well with both those in the literature and newly computed gdq results furthermore it is observed that the dsc lk performs much better for plates vibrating at higher order eigenfrequencies unlike the gdq the dsc lk is numerically stable for problems which require a large number of grid points

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

highly accurate numerical schemes for multi dimensional space variable order fractional schrdinger equations as a natural generalization of the fractional schrdinger equation the variable order fractional schrdinger equation has been exploited to study fractional quantum phenomena in this paper we develop an exponentially accurate jacobigausslobatto collocation jgl c method to solve the variable order fractional schrdinger equations in one dimension 1d and two dimensions 2d in this method the aforementioned problem is reduced to a system of ordinary differential equations odes in the time variable as a result we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations one based on the implicit rungekutta irk method of fourth order and the other based on jacobigaussradau collocation jgr c method the validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1d and 2d the convergence of the methods is graphically analyzed the results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable order nonlinear partial differential equations

collocation approach to the helmholtz eigenvalue problem on multiply connected domains we present a collocation approach to the numerical solution of the helmholtz eigenvalue problem on multiply connected domains of arbitrary shape in two dimensions a suitable representation of the helmholtz equation on an uniform grid is obtained and the problem is converted to the calculation of the first few eigenvalues and eigenvectors of a large sparse matrix the accuracy of this approach is tested by performing a comparison with results available in the literature the flexibility and simplicity of the method make easy the application to arbitrary geometries

eigenvalue problems for fractional differential equations with right and left fractional derivatives this paper studies the eigenvalue problem of a class of fractional differential equations with right and left fractional derivatives with the aid of the spectral theory of compact self adjoint operators in hilbert spaces we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete orthogonal system furthermore the lower bound of the eigenvalues is obtained

the fundamental frequency of non homogeneous rectangular membranes the present study deals with the case in which the density of the membrane is a linear function of the x variable two independent solutions are obtained using the optimized galerkin kantorovich approach and the differential quadrature method

variational collocation on finite intervals in this paper we study a set of functions defined on an interval of finite width which are orthogonal and which reduce to the sinc functions when the appropriate limit is taken we show that these functions can be used within a variational approach to obtain accurate results for a variety of problems we have applied them to the interpolation of functions on finite domains and to the solution of the schrodinger equation and we have compared the performance of the present approach with others

a fast linearized conservative finite element method for the strongly coupled nonlinear fractional schrodinger equations abstract in this paper a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional schrodinger equations we prove that the scheme preserves both the mass and energy which are defined by virtue of some recursion relationships using the sobolev inequalities and then employing the mathematical induction the discrete scheme is proved to be unconditionally convergent in the sense of l 2 norm and h 2 norm which means that there are no any constraints on the grid ratios then the prior bound of the discrete solution in l 2 norm and l norm are also obtained moreover we propose an iterative algorithm by which the coefficient matrix is independent of the time level and thus it leads to toeplitz like linear systems that can be efficiently solved by krylov subspace solvers with circulant preconditioners this method can reduce the memory requirement of the proposed linearized finite element scheme from o m 2 to o m and the computational complexity from o m 3 to o m log m in each iterative step where m is the number of grid nodes finally numerical results are carried out to verify the correction of the theoretical analysis simulate the collision of two solitary waves and show the utility of the fast numerical solution techniques

on the boundary strichartz estimates for wave and schrodinger equations abstract we consider the l t 2 l x r estimates for the solutions to the wave and schrodinger equations in high dimensions for the homogeneous estimates we show l t 2 l x estimates fail at the critical regularity in high dimensions by using stable levy process in r d moreover we show that some spherically averaged l t 2 l x estimate holds at the critical regularity as a by product we obtain strichartz estimates with angular smoothing effect for the inhomogeneous estimates we prove double l t 2 type estimates

extremal theory for spectrum of random discrete schrodinger operator iii localization properties in this paper we study the asymptotic localization properties with high probability of the kth eigenfunction associated with the kth largest eigenvalue k 1 fixed of the multidimensional anderson hamiltonian in torus v increasing to the whole of lattice denote by z k v v the site at which the kth largest value of potential is attained it is well known that if the tails of potential distribution are heavier than the double exponential function and satisfies additional regularity and continuity conditions at infinity then the kth eigenfunction is asymptotically delta function at the site z k v localization centre for some random k v k 1 we study the asymptotic behavior of the index v k by distinguishing between three cases of the tails of potential distribution i for the heavy tails including gaussian v k is asymptotically bounded ii for the light tails but heavier than the double exponential the index v k unboundedly increases like v o 1 iii finally for the double exponential tails with high disorder the index v k behaves like a power of v for weibull s and fractional double exponential types distributions associated with the case ii we obtain the first order expansion formulas for log v k

solving the helmholtz equation for membranes of arbitrary shape numerical results i calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on little sinc functions the matrix representation of the pde obtained using this method is explicit and does not require the calculation of integrals to illustrate the virtues of this approach i have considered a large number of examples part of them are taken from the literature and part of them new when possible i have tested the accuracy of these results by comparing them with the exact results when available or with results from the literature in particular in the case of the l shaped membrane the first example discussed in the paper i show that it is possible to extrapolate the results obtained with different grid sizes to obtain highly precise results finally i also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results

exact free vibration analysis for membrane assemblies with general classical boundary conditions abstract this paper proposes exact dynamic stiffness formulations for membranes and their assemblies under any arbitrary classical boundary conditions first by taking exact solutions in one direction satisfying all possible opposite edge supports we can derive exact general solutions of the helmholtz equation for membrane vibration then generic force and displacement boundary conditions in the other direction are expressed in terms of the general solutions finally the dynamic stiffness matrices of rectangular membrane elements are formulated which can be assembled directly and allows applications of arbitrary boundary conditions as an accurate and efficient modal solution technique the wittrick williams ww algorithm is applied onto the global dynamic stiffness matrix of the final structure the most important issue of the ww algorithm j 0 count has been resolved with the analytical expressions derived for all possible cases the proposed dynamic stiffness method dsm is then applied to several examples including individual membranes and their assemblies high accuracy and exactness of the proposed method within the whole frequency range is demonstrated by comparison with the finite element method besides interesting findings have been observed on repeated eigenvalues with distinct mode shapes corresponding to certain aspect ratio and tension ratio where physical and mathematical understanding has been provided

a new method for studying the vibration of non homogeneous membranes abstract we present a method to solve the helmholtz equation for a non homogeneous membrane with dirichlet boundary conditions at the border of arbitrary two dimensional domains the method uses a collocation approach based on a set of localized functions called little sinc functions which are used to discretize two dimensional regions we have performed extensive numerical tests and we have compared the results obtained with the present method with the ones available from the literature our results show that the present method is very accurate and that its implementation for general problems is straightforward

pseudo orbit shadowing in the family of tent maps on montre que les applications en tente ont la propriete d ombrage pour presque tous les parametres s mais elles n ont pas cette propriete pour un ensemble dense non denombrable des parametres on montre que pour une application tente chaque pseudo orbite peut etre approchee par une orbite actuelle d une application en tente avec une boucle peut etre legerement plus grande

statistical properties of discretizations of a class of chaotic dynamical systems computer simulations of dynamical systems contain discretizations where finite machine arithmetic replaces continuum state spaces in some circumstances complicated theoretical behavior has a tendency to collapse to trivial and degenerate behavior as a result of discretizations various statistical estimators associated with such collapsing effects often seem to depend on the corresponding discretization in a random way behavior of some statistical properties of collapse is discussed results of computer modeling of mappings x 1 1 2x l x 0 1 l 2 are presented

maximum norm error analysis of a difference scheme for the space fractional cnls the difference method for the space fractional coupled nonlinear schrodinger equations cnls is studied the fractional centered difference is used to approximate the space fractional laplacian this scheme conserves the discrete mass and energy due to the nonlocal nature of fractional laplacian in the classic sobolev space it is hard to obtain the error estimation in l to overcome this difficulty the fractional sobolev space h 2 and a fractional norm equivalence in h 2 are introduced then the convergence of order o h 2 2 in l is proved by fractional sobolev inequality where h is the mesh size and is the time step numerical examples are given to illustrate the theoretical results at last

a new algorithm for solving some mechanical problems this paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems benchmark mechanical systems including plate vibrations and incompressible flows are employed to illustrate the robustness and to test accuracy of the present algorithm numerical results indicate that the present approach is very accurate efficient and reliable for solving the aforementioned problems

split step alternating direction implicit difference scheme for the fractional schrodinger equation in two dimensions in this paper we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space fractional schrodinger equation with the riesz fractional derivative the scheme is constructed by combining the split step method for handling the nonlinearity with the alternating direction implicit adi method for resolving the multi dimensions difficulty the riesz space fractional derivative is approximated by the second order accurate fractional centered difference based on matrix analysis we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems the unconditional convergence is proved rigorously in the linear case numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme

a generalization of the membrane plate analogy to non homogeneous polygonal domains consisting of homogeneous subdomains the well known membrane plate analogy that relates the natural frequencies when dealing with polygonal homogeneous domains is herein extended to non homogeneous systems comprised of homogeneous subdomains the analogy is generalized and demonstrated and it is shown that certain restrictions among the frequency parameters of the membranes and plates arise several examples of membranes and plates with interfaces separating areas with different material properties are numerically solved with different approaches the subdomains are separated by straight curved and closed line interfaces it is shown that the analogy is verified provided that the restrictions are satisfied the analogy is first demonstrated and presented as a practical methodology to find the natural frequencies of membranes knowing the corresponding ones of the plates or vice versa second the plate and membrane vibration problems governed by the bi laplacian and laplacian differential operators respectively can be solved without distinction though under certain conditions i e solve one of them and deduce the other using the analogy various numerical examples validate the analogy

what good are numerical simulations of chaotic dynamical systems abstract numerical simulations of mathematical models can suggest that the models are chaotic for example one can compute an orbit and its associated finite time lyapunov exponents and these computed exponents can be positive it is not clear how far these suggestions can be trusted because as is well known numerical methods can introduce spurious chaos or even suppress actual chaos this focused review examines the fidelity of numerical methods we look at the didactic example of the gauss map from the theory of continued fractions which allows a simple examination of backward error analysis for discrete dynamical systems and gives a clear picture of the effects of floating point arithmetic a similar use of backward error analysis in the form of defect control gives a useful understanding in the case of continuous dynamical systems finally we discuss limitations of this backward point of view

a variational sinc collocation method for strong coupling problems we have devised a variational sinc collocation method vscm which can be used to obtain accurate numerical solutions to many strong coupling problems sinc functions with an optimal grid spacing are used to solve the linear and nonlinear schrodinger equations and a lattice 4 model in 1 1 our results indicate that errors decrease exponentially with the number of grid points and that a limited numerical effort is needed to reach high precision

stability of standing waves for the fractional schrodinger hartree equation abstract in this paper we consider the stability of standing waves for the fractional schrodinger hartree equation by using the profile decomposition theory and variational methods we firstly obtain the best constant of a generalized gagliardo nirenberg inequality and then prove that the standing waves are orbitally stable

some physical applications of fractional schrodinger equation the fractional schrodinger equation is solved for a free particle and for an infinite square potential well the fundamental solution of the cauchy problem for a free particle the energy levels and the normalized wave functions of a particle in a potential well are obtained in the barrier penetration problem the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined in the quantum scattering problem according to the fractional schrodinger equation the green s function of the lippmann schwinger integral equation is given

lifschitz tails for the anderson model we present in an expository way an elementary rigorous proof patterned after an argument of kirsch martinelli that the anderson model has lifschitz tails in very great generality

rigorous shadowing of numerical solutions of ordinary differential equations by containment an exact trajectory of a dynamical system lying close to a numerical trajectory is called a shadow we present a general purpose method for proving the existence of finite time shadows of numerical ode integrations of arbitrary dimension in which some measure of hyperbolicity is present and there are either 0 or 1 expanding modes or 0 or 1 contracting modes much of the rigor is provided automatically by interval arithmetic and validated ode integration software that is freely available the method is a generalization of a previously published containment process that was applicable only to two dimensional maps we extend it to handle maps of arbitrary dimension with the above restrictions and finally to odes the method involves building n cubes around each point of the discrete numerical trajectory through which the shadow is guaranteed to pass at appropriate times the proof consists of two steps first the rigorous computational verification of a simple geometric property which we call the inductive containment property and second a simple geometric argument showing that this property implies the existence of a shadow the computational step is almost entirely automated and easily adaptable to any ode problem the method allows for the rescaling of time which is a necessary ingredient for successfully shadowing odes finally the method is local in the sense that it builds the shadow inductively requiring information only from the most recent integration step rather than more global information typical of several other methods the method produces shadows of comparable length and distance to all currently published results finally we conjecture that the inductive containment property implies the existence of a shadow without restriction on the number of expanding and contracting modes although proof currently eludes us

computing the topological entropy of general one dimensional maps a matrix theoretic method for computing the topological entropy of continuous piecewise monotonic maps of the interval is presented the method results in a constructive procedure which is easily implemented on the computer examples for families of unimodal nonunimodal and discontinuous maps are presented

vibration of non homogeneous rectangular membranes with arbitrary interfaces abstract the study deals with the generalized solution of the title problem the free vibration problem of a rectangular membrane with partial domains each of uniform density and arbitrary interface is tackled previous studies by other researchers include straight parallel to the borders and oblique interfaces and bent ones the solution is found by means of a direct variational method with a series composed with a complete set of functions two alternative sets are explored trigonometric and power series such series are uniformly convergent to the exact solution the approach is straightforward and very efficient from the computational viewpoint a determinant factorization method is employed to automatically eliminate eventual spurious frequencies the well known analogy between plates and membranes does not hold in this problem and a demonstration is included diverse illustrations are worked out as the cases of an oblique straight line interface and an open curve line which divides the membrane in two domains each of different density are first presented also a rectangular membrane with an interior closed domain is stated and the numerical example of a circular interior zone is included comparison between the two alternatives and with other authors results show excellent agreement in all cases the computational cost is very low