a near tight lower bound on the time complexity of distributed minimum weight spanning tree construction this paper presents a lower bound of omega d sqrt n log n on the time required for the distributed construction of a minimum weight spanning tree mst in weighted n vertex networks of diameter d omega log n in the bounded message model this establishes the asymptotic near optimality of existing time efficient distributed algorithms for the problem whose complexity is o d sqrt n log n

arbitrary l state solutions of the rotating morse potential by the asymptotic iteration method for non zerovalues we present an analytical solution of the radial schr odinger equation for the rotating morse potential using the pekeris approximation within the framework of the asymptotic iteration method the bound state energy eigenvalues and corresponding wavefunctions are obtained for a number of diatomic molecules and the results are compared with the findings of the super symmetry the hypervirial perturbation the nikiforov uvarov the variational the shifted 1 n and the modified shifted 1 n expansion methods

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

construction of exact solutions to eigenvalue problems by the asymptotic iteration method we apply the asymptotic iteration method aim ciftci hall and saad 2003 j phys a math gen 36 11807 to solve new classes of second order homogeneous linear differential equation in particular solutions are found for a general class of eigenvalue problems which includes schrodinger problems with coulomb harmonic oscillator or poschl teller potentials as well as the special eigenproblems studied recently by bender et al 2001 j phys a math gen 34 9835 and generalized in the present paper to arbitrary dimension

adaptive weak approximation of diffusions with jumps this work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented monte carlo euler maruyama method which achieve a prescribed precision the main result is the derivation of new expansions for the time discretization error with computable leading order term in a posteriori form which are based on stochastic flows and discrete dual backward functions combined with proper estimation of the statistical error they lead to efficient and accurate computation of global error estimates extending the results by a szepessy r tempone and g e zouraris comm pure appl math 54 2001 pp 1169 1214 adaptive algorithms for either deterministic or trajectory dependent time stepping are proposed numerical examples show the performance of the proposed error approximations and the adaptive schemes

convergence and stability of the compensated split step theta method for stochastic differential equations with piecewise continuous arguments driven by poisson random measure abstract this paper deals with the numerical solutions of stochastic differential equations with piecewise continuous arguments sdepcas driven by poisson random measure in which the coefficients are highly nonlinear it is shown that the compensated split step theta csst method with 0 1 is strongly convergent in p th p 2 moment under some polynomially lipschitz continuous conditions it is also obtained that the convergence order is close to 1 p in terms of the stability it is proved that the csst method with 1 2 1 reproduces the exponential mean square stability of the underlying system under the monotone condition and some restrictions on the step size without any restriction on the step size there exists 1 2 1 such that the csst method with 1 is exponentially stable in mean square moreover if the drift and jump coefficients satisfy the linear growth condition the csst method with 0 1 2 also preserves the exponential mean square stability some numerical simulations are presented to verify the conclusions

minimum weight spanning tree construction in o log log n communication rounds we consider a simple model for overlay networks where all n processes are connected to all other processes and each message contains at most o log n bits for this model we present a distributed algorithm which constructs a minimum weight spanning tree in o log log n communication rounds where in each round any process can send a message to every other process if message size is theta n epsilon for some epsilon 0 then the number of communication rounds is o log 1 over epsilon

asymptotic iteration method solutions to the relativistic duffin kemmer petiau equation a simple exact analytical solution of the relativistic duffin kemmer petiau equation within the framework of the asymptotic iteration method is presented exact bound state energy eigenvalues and corresponding eigenfunctions are determined for the relativistic harmonic oscillator as well as the coulomb potentials as a nontrivial example the anharmonic oscillator is solved and the energy eigenvalues are obtained within the perturbation theory using the asymptotic iteration method

solutions of the spatially dependent mass dirac equation with the spin and pseudospin symmetry for the coulomb like potential we study the effect of spatially dependent mass function over the solution of the dirac equation with the coulomb potential in the 3 1 dimensions for any arbitrary spin orbit k state in the framework of the spin and pseudospin symmetry concept the analytic bound state energy eigenvalues and the corresponding upper and lower two component spinors of the two dirac particles are obtained by means of the nikiforov uvarov method in closed form this physical choice of the mass function leads to an exact analytical solution for the pseudospin part of the dirac equation the special cases k 1 l l 0 i e s wave the constant mass and the non relativistic limits are briefly investigated

convergence rates for adaptive approximation of ordinary differential equations this paper constructs an adaptive algorithm for ordinary differential equations and analyzes its asymptotic behavior as the error tolerance parameter tends to zero an adaptive algorithm based on the error indicators and successive subdivision of time steps is proven to stop with the optimal number n of steps up to a problem independent factor defined in the algorithm a version of the algorithm with decreasing tolerance also stops with the total number of steps including all refinement levels bounded by mathcal o n the alternative version with constant tolerance stops with mathcal o n rm log n total steps the global error is bounded by the tolerance parameter asymptotically as the tolerance tends to zero for a p th order accurate method the optimal number of adaptive steps is proportional to the p th root of the l frac 1 p 1 quasi norm of the error density while the number of uniform steps with the same error is proportional to the p th root of the larger l 1 norm of the error density

analytical approximations of scattering states to the l wave solutions for the schrodinger equation with the eckart potential the scattering state solutions of the schrodinger equation for the eckart potential with the centrifugal term are obtained approximately it is shown that the solutions can be expressed in terms of the generalized hypergeometric functions 2f1 a b c z the normalized radial wave functions of scattering states on the k 2 scale and the calculation formula of phase shifts are also derived three special cases for l 0 0 and r0 are also studied briefly

matrix elements for a generalized spiked harmonic oscillator closed form expressions for the singular potential integrals m x n are obtained with respect to the gol dman and krivchenkov eigenfunctions for the singular potential bx2 a x2 b 0 a 0 the formulas obtained are generalizations of those found earlier by use of the odd solutions of the schrodinger equation with the harmonic oscillator potential aguilera navarro et al j math phys 31 99 1990

weak approximations and extrapolations of stochastic differential equations with jumps numerical discretization schemes are developed to approximate functionals of stochastic differential equations with jumps and the convergence is shown to have an appropriate order for the euler scheme and the second order weak scheme the leading coefficient of their global errors are determined by the stochastic taylor expansion based on the error expression the extrapolation technique can be applied to get a higher order convergence numerical examples are provided to compare various weak schemes and extrapolations

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

direct calculation of accurate siegert eigenvalues the roots of the hankel determinant built with the taylor coefficients of the regularized logarithmic derivative of the eigenfunction give both the bound state and siegert eigenvalues resonances of separable quantum mechanical systems we obtain accurate positions and widths for the resonances of some commonly chosen test examples

orientation of particle attachment and local isotropy in diffusion limited aggregation dla we simulate 50 off lattice diffusion limited aggregation dla clusters of one million particles each the probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically for increasing cluster size n the distribution crosses over extremely accurately to a cosine with an amplitude which decreases towards zero as a power law in n from this specific viewpoint asymptotically large dla clusters are locally isotropic

runge kutta methods for jump diffusion differential equations in this paper we consider runge kutta methods for jump diffusion differential equations we present a study of their mean square convergence properties for problems with multiplicative noise we are concerned with two classes of runge kutta methods first we analyse schemes where the drift is approximated by a runge kutta ansatz and the diffusion and jump part by a maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the runge kutta ansatz for the drift the second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise we present results showing when the implicit stochastic equations defining the stage values of the runge kutta methods are uniquely solvable finally simulation results illustrate the theoretical findings

the asymptotic iteration method for the eigenenergies of the schrodinger equation with the potential v r z r gr λr2 the eigenenergies of the schrodinger equation with coulomb potential plus linear and harmonic radial terms are investigated using the perturbation technique within the framework of the asymptotic iteration method the obtained eigenenergies are compared with those obtained from the shifted 1 n expansion method the moment method and also with the exact values obtained from the super symmetric method it is found that asymptotically this method gives accurate results for arbitrary g z and

any l state solutions of the hulthen potential by the asymptotic iteration method in this paper we present the analytical solution of the radial schrodinger equation for the hulthen potential within the framework of the asymptotic iteration method by using an approximation to the centrifugal potential for any l states we obtain the energy eigenvalues and corresponding eigenfunctions for different screening parameters the wavefunctions are physical and energy eigenvalues are in good agreement with the results obtained by other methods for different values in order to demonstrate this the results of the asymptotic iteration method are compared with the results of the supersymmetry numerical integration variational and shifted 1 n expansion methods

spiked harmonic oscillators a complete variational treatment is provided for a family of spiked harmonic oscillator hamiltonians h d2 dx2 bx2 x b 0 0 for arbitrary 0 a compact topological proof is presented that the set s n of known exact solutions for 2 constitutes an orthonormal basis of the hilbert space l2 0 closed form expressions are derived for the matrix elements of h with respect to s these analytical results and the inclusion of a further free parameter facilitate optimized variational estimation of the eigenvalues of h to high accuracy

comment on the asymptotic iteration method for polynomial potentials it is shown that polynomial potentials can be treated by a method which cuts out most of the complications of the popular aim approach several examples are given to show how expectation values and resonance energies can be calculated using the simplified approach

solution of ordinary differential equations via nonlocal transformations the classical lie analysis provides a useful technique for the solution of ordinary differential equations via point symmetries transformations unfortunately the requirement that an nth order equation possesses at least an n dimensional solvable lie algebra of symmetries is only satisfied by a select number of equations we provide a class of second order ordinary differential equations with less than the required two lie point symmetries that can be solved via nonlocal transformations

closed form sums for some perturbation series involving hypergeometric functions infinite series of the type are investigated closed form sums are obtained for a positive integer 1 2 3 the limiting case of b after y is replaced with x2 b leads to this type of series appears in the first order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator hamiltonian these results have immediate applications to perturbation series for the energy and wavefunction of the spiked harmonic oscillator hamiltonian

strong approximations of stochastic differential equations with jumps this paper is a survey of strong discrete time approximations of jump diffusion processes described by stochastic differential equations sdes it also presents new results on strong discrete time approximations for the specific case of pure jump sdes strong approximations based on jump adapted time discretizations which produce no discretization error in the case of pure jump processes are analyzed the computational complexity of these approximations is proportional to the jump intensity by exploiting a stochastic expansion for pure jump processes higher order discrete time approximations whose computational complexity is not dependent on the jump intensity are proposed for the specific case of pure jump sdes the strong order of convergence of strong taylor schemes is established under weaker conditions than those currently known in the literature

lessons from the congested clique applied to mapreduce the main result of this paper is a simulation algorithm which under quite general constraints transforms algorithms running in the congested clique model into algorithms running in the mapreduce model as a case study of applying this simulation algorithm we first present a distributed o coloring algorithm running on the congested clique in o 1 rounds if log 2 n and o log log log n rounds otherwise applying the simulation theorem to this congested clique o coloring algorithm yields an o 1 round o coloring algorithm in the mapreduce model we apply our simulation algorithm to other congested clique algorithms including the 2 ruling set and metric facility location algorithms of berns et al icalp 2012 our simulation algorithm illustrates a natural correspondence between per node bandwidth in the congested clique model and memory per machine in the mapreduce model in the congested clique and more generally any network in the congest model the major impediment to constructing fast algorithms is the o log n restriction on message sizes similarly in the mapreduce model the combined restrictions on memory per machine and total system memory have a dominant effect on algorithm design in showing a fairly general simulation algorithm we highlight the similarities and differences between these models

criterion for polynomial solutions to a class of linear differential equations of second order we consider the differential equations y 0 x y s0 x y where 0 x s0 x are c functions we prove i if the differential equation has a polynomial solution of degree n 0 then n nsn 1 n 1sn 0 where n n 1 sn 1 0 n 1andsn s n 1 s0 k 1 n 1 2 conversely ii if n n 1 0 and n 0 then the differential equation has a polynomial solution of degree at most n we show that the classical differential equations of laguerre hermite legendre jacobi chebyshev first and second kinds gegenbauer and the hypergeometric type etc obey this criterion further we find the polynomial solutions for the generalized hermite laguerre legendre and chebyshev differential equations

the asymptotic iteration method for the angular spheroidal eigenvalues the asymptotic iteration method is applied to calculate the angular spheroidal eigenvalues ml c it is shown that this method asymptotically gives accurate results over the full range of parameter values l m and c

optimal simulation schemes for levy driven stochastic differential equations we consider a general class of high order weak approximation schemes for stochastic differential equations driven by levy processes with infinite activity these schemes combine a compound poisson approximation for the jump part of the levy process with a high order scheme for the brownian driven component applied between the jump times the overall approximation is analyzed using a stochastic splitting argument the resulting error bound involves separate contributions of the compound poisson approximation and of the discretization scheme for the brownian part and allows on one hand to balance the two contributions in order to minimize the computational time and on the other hand to study the optimal design of the approximating compound poisson process for driving processes whose levy measure explodes near zero in a regularly varying way this procedure allows to construct discretization schemes with arbitrary order of convergence

on the elementary schrodinger bound states and their multiplets the problem of the existence of elementary bound states is discussed a trivial observation that every elementary wave function i r is an exact bound state for an appropriate potential v r v i r r is shown to lead to a very transparent form of the quasiexact qe solvability condition v i v j for doublets and multiplets of the s in this sense the particular class of elementary ansatze i r r polynomial r2 exp r polynomial r2 also defines the particular class of qe solvable potentials they have an elementary nonpolynomial rational form possibly also with a strongly singular repulsive core at the origin the properties of these forces are discussed in detail

on an iteration method for eigenvalue problems we discuss a recently proposed asymptotic iteration method for eigenvalue problems we analyse its rate of convergence the use of adjustable parameters to improve it and the relationship with an alternative method based on the same ideas

on the eigenvalues of the s state radial equation of a spiked harmonic oscillator eigenvalues of the ground state of the radial schrodinger equation for a spiked harmonic oscillator potential have been evaluated employing two methods numerically via the lanczos grid technique and by means of standard pade approximants constructed from an expansion of large coupling parameter series for the energy numerical results are compared for several values of the parameters characterizing the spiked singular potential

simple bound state energy spectra solutions of dirac equation for hulthen and eckart potentials within the framework of simple similarity transformation method we present simple schrodinger like equation for dirac equation with hulthen and eckart lorentz vectors potentials using the asymptotic iteration method we have managed to obtain the explicit analytical solution of the bound state energy spectra as well as the eigenfunctions for both potentials discussion related to the eigenfunction for both cases is also considered we have shown that the wave function is sensitive for both potentials as well as the parameter

bosonic oscillator in the presence of minimal length we present an exact solution of the one dimensional bosonic oscillator for spin 1 and spin 0 in the momentum space with the presence of minimal length uncertainty the energy eigenvalues and eigenfunctions are then determined for both cases

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

an approximate κ state solutions of the dirac equation for the generalized morse potential under spin and pseudospin symmetry by using an improved approximation scheme to deal with the centrifugal pseudo centrifugal term we solve the dirac equation for the generalized morse potential with arbitrary spin orbit quantum number in the presence of spin and pseudospin symmetry the analytic bound state energy eigenvalues and the associated upper and lower spinor components of two dirac particles are found by using the basic concepts of the nikiforov uvarov method we study the special cases when 1 l l 0 s wave the non relativistic limit and the limit when becomes zero kratzer potential model the present solutions are compared with those obtained by other methods

the klein gordon equation with ring shaped potentials asymptotic iteration method this study presents the solutions of three dimensional klein gordon equation for the spherically and non spherically harmonic oscillatory ring shaped potentials within the framework of asymptotic iteration method using the method of variable separation this study obtains the radial and angular equations and then the bound states energy eigenvalues and corresponding eigenfunctions are obtained analytically

pseudospin symmetry for a new oscillatory ring shaped noncentral potential a new oscillatory ring shaped noncentral potential is proposed and the pseudospin symmetry for this potential is investigated by solving the dirac equation with equally mixed scalar and vector potentials with opposite signs the spinor wave functions and the energy spectrum for the bound states are obtained by using the nikiforov uvarov method the effect of angle dependent part on the radial solutions and the algebraic property of the energy equation are discussed

a solution of the bosonic and algebraic hamiltonians by using an aim we apply the notion of an asymptotic iteration method aim to determine the eigenvalues of the bosonic hamiltonians that include a wide class of quantum optical models we consider the solutions of the hamiltonians which are even the polynomials of the fourth order with respect to the boson operators we also demonstrate the applicability of the method for obtaining the eigenvalues of the simple lie algebraic structures the eigenvalues of the multi boson hamiltonians have been obtained by transforming them into the form of the single boson hamiltonian in the framework of the aim

a simple deterministic distributed mst algorithm with near optimal time and message complexities the distributed minimum spanning tree mst problem is one of the most central and fundamental problems in distributed graph algorithms kutten and peleg devised an algorithm with running time o d n log n where d is the hop diameter of the input n vertex m edge graph and with message complexity o m n3 2 peleg and rubinovich showed that the running time of the algorithm of kutten and peleg is essentially tight and asked if one can achieve near optimal running time together with near optimal message complexity in a recent breakthrough pandurangan et al answered this question in the affirmative and devised a randomized algorithm with time o d n and message complexity o m they asked if such a simultaneous time and message optimality can be achieved by a deterministic algorithm in this article building on the work of pandurangan et al we answer this question in the affirmative and devise a deterministic algorithm that computes mst in time o d n log n using o m log n n log n cdot log n messages the polylogarithmic factors in the time and message complexities of our algorithm are significantly smaller than the respective factors in the result of pandurangan et al in addition our algorithm and its analysis are very simple and self contained as opposed to rather complicated previous sublinear time algorithms finally we use our new algorithm to devise a randomized mst algorithm with running time o g n and message complexity o e where radius g d is a graph parameter which is typically much smaller than d this improves a previous bound from elkin

a sublinear time distributed algorithm for minimum weight spanning trees this paper considers the question of identifying the parameters governing the behavior of fundamental global network problems many papers on distributed network algorithms consider the task of optimizing the running time successful when an o n bound is achieved on an n vertex network we propose that a more sensitive parameter is the network s diameter diam this is demonstrated in the paper by providing a distributed minimum weight spanning tree algorithm whose time complexity is sublinear in n but linear in diam specifically o diam n varepsilon cdot log n for varepsilon frac ln 3 ln 6 0 6131 our result is achieved through the application of graph decomposition and edge elimination by pipelining techniques that may be of independent interest

the pseudospin symmetric solution of the morse potential for any κ state in this paper the energy eigenvalues and the corresponding spinors of the dirac particles are obtained and the pseudospin symmetric solution of the attractive scalar and repulsive vector morse potential for any spin orbit quantum number is provided by using the pekeris approximation within the framework of the asymptotic iteration method

adaptive weak approximation of stochastic differential equations adaptive time stepping methods based on the monte carlo euler method for weak approximation of ito stochastic differential equations are developed the main result is new expansions of the computational error with computable leading order term in a posteriori form based on stochastic flows and discrete dual backward problems the expansions lead to efficient and accurate computation of error estimates adaptive algorithms for either stochastic time steps or deterministic time steps are described numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps stochastic time steps use brownian bridges and require more work for a given number of time steps deterministic time steps may yield more time steps but require less work for example in the limit of vanishing error tolerance the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps c 2001 john wiley sons inc

the asymptotic iteration method revisited the asymptotic iteration method is a technique for solving analytically and approximately the linear second order differential equation especially the eigenvalue problems that frequently appear in theoretical and mathematical physics the analysis and mathematical justifications of the success and failure of the asymptotic iteration method are detailed in this work a theorem explaining why the asymptotic iteration method works for the eigenvalue problem is presented as a byproduct a new procedure to generate unlimited classes of exactly solvable differential equations is also introduced

variational sturmian approximation a nonperturbative method of solving time independent schrodinger equation a variationally improved sturmian approximation for solving time independent schrodinger equation is developed this approximation is used to obtain the energy levels of a quartic anharmonic oscillator a quartic potential and a gaussian potential the results are compared with those of the perturbation theory the wkb approximation and the accurate numerical values

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

a variational principle for adaptive approximation of ordinary differential equations a variational principle inspired by optimal control yields a simple derivation of an error representation global error local error weight for general approximation of functions of solutions to ordinary differential equations this error representation is then approximated by a sum of computable error indicators to obtain a useful global error indicator for adaptive mesh refinements a uniqueness formulation is provided for desirable error representations of adaptive algorithms

the euler scheme for levy driven stochastic differential equations in relation with monte carlo methods to solve some integro differential equations we study the approximation problem of ee g x t by ee g bar x t n where x t 0 leq t leq t is the solution of a stochastic differential equation governed by a levy process z t bar x n t is defined by the euler discretization scheme with step fractn with appropriate assumptions we show that the error ee g x t ee g bar x t n can be expanded in powers of frac1n if the levy measure of z has finite moments of order high enough otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the levy measure the simulation of the increments of z t is also discussed

comment on the asymptotic iteration method revisited j math phys 61 033501 2020 in this comment we show that the eigenvalues of a quartic anharmonic oscillator obtained recently by means of the asymptotic iteration method may not be as accurate as the authors claim them to be

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

comment on an application of the asymptotic iteration method to a perturbed coulomb model we discuss a recent application of the asymptotic iteration method aim to a perturbed coulomb model contrary to what was argued before we show that the aim converges and yields accurate energies for that model we also consider alternative perturbation approaches and show that one of them is equivalent to that recently proposed by another author

strong convergence rates for backward euler on a class of nonlinear jump diffusion problems we generalise the current theory of optimal strong convergence rates for implicit euler based methods by allowing for poisson driven jumps in a stochastic differential equation sde more precisely we show that under one sided lipschitz and polynomial growth conditions on the drift coefficient and global lipschitz conditions on the diffusion and jump coefficients three variants of backward euler converge with strong order of one half the analysis exploits a relation between the backward and explicit euler methods

convergence of a discretization scheme for jump diffusion processes with state dependent intensities this paper proves a convergence result for a discretization scheme for simulating jump diffusion processes with state dependent jump intensities with a bound on the intensity the point process of jump times can be constructed by thinning a poisson random measure using state dependent thinning probabilities between the jump epochs of the poisson random measure the dynamics of the constructed process are purely diffusive and may be simulated using standard discretization methods under conditions on the coefficient functions of the jump diffusion process we show that the weak convergence order of this method equals the weak convergence order of the scheme used for the purely diffusive intervals the construction of jumps does not degrade the convergence of the method

variational and perturbative schemes for a spiked harmonic oscillator a variational analysis of the spiked harmonic oscillator hamiltonian operator d2 dx2 x2 l l 1 x2 x where is a real positive parameter is reported in this work the formalism makes use of the functional space spanned by the solutions of the schrodinger equation for the linear harmonic oscillator hamiltonian supplemented by a dirichlet boundary condition and a standard procedure for diagonalizing symmetric matrices the eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system valid for positive and relatively large values of the coupling parameter additionally a large coupling perturbative expansion is carried out and the contributions up to fourth order to the ground state energy are explicitly evaluated numerical results are compared for the special case 5 2

large order shifted 1 n expansions through the asymptotic iteration method the perturbation technique within the framework of the asymptotic iteration method is used to obtain large order shifted 1 n expansions where n is the number of spatial dimensions this method is contrary to the usual rayleigh schrodinger perturbation theory no matrix elements need to be calculated the method is applied to the schrodinger equation and the non polynomial oscillator potential is discussed as an illustrative example

application of the exact quantization rule to the relativistic solution of the rotational morse potential with pseudospin symmetry we present an analytical solution of the radial dirac equation for the rotational morse potential through the pekeris approximation the bound state energy eigenvalues are obtained by using an exact quantization rule for non zero values of the dirac equation as an application of the rule we give the numerical solutions of the results for special values of the potential parameters

hypervirial perturbation theory revisited a wide range of topics in the area of hypervirial perturbation theory hvpt is discussed it is shown that with the use of a few simple procedures hvpt is capable of high accuracy for many problems results from many previous works in the literature are found to be improvable by the careful use of hvpt with appropriate choice of the unperturbed potential and of the origin at which the energy expansion is carried out two multi well problems from the literature are analysed in detail to show the value of a combination of hvpt and finite difference methods

duffin kemmer petiau oscillator with snyder de sitter algebra we present an exact solution of the one dimensional bosonic oscillator for spin 1 and spin 0 particles with the snyder de sitter model where the energy eigenvalues and eigenfunctions are determined for both cases the wave functions can be given in terms of gegenbauer polynomials we also comment on the thermodynamic properties of the system

a randomized linear time algorithm to find minimum spanning trees we present a randomized linear time algorithm to find a minimum spanning tree in a connected graph with edge weights the algorithm uses random sampling in combination with a recently discovered linear time algorithm for verifying a minimum spanning tree our computational model is a unit cost random access machine with the restriction that the only operations allowed on edge weights are binary comparisons

asymptotic iteration method for eigenvalue problems an asymptotic iteration method for solving second order homogeneous linear differential equations of the form y 0 x y s0 x y is introduced where 0 x 0 and s0 x are c functions applications to schrodinger type problems including some with highly singular potentials are presented