Differential algebraic method for time of flight property of electrostatic electron lenses

Differential algebraic method is a powerful technique in computer numerical analysis. It presents a straightforward method for computing arbitrary order derivatives of functions with extreme high accuracy limited only by the truncation error of the computer. When applied to nonlinear dynamical systems, the arbitrary high-order transfer properties of the systems can be computed directly with high precision. In this article, the time of flight (TOF) property of electrostatic electron lens systems is studied by differential algebraic method and their arbitrary order TOF transfer properties can be numerically calculated. As an example, Schiske's model electrostatic lens has been studied by the efficient differential algebraic TOF method.     

Explicit spheroidal wave functions of the hydrogen atom

A simple and straightforward calculating scheme is suggested for finding wave functions of the hydrogen atom in prolate spheroidal coordinates. The wave functions are found in an explicit form by the direct solution of appropriate one-dimensional equations. The suggested calculating scheme allows us to carry out simple calculations and to obtain spheroidal wave functions in principle for arbitrary eigenstates of the hydrogen atom. Expansions are found for the obtained spheroidal wave functions over a spherical basis.     

An exact noniterative linear method for locating sources based on measuring receiver arrival times

In this paper an exact, linear solution to the source localization problem based on the time of arrival at the receivers is presented. The method is unique in that the source's position can be obtained by solving a system of linear equations, three for a plane and four for a volume. This simplification means adding an additional receiver to the minimum mathematically required (3+1 in two dimensions and 4+1 in three dimensions). The equations are easily worked out for any receiver configuration and their geometrical interpretation is straightforward. Unlike other methods, the system of reference used to describe the receivers' positions is completely arbitrary. The relationship between this method and previously published ones is discussed, showing how the present, more general, method overcomes nonlinearity and unknown dependency issues.     

Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer-Kaup-Kupershmidt equations

In this Letter, the Exp-function method is used to seek generalized solitonary solutions of Riccati equation. Based on the Riccati equation and its generalized solitonary solutions, new exact solutions with three arbitrary functions of the (2+1)-dimensional Broer-Kaup-Kupershmidt equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.     

A Poisson–Boltzmann solver on irregular domains with Neumann or Robin boundary conditions on non-graded adaptive grid

We introduce a second-order solver for the Poisson–Boltzmann equation in arbitrary geometry in two and three spatial dimensions. The method differs from existing methods solving the Poisson–Boltzmann equation in the two following ways: first, non-graded Quadtree (in two spatial dimensions) and Octree (in three spatial dimensions) grid structures are used; Second, Neumann or Robin boundary conditions are enforced at the irregular domain’s boundary. The irregular domain is described implicitly and the grid needs not to conform to the domain’s boundary, which makes grid generation straightforward and robust. The linear system is symmetric, positive definite in the case where the grid is uniform, nonsymmetric otherwise. In this case, the resulting matrix is an M-matrix, thus the linear system is invertible. Convergence examples are given in both two and three spatial dimensions and demonstrate that the solution is second-order accurate and that Quadtree/Octree grid structures save a significant amount of computational power at no sacrifice in accuracy.     

Traveling-wave-type gravitational soliton solutions

With the traveling-wave condition of Yan and Ge, the traveling-wave-type gravitational two-soliton solutions are generated from a flat metric by using the inverse scattering method (ISM) of Belinsky and Zakharov (BZ). It is shown that when the traveling-wave condition is added to the condition required by the BZ technique the exact general solutions of the vacuum gravitational field equations can be found by straightforward integration; in the general solutions there are three arbitrary functions. A special solution with solitonic character in the ordinary sense is also given.     

Uncertainty relations for general phase spaces

We describe a setup for obtaining uncertainty relations for arbitrary pairs of observables related by a Fourier transform. The physical examples discussed here are the standard position and momentum, number and angle, finite qudit systems, and strings of qubits for quantum information applications. The uncertainty relations allow for an arbitrary choice of metric for the outcome distance, and the choice of an exponent distinguishing, e.g., absolute and root mean square deviations. The emphasis of this article is on developing a unified treatment, in which one observable takes on values in an arbitrary locally compact Abelian group and the other in the dual group. In all cases, the phase space symmetry implies the equality of measurement and preparation uncertainty bounds. There is also a straightforward method for determining the optimal bounds.     

Theory of a three-level gas laser amplifier

A laser amplifier to be treated in this work consists of an ensemble of atoms three energy levels of which form two coupled transitions of arbitrary frequencies. Two classical monochromatic travelling light waves are to be close to resonance with the transitions. The gain profile (or spontaneous emission) on the transition corresponding to the weak “probe” wave, modified by the perturbing field on the other transition, is calculated via a susceptibility. Within this framework, the atoms are described by an ensemble-averaged density matrix with full account of level degeneracies, light polarizations, and inelastic and dephasing collisions; an extension to elastic collisions and disorientation is straightforward. An integration over the thermal velocity distribution gives results applicable to gas discharges: directionally anisotropic narrow structures superimposed on the Doppler-broadened probe-gain profile due to non-linear interference effects in addition to saturation. At alower probe frequency, a peculiar non-Lorentzian signal appears even with transparency on the perturbing transition. At low intensities a distinction is reasonable of frequency correlations due to generalized two-quantum processes, and of a dynamic Stark splitting. These effects permit an information on the linewidth of the third forbidden transition. The connection with numerous related approaches is pointed out.     

Entangled Property of a Multipartite State Superposed by Three Dicke States with Arbitrary Relative Phases

With spin squeezing parameter we investigate the entangled property of a multipartite state superposed by three Dicke states with arbitrary relative phases. We first derive the mean spin direction, the optimally squeezed angle, and then numerically calculate the dependence of spin squeezing parameter on the superposition coefficients, the relative phases and the forms of the Dicke states. It’s shown that the entangled property depends on these parameters.     

A rational evaluation of fibre textures of thin solid films from electron diffraction patterns

A simple algebraic criterion based on reciprocal lattice geometry was established. It allows a straightforward construction of the distribution of intensity peaks on an electron diffraction pattern along layer lines associated with any crystallographic direction. It can be used to determine the fibre texture axis in materials composed of crystals of arbitrary symmetry. The method was applied to the determination of the fibre texture of thin films of PbO.     

Shape invariant potentials in higher dimensions

In this paper we investigate the shape invariance property of a potential in one dimension. We show that a simple ansatz allows us to reconstruct all the known shape invariant potentials in one dimension. This ansatz can be easily extended to arrive at a large class of new shape invariant potentials in arbitrary dimensions. A reformulation of the shape invariance property and possible generalizations are proposed. These may lead to an important extension of the shape invariance property to Hamiltonians that are related to standard potential problems via space time transformations, which are found useful in path integral formulation of quantum mechanics.     

General formulation of Griffiths' inequalities

We present a general framework in which Griffiths inequalities on the correlations of ferromagnetic spin systems appear as natural consequences of general assumptions. We give a method for the construction of a large class of models satisfying the basic assumptions. Special cases include the Ising model with arbitrary spins, and the plane rotator model. The general theory extends in a straightforward way to the non-commutative (quantum) case, but non-commutative examples satisfying all the assumptions are lacking at the moment.     

Designing arbitrary nanoscale patterns by a nanocavity waveguide with omnidirectional illumination

We report one- and two-dimensional arbitrary patterns which are achieved by nanocavity waveguides made of a quasi-metal?Cdielectric-metal heterostructure with omnidirectional illumination. This proposed heterostructure supports the surface plasmon polaritons whose phase and group velocities have opposite sign for given frequency. Negative refraction and reflection in the waveguide result in imaging nanolithography and the omnidirectional property can be well understood by the dispersive relation of the nanocavity waveguide. Numerical results demonstrate that such an omnidirectional nanolithography scheme is feasible for arbitrary 1D gratings and 2D linearly chirped gratings with TM and circular polarized incidence, respectively, at 365?nm.     

On clustering property

A new clustering property for an arbitrary primary state is proved.     

On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

We construct uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics. The same framework also applies to high order accurate finite volume (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO)) schemes. Motivated by Perthame and Shu (1996) [20] and Zhang and Shu (2010) [26], a general framework, for arbitrary order of accuracy, is established to construct a positivity preserving limiter for the finite volume and DG methods with first order Euler forward time discretization solving one-dimensional compressible Euler equations. The limiter can be proven to maintain high order accuracy and is easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the positivity property. Following the idea in Zhang and Shu (2010) [26], we extend this framework to higher dimensions on rectangular meshes in a straightforward way. Numerical tests for the third order DG method are reported to demonstrate the effectiveness of the methods.     

Composition of propagated,reflected and transmitted waves in arbitrary and continuously stratified environments

First, a solution is presented for a canonical problem in wave propagation. Second, illustrations and applications of the results are carried out to study cases which are relevant to the propagation problem in the ocean and atmosphere.The canonical problem consists of a plane wave incident on an arbitrary and continuously stratified region with planar boundaries. The explicit composition of the reflected, transmitted and propagated waves are derived. The solution is systematic and allows for (i) discontinuities in the acoustic properties at boundaries and arbitrary variation within, (ii) attenuation, (iii) all angles of incidence. The general expressions are obtained by using an alternate procedure to one recently devised [1]. The present approach is straightforward and plainly amenable to physical interpretation of its auxiliary mathematical constants. The discontinuities at the boundaries are satisfied at the outset. The reflected and transmitted waves are directly and explicitly specified. Comparison to widely used techniques in both analytical and numerical works is made to demonstrate the viability of the present approach.A series of cases relevant to the problem at hand are considered. These cases illustrate the mechanics involved in use of the method, and expand its application to problems that appear to be at variance with the formulation of the canonical problem. The illustrations include attenuation in the medium, effect on the solution of different acoustic discontinuities at the boundaries, and use of an inhomogeneous background profile with known independent solutions. The expanded applications treat formally three types of problems: (i) the exact solution for plane waves in continuously stratified media where the well-used ray theory or W-K-B approximation serves only as a first approximation in a correct iterative solution; (ii) the scattering of a plane wave by non-planar boundaries, i.e., spherical or cylindrical acoustic lens with the stratification along the radial direction; (iii) the field due to a point source in a continuously stratified wave guide, like the ocean or atmosphere.     

Investigating RF and microwave nonlinear magnetoelastic interactions in a three-layer structure

The problem of hypersound excitation in a structure consisting of three magnetic layers is considered. Motion equations and boundary conditions for the components of magnetization and elastic displacement when there is an arbitrary angle of magnetization vector precession are obtained. The development of oscillations over time in an alternating field is considered.     

Exact diffraction calculation from fields specified over arbitrary curved surfaces

Calculation of the scalar diffraction field over the entire space from a given field over a surface is an important problem in computer generated holography. A straightforward approach to compute the diffraction field from field samples given on a surface is to superpose the emanated fields from each such sample. In this approach, possible mutual interactions between the fields at these samples are omitted and the calculated field may be significantly in error. In the proposed diffraction calculation algorithm, mutual interactions are taken into consideration, and thus the exact diffraction field can be calculated. The algorithm is based on posing the problem as the inverse of a problem whose formulation is straightforward. The problem is then solved by a signal decomposition approach. The computational cost of the proposed method is high, but it yields the exact scalar diffraction field over the entire space from the data on a surface.