One-point Approximation Research Articles

Bound states and point interactions of the one-dimensional pseudospin-one Hamiltonian

The spectrum of a one-dimensional pseudospin-one Hamiltonian with a three-component potential is studied for two configurations: (i) all the potential components are constants over the whole coordinate space and (ii) the profile of some components is of a rectangular form. In case (i), it is illustrated how the structure of three (lower, middle and upper) bands depends on the configuration of potential strengths including the appearance of flat bands at some special values of these strengths. In case (ii), the set of two equations for finding bound states is derived. The spectrum of bound-state energies is shown to depend crucially on the configuration of potential strengths. Each of these configurations is specified by a single strength parameter V. The bound-state energies are calculated as functions of the strength V and a one-point approach is developed realizing correspondent point interactions. For different potential configurations, the energy dependence on the strength V is described in detail, including its one-point approximation. From a whole variety of bound-state spectra, four characteristic types are singled out.

Point Interactions With Bias Potentials

We develop an approach on how to define single-point interactions under the application of external fields. The essential feature relies on an asymptotic method based on the one-point approximation of multi-layered heterostructures that are subject to bias potentials. In this approach , the zero-thickness limit of the transmission matrices of specific structures is analyzed and shown to result in matrices connecting the two-sided boundary conditions of the wave function at the origin. The reflection and transmission amplitudes are computed in terms of these matrix elements as well as biased data. Several one-point interaction models of two- and three-terminal devices are elaborated. The typical transistor in the semiconductor physics is modeled in the "squeezed limit'' as a $\delta$- and a $\delta'$-potentialand referred to as a "point" transistor. The basic property of these one-point interaction models is the existence of several extremely sharp peaks as an applied voltage tunes, at which the transmission amplitude is non-zero, while beyond these resonance values, the heterostructure behaves as a fully reflecting wall. The location of these peaks referred to as a "resonance set" is shown to depend on both system parameters and applied voltages. An interesting effect of resonant transmission through a $\delta$-like barrier under the presence of an adjacent well is observed. This transmission occurs at a countable set of the well depth values.

Metric Max-Distance Function in Max-Approximation Problems

The paper is concerned with approximation properties of sets in the problem of one-point approximation. The Chebyshev radius of a set is estimated depending on the behavior of the metric max-function and the diameter of the set of furthest points.

Generation of Supramolecular Chirality around Twofold Rotational or Helical Axes in Crystalline Assemblies of Achiral Components

A multi-point approximation method clarifies supramolecular chirality of twofold rotational or helical assemblies as well as bundles of the one-dimensional (1D) assemblies. While one-point approximation of materials claims no chirality generation of such assemblies, multi-point approximations do claim possible generation in the 1D assemblies of bars and plates. Such chirality derives from deformations toward three-axial directions around the helical axes. The chiral columns are bundled in chiral ways through symmetry operations. The preferable right- or left-handed columns are bundled together to yield chiral crystals with right- or left-handedness, respectively, indicating that twofold helix symmetry operations cause chiral crystals composed of achiral components via a three-stepwise and three-directional process.

Optimal global approximation of SDEs with time-irregular coefficients in asymptotic setting

We investigate strong approximation of solutions of scalar stochastic differential equations (SDEs) with irregular coefficients. In Przybyłowicz (2015) [23], an approximation of solutions of SDEs at a single point is considered (such kind of approximation is also called a one-point approximation). Comparing to that article, we are interested here in a global reconstruction of trajectories of the solutions of SDEs in a whole interval of existence. We assume that a drift coefficient a:[0,T]×R→R is globally Lipschitz continuous with respect to a space variable, but only measurable with respect to a time variable. A diffusion coefficient b:[0,T]→R is only piecewise Hölder continuous with Hölder exponent ϱ ∈ (0, 1]. The algorithm and results concerning lower bounds from Przybyłowicz (2015) [23] cannot be applied for this problem, and therefore we develop a suitable new technique. In order to approximate solutions of SDEs under such assumptions we define a discrete type randomized Euler scheme. We provide the error analysis of the algorithm, showing that its error is O(n−min{ϱ,1/2}). Moreover, we prove that, roughly speaking, the error of an arbitrary algorithm (for fixed a and b) that uses n values of the diffusion coefficient, cannot converge to zero faster than n−min{ϱ,1/2} as n→+∞. Hence, the proposed version of the randomized Euler scheme achieves the established best rate of convergence.

Minimal asymptotic error for one-point approximation of SDEs with time-irregular coefficients

We consider strong one-point approximation of solutions of scalar stochastic differential equations (SDEs) with irregular coefficients. The drift coefficient a:[0,T]×R→R is assumed to be Lipschitz continuous with respect to the space variable but only measurable with respect to the time variable. For the diffusion coefficient b:[0,T]→R we assume that it is only piecewise Hölder continuous with Hölder exponent ϱ∈(0,1]. We show that, roughly speaking, the error of any algorithm, which uses n values of the diffusion coefficient, cannot converge to zero faster than n−min{ϱ,1/2} as n→+∞. This best speed of convergence is achieved by the randomized Euler scheme.

One-point pseudospectral collocation for the one-dimensional Bratu equation

The one-dimensional planar Bratu problem is uxx+λ exp(u)=0 subject to u(±1)=0. Because there is an analytical solution, this problem has been widely used to test numerical and perturbative schemes. We show that over the entire lower branch, and most of the upper branch, the solution is well approximated by a parabola, u(x)≈u0 (1−x2) where u0 is determined by collocation at a single point x=ξ. The collocation equation can be solved explicitly in terms of the Lambert W-function as u(0)≈−W(−λ(1−ξ2)/2)/(1−ξ2) where both real-valued branches of the W-function yield good approximations to the two branches of the Bratu function. We carefully analyze the consequences of the choice of ξ. We also analyze the rate of convergence of a series of even Chebyshev polynomials which extends the one-point approximation to arbitrary accuracy. The Bratu function is so smooth that it is actually poor for comparing methods because even a bad, inefficient algorithm is successful. It is, however, a solution so smooth that a numerical scheme (the collocation or pseudospectral method) yields an explicit, analytical approximation. We also fill some gaps in theory of the Bratu equation. We prove that the general solution can be written in terms of a single, parameter-free β(x) without knowledge of the explicit solution. The analytical solution can only be evaluated by solving a transcendental eigenrelation whose solution is not known explicitly. We give three overlapping perturbative approximations to the eigenrelation, allowing the analytical solution to be easily evaluated throughout the entire parameter space.

Some one-parameter families of third-order methods for solving nonlinear equations

In this paper, two one-parameter families of irrational and rational iterative methods for solving nonlinear equations are constructed. The construction of the iterative process is based on one-point approximation by the quadratic equation x 2 + ay + bx + cxy + d = 0 . Euler, Chebyshev, Halley, Super-Halley methods can be seen as the special cases of the family. And by analysis of asymptotic errors, we prove the two families are all cubically convergent. Furthermore, the rational iterative family can be extended to Banach spaces easily.

A family of third-order methods to solve nonlinear equations by quadratic curves approximation

A one-parameter family of iteration functions for finding the simple roots of nonlinear equations is presented. The iteration process is based on one-point approximation by the quadratic equation x 2 + ay 2 + bx + cy + d = 0, where the unknowns b, c and d are determined in terms of a. Different choices of a correspond to different approximating quadratic curves, viz. parabola, circle, ellipse and hyperbola. Euler, Chebyshev, Halley, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family are cubically convergent except Newton’s which is quadratically convergent.

A family of one-point iteration formulae for finding roots

In this paper a one parameter family of one-point iteration formulae for solution of a single variable equation is presented. The formulae are derived on the basis of one-point approximation by the four parameter function . The family includes the Halley's, Chebyshev's and Cauchy's formulae, and, as a limiting case, Newton's formula. All formulae of the family are cubically convergent to simple roots (except Newton's which is quadratically convergent).

Determination of small reactivity in real time by the inverse kinetics method

This article describes measurements of small reactivity which have been carried out on-line and in real time. Calculations were made with use of a one-point approximation and the inverse kinetics method. Experiments were performed on the fast-thermal critical assembly at the Institute of Nuclear Research in Świerk. Having in mind the significance of precision in reactivity measurements applied to reactor diagnostics, the authors tested the accuracy of the inverse kinetics method algorithm. The measurements carried out in real time on a reactivity meter designed on the basis of this method are influenced by an error of ≲ ± 0.06¢, and the measurements of the steady reactivity executed during 80 sec after perturbation are determined with an error of ∼ ± 0.012¢.

Improved treatment of Pauli operator in calculation of saturation diagrams in 16O

The free parameter kF for Fermi momentum appearing in the Kerman–Pal procedure for renormalizing the nucleon–nucleon interaction to second order in nuclei has been eliminated by making use self-consistently of Wong's one-point approximation. Calculations of the saturation diagrams in 16O with this approach reproduce very closely our earlier results with a fixed estimated value for kF.