Approximate solution of bound state problems through continued fractions

A method to solve ordinary linear differential equations through continued fractions is applied to several physical systems. In particular, results for the Schrödinger equation give a good accuracy for the eigenvalues of bound states in theS-wave Yukawa potential, and the lowest order approximations provide exact-values for the harmonic oscillator and Coulomb potential eigenvalues and eigenfunctions.     

Exact solutions of discrete master equations in terms of continued fractions

We present the continued fraction solution for the stationary probability of discrete master equations of one-variable processes. After we elucidate the method for simple birth and death processes we focus the study on processes which introduce at least two-particle jumps. Consequently, these processes do in general not obey a detailed balance condition. The outlined method applies as well to solutions of eigenmodes of the stochastic operator. Further we derive explicit continued fraction solutions for the Laplace transform of conditional probabilities. All the various continued fraction coefficients are given directly in terms of the transition rates and they obey recursion relations. The method is illustrated for the stationary solution of a simple nonlinear chemical reaction scheme originated by Nicolis.     

Block spin approach to the singularity properties of the continued fractions

The massless singularity of a ferromagnetic Gaussian measure on ₊ is studied by means of the coarse graining renormalization group method. The result gives information about a singularity behavior of a continued fraction and a time decay rate of a diffusion (random walk) on ₊.     

Method of matrix continued fractions for the analysis of nonlinear stochastic systems

Structural Engineering Institute, Gorki. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 32, No. 2, pp. 169–175, February, 1989.     

Thermoluminescence glow curve deconvolution functions by continued fractions for different orders of kinetics

The shape of the peaks in thermoluminescence (TL) dosimetry can be represented by the so-called temperature integral. In this article, we present a very efficient method, based on a continued fraction approach to the incomplete gamma function, intended to calculate the overall temperature integral which includes the frequency factor ∝ T ^a . The single glow-peak algorithm for linear and exponential heating rates is derived. In the first case, the method provides a good approximation with a maximum relative error of 1.1×10^?5 within the 0.1≤E/kT≤90 range in the case of a=0. It is shown that, in general, the method is efficient, converges quickly and can be adopted in the numerical fitting of glow lines in order to obtain the parameters relevant to thermoluminescence (TL). The utility of this approach is exemplified by adjusting the standard LiF: Mg, Ti (TLD-100) using five and six TL peaks, determining that peak 6 is present and observable in the analysed spectrum. Finally, methods such as asymptotic expansion of the temperature integral by asymptotic series, convergent series, Lagrange continued fractions and a new obtained continued fraction approximations are compared to the method proposed here, in case of linear heating.     

混沌时间序列全局预测新方法——连分式法

拓展了多项式逼近理论，利用连分式法建立了混沌时间序列非线性全局预测模型，此模型替代混沌序列的动力学方程，实现对其动力学特性分析，达到预测目的. 理论分析和仿真实验表明，连分式法能够有效预测一些混沌序列，该方法预测精度高，并且能得到显式的预测表达式. 关键词： 混沌时间序列 全局预测 连分式     

Analytical characterization of gold nanoparticle primary particles,aggregates, agglomerates,and agglomerated aggregates

Gold nanoparticles have been studied for many biomedical applications. However, alterations in the gold nanoparticles’ environment frequently lead to the formation of aggregates and agglomerates, which have not been well characterized. These new structures could significantly change the biological impact of the nanoparticles, so the appropriate characterization of these structures prior to biological administration is vital for the correct interpretation of toxicology results. By varying the solvent or heating under pressure, four reproducible gold nanoparticles structures were created: 10 nm primary particles, aggregates of the primary particles that contain non-reversible bonds between the individual nanoparticles, agglomerates of primary particles that contain reversible interactions between the individual nanoparticles, and agglomerated aggregates that have reversible bonds linking individual aggregates. Ultraviolet–visible (UV–Vis) spectroscopy, thermal gravitational analysis, and neutron activation analysis were each found to accurately measure the concentration of the primary particles. The primary particles measured 10 nm by dynamic light scattering (DLS) and had a spherical morphology by transmission electron microscopy (TEM) while the aggregates measured 110 nm by DLS and had a distorted morphology by TEM. The agglomerate and aggregated agglomerate samples both measured >1,000 nm by DLS, but the individual particles had significantly different morphologies by TEM. Multiple other analytical techniques, including ultracentrifugation, gel electrophoresis, and X-ray diffraction, also showed unique traits for each structure. The structural differences did not change in the presence of cell culture media or rat serum. In addition, the primary particles, aggregates, and agglomerates each had a unique UV–Vis spectrum, allowing for an inexpensive, rapid method to differentiate between the structures.     

Solutions of the Fokker-Planck equation for a double-well potential in terms of matrix continued fractions

Solutions of the Fokker-Planck (Kramers) equation in position-velocity space for the double-well potentiald ₂x²/2+d₄x⁴/4 in terms of matrix continued fractions are derived. It is shown that the method is also applicable to a Boltzmann equation with a BGK collision operator. Results of eigenvalues and of the Fourier transform of correlation functions are presented explicitly. The lowest nonzero eigenvalue is compared with the escape rate in the weak noise limit for various damping constants and the susceptibility is compared with the zero-friction-limit result.     

Self-assembly of twisted,multi-sheet aggregates

ABSTRACT

Hierarchical self-assembly underpins much of the diversity of form and function seen in soft systems, yet the pathways by which they achieve their final form are not always straightforward – intermediate steps, kinetic effects and finite sizes of aggregates all influence the self-assembly pathways of these systems. In this paper, we use molecular dynamics simulations of binary mixtures of spheres and ellipsoidal discs to investigate the self-assembly of anisotropic aggregates with internal structures. Through this, the full aggregation pathways of spontaneously chiral, multi-bilayer and multi-layer assemblies have been tracked and characterised via a semi-qualitative analysis. This includes the unambiguous identification of first-, second- and third-generation hierarchical assemblies within a single simulation. Given the significant challenge of tracking full aggregation pathways in experimental systems, our findings strongly support the notion that molecular simulation has much to contribute to improving our understanding of hierarchical self-assembling systems.     

On the approximate solution through continued fractions of the schrödinger equation with central potentials for positive energies

We present an extension to positive energies of an approximation method based on continued fractions already used to provide approximate energies and wave functions for bound states. A formalism particularly adapted for numerical computation is proposed and its applications are shown for the free particle case and short and long range central potentials, for the lowest angular momenta. The domain of convergence is displayed; especially for low and medium energies, the results are quite good. Some care is devoted to the successful determination of theS-wave scattering length for the Yukawa potential. The mathematical relationship with Calogero's formulation of the problem is established.     

Growth and shape of branched polymers,aggregates and percolating clusters

Fractals are described as branching chains, growing stochastically in “time” t. This relates critical branching to conductivity and leads to a generalized Flory approximation describing radius and mass versus t, for branched polymers, percolating clusters and random aggregates.     

Multicomponent composites,electrical networks and new types of continued fraction II

The outstanding problem of systematically developing rigorous bounds on the complex effective conductivity tensor * ofd-dimensional,n-component composites withn>2 is solved. The bounds incorporate information contained in successively higher order correlation functions which reflect the composite geometry. Explicit expressions are given for many of the bounds and some, but not all of them, are represented by nested sequences of circles in the complex plane that enclose, and in fact converge to, each diagonal element of *. They are derived from the fractional linear matrix transformations found in Part I that recursively link * with a hierarchy of complex effective tensors ^(j),j=0, 1, 2, ..., of increasing dimension,d(n–1) ^j . Elementary bounds on ^(j) confining the diagonal elements of ^(j) or its inverse to half-plane, wedge or open polygon regions of the complex plane, imply narrow bounds on * which converge to the exact value of * in the limit asj . When the component conductivities are real these bounds are more restrictive than the corresponding variational bounds. Besides applying to the effective conductivity *, the bounds extend to a wide class of matrix-valued multivariate functions called -functions, and thereby to conduction in polycrystalline media, viscoelasticity in composites, and conduction in multi-component, multiterminal, linear electrical networks. The analytic and invariance properties of -functions are explored and within this class of function most of the bounds are found to be optimal or at least attainable. The bounds obtained here are essentially a generalization to matrix-valued, multivariate functions of the nested sequence of lens-shaped bounds in the complex plane derived by Gragg and Baker for single variable Stieltjes functions.     

Multicomponent composites,electrical networks and new types of continued fraction I

The development of bounds on the complex effective conductivity tensor * (that relates the average current to the average electric field in a multicomponent composite) has been hindered by lack of a suitable continued-fraction representation for *. Here a new field equation recursion method is developed which gives an expression for * as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry. A hierarchy of field equations is set up such that the solutions of the (j+1)th-order equation generate the solutions of thejth-order equation. Consequently the effective tensor ^(j) associated with thejth-order field equation is expressible as a fractional linear matrix transformation of ^(j+1). These transformations combine to form the continued fraction expansion for *=⁽⁰⁾ which is exploited in the following paper, Part II, to obtain bounds: crude bounds on ^(j), forj1, give narrow bounds on *. The continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable Stieltjes functions that proved important in the development of the theory of Páde approximants, asymptotic analysis, and the theory of orthogonal polynomials in the last century. The results extend to other transport problems, including conduction in polycrystalline media, the viscoelasticity of composites, and the response of multicomponent, multiterminal linear electrical networks.     

Synthesis, structure, and dielectric properties of KH2PO4 fractal aggregates

Macroscopic fractal aggregates of KH₂PH₄ (KDP) measuring up to 500 μm have been obtained. The fractal structure forms as a result of the precipitation of KDP particles from a supersaturated aqueous solution in the presence of a temperature gradient followed by a diffusioncontrolled mechanism of aggregation. The electron-microscopic analysis performed has shown that the fractals are formed predominantly from crystallites of the tetragonal modification measuring ∼1 μm. The dielectric constant (ɛ) of fractal KH₂PO₄ has been measured in the temperature range 80–300 K. A characteristic anomaly has been discovered on the ɛ(T) curve in the vicinity of 122 K, which attests to a ferroelectric phase transition. The absolute value of ɛ is significantly smaller than the components ɛ ₁₁ and ɛ ₃₃ for KH₂PO₄. Fiz. Tverd. Tela (St. Petersburg) 41, 2059–2061 (November 1999)     

Workplace air measurements and likelihood of exposure to manufactured nano-objects,agglomerates, and aggregates

Workplace exposure to nanoparticles from gas metal arc welding (GMAW) process in an automobile manufacturing factory was investigated using a combination of multiple metrics and a comparison with background particles. The number concentration (NC), lung-deposited surface area concentration (SAC), estimated SAC and mass concentration (MC) of nanoparticles produced from the GMAW process were significantly higher than those of background particles before welding (P < 0.01). A bimodal size distribution by mass for welding particles with two peak values (i.e., 10,000–18,000 and 560–320 nm) and a unimodal size distribution by number with 190.7-nm mode size or 154.9-nm geometric size were observed. Nanoparticles by number comprised 60.7 % of particles, whereas nanoparticles by mass only accounted for 18.2 % of the total particles. The morphology of welding particles was dominated by the formation of chain-like agglomerates of primary particles. The metal composition of these welding particles consisted primarily of Fe, Mn, and Zn. The size distribution, morphology, and elemental compositions of welding particles were significantly different from background particles. Working activities, sampling distances from the source, air velocity, engineering control measures, and background particles in working places had significant influences on concentrations of airborne nanoparticle. In addition, SAC showed a high correlation with NC and a relatively low correlation with MC. These findings indicate that the GMAW process is able to generate significant levels of nanoparticles. It is recommended that a combination of multiple metrics is measured as part of a well-designed sampling strategy for airborne nanoparticles. Key exposure factors, such as particle agglomeration/aggregation, background particles, working activities, temporal and spatial distributions of the particles, air velocity, engineering control measures, should be investigated when measuring workplace exposure to nanoparticles.