Majorana solution of the Thomas-Fermi equation

We report on an original method, due to Majorana, leading to a semi-analytical series solution of the Thomas-Fermi equation, with appropriate boundary conditions, in terms of only one quadrature. We also deduce a general formula for such a solution which avoids numerical integration, but is expressed in terms of the roots of a given polynomial equation.Comment: RevTex, 5 pages, 1 figur

Characterization of Maximally Random Jammed Sphere Packings: II. Correlation Functions and Density Fluctuations

In the first paper of this series, we introduced Voronoi correlation functions to characterize the structure of maximally random jammed (MRJ) sphere packings across length scales. In the present paper, we determine a variety of correlation functions that can be rigorously related to effective physical properties of MRJ sphere packings and compare them to the corresponding statistical descriptors for overlapping spheres and equilibrium hard-sphere systems. Such structural descriptors arise in rigorous bounds and formulas for effective transport properties, diffusion and reactions constants, elastic moduli, and electromagnetic characteristics. First, we calculate the two-point, surface-void, and surface-surface correlation functions, for which we derive explicit analytical formulas for finite hard-sphere packings. We show analytically how the contacts between spheres in the MRJ packings translate into distinct functional behaviors of these two-point correlation functions that do not arise in the other two models examined here. Then, we show how the spectral density distinguishes the MRJ packings from the other disordered systems in that the spectral density vanishes in the limit of infinite wavelengths. These packings are hyperuniform, which means that density fluctuations on large length scales are anomalously suppressed. Moreover, we study and compute exclusion probabilities and pore size distributions as well as local density fluctuations. We conjecture that for general disordered hard-sphere packings, a central limit theorem holds for the number of points within an spherical observation window. Our analysis links problems of interest in material science, chemistry, physics, and mathematics. In the third paper, we will evaluate bounds and estimates of a host of different physical properties of the MRJ sphere packings based on the structural characteristics analyzed in this paper.Comment: 25 pages, 13 Figures; corrected typos, updated reference

Characterization of Maximally Random Jammed Sphere Packings. III. Transport and Electromagnetic Properties via Correlation Functions

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here, we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.Comment: 19 pages, 16 Figure

Demonstration of down-chirped and chirp-free pulses from high-repetition-rate passively mode-locked lasers

Knowledge and control of the chirp parameters of semiconductor lasers is a prerequisite to obtaining transform-limited pulses and/or to compensate for group velocity dispersion in fiber. Here, we report measurements of the sign and magnitude of chirp in high-repetition-rate mode-locked semiconductor lasers. The chirp of these monolithic lasers is measured in the frequency domain, using filtering and cross-correlation techniques. For different injection currents, a range of different chirp values is measured, including strongly down-chirped pulses at higher injection currents and transform-limited pulses to slightly up-chirped pulses at lower injection currents. The pulse chirp and the resulting broadening are due to the algebraic addition of opposite-signed chirps due to saturation of the absorption section and the gain section. These may cancel each other under some conditions, leading to a soliton-like transform-limited pulse

Retardation and reduction of pulse distortion by group-velocity dispersion through pulse shaping

We show that a reduction in the pulse distortion caused by chromatic dispersion can be achieved through pulse shaping. We argue that a simple binary phase mask in the Fourier plane of the laser spectrum can improve the transmission of short pulses in a dispersive channel through reduced broadening. The argument was tested experimentally, and a good agreement was found with the theory

Puberty: Is Your Gingiva Having Mood Swings?

Objectives/aim: The purpose of this paper is to explore the effects on the different pathological changes in the oral cavity due to puberty, in both males and females. Hormonal changes caused by menstrual cycles, ovulation, the use contraceptives, and increased testosterone and estrogen levels. Methods: This topic will be analyzed by thoroughly reviewing research on articles that relate to the oral health of individuals specifically between the ages of 12-18 years old. Results: Research presents significant evidence that supports changes occurring in the oral cavity during an individual’s stage of puberty. These stages include ovulation, pre-menstruation, menstruation and males transitioning through puberty. During the puberty stage adolescents are more prone to have increased gingival crevicular fluid (GCF), gingival index, and bleeding on probing while research has shown no significant findings on plaque indexes or probing depths. Changes occurring during the menstrual cycle tend to influence the periodontium and induce inflammatory conditions as well. While the periodontium and inflammatory cytokines play a major role in the effects during puberty, changes in diet during this phase can increase the risk of developing caries as well. Conclusion: When adolescents are transitioning into adulthood, there are multiple changes their body goes through. During the literature review, many changes happen during puberty significantly affecting the oral cavity were discovered. These changes have both positive and negative effects. Variations in hormone levels and diet greatly influence the health of the oral cavity and can be a deciding factor on development or severity of oral disease.https://scholarscompass.vcu.edu/denh_student/1008/thumbnail.jp