Exact Analytical Methods Research Articles

Exact and Approximate Analytic Methods to Calculate Maximum Heat Flow in Annular Fin Arrays with a Rectangular Step Profile

Exact and approximate techniques to determine the performance of annular fins with a step rectangular profile are developed. A simple approximate method is proposed to analyze the prevalent heat transfer characteristics of the fin array. An algebraic expression based on the mean value approach is used as an approximation tool, and the temperature distribution in the fins is determined using an exponential function. A method based on a modified Bessel function formulation is employed for exact analysis. The analyses are extended to optimize fins based on the principle of maximizing heat transfer rate for a given volume. The results obtained from the exact and approximate analyses are presented in a one-to-one comparative manner to allow for a wide range of practical design variables. The error in the approximate analysis calculations is investigated, and it is found to be well within engineering accuracy requirements. It is expected that the approximate analytical tool designed here will be extremely useful for designers who want to easily determine design parameters. Because the approximate methodology is so simple, all calculations can easily be done.

Self-trapping of scalar and vector dipole solitary waves in Kerr media

We report solutions for expanding dipole-type optical solitary waves in two-dimensional Kerr media with the self-focusing nonlinearity, using exact analytical (Hirota) and numerical methods. Such localized beams carry intrinsic vorticity and exhibit symmetric shapes for both scalar and vector solitary modes. When vector beams are close to the scalar limit, simulations demonstrate their stability over propagation distances exceeding 50 diffraction lengths. In fact, the continuous expansion helps the vortical beams avoid the instability against the splitting, collapse, or decay, making them ``convectively stable'' patterns.

Analytical studies of fronts, colonies, and patterns: Combination of the Allee effect and nonlocal competition interactions

We present an analytic study of traveling fronts, localized colonies, and extended patterns arising from a reaction-diffusion equation which incorporates simultaneously two features: the well-known Allee effect and spatially nonlocal competition interactions. The former is an essential ingredient of most systems in population dynamics and involves extinction at low densities, growth at higher densities, and saturation at still higher densities. The latter feature is also highly relevant, particularly to biological systems, and goes beyond the unrealistic assumption of zero-range interactions. We show via exact analytic methods that the combination of the two features yields a rich diversity of phenomena and permits an understanding of a variety of issues including spontaneous appearance of colonies.

Statistical Theories of Income and Wealth Distribution

AbstractThe distributions of income and wealth in countries across the world are found to possess some robust and stable features independent of the specific economic, social and political conditions of the countries. We discuss a few physics-inspired multi-agent dynamic models along with their microeconomic counterparts, that can produce the statistical features of the distributions observed in reality. A number of exact analytical methods and solutions are also provided.

Disparities in Adequate Mental Health Care for Past-Year Major Depressive Episodes Among Caucasian and Hispanic Youths

Disparities in Adequate Mental Health Care for Past-Year Major Depressive Episodes Among Caucasian and Hispanic Youths

Disparities in Adequate Mental Health Care for Past-Year Major Depressive Episodes Among Caucasian and Hispanic Youths

Following efforts made in recent years to provide effective mental health treatments based on evidence-based guidelines, a working definition was developed in the literature detailing a minimum level of "adequate mental health care" for serious mental illness. However, little is known about racial or ethnic disparities in receipt of adequate mental health care for individuals affected with serious mental illness. The objective of this study was to examine disparities among Caucasian and Hispanic youths in receipt of adequate mental health care for past-year major depressive episodes. Data for this study were drawn from the 2005 National Survey on Drug Use and Health. The study sample was composed of 1,169 Caucasian youths and 316 Hispanic youths aged 12 to 17 with past-year major depressive episodes. The percentages of youths in the sample who received adequate mental health care for past-year major depressive episodes were estimated, and the correlates of receipt of adequate mental health care were examined. Thirty-four percent of the full sample received adequate mental health care for past-year major depressive episodes, but separate analyses indicated that adequate mental health care was received by a significantly higher proportion of Caucasian youths (36%) than Hispanic youths (27%). The odds of receiving adequate mental health care for past-year major depressive episodes for Caucasians were 1.55 times that of Hispanics (p=.01). Having Medicaid or coverage via the State Children's Health Insurance Program significantly increased the odds of receiving adequate mental care for past-year major depressive episodes for both Hispanics and Caucasians. As mental health problems of adolescents from diverse racial or ethnic backgrounds become more easily identified and a larger proportion of these groups is referred to mental health treatment services, it is important to examine the degree to which treatment should be tailored to engage and retain specific racial or ethnic groups so that they will receive the minimum of adequate mental health care.

Mathematical formulation for zero reflection from multilayer metamaterial structures and their notable applications

The behaviour of bilayer structures composed of common materials and metamaterials (MTMs) under oblique incidence of plane waves is investigated by exact analytical methods. The TE, TM and elliptical polarisations are analysed. There are several combinations of double positive (DPS), double negative (DNG), epsilon negative (ENG) and mu negative (MNG) media for the bilayer structures, but only DPS-DPS, DPS-DNG and ENG-MNG bilayers with TE, TM and circular polarisations are analysed in detail. For homogeneous and isotropic MTM media, exact mathematical relations are derived for the design of reflectionless bilayer structures as a function of their geometry (thickness) and electric and magnetic parameters. Frequency dispersion is included in the formulations. It is shown that bilayers composed of common materials are not effective for the construction of zero reflection bilayer surfaces, whereas the application of MTMs is required to realise reflectionless phenomena. For the design of zero reflection bilayer structures, their thicknesses and values of epsiv and mu are determined. Finally, the performance of forward and backward notch filters observed by MTM bilayer structures are studied in detail and their designs and applications are investigated. The bandwidth of lossy MTMs increases considerably.

Statistical Theories of Income and Wealth Distribution

The distributions of income and wealth in countries across the world are found to possess some robust and stable features independent of the specific economic, social and political conditions of the countries. We discuss a few physics-inspired multi-agent dynamic models along with their microeconomic counterparts, that can produce the statistical features of the distributions observed in reality. A number of exact analytical methods and solutions are also provided.

Specific Heat of the Mixed Spin-1/2 and Spin-S Ising Model with a Rope Ladder Structure

The mixed spin-1/2 and spin-S (S>1/2) Ising model on a rope ladder is examined by combining two exact analytical methods. By the decoration-iteration mapping transformation, this mixed-spin system is firstly transformed to a simple spin-1/2 Ising model on the two-leg ladder, which is then exactly solved by the standard transfer-matrix method. The thermal variations of the zero-field specific heat are discussed in particular.

Joint neighbors approximation of macromolecular solvent accessible surface area

A new method for approximate analytical calculations of solvent accessible surface area (SASA) for arbitrary molecules and their gradients with respect to their atomic coordinates was developed. This method is based on the recursive procedure of pairwise joining of neighboring atoms. Unlike other available methods of approximate SASA calculations, the method has no empirical parameters, and therefore can be used with comparable accuracy in calculations of SASA in folded and unfolded conformations of macromolecules of any chemical nature. As shown by tests with globular proteins in folded conformations, average errors in absolute atomic surface area is around 1 A2, while for unfolded protein conformations it varies from 1.65 to 1.87 A2. Computational times of the method are comparable with those by GETAREA, one of the fastest exact analytical methods available today.

Constructal multi-scale design of compact micro-tube heat sinks and heat exchangers

A constructal multi-scale design approach is examined for micro-tube heat sinks and heat exchangers. Heat transfer per unit volume is increased by considering the use of additional micro-tubes placed in the intersticial regions of a circular tube array. Three constructs are considered in the proposed analysis. As the system complexity increases, the heat transfer rate increases, and exceeds the theoretical value for a volume of similar size, composed of parallel plates. Approximate solutions for the diameter of the principal construct are obtained for each case using Bejan's intersection of asymptotes method. Exact analytical methods are applied to determine the relative increase in heat dissipation per unit volume as compared with systems containing parallel plates.

Analytical solution of the problem on thermal slip of second order for molecular gases

Results obtained using exact analytical methods in the problem on thermal slip of second order for molecular gases with allowance for the rotational degrees of freedom of molecules have been presented. Numerical calculations of the thermal-slip coefficient for a number of molecular gases have been carried out. The dependence of the velocity of thermal slip of second order of a molecular gas on the Prandtl number has been shown. The found value of the coefficient of thermal slip of second order theoretically confirms the existence of negative (in the direction of the temperature gradient) thermophoresis for molecular gases.

Calculation of the velocity of a molecular gas slipping over a sphere with regard to accommodation coefficients

The velocity of molecular gas slipping over a spherical surface is calculated by exact analytical methods including the accommodation coefficients for the first two moments of the distribution function. An extension of the BGK approach to the Boltzmann kinetic model for the case of rotational degrees of freedom is used as the basic equation.

Randomly accelerated particle in a box: Mean absorption time for partially absorbing and inelastic boundaries

Consider a particle which is randomly accelerated by Gaussian white noise on the line segment 0<x<1 and is absorbed as soon as it reaches x=0 or x=1. The mean absorption time T(x,v), where x and v denote the initial position and velocity, was calculated exactly by Masoliver and Porrà in 1995. We consider a more general boundary condition. On arriving at either boundary, the particle is absorbed with probability 1-p and reflected with probability p. The reflections are inelastic, with coefficient of restitution, r. With exact analytical and numerical methods and simulations, we study the mean absorption time as a function of p and r.

Soft handoff and uplink capacity in a two-tier CDMA system

This paper examines the effect of soft handoff on the uplink user capacity of a CDMA system consisting of a single macrocell in which a single hotspot microcell is embedded. The users of these two base stations operate over the same frequency band. In the soft handoff scenario studied here, both macrocell and microcell base stations serve each system user and the two received copies of a desired user's signal are summed using maximal ratio combining. Exact and approximate analytical methods are developed to compute uplink user capacity. Simulation results demonstrate a 20% increase in user capacity compared to hard handoff. In addition, simple, approximate methods are presented for estimating soft handoff capacity and are shown to be quite accurate.

Particle statistics and population dynamics

We study a master equation system modelling a population dynamics problem in a lattice. The problem is the calculation of the minimum size of a refuge that can protect a population from hostile external conditions, the so-called critical patch size problem. We analyse both cases in which the particles are considered fermions and bosons and show using exact analytical methods that, while the Fermi–Dirac statistics lead to certain extinction for any refuge size, the Bose–Einstein statistics allow survival even for the minimal refuge.

Analytical Solution of the Problem of Second-Order Thermal Slip with Allowance for the Rotational Degrees of Freedom of Gas Molecules

Exact analytical methods are applied to description of second-order thermal slip for a polyatomic gas (with the number of atoms in the molecule N ≥ 2) with allowance for the rotational degrees of freedom of the gas molecules. The thermal slip coefficients are determined for some particular molecular gases. It is shown that the second-order thermal slip coefficients of polyatomic gases depend on the Prandtl number.

Calculation of the Velocity of a Molecular Gas Slipping Round a Spherical Surface of Small Curvature

Exact analytical methods are applied to description of the thermal and isothermal slip of a molecular gas along a hard spherical surface with a relatively small curvature. The thermal and isothermal slip coefficients of the gas are determined in an approximation linear with respect to the Knudsen number. The problem is solved within the framework of the generalized Bhatnagar-Gross-Kruck (BGK) model of the Boltzmann kinetic equation taking into account the rotational degrees of freedom. The values of the thermal and isothermal slip coefficients have been numerically calculated for several molecular gases. It is shown that the obtained slip coefficients depend on the Prandtl number.

ARVO: A Fortran package for computing the solvent accessible surface area and the excluded volume of overlapping spheres via analytic equations

In calculating the solvation energy of proteins, the hydration effects, drug binding, molecular docking, etc., it is important to have an efficient and exact algorithms for computing the solvent accessible surface area and the excluded volume of macromolecules. Here we present a Fortran package based on the new exact analytical methods for computing volume and surface area of overlapping spheres. In the considered procedure the surface area and volume are expressed as surface integrals of the second kind over the closed region. Using the stereographic projection the surface integrals are transformed to a sum of double integrals which are reduced to the curve integrals. MPI Fortran version is described as well. The package is also useful for computing the percolation probability of continuum percolation models. Program summary Title of program: ARVO Catalogue identifier: ADUL Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUL Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Operating system under which the program has been tested: LINUX system and Windows system Programming language used: FORTRAN Computer: PC Pentium; SPP'2000; Number of bytes in distributed program, including test data, etc.: 322 633 Number of lines in distributed program, including test data, etc.: 5051 Distribution format: tar.gz Card punching code: ASCII Nature of physical problem: Molecular mechanics computations, continuum percolations. Method of solution: Numerical algorithm based on the analytical formulas, after using the stereographic transformation. Restriction on the complexity of the problem: The program does not account explicitly for cavities inside the molecule. Typical running time: Depends on the size of the molecule under consideration. Unusual features of the program: No

Equilibrium of a confined, randomly accelerated, inelastic particle: is there inelastic collapse?

We consider the one-dimensional motion of a particle randomly accelerated by Gaussian white noise on the line segment 0<x<1. The reflections of the particle from the boundaries at x=0 and 1 are inelastic, with velocities just after and before reflection related by v(f) =-r v(i). Cornell et al. have predicted that the particle undergoes inelastic collapse for r< r(c) = e(-pi/sqrt[3]) =0.163, coming to rest at the boundary after an infinite number of collisions in a finite time and remaining there. This has been questioned by Florencio et al. and Anton on the basis of simulations. We have solved the Fokker-Planck equation satisfied by the equilibrium distribution function P(x,v) with a combination of exact analytical and numerical methods. Throughout the interval 0<r<1, P(x,v) remains extended, as opposed to collapsed. There is no transition in which P(x,v) collapses onto the boundaries. However, for r< r(c) the equilibrium boundary collision rate is infinite, as predicted by Cornell et al., and all moments [v](q);, q>0 of the velocity just after reflection from the boundary vanish.