A Formula for the General Solution of a Constant-coefficient Difference Equation

AbstractWe give a formula for the general solution of a d th-order linear difference equation with constant coefficients in terms of one of the solutions of its associated homogeneous equation. The formula neither uses the roots of the characteristic equation nor their multiplicities. It can be readily generalized to the case where the domain of the difference equation is the real numbers, and the initial values are given by a function defined on the interval [0,d ). In both cases, we express the general solution of the difference equation in terms of a single solution of its associated homogeneous equation at integer arguments

Rates of evaporation of small drops at low pressures

Thesis (Ph.D.)--Boston UniversityIn this research the rates of evaporation of small drops of diamyl sebacate were determined in the range of radii of 0.5 - 2.0 microns and for a range of pressures of 0.5 - 10 cm Hg. An expression is derived for the rate of evaporation and is compared with the rates determined experimentally. From the method of approach, the degree of approximation of earlier treatments may be estimated. In the process of evaporation from a liquid surface, the determining factor is the rate of diffusion of vapor away from the surface of a spherical drop of radius H is proportional to the gradient of the concentration, n1. To determine the rate of evaporation, the gradient at the surface must be calculated from the solution to Fick's law of diffusion and the appropriate set of boundary conditions. This was first done in an approximate manner by Langmuir, and later with more accuracy by Bradley, Evans, and Whytlaw-Gray and by Frisch and Collins. In this research, the appropriate boundary condition at the surface was determined by equating the flux given by Fick's law to the flux calculated using the kinetic theory of gases. From the kinetic theory of gases, the flux of molecules condensing on a spherical surface per unit time from the gas phase is given by an integral involving the product of the velocity distribution (the density of molecules in velocity space) and the condensation coefficient; the probability or a molecule condensing on impact with the surface. In the presence of a concentration gradient, according to Chapman and Enskog's theory of non-uniform gases, the velocity distribution function will differ from the Maxwell-Boltzmann distribution, the distribution which is valid under equilibrium conditions, by a term proportional to the concentration gradient. The contribution of this term to the total flux was found to be appreciable. In general, the condensation coefficient may be considered to be a function of both the magnitude and the direction of the molecular velocity. However, very little is known about the functional dependence. Lennard-Jones and Devonshire have shown by calculation that for the case of hydrogen and deuterium adsorbing on a crystalline copper surface, the condensation coefficient has a long flat maximum. It may be assumed, therefore, that it is only a slowly varying function or the non-uniformity of the gas. In the subsequent development, the condensation coefficient was replaced by its average value, α, a constant which can be determined by experiment. If the number of molecules evaporating from the surface is assumed to be independent of the number arriving, the flux of evaporating molecules is easily evaluated by setting it equal to the flux of molecules that would be condensing from the gas phase if the system were at equilibrium. Equating the difference of the two fluxes to the total flux given by Fick's law, the boundary condition at the surface is shown to be of the form (∂n/∂r)s = h (n-(n(eq)) where h is a function of α, the temperature and the pressure. Using the stochastic formulation or Frisch and Collins, this boundary condition was rederived in an independent manner using as specialized conditions that the stochastic transition probability is isotropic, that the concentration is the steady-state concentration, and that the mean stochastic jump length is equal in magnitude to the mean free path. Using certain results of Frisch and Collins. an approximate solution to the differential equation expressing Fick's law and the appropriate boundary conditions may be obtained. To a high degree of accuracy, the rate of evaporation may then be shown to be given by the expression (dR/dt) - (hDn(eq))/(1+hR) The rates of evaporation of small diamyl sebacate droplets were determined in a Millikan oil drop chamber, which was thermostated by a built-on water bath. To measure the radii the law relating the radius to the velocity of free fall v, must be known. This was determined in the following manner. The law of fall was assumed to obey the Stokes-Cunningham formula relating the radius to the velocity, the viscosity and the pressure. A drop of dicapryl sebacate, whose rate of evaporation was measurably small was suspended between the plates by a voltage, V, and the velocities of free fall determined at a series of pressures. By a suitable extrapolation procedure, a rough value of R was determined. The charge was then calculated, corrected to the nearest integer, and R recalculated. b, the Cunningham coefficient, was now calculated, and for a series of drops, was found to obey the relation b = 7.2 x 10^-4 x (0.864+0.28 exp(-.18Rp) cm-cm Hg. This agrees to within 1% of the form given by Millikan for clock oil and air. Due to the similarity in nature of diamyl sebacate and dicapryl sebacate, the same b was assumed to hold. The rate or evaporation was determined by measuring the voltage, V, just required to suspend the drop as a function of time. At any particular value of V, a measurement of the velocity of fall and thus of R, yielded a rough value of the charge. When this was corrected to the nearest integer, the radius was known as a function of time. When 1/(dR/dt), corrected for the effect of radius on vapor pressure, was plotted versus Rp, since the diffusion coefficient varies inversely as the pressure, a straight line was obtained as predicted by equation (4). Using the experimental vapor pressures as determined by Perry and Weber, values of the condensation coefficient, and D, the diffusion coefficient were calculated from the intercept and slope. Using a hard sphere model, the radius calculated from the diffUsion coefficient corresponded to the state where the organic molecule was coiled up tightly. The condensation coefficient at the temperatures 25.16°C and 34.94°C was found to be 0.50 and temperature independent. Probably, higher degrees of accuracy could be obtained from a law derived using higher order terms in the velocity distribution function or taking into account the perturbation of the surface on f. However, it is shown that for the larger part of experimental conditions, these effects are small. The values of obtained by the application of equation (4) compare very well with those obtained by independent means. Condensation coefficients obtained from the application of a formula developed by Bradley, Evans, and Whytlaw-Gray do not agree as well. There seems to be negligible difference in the values of the diffusion coefficient deduced by the two methods. It may be concluded that this method constitutes a fairly accurate method of determining α and D. The equations developed hold strictly only for the case of non-polar substances. In the case of polar liquids, where a deep transition zone due to long range forces may exist, the model proposed may break down quite seriously. This may account for such effects as very low condensation coefficient and large thermal accommodation coefficient (1.0), as has been found experimentally for water

A High-Order Method for Stiff Boundary Value Problems with Turning Points

This paper describes some high-order collocation-like methods for the numerical solution of stiff boundary-value problems with turning points. The presentation concentrates on the implementation of these methods in conjunction with the implementation of the a priori mesh construction algorithm introduced by Kreiss, Nichols and Brown [SIAM J. Numer. Anal., 23 (1986), pp. 325–368] for such problems. Numerical examples are given showing the high accuracy which can be obtained in solving the boundary value problem for singularly perturbed ordinary differential equations with turning points

Closed form asymptotics for local volatility models

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5, 6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family of general closed-form approximate solutions for both the pricing kernel and derivative price. A bootstrap scheme allows us to extend our method to large time. We also perform analytic as well as a numerical error analysis, and compare our results to other known methods.Comment: 30 pages, 10 figure

Modeling style rotation: switching and re-switching

The purpose of this paper is to investigate the dynamics and statistics of style rotation based on the Barberis-Shleifer model of style switching. Investors in stocks regard the forecasting of style-relative performance, especially style rotation, as highly desirable but difficult to achieve in practice. Whilst we do not claim to be able to do this in an empirical sense, we do provide a framework for addressing these issues. We develop some new results from the Barberis-Shleifer model which allows us to understand some of the time series properties of style relative price performance and determine the statistical properties of the time until a switch between styles. We apply our results to a set of empirical data to get estimates of some of the model parameters including the level of risk aversion of market participants

Explicit Relativistic Vortex Solutions for Cool Two-Constituent Superfluid Dynamics

We give a class of explicit solutions for the stationary and cylindrically symmetric vortex configurations for a ``cool'' two-component superfluid (i.e. superfluid with an ideal gas of phonons). Each solution is characterized only by a set of (true) constants of integration. We then compute the effective asymptotic contribution of the vortex to the stress energy tensor by comparison with a uniform reference state without vortex.Comment: 33 pages, RevTeX, no figures, to appear in Phys. Rev.