Laplace distribution

Equation (3) reveals that fluctuations in the Laplace distribution summed over all space vanish. The more negative eharge is concentrated in the valence shell, the deeper is the surrounding depletion shell. This can be seen for He and Ne, where in the latter case the extension of the valence shell from two 2s electrons to additional six 2p electrons has a distinct effect on both the value of charge concentration and charge depletion in the valence shell. 

Using real data, we are trying to find the suitable stochastic model. The collected dataset consists of the daily closing prices for 13 equity indexes from different countries. Starting from 1th January 1993 until 11th January 2013, the corresponding data series for each index forms 5000 daily points. In a preliminary study we found that the probabiUty distribution of the log-returns follows a combined density function of Normal and Laplace distribution, which is consistent with the previously mentioned proprieties of real data. In section 2, the model that we propose to use for the description of the dynamic of equity indexes is presented. First the data evolution is represented and briefly descried in order to clarify the chosen model motivation. Afterwards the two essential parts of the proposed model are represented the economic environmental is divided into three states (calm, normal and agitated) controlled by an external covariate that follows a Markov Chain, and the price evolution of different index at each state is considered to follow a log-normal diffusion, log-uniform jump... 

Laplace s equation is appHcable to many electrochemical systems, and solutions are widely available (8). The current distribution is obtained from Ohm s law... 

The distribution of current (local rate of reaction) on an electrode surface is important in many appHcations. When surface overpotentials can also be neglected, the resulting current distribution is called primary. Primary current distributions depend on geometry only and are often highly nonuniform. If electrode kinetics is also considered, Laplace s equation stiU appHes but is subject to different boundary conditions. The resulting current distribution is called a secondary current distribution. Here, for linear kinetics the current distribution is characterized by the Wagner number, Wa, a dimensionless ratio of kinetic to ohmic resistance. 

For many particles, the diffuse-charge layer can be characterized adequately by the value of the zeta potential. For a spherical particle of radius / o which is large compared with the thickness of the diffuse-charge layer, an electric field uniform at a distance from the particle will produce a tangential electric field which varies with position on the particle. Laplace s equation [Eq. (22-22)] governs the distribution... 

The combination of reac tor elements is facihtated by the concept of transfer functions. By this means the Laplace transform can be found for the overall model, and the residence time distribution can be found after inversion. Finally, the chemical conversion in the model can be developed with the segregation and maximum mixed models. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

As the corrosion rate, inclusive of local-cell corrosion, of a metal is related to electrode potential, usually by means of the Tafel equation and, of course, Faraday s second law of electrolysis, a necessary precursor to corrosion rate calculation is the assessment of electrode potential distribution on each metal in a system. In the absence of significant concentration variations in the electrolyte, a condition certainly satisfied in most practical sea-water systems, the exact prediction of electrode potential distribution at a given time involves the solution of the Laplace equation for the electrostatic potential (P) in the electrolyte at the position given by the three spatial coordinates (x, y, z). 

By means of Laplace transforms of the foregoing three equations mating use of the convolution theorem and the assumptions Pf(t) — Pt a constant which is the ratio of the in use time (t the total operating time of the 4th component), Gt(t) si — exp ( — t/dj (note that a double transform is applied to Ff(t,x)), we obtain an expression in terms of the lifetime distribution, i.e.,... 

Newman, J. Determination of Current Distributions Governed by Laplace s Equation 23... 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

Equation (15.39) allows moments of a distribution to be calculated from the Laplace transform of the dilferential distribution function without need for finding f t). It works for any f t). The necessary algebra for the present case is formidable, but finally gives the desired relationship ... 

Thus, the fraction unreacted is the Laplace transform with respect to the transform parameter k of the differential distribution function. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

Determination of Current Distributions Governed by Laplace s Equation Direct Methanol Fuel Cells From a Twentieth Century Electrochemist s Dream to a Twenty-First Century Emerging Technology West, A. C. Newman, J. Lamy, C. Ueger, J.-M. Srinivasan, S. 23... 

Then, taking into account that the distribution of masses inside the spheroid is independent of the azimuth coordinate cp, we have for Laplace s equation... 

The trace vanishes because only p- and /-electrons contribute to the EFG, which have zero probability of presence at r = 0 (i.e. Laplace s equation applies as opposed to Poisson s equation, because the nucleus is external to the EFG-generating part of the electronic charge distribution). As the EFG tensor is symmetric, it can be diagonalized by rotation to a principal axes system (PAS) for which the off-diagonal elements vanish, = 0. By convention, the principal axes are chosen such that... 

The spatial current distribution can be simulated by solving Laplace s equation with Neumann boundary conditions at the pore walls,... 

The Laplace inversion (LI) is the key mathematical tool of the DDIF experiment. The ability to convert the measured multi-exponential decay into a distribution of decay times is crucial to the DDIF pore size distribution application. However, unlike other mathematical operations, the Laplace inversion is an ill-conditioned problem in that its solution is not unique, and is fairly sensitive to the noise in the input data. In this light, significant research effort has been devoted to optimizing the transform and understanding its boundaries [17, 53, 54],... 

An example of DDIF data on a Berea rock sample is shown in Figure 3.7.1 illustrating the decay data (A), the pore size distribution after Laplace inversion... 

Experimental considerations Frequently a numerical inverse Laplace transformation according to a regularization algorithm (CONTEST) suggested by Provencher [48,49] is employed to obtain G(T). In practice the determination of the distribution function G(T) is non-trivial, especially in the case of bimodal and M-modal distributions, and needs careful consideration [50]. Figure 10 shows an autocorrelation function for an aqueous polyelectrolyte solution of a low concentration (c = 0.005 g/L) at a scattering vector of q — 8.31 x 106 m-1 [44]. 

Mw = 2.1 x 106g/mol) in water, which is denoted Cw(t) in the original work [44]. The subscript indicates that both the incoming beam and the scattered light are vertically polarized. The correlation function was recorded for a solution with a concentration of c = 0.005 g/L at a scattering vector of q = 8.31 x 106m-1. The inset shows the distribution function of the relaxation times determined by an inverse Laplace transformation. 

Equation (51)) contains all information about the distribution function G(r) of the decay rates T — Dq2 (Equation (52)). G(F) is obtained from an inverse Laplace transform of Equation (51). The computation of G(T) is a rather difficult task and a short discussion has been given in Section 5.2. 

The Laplace equation also applies to the distribution of electrical potential and current flow in an electrically conducting medium as well as the temperature distribution and heat flow in a thermally conducting medium. For example, if => E, V => i, and fi/K => re, where re is the electrical resistivity (re = RA/Ax), Eq. (13-22) becomes Ohm s law ... 

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