Laplace principle

The Laplace principle, sometimes known as the principle of insufficient reason, assumes that the probabilities of future events occurring are equal. That is, in the absence of any information to the contrary, it is assumed that all future outcomes are equally likely to occur. The expected cost (or profit/revenue) of each alternative is then computed, and the alternative that yields the minimum expected cost (or maximum expected profit/revenue) is selected. The mathematical expression for this principle is... 

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. 

A real foam has further degrees of freedom available for estabHshing local mechanical equiHbrium the films and Plateau borders may curve. In fact, curvature can be readily seen in the borders of Figure 1. In order to maintain such curvature, there must be a pressure difference between adjacent bubbles given by Laplace s law according to the surface free energy of the film and the principle radii of curvature of the film AP = ) Note that the... 

The emulsification process in principle consists of the break-up of large droplets into smaller ones due to shear forces (10). The simplest form of shear is experienced in lamellar flow, and the droplet break-up may be visualized according to Figure 4. The phenomenon is governed by two forces, ie, the Laplace pressure, which preserves the droplet, and the stress from the velocity gradient, which causes the deformation. The ratio between the two is called the Weber number. We, where Tj is the viscosity of the continuous phase, G the velocity gradient, r the droplet radius, and y the interfacial tension. 

Laplace s equation, 146 Least action, principle of, 69, 304 Line of heterogeneous states, 172 Liquefaction of gases, 167, 173 of mixtures, 428... 

Equation (15.38) gives the Laplace transform of the outlet response to an inlet delta function i.e., a utik) = k[f t)]- In principle. Equation (15.38) could be inverted to obtain/(r) in the time domain. This daunting task is avoided by... 

In principle, numerical methods can be employed to evaluate inverse Laplace transforms. However, the procedure is subject to errors that are often very laige-even catastrophic. 

The principles of complex variables are useful in the solution of a variety of applied problems, including Laplace transforms and process control (Sec. 8). 

An interesting approach has been employed in paper [74] to find the distribution f(li, l2) of copolymer chains for numbers l and h of monomeric units Mi and M2. This distribution is evidently equivalent to the SCD, because the pair of numbers k and I2 unambiguously characterizes chemical size (l = h + l2) and composition ( 1 = l] //, 2 = h/l) of a macromolecule. The essence of this approach consists of invoking the Superposition Principle [81] that enables the problem of finding the Laplace transform G(pi,p2) of distribution f(li,k) to be reduced to the solution of two subsidiary problems. The first implies the derivation of the expression for the generating function [/(z1",z 2n ZjX,z ) of distribution P(ti, M2 mt, m2), and the second is concerned with finding the Laplace transforms g (pi,p2) and (pi,p2) of distributions (Eq. 91). With these two problems solved, it is possible to obtain the characteristic function G(pi,p2) of distribution f(li,h) using the Superposition Principle formula... 

The sensoric principle of the dynamic bubble-pressure tensiometer is based on the differential Laplace pressure between two capillaries from which a controlled gas flow is released. At the lower end of a capillary which points into the liquid, a gas bubble is formed which increases its radius with increasing gas pressure (see... 

The basic principles are taken from Zwillinger (1989). Duhamel s principle enables solutions for surface conditions being functions of time to be calculated from solutions with permanent surface conditions. Although this principle is most easily derived through the use of Laplace transforms, more conventional demonstrations, not repeated here, can be found in Sneddon (1957) or Carslaw and Jaeger (1959). 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

Notice that the right-hand side of Eq. (34) is equal to the ratio of the transformed concentration at the second measurement point to the transformed concentration at the first measurement point. In the terminology of control engineering, this quantity is the transfer function of the system between Xo and Xm- The Laplace-transform method is possible because the diffusion equation is a linear differential equation. Thus, the right-hand side of Eq. (34) could in principle be used in a control-system analysis of an axial-dispersion process. 

The Young Equation. The principle of balancing forces used in the derivation of the Laplace equation can also be used to derive another important equation in surface thermodynamics, the Young equation. Consider a liquid droplet in equilibrium... 

First of all, the mathematical background will be developed for the case of a simple electrode reaction O + n e = R. In this treatment, contrasts like potential versus current perturbation, large amplitude versus small amplitude, and single step versus periodical perturbation are emphasized. While discussing these principles, the most common methods derived from them will be briefly mentioned. On the other hand, it will be shown that, by virtue of the method of Laplace transformation, these methods have much in common and contain, in principle, the same information if the detected cell response is of the same order. 

A most convenient way to solve the differential equations describing a mass transport problem is the Laplace transform method. Applications of this method to many different cases can be found in several modern and classical textbooks [21—23, 53, 73]. In addition, the fact that electrochemical relationships in the so-called Laplace domain are much simpler than in the original time domain has been employed as an expedient for the analysis of experimental data or even as the basic principle for a new technique. The latter aspect, especially, will be explained in the present section. 

In the limiting case where we have a thin-gap thin-ring electrode and consequently normal convection may be neglected, this problem can, in principle, be solved analytically. However, inversion from Laplace space is difficult and polynomial expansion is necessary except for small k,. If we define... 

Calorimeters of Historical and Special Interest Around 1760 Black realized that heat applied to melting ice facilitates the transition from the solid to the liquid stale at a constant temperature. For the first time, the distinction between the concepts of temperature and heat was made. The mass of ice that melted, multiplied by the heal of fusion, gives the quantity of heal. Others, including Bunsen, Lavoisier, and Laplace, devised calorimeters based upon this principle involving a phase transition. The heat capacity of solids and liquids, as well as combustion heats and the production of heat by animals were measured with these caloritnelers. 

Although the principles of heat flow have lieen understood and treated mathematically since the early 19th century (Fourier, LaPlace. Poisson, Peclel. Lord Kelvin. Riemann. and many otherst. it was not until nearly... 

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