Neuman equation

The state of an NMR-relevant physical system changes over time as described by the Schrodinger equation which within a statistical density operator formalism may be recast in form of the so-called Liouville-von Neuman equation... 

Equipped with the Hamiltonian at any time of the experiment, the next step in a numerical evaluation is to solve the Liouville-von Neuman equation in Eq, (1), The formal solution is given by... 

When the material of solid surface is to a significant extent soluble in the wetting liquid, one also observes the hysteresis phenomena. In this case changes in the profile of the solid surface occur due to a contact with the liquid phase. To understand this, let us recall Fig. III-19, from which one can see that the vertical component of the oLG vector can not be fully balanced by the surface tensions of the two other surfaces. If a liquid is in contact with a solid phase that is insoluble, this vertical component, oLG, is balanced by the elastic resistance of the solid surface. The situation is entirely different when a drop of liquid (L,) is placed onto the surface of another liquid (L2) in this case all phases are highly mobile, and the state of equilibrium is described by the vector Neuman equation ... 

Similar conditions exist if a drop of liquid is placed on the surface of a solid which is soluble in this liquid. Due to diffusion through the liquid, the system may come to a true equilibrium described by the Neuman equation ... 

The term decoherence describes the process by which the off-diagonal elements of the reduced density matrix tend to zero when evolving with time. Our objective is to reach an understanding of the molecular mechanisms governing decoherence with an atomic resolution. In addition we wish to be in a position to treat systems consisting of tens to thousands of atoms since the brute force simulation of the time evolution of p t) by the Liouville-von Neuman equation (p (i) = ih [H, p ]), the equivalent of the TDSE in the density matrix formalism, is out of question for such molecular systems. 

Neuman and Jonas (38) summarized these results, although the number of examples they considered was rather small. According to their study, tire difference in barriers to rotation between a given substituent and a methyl group in the acyl part of Ar,Y-dimethy lamides is given by the following equation ... 

Determination of Current Distributions Governed by Laplace s Equation West, A. C. Neuman, J. 23... 

Neuman boundary condition, when the derivatives of the generic variable on the boundary are known and this yields an extra equation. 

Some authors describe the kinetic of the adsorption process as a linear combination of exponential terms analogous to a series of simultaneous first-order reactions. Neuman and Neuman (1958) described the kinetics of the adsorption process by the following equation ... 

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

Compatibilization of polymer blends aims to improve the interaction between phases, ascertaining the appropriate, stable morphology and improved performance. Blends have been compatibilized mainly by addition of a compatibilizer, a co-solvent, or in a reactive process, where the compatibilizing molecules are formed within the interphase [1, 73, 302]. About 20 years ago a note in a USSR technological journal reported that the addition of a small amount of PMMA to a PE/PS blend reduced the PS drop diameter by a factor of ten. The effect was later explained by a balance of the three interfacial tension coefficients in the blends, inter-related by Neuman s triangle equation [1,352,353]. In simple terms, the PMMA, immiscible in PE and PS, formed a layer around PS drops, preventing coalescence. In a sense, addition of nanoparticles to polymer blends acts similarly. 

To determine the threshold behavior of the cross-sections for scattering in collision channels with m > 0, we express the Hankel functions in Equation 4.20 in terms of the Bessel and Neuman functions. Using the asymptotic expansions of the Bessel and Neuman functions, we find thata kf as ks 0. This yields the following energy dependence of the cross-section near threshold ... 

Neuman [42] simplifled the analysis by using a simple equation of state approach. He showed that a plot of yiy cos 9 versus ypy gives a smooth curve (Figure 11.43). This analysis allows one to obtain yg from a single contact angle measurement. 

A mutual impedance of a nonparallel conductor above an imperfectly conducting earth illustrated in Figure 1.60 is obtained in the following equation by applying the concept of the complex penetration depth to Neuman s inductance formula [37,42] ... 

The applicable mathematical form of the flux equation is model dependent. For example, the first type model consists of differential equations (DEs). They are developed to yield concentration profiles in the sediment layers as well as the flux. These DEs typically use Equation 4.1 as a boundary condition. The solutions to these DEs require one or more of the following boundary condition categories the Dirichlet condition, the Neuman condition, or a third condition. The first two types are the most common these require mathematical functions containing gradients of the dependent variable (i.e., Cw) as well as functions of the dependent variable itself. For these diffusive-type fluxes, the transport parameter is a diffusion coefficient such as Dg. Several other transport parameters are commonly used and represent diffusion in air and the biodiffusion or bioturbation of soil/sediment particles. 

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