Gauss equation

Gaussian profile "Bell-shaped" profile following the Gauss equation (Eq. 6.5). 

In order to find out which of the Trp residues that is the most affected by the presence of calcofluor, two deconvolution methods were applied. In the first method, we fitted the spectra obtained as a sum of Gaussian bands. In the second one, we used the method described by Siano and Metzler (1969), and that was then applied by Burstein and Emelyanenko (1996) using a four parametric log-normal function (a skewed Gauss equation). The analysis were done at the three excitation wavelengths (295, 300 and 305 nm), although the results shown are those obtained at = 295 nm. 

The Nemst equation (Eq. 7.9) defines potential differences as a function of ionic activities, and the Maxwell/Gauss equation defines potential changes as a function of charge densities. We may hypothesize that there are DC potential differences between some of the... 

For the long-range interactions such as gravitational and electrostatic the forces acting on each particle i can be computed by using Particle-Mesh (PM) approximation of the Gauss equation... 

The realistic modefing, simulating millions of atoms and molecules, involves more heuristic approach to define the interparticle forces. Instead of integrating the Gauss equation (which is very demanding computationally for large systems), the potential function is... 

In the limit where (kg - k ) goes to zero, if we make ft(kB - k ) go to a nonzero limit of a, the surface charge density, the Gauss equation results ... 

Dielectric relaxation is a standard name for conductive-capacitive processes occurring in materials submitted to electric field and current under a dynamic regime. The two system constitutive properties involved are the permittivity e, also called dielectric constant, which is the spatially reduced capacitance, and the conductivity a, which is the spatially reduced conductance. The permittivity has been studied in Chapter 5 (dealing with space-distributed poles) in case studies B3 Electric Space Charges, B4 Poisson Equation, and B5 Gauss Equation. It relates the electric field E to the electric displacement ( electrization ) D... 

In developing the first item, the fact that, according to Gauss equation H8.2, the divergence of the electric field is proportional to the charge concentration has been taken into account, so its gradient is zero because the charge concentration is supposed to be uniform or equal to zero. For the second item, the absorption factor has been taken as a constant. With these developments, the curl of the potential density becomes... 

In the random walk conformation, the polymer molecule attains maximum rotational freedom along the chain resulting in a root mean square end-to-end distance given by Equation 12.9. The probability Q for a polymer chain to deviate from the random coil conformation and, therefore, having an end-to-end distance deviating from e is given by a Gauss equation. 

The present paper cuts the Gordian knot. It crosses from the right-hand side of Gauss equations to the left-hand side immediately and completely at equation (11) rather than at equation (6), the traditional location. The new strategy is simply... 

Solution of Eq. (14.9) requires knowledge of the electrostatic potential gradient, which is provided by the Gauss equation ... 

The equation system of eq.(6) can be used to find the input signal (for example a crack) corresponding to a measured output and a known impulse response of a system as well. This way gives a possibility to solve different inverse problems of the non-destructive eddy-current testing. Further developments will be shown the solving of eq.(6) by special numerical operations, like Gauss-Seidel-Method [4]. 

GAUSSP. Gives Gauss point coordinates and weights required in the numerical integration of the members of the elemental stiffness equations. 

The purpose of this projeet is to gain familiarity with the strengths and limitations of the Gauss-Seidel iterative method (program QGSEID) of solving simultaneous equations. 

If P = I, this is the Gauss-Seidel method. If > I, it is overrelaxation if P < I it is underrelaxation. The value of may be chosen empirically, 0 < P < 2, but it can be selected theoretically tor simple problems hke this (Refs. 106 and 221). In particular, these equations can be programmed in a spreadsheet and solved using the iteration feature, provided the boundaries are all rectangular. 

These early results of Coulomb and his contemporaries led to the full development of classical electrostatics and electrodynamics in the nineteenth cenmry, culminating with Maxwell s equations. We do not consider electrodynamics at all in this chapter, and our discussion of electrostatics is necessarily brief. However, we need to introduce Gauss law and Poisson s equation, which are consequences of Coulomb s law. 

Although the continuum model of the ion could be analyzed by Gauss law together with spherical symmetry, in order to treat more general continuum models of electrostatics such as solvated proteins we need to consider media that have a position-specific permittivity e(r). For these a more general variant of Poisson s equation holds ... 

If the Al value is not given for a partieular eore/air-gap eombination. Equation 3.30 ean be used. Do not mix CGS and MKS units (gauss and eentimeters with teslas and meters). 

Application of the Gauss error-function equation for velocity profile in the form proposed by Shepelev (Table 7.12) in Eq. (7.39) results in the following formula for the centerline velocity in Zone 3 of the compact jet ... 

The equation for the centerline temperature differential in Zone 3 of the compact jet derived" from Eq. (7.61) using the Gauss error-function temperature profile (Table 7.14) is... 

Gauss-Siedel method is an iterative technique for the solution of sets of equations. Given, for example, a set of three linear equations... 

Note that Laplace (not Gauss) first derived the equation for the Gaussian (normal) error curves, which need not be normal in the sense that they normally apply to errors encountered in practice (text above). 

With the aid of effective Gauss method for solving linear equations with such matrices a direct method known as the elimination method has been designed and unveils its potential in solving difference equations,... 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

This equation describes the second fundamental feature of the attraction field, and it can be also treated as the bridge between sources (masses) and the field. It may be appropriate to notice that the similar relationship holds for any field and is often called Gauss s formula. 

At the same time it is worth to notice that in modern numerical methods of a solution of boundary value problems, based on replacement of differential equations by finite difference, these steps are performed simultaneously. In accordance with the theorem of uniqueness, the field inside the volume V is defined by a distribution of masses inside this volume and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind we will again proceed from Gauss s theorem,... 

This relationship is called the second Green s formula and it represents Gauss s theorem when the vector X is given by Equation (1.98). In particular, letting ij/ — constant we obtain the first Green s formula ... 

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