Gaussian characteristics

On the fractal lattice, Dv is a constant, and holes exhibit the Gaussian characteristics, The self-similarity of the fractal has dilation symmetry shown in Eq. (8). Using the Fourier-Laplace transformation in time,... 

We have described step 1 already. Step 2 is accomplished by using at time r = 0 a perturbation of the disk particle angular velocity of Dirac delta form in time. As we wanted a a (u, t = 0) not too far from equilibrium Gaussian characteristics (see Section V), we gave a ,.(w) a Gaussian shape with a width larger than the equilibrium one, that is. 

For a detailed description of this method it is referred to the literature Numerical investigations show that for low excitation levels the method of equivalent linearization leads to quite accurate results. However, for high excitation levels considerable non-Gaussian characteristics of the response processes arise. 

Parameter k of Equation (4.10) is an expression of the breadth of the Gaussian distribution of the cumulative micropore volume IF over the normalized work of adsorption sfifi, and is therefore determined by the pore structure. Thus B also (cf. Equation (4.13)) is characteristic of the pore structure of the adsorbent, and has accordingly been termed the structural constant of the adsorbent. ... 

Characterization of Chance Occurrences To deal with a broad area of statistical apphcations, it is necessary to charac terize the way in which random variables will varv by chance alone. The basic-foundation for this characteristic is laid through a density called the gaussian, or normal, distribution. 

While being very attractive in view of their similarity to CLTST, on closer inspection (3.61)-(3.63) reveal their deficiency at low temperatures. When P -rcc, the characteristic length Ax from (3.60b) becomes large, and the expansion (3.58) as well as the gaussian approximation for the centroid density breaks down. In the test of ref. [Voth et al. 1989b], which has displayed the success of the centroid approximation for the Eckart barrier at T> T, the low-temperature limit has not been reached, so there is no ground to trust eq. (3.62) as an estimate for kc ... 

A very important characteristic of laser radiation is the beam shape. So far most LA experiments have been performed with Gaussian laser beams. Lasers with uniform distribution of the beam cross-section have been used only recently to achieve high lateral and depth resolution. Specially designed beam homogenizers must be used for this purpose [4.226-4.228]. The Cetac LSX-200 system has a flat-top distribution of the laser beam. 

When an element is present on the surface of a sample in several different oxidation states, the peak characteristic of that element will usually consist of a number of components spaced close together. In such cases, it is desirable to separate the peak into its components so that the various oxidation states can be identified. Curve-fitting techniques can be used to synthesize a spectrum and to determine the number of components under a peak, their positions, and their relative intensities. Each component can be characterized by a number of parameters, including position, shape (Gaussian, Lorentzian, or a combination), height, and width. The various components can be summed up and the synthesized spectrum compared to the experimental spectrum to determine the quality of the fit. Obviously, the synthesized spectrum should closely reproduce the experimental spectrum. Mathematically, the quality of the fit will improve as the number of components in a peak is increased. Therefore, it is important to include in a curve fit only those components whose existence can be supported by additional information. 

The surface dividing the components of the mixture formed by a layer of surfactant characterizes the structure of the mixture on a mesoscopic length scale. This interface is described by its global properties such as the surface area, the Euler characteristic or genus, distribution of normal vectors, or in more detail by its local properties such as the mean and Gaussian curvatures. 

The Gaussian (ii (r)) and mean curvatures (//(r)), see Fig. 1, present another characteristic of internal surfaces. By definition we have... 

The Euler characteristic for a closed surface is related to the Gaussian K curvature and genus g of this surface in the following way [33,29]... 

FIG. 12 The behavior of the internal energy U (per site), heat capacity Cy (per site), the average Euler characteristic (x) and its variance (x") — (x) close to the transition line and at the transition to the lamellar phase for/o = 0. The changes are small at the transition and the transition is very weakly first-order. The weakness of the transition is related to the proliferation of the wormhole passages, which make the lamellar phase locally very similar to the microemulsion phase (Fig. 13). Note also that the values of the energy and heat capacity are not very much different from their values (i.e., 0.5 per site) in the Gaussian approximation of the model [47]. (After Ref. 49.)... 

Not every model can completely achieve all of these ideals. We ll look at the characteristics of the various methods in Gaussian in Appendix A. 

As an example of these techniques, we shall calculate the characteristic function of the gaussian distribution with zero mean and unit variance and then use it to calculate moments. Starting from the definition of the characteristic function, we obtain18 ... 

This result checks with our earlier calculation of the moments of the gaussian distribution, Eq. (3-66). The characteristic function of a gaussian random variable having an arbitrary mean and variance can be calculated either directly or else by means of the method outlined in the next paragraph. 

Equation (3-88) enables us to calculate the characteristic function of the unnormalized random variable from a knowledge of the characteristic function of . For example, the characteristic function of a gaussian random variable having arbitrary mean and variance can be written down immediately by combining Eqs. (3-83) and (3-88)... 

The right-hand side of Eq. (3-205) is the characteristic function of the gaussian distribution having zero mean and unit variance, and this leads us to conclude that the distribution function of sj approaches... 

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

The Gaussian Process.—A gaussian process was defined in the last section to be a process all of whose finite-order distributions are multi-dimensional gaussian distributions. This means that the multi-dimensional characteristic function of Px.fK must be of the form... 

Equation (3-325), along with the fact that Y(t) has zero mean and is gaussian, completely specifies Y(t) as a random process. Detailed expressions for the characteristic function of the finite order distributions of Y(t) can be calculated by means of Eq. (3-271). A straightforward, although somewhat tedious, calculation of the characteristic function of the finite-order distributions of the gaussian Markov process defined by Eq. (3-218) now shows that these two processes are in fact identical, thus proving our assertion. 

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