Gauss function

The popular radial basis function nets (RBF nets) model nonlinear relationships by linear combinations of basis functions (Zell [1994] Jagemann [1998] Zupan and Gasteiger [1993]). Functions are called to be radial when their values, starting from a central point, monotonously ascend or descend such as the Cauchy function or the modified Gauss function at Eq. (6.125) ... 

Quite illustrative is the case of a finite amount of a diffusing substance deposited initially at a given position. The Gauss function... 

Figure 8.13 Dispersion of an instantaneous point source [equation (8.5.1)]. A quantity of the diffusing species equivalent to the surface of whatever curve on the diagram is deposited initially at x = 0. The curves are Gauss functions.

The choice of the specific orthogonal polynomial is determined by the convergence. If the signal to be approximated is a bell-shaped function, it is evident to use a polynomial derived from the Gauss function, i.e. one of the so-called classical polynomials, the Hermite polynomial. Widely used is the Chebyschev polynomial one of the special features of this polynomial is that the error will be spread evenly over the whole interval. 

Different types of functions have been used as trial functions for determining / r).263 266"269 The best agreement with the dependence of Gfi on % has been obtained with the exponential function F(r) = (l/r0) exp (-r/r0) and with the Gauss function F(r) = (4r2/7rl,2ro) exp (-r2lr ), and in certain cases with a combination of the latter with a power function.268"273... 

Figure 6.1 Comparison of 26 — 6 scan profiles obtained by a monochromatized (pure Cu kal) parallel beam configuration (hybrid x-ray mirror) and a conventional parallel beam configuration achieved by divergence slit (ds) module measured at 001/100 (a), 002/200 (b), 003/300 (c), 004/400 (d) of 500nm-thick Pb(Zro.B4Tio.46)03 thin film. Dotted lines represent the second derivative of the profiles, indicating the peak positions. Note that the profiles are simulated fitted profiles for obtained spectrum using pseudo-Voight function (mixed Lorentz and Gauss function).

Electron transition probabilities (hereafter ETPs) from the Is orbital of Al to the unoccupied orbitals are calculated from the SXS code [17] on the basis of the dipole approximation using electron state data of MOs. The theoretical Al K-edge XANES spectra of the models were obtained from convoluting a Gauss function to ETPs. 

The foregoing discussion makes it clear that the vertical energy difference -I p) forms an electronic energy coordinate axis with a characteristic reference point, "(Ox/Red), which is experimentally accessible. This means that the probability functions can be used directly in models for electrochemical electron transfer between a solid and a simple redox system (see Sections 4.8 and 4.9). In the literature, the electronic energy coordinate axis has been denoted as E, and the characteristic points as /z°(Ox/Red) (or Fp), and and E denoted the energies corresponding to the maximum Wox E) and IERed( ) values, respectively. For quantitative purposes (see Sections 4.8 and 4.9), the probability functions are expressed as normalized Gauss functions ... 

The two simplest peak shape functions (Eqs. 2.49 and 2.50) represent Gaussian and Lorentzian distributions, respectively, of the intensity in the Bragg peak. They are compared in Figure 2.42, from which it is easy to see that the Lorentz function is sharp near its maximum but has long tails on each side near its base. On the other hand, the Gauss function has no tails at the base but has a rounded maximum. Both functions are centrosymmetric, i.e. G(x) = G -x) and L x) = L -x). 

Figure 2.45. The schematic illustrating the asymmetric Bragg peak (solid line) when compared with the symmetric peak composed of the dash-dotted line (left slope) and the solid line (right slope). Both peaks are modeled by the pure Gauss function (Eq. 2.49) using two different FWHM s on different sides of the peak maximum in the asymmetric case.

Hence, high-temperature spectra bands of all structural units could not be observed, as it is the case of low-temperature spectra in molecular systems, and moreover, only one band averaging contributions from all structural units that can be expressively wide-spread. In the case of bands with higher half-width, it is thus necessary to use for their mathematical description, the weighted combination of the Lorentzian and Gauss functions. 

A Gauss pulse and a polynomial multiplied by a Gauss pulse satisfy the requirements of rapid fall-off in both the time and the frequency domains. A pulse the shape of which is determined by a product of a Hermitian polynomial and a Gauss function is a Hermite pulse [Sill, War2]. Its width can be more narrow than that of a Gauss pulse. A fairly uniform rotation of z magnetization can be generated if the sum of a zeroth- and a... 

Asymmetric peaks can be described by an exponentially modified Gauss function (EMG). The deviation from Gauss shape needs the expansion of the function by a time constant term r. 

Tli( (listril)ution of the elementary waves composing the wave packet f(x) is again a Gauss function with the half width b Putting... 

Apodization is the process of multiplying the FID prior to Fourier transformation by a mathematical function. The type of mathematical or window function applied depends upon the enhancement required the signal-to-noise ratio in a spectrum can be improved by applying an exponential window function to a noisy FID whilst the resolution can be improved by reducing the signal linewidth using a Lorentz-Gauss function. ID WIN-NMR has a variety of window functions, abbreviated to wdw function, such as exponential (EM), shifted sine-bell (SINE) and sine-bell squared (QSINE). Each window function has its own particular parameters associated with it LB for EM function, SSB for sine functions etc. 

The overlap parameter 2yAf y is likewise not sharp, but rather statistically distributed. The distribution is assumed to be a Gauss function with a width S. The distribution of the overlap parameters is termed the non-diagonal disorder, a and S are the two important materials parameters in the Bassler model for hopping transport in disordered semiconductors. 

Fig. 8.46 A Monte-Carlo simulation of the time dependence of the distribution of charge-carrier energies as a Gauss function of the states (DOS) with a width cr. The centre of the energy of...

Figure 4-4. Difference spectra between on- and off-resonance photoemission spectra of La203, Pr203 and NdiOa. (The vertical bars are the multiplet structure of final state. The solid lines are convolutions of the multiplet stmctures with Lorentz and Gauss functions. (Reproduced with permission firom ref 11. Copy right 2000 J. Phys. Soc. Japan)...

Best results are achieved using Lorentz-Gauss functions for representing NMR peaks (Eq. (13.2)). The function includes four adjustable parameters such as the maximum intensity of the peak, 1, the chemical shift at maximum intensity of the peak, and the Lorentz and Gaussian parameters a and b, respectively. 

As a starting structure we used a crystobalite-like structure as this crystalline form is more stable at the high temperature of the flame synthesis. As is well known, the particle surface is covered by hydroxyl groups which terminate all broken bonds on a particle body. To present the particle structure as a whole the interatomic distance distribution function as a sum of sets of Gauss functions with a small broadening parameter of 0.05 A was used. To present the particle chemical behavior as a whole the coordination number distribution for all atom types has been used. 

Related with the real numbers A, B from (1) and the position of w on the Gauss function there are three possibilities... 

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