Sums of products model

The prosodic part of the Campbell model uses a neural network to calculate a syllable duration. This is an attractive approach as the neural network, unlike the Klatt or sums-of-products model, can model the interactions between features. A somewhat awkward aspect of the model however is the fact that the syllable duration itself is of course heavily influenced by its phonetic content If left as is, the phonetic variance in model duration may swamp the prosodic variance that the neural network is attempting to predict. To allow for this, Campbell also includes some phonetic features in the model. Campbell maps from syllable to phone durations using a model based on his elasticity hypothesis which states that each phone in a syllable e q)ands or contracts according to a constant factor, normalised by the variance of the phone class. This operates as follows. A mean syllable is created containing the correct phones with their mean durations. This duration is then compared to the predicted duration, and the phones are either expanded or contracted until the two durations match. This expansion/contraction is performed with a constant variance, meaning that if the vowel is expanded by 1.5 standard deviations of its variance, the constant before it will be expanded by 1.5 standard deviations of its variance. 

S v are elements of the overlap matrix. Similar types of expressions may be constructed for density functional and correlated models, as well as for semi-empirical models. The important point is that it is possible to equate the total number of electrons in a molecule to a sum of products of density matrix and overlap matrix elements. ... 

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

The model underlying PCA exactly corresponds to equations (3) and (4) the elements of the standardized data matrix Y (see equation (1)) are described by a sum of product terms where, in each term, one factor is characteristic of the molecules and the other of the biological test. In the terminology of PCA this model reads ... 

A reliable way to obtain such distributed multipole models is the distributed multipole analysis (DMA) method[8]. This takes advantage of the special properties of the Gaussian basis functions which are used in almost all modern wavefunction calculations. The electron density is a sum of products of such functions, and ai product of Gaussian functions can be represented as a Gaussian centred at some intermediate point[4l] ... 

In bond orbital theory the wave functions of the molecule are derived from the wave functions associated with the different bonds of a molecule, i.e. from the bond orbitals. In the LCBO model the molecular orbitals are a linear combination of bond orbitals which in turn are a combination of the atomic orbitals or hybrids forming the bond in question [2b]. In the HLSP-method the many-electron wave functions are a sum of product functions which contain a Heitler and London type of factor (space function and spin function) for each bond of the molecule. 

In principle, the reaction cross section not only depends on the relative translational energy, but also on individual reactant and product quantum states. Its sole dependence on E in the simplified effective expression (equation (A3.4.82)) already implies unspecified averages over reactant states and sums over product states. For practical purposes it is therefore appropriate to consider simplified models for tire energy dependence of the effective reaction cross section. They often fonn the basis for the interpretation of the temperature dependence of thennal cross sections. Figure A3.4.5 illustrates several cross section models. 

The intennolecular Hamiltonian of the product fragments is used to calculate the sum of states of the transitional modes, when they are treated as rotations. The resulting model [28] is nearly identical to phase space theory [29],... 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

For opaque materials, the reflectance p is the complement of the absorptance. The directional distribution of the reflected radiation depends on the material, its degree of roughness or grain size, and, if a metal, its state of oxidation. Polished surfaces of homogeneous materials reflect speciilarly. In contrast, the intensity of the radiation reflected from a perfectly diffuse, or Lambert, surface is independent of direction. The directional distribution of reflectance of many oxidized metals, refractoiy materials, and natural products approximates that of a perfectly diffuse reflector. A better model, adequate for many calculational purposes, is achieved by assuming that the total reflectance p is the sum of diffuse and specular components p i and p. ... 

In the CHS model only nearest neighbors interact, and the interactions between amphiphiles in the simplest version of the model are neglected. In the case of the oil-water symmetry only two parameters characterize the interactions b is the strength of the water-water (oil-oil) interaction, and c describes the interaction between water (oil) and an amphiphile. The interaction between amphiphiles and ordinary molecules is proportional to a scalar product between the orientation of the amphiphile and the distance between the particles. In Ref. 15 the CHS model is generalized, and M orientations of amphiphiles uniformly distributed over the sphere are considered, with M oo. Every lattice site is occupied either by an oil, water, or surfactant particle in an orientation ujf, there are thus 2 + M microscopic states at every lattice site. The microscopic density of the state i is p.(r) = 1(0) if the site r is (is not) occupied by the state i. We denote the sum and the difference of microscopic oil and water densities by and 2 respectively and the density of surfactant at a point r and an orientation by p (r) = p r,U(). The microscopic densities assume the values = 1,0, = 1,0 and 2 = ill 0- In close-packing case the total density of surfactant ps(r) is related to by p = Ylf Pi = 1 - i i. The Hamiltonian of this model has the following form [15]... 

Impurities travel from atmosphere to ice sheet surface either attached to snowflakes or as independent aerosols. These two modes are called wet and dry deposition, respectively. The simplest plausible model for impurity deposition describes the net flux of impurity to ice sheet (which is directly calculated from ice cores as the product of impurity concentration in the ice, Ci, and accumulation rate, a) as the sum of dry and wet deposition fluxes which are both linear functions of atmospheric impurity concentration Ca (Legrand, 1987) ... 

Intermediate liquid 8 values are obtained by mixing liquids of known solubility parameter SPS makes use of this. The 8 value of the mixture is equal to the volume-weighted sum of the individual component liquid 8 values. Thus, the mass uptake of a miscible liquid mixture by an elastomer may be very much greater than the swelling which would occur in the presence of either one of the constituent liquids alone. The mixture could of course comprise more than two liquid components, and an analogous situation would apply MERL have applied this approach for the offshore oil-production industry to allow realistic hydrocarbon model oils to be developed,basically by mixing one simple aliphatic (paraffinic) hydrocarbon, one naphthenic, and one aromatic to proportions that meet two criteria, namely, that... 

FI = continued product where any term is defined as equal to 1 when the index takes a forbidden value, i.e., i = 1 in the numerator or m = j in the denominator X = summation where any term is defined as equal to zero when the index j takes a forbidden value, i.e., j = 1 ky, kji = first-order intercompartmental transfer rate constants Eh Em = sum of exit rate constants from compartments i or m n = number of driving force compartments in the disposition model, i.e., compartments having exit rate constants... 

The HMO model further assumes that the wave function is a product of one-electron functions and that Hn is the sum of one-electron operators ... 

This more detailed model is necessary for target stock calculations where productions may overlap, the lengths of quants differ significantly and quants have multiple predecessors and successors (see Figure 4.14). Calculating lot sizes with for example the formula of Andler yields completely different results and the sum of changeover costs and stock costs are much higher. 

Compound densities, also called random sums, are typically applied in modeling random demands or random claim sums in actuarial theory. The reason for this is simple. Assume that customers randomly order different quantities of a product. Then the total quantity ordered is the random sum of a random number of orders. The conversion to the actuarial variant is obvious The total claim is made from the individual claims of a random number of damage events. Zero quantities usually are neither ordered nor claimed. This leads to the following definition. 

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