Functional dependence

An alternative to obtaining 0 directly involves defining some more convenient shape-dependent function, and an early but still very practical method is the following. We define a shape-dependent quantity as S = dsldg, as indicated in Fig. 11-16, de is the equatorial diameter and ds is the diameter measured at a distance de up from the bottom of the drop. The hard-to-measure size parameter h in Eq. 11-17 is combined with 0 by defining the quantity H = -0(defb). Thus... 

Corresponding to any collection of n particles, there exists a tune-dependent function T (q ... 

The residual, Rq, is a function of position in Q. The weighted residual method is based on the elimination of this residual, in some overall manner, over the entire domain. To achieve this the residual is weighted by an appropriate number of position dependent functions and a summation is carried out. This is written as... 

A second advantage is that the procedure, applied for infinite dilution of each species, yields two values of kj from which a composition-dependent function can be generated, a simple linear relation proving fully satisfactory ... 

In time-dependent problems, all the above conditions can be time-dependent functions, such as a machine heating cycle or another time-dependent process under study. 

In comparison with the ordinary OZ equation, there is an additional integration with respect to the particle size. To proceed further, this integration can be removed by applying the method proposed by Lado [81,82]. Thus, we expand the (j-dependent functions in orthogonal polynomials 7 = 0,1,2,..., defined such that... 

Figure A1.2 Models schematically illustrating the angular dependence functions A ,(6, 0). There is no unique way of representing the angular dependence functions of all seven f orbitals. An alternative to the set shown is one f 3, three f i, fy i, and three fx(i2 3j,2), fj.(j,2 3x2), and fj(x2 3x2).

Orbital Angular dependence function Orbital Angular dependence function... 

For a nonassociative desorption, evidently = 1 and fcd" is configuration-independent. In other cases, the functions gi(6 y conf) and (conf y) can be combined into a single coverage-dependent function... 

Fourier transform of a time dependent function that involves dynamical motions on the initial and final electronic states potential energy surfaces. 

To evaluate the time-dependent function, X(t), a simple model of diffusion is proposed. Starting from Langmuir adsorption theory, we consider that liquid molecules having diffused into the elastomer are localized on discrete sites (which might be free volume domains). In these conditions, we can deduce the rate of occupation of these sites by TCP with time. Only the filhng of the first layer of the sites situated below the liquid/solid interface at a distance of the order of the length of intermolecular interaction, i.e., a few nanometers, needs to be considered to estimate X(t). 

The water-oil ratio is a complex time-dependent function of the state variables since a well can produce oil from several grid cells at the same time. In this case the relationship of the output vector and the state variables is nonlinear of the form y(t,)=h(x(t,)). 

Monomers employed in a polycondensation process in respect to its kinetics can be subdivided into two types. To the first of them belong monomers in which the reactivity of any functional group does not depend on whether or not the remaining groups of the monomer have reacted. Most aliphatic monomers meet this condition with the accuracy needed for practical purposes. On the other hand, aromatic monomers more often have dependent functional groups and, thus, pertain to the second type. Obviously, when selecting a kinetic model for the description of polycondensation of such monomers, the necessity arises to take account of the substitution effects whereas the polycondensation of the majority of monomers of the first type can be fairly described by the ideal kinetic model. The latter, due to its simplicity and experimental verification for many systems, is currently the most commonly accepted in macromolecular chemistry of polycondensation processes. 

Jaramillo, J., Scuseria, G. E., 1999, Performance of a Kinetic Energy Density Dependent Functional (VSXC) for Predicting Vibrational Frequencies , Chem. Phys. Lett., 312, 269. 

Recall from the examples of partial fraction expansion that the polynomial Q(s) in the numerator, or the zeros, affects only the coefficients of the solution y(t), but not the time dependent functions. That is why for qualitative discussions, we focus only on the poles. 

A signal function y(z), which is mostly treated as a time-dependent function, /(f), can be regarded as consisting of the original true signal function yt rue (S ) which is superposed by a noise function n(t) see Fig. 2.15i... 

Given the density matrix, the average of a coordinate-dependent function F(x) is then... 

The s-states have spherical symmetry. The wave functions (probability amplitudes) associated with them depend only on the distance, r from the origin (center of the nucleus). They have no angular dependence. Functionally, they consist of a normalization coefficient, Nj times a radial distribution function. The normalization coefficient ensures that the integral of the probability amplitude from 0 to °° equals unity so the probability that the electron of interest is somewhere in the vicinity of the nucleus is unity. 

Using this factorizability of response into a time-dependent and a strain-dependent function. Landel et ai. (61,62) have proposed a theory that would express tensile stress relaxation in the nonlinear regime as the product of a time-dependent modulus and a function of the strain ... 

The kinetics of the CTMAB thermal decomposition has been studied by the non-parametric kinetics (NPK) method [6-8], The kinetic analysis has been performed separately for process I and process II in the appropriate a regions. The NPK method for the analysis of non-isothermal TG data is based on the usual assumption that the reaction rate can be expressed as a product of two independent functions,/ and h(T), where f(a) accounts for the kinetic model while the temperature-dependent function, h(T), is usually the Arrhenius equation h(T) = k = A exp(-Ea / RT). The reaction rates, da/dt, measured from several experiments at different heating rates, can be expressed as a three-dimensional surface determined by the temperature and the conversion degree. This is a model-free method since it yields the temperature dependence of the reaction rate without having to make any prior assumptions about the kinetic model. 

We want to approximate AM(s) by a Pade approximant Am s). The functional form of Am(s) should be such that the corresponding time-dependent function Am(t) is a series of N exponentials describing the relation of AM(t) to AM(oo) = 0. This implies that Am(s) is an [N — 1. /V]-Pade approximant that can be written in the form... 

Ajoene has antitumor activity, inhibits cholesterol biosynthesis, modulates membrane-dependent functions of immune cells, inhibits protein prenylation83 and is an anti-leukaemia agent for acute myeloid leukaemia.85 In antithrombotic assays, the Z isomer is more active than the E isomer.84... 

Cooper, K. D. et al., Effects of UVR on human epidermal cell alloantigen presentation Initial depression of Langerhans cell dependent function is followed by the appearance of T6-DR+ cells that enhance epidermal alloantigen presentation, J. Immunol. 134,129-137, 1985. 

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