Differential approximants

Differentials of higher orders are of little significance unless dx is a constant, in which case the first, second, third, etc. differentials approximate the first, second, third, etc. differences and may be used in constructing difference tables (see Algebra ). 

The method used for the localization of the orbitals is to be carefully chosen. It is natural to expect that if the orbitals are localized into different spatial regions, for the matrix elements ij kf) the zero differential approximation can be applied all terms containing at least one factor ij kl) in which tj/itj/,-and/or are localized to different spatial regions can be neglected. Thus the summation in a closed loop in evaluating a perturbation correction should only be extended over indices of orbitals which are localized into the same region of space. 

A spatiotemporal interpolation algorithm based on the differential approximation method... 

Thus, at each iteration of (5.8), system (5.14) should be solved. The rate of convergence of this procedure depends on the correct choice of initial conditions. The method of differential approximation refers to universal approaches in the function approximation theory to the analysis of dynamic systems. Under remote monitoring conditions, the use of this method can be justified by allowing aircraft and satellite measurements to be spaced in time with respect to the objects to be monitored and, hence, in processing the readings from measuring instruments it is necessary to take into account possible changes in the object between moments of measurement. 

Without breaking integrity, we apply this method together with the method of differential approximation to the procedure of data retrieval from the route measurement and mapping of territory G at time moment t. Let remote measurements be made in the time interval 70, //] at a discrete number of points A,- (i = 1,N) at boundary T. Assume that during the time of measurements At the level of time dependence of observational data is negligibly small that is, the whole series of... 

Tj(i,j). The method of differential approximation makes it possible to reduce all lines in this matrix to moment t and then, following the method described above to retrieve Tj in territory G. 

Table 5.5. Comparison of the accuracies of the Method of Self-Organizing Models (MSOM) and differential approximation algorithms from results of retrieval of water level oscillations at the boundary of the Nyok Ngot lagoon (South Vietnam) with the South China Sea. From Bui (2001). Notation At is the time step, and e is the error (%).

GMDH Error, (%) Method of differential approximation Error, (%) Evolu- tionary method Error, (%)... 

Modest, M. F. (1989). Modified Differential Approximation for Radiative Transfer in General Three-Dimensional Media. J. Thermophysics, 3, 283. 

E. B. Krissinel and N. V. Shokhirev, Differential approximation of spin-controlled and anisotropic diffusional kinetics (Russian), in Siberian Academy Mathematical Methods in Chemistry, Preprint 30 (1989) Diffusion-controlled reactions 22, in DCR User s Manual 11-20-1990. 

Fig. 12. Distribution curve of total protein precipitated, plotted from curve A of Pig. 11 by graphical differentiation (approximate only). Ordinates expressed in terms of grams of protein per liter.

This latter approximation shows that the strain dependence of the Doi-Edwards equation is softer than that of the temporary network model roughly by the factor 1 -p (7i — 3)/5, There is also a differential approximation to the Doi-Edwards equation (Marmcci 1984 Larson 1984b) ... 

Problem 3.8 Numerically solve Eq. (3-78), the differential approximation to the Doi-Edwards equation for entangled linear melts, in a steady-state shearing flow. Plot the dimensionless shear stress ayijG against Weissenberg number W/ = )>r for Wi between 0.1 and 100. 

Various heteroantisera and monoclonal antibodies to OCN have been used in reported immunohisto-chemical studies, " which indicate that polyclonal anti-OCN reagents are inferior to monoclonal antibodies for diagnostic work because of problems with specificity. Although studies have shown that OCN has a reasonable level of sensitivity for osteoblastic differentiation (approximately 70%) and is, for practical purposes, apparently virtually completely specific for bone-forming cells, 740 rarely used in clinical practice. 

J. Higenyi and Y. Bayazitoglu, Differential Approximation of Radiative Heat Transfer in a Gray Medium Axially Symmetric Radiation Field, ASME Journal of Heat Transfer, 102, pp. 719-723, 1980. 

D. B. Olfe, A Modification of a Modified Differential Approximation for Radiative Transfer, AIAA Journal, 5, pp. 638-643,1967. 

P. Cheng, Exact Solutions and Differential Approximation for Multidimensional Radiative Transfer in Cartesian Coordinate Configuration, Progress in Astronautics and Aeronautics, vol. 31, pp. 269-308,1972. 

If so, the generating function 77(x) = Rmx has a radius of convergence Xc = 1/A, which they estimated using differential approximants [38]. For the crossing problem they found Xc = 0.32858(5), for the spanning problem Xc = 0.3282(6) and for the cow-patch problem Xc = 0.328574(2). It can be proved that these three problems have the same growth constant, so taking the most precise estimate, they obtained A = 1.744550(5). 

They also used the method of differential approximants (DAs) [38]. The underlying idea is to fit a linear differential equation with polynomial coefficients to the generating function of the sequence, truncated at some order no- In Table 6 the results for the DA analysis are listed. 

The analysis of SAP data followed the same lines. However, the estimates suffered from large finite size errors due to the low number (11) of available coefficients. First order differential approximants for the mean number of SAPs yielded Xc = 0.3688(41). 

The above analysis has also been applied to the rhombic Penrose tiling data. That data displays qualitatively the same finite size behaviour as the Ammann-Beenker tiling data. In Table 7 estimates of Xc and 7 obtained by analysing first order differential approximants are given. 

N (which is even rather small in the case of the EE technique) and one has to apply sophisticated finite-size scaling schemes. One of the simplest techniques is the so-called successive slopes method, in which the local slope of the quantity of interest versus N (usually plotted in log-log scale) is obtained numerically and plotted versus 1/N. The thermodyuamic limit is then determined as the limit of 1/N -i- 0. For excunples of the successive slopes method see Figs. 3, 4, 7, 8, and 14. More elaborated methods are the ratio method and Fade and differential approximants [36-38]. 

Homogeneous chemical kinetics can be treated in a number of ways in the context of simulation. The simplest way is to use a simple differential approximation. For instance, the change in concentration due to first-order chemical kinetics, A = > B, can be calculated as follows ... 

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