Numerical approximations to some expressions

These equations, supplemented by the expression for the liquid density and vapor pressure, may be integrated into the general case only numerically. However, for some important particular cases, reasonable approximations can be introduced which simplify the system of equations for the average parameters to a form that can be integrated analytically. This approach, developed below, yields expressions for a set of first-order integral equations of the average parameters. 

This treatment, which is due to Semenov, includes two assumptions, a uniform reactant temperature and heat loss by convection. While these may be reasonable approximations for some situations, e.g. a well-stirred liquid, they may be unsatisfactory in others. In Frank-Kamenetskii s theory, heat transfer takes place by conduction through the reacting mixture whose temperature is highest at the centre of the vessel and falls towards the walls. The mathematics of the Frank-Kamenetskii theory are considerably more complicated than those of the simple Semenov treatment, but it can be shown that the pre-explosion temperature rise at the centre of the vessel is given by an expression which differs from that already obtained by a numerical factor, the value of which depends on the geometry of the system (Table 7). 

The main advantage of analytical radial orbitals consists in the possibility to have analytical expressions for radial integrals and compact tables of numerical values of their parameters. There exist computer programs to find analytical radial orbitals in various approximations. Unfortunately, the difficulties of finding optimal values of their parameters grow very rapidly as the number of electrons increases. Therefore they are used only for light, or, to some extent, for middle atoms. Hence, numerical radial orbitals are much more universal and powerful. 

The fundamental difficulty in solving DEs explicitly via finite formulas is tied to the fact that antiderivatives are known for only very few functions / M —> M. One can always differentiate (via the product, quotient, or chain rule) an explicitly given function f(x) quite easily, but finding an antiderivative function F with F x) = f(x) is impossible for all except very few functions /. Numerical approximations of antiderivatives can, however, be found in the form of a table of values (rather than a functional expression) numerically by a multitude of integration methods such as collected in the ode... m file suite inside MATLAB. Some of these numerical methods have been used for several centuries, while the algorithms for stiff DEs are just a few decades old. These codes are... 

Brief details are given below of some numerical approximations used to compute the posterior distribution of the fold change 9k (3), for k = 1,..., g. We assume that, given the model parameters, the expression data ykj, are independent for different genes and samples. 

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