The probability density for T= X/ Y, where,, and ρ=Corr X, Y)≠±1 is given by the product of two terms (11) or (12) where (13) (14) and (15) Thus, anytime we can approximate the distribution of the intersection and the union by two correlated Normal distributions, the distribution of the Tanimoto scores can be approximated using Equations (11)–(15) with X= I and Y= U.
The behavior of distillation column sections can be approximated using differential equations.
For positively selected alleles, the allele frequency trajectories can either be generated by stochastic simulation or be approximated using deterministic equations.
Therefore, the first characteristic function can be approximated using the following equation: Φ ̂ X u = 1 n ∑ i = 1 n e j 2 πu x i. (18).
Knowing that the propagation velocity in air C air is, about 300 mm/ns, V m in a material can be approximated using the following equation: V_{text{m}} = frac{{C_{text{air}} }}{{sqrt {varepsilon_{text{r}} } }} (1).
The decision function f x, w) can be approximated using the following equation (Kecman 2005): f x,w) = w^{T} x + b, (7)in which w and b denote the weight vector and the bias term, respectively, and w T denotes the transposition of weight matrices.
The length scale ℓ B can be approximated using the magnetic induction equation: frac{partial mathbf{B}}{partial t} = nabla times (mathbf{u} times mathbf{B}) + eta nabla^{2} mathbf{B} (9).
It was found that natural frequencies of a fixed string under constant tension T with linear variation in the mass density, ρ(x)="ρ0(1+αx/L), can be approximated using a very simple equation, ωn="(nπ/L)[T/{ρ0(1+0.5α)}]1/2 and results reasonably agree with previous results for a wide range of α.
The activity coefficients of Pb(PO4 3/5OH1/5 and Ca(PO4 3/5OH1/5 in the solid solution (PbxCa1−x)(PO4)3/5OH1/5 can be approximated using the Redlich and Kister equation.
The second factor in the last equation may be approximated using the ancestral trajectories (left (a^{k}_{1:i}right)_{1le k le N}) and the associated importance weights (left (omega ^{k}_{i}right)_{1le k le N}) produced by the forward filter.
