More specifically, the actual gas and liquid distribution in parallel channels is determined by the stability of mathematical solutions of mass and momentum balance equations and also the flow history.
In particular, on the basis of the mathematical solutions of these equations, we obtain the equation of an 'inversion curve', implicit function of the irradiation parameters, which gives the dose values enhancing chain branching and molecular weight increase at fixed dose rate and vice versa.
We provide mathematical solutions of the model, and explain how to estimate the parameters from real data (see Method for details).
We first present mathematical solutions of this model using mean-field approximation, and demonstrate that this model can reproduce structural properties, such as heterogeneous connectivity and hierarchical modularity, in real metabolic networks both qualitatively and quantitatively.
Using mean-field approximation, we can obtain mathematical solutions of the model's degree distribution, degree-dependent clustering coefficient, and average clustering coefficient, which were observed to depend on temperature in Reference [ 18].
It is worth stressing that we did not explore the intricacies of the mathematical solutions of this new epidemiological model, but, rather, tried to test with very crude hypotheses whether a new mode of transmission might not account for surprising aspects of some epidemics.
The mathematical solution of a control problem may not always exist.
Appendix A.2 Proof of Lemma 1 shows how the closed-form expression (3), mathematical solution of (2), can be obtained.
Mathematical solution of the pressure boundary-value problem of linear elasticity is the theoretical underpinning of this method.
CMAC is an adaptive system by which defect types for partial discharge can be identified by referring to a table rather than by mathematical solution of simultaneous equations.
Maxwell's theory of electromagnetism, for instance, admits two kinds of mathematical solutions for the equations describing the radiation of energy in an electromagnetic field.
