In general, such an equation takes the form Systematic methods have been developed for the solution of these equations and for those in which, for example, second-order differences are involved.
As mentioned in 1.2 above, one feature of the deductivist style is that paradigmatic mathematical proofs are expressed entirely in some appropriate formal language (for example, first-order predicate logic with identity).
The first is an extension problem: the approach does produce some intuitively acceptable results in a finite propositional framework, but it needs to be extended to more realistic frameworks — for example, first-order and higher-order frameworks.
In order to assure asymptotic stability, several components were previously added to the classic linear PID controllers, for example, fourth-order filter [101], nonlinear derivative term [4], nonlinear integral term (saturated function) [71], input saturation, and nonlinear observer [2].
(E ) Example third-order spatial correlation function involving two points in space.
In some examples, third order interpretations seemed to be derived from first and second order constructs reported in the primary studies [ 4, 14, 33], but in most cases the process for doing this was not clearly defined.
For example, the first order tumour growth rate (Kgrow) from the TGI model provides tumour doubling times for various cancer types.
The formation of Smog is an example of first order chemical reaction.
For example, the first order polarizabilities enable to calculate refractive indices, applying the Clausius-Mossotti equation [88],[89] or, taking into account long range interactions, the anisotropic Lorentz field factor approach [90].
Consider, for example, distorted first order hexahedral elements or distorted second order elements.
Consider, for example, a first order quadrilateral element where all element faces are on the boundary.
