Sentence examples for index law from inspiring English sources

for any constants k 1 and k 2. Lemma 1.2 (Index law).

On p. 17 Boole added the commutative law xy = yx and the idempotent law x2 = x (which Boole called the index law).

In [6] and [18], the authors proved many time scales analogues of important properties of fractional operators known from continuous analysis, e.g., the index law ∇ − γ a a ∇ − β f ( t ) = a ∇ − f ( t ) ( γ, β > 0 ).

Note that the nonlinear operator (D_{0^^{beta}phi_{p}(D_{0^^{alpha})) reduces to the linear operator (D_{0^^{beta}D_{0^^{alpha}) when (p=2) and the additive index law D_{0^^{beta}D_{0^^{alpha}u(t)=D_{0^^{alpha+beta}u(t) holds under some reasonable constraints on the function u (see [26]).

Note that the nonlinear operator (D_{0^^{beta}phi _{p}(D_{0^^{alpha})) reduces to the linear operator (D_{0^^{beta}D_{0^^{alpha}) when (p=2) and the additive index law begin{aligned} D_{0^^{beta}D_{0^^{alpha}u(t)=D_{0^^{alpha+beta}u(t) end{aligned} holds under some reasonable constraints on the function u (see [32]).

For example, it loses its geometric or physical interpretation but the index law is only valid when working on very specific function spaces and the derivative of the product of two functions is difficult to obtain and the chain rule is not straightforward to apply.

Especially, paying attention to the nonlinear operator (D^{beta}_{0^ phi(D^{alpha}_{0^))) with the discussion in (1), we can convert it to the linear operator (D^{beta}_{0^D^{alpha}_{0^), if (phi u)=u), and the additive index law D^{beta}_{0^D^{alpha}_{0^u(t)=D^{alpha+beta}_{0^u(t) holds under some reasonable constraints on the function u (see [24]).

In fact, varying the xanthan gum concentration changes the values of all rheological parameters (i.e. consistency index (K), power law index (n) and fluid yield stress (τy)) simultaneously.