Sentence examples for whenever the initial from inspiring English sources

This paper deals with the problem of positive observation for linear positive time-lag systems for which the states take nonnegative values whenever the initial conditions are nonnegative.

The proposed stabilizing control laws are required to preserve the positivity of the system, that is the state variable take nonnegative values whenever the initial state is nonnegative.

This kind of systems has the property that the states take nonnegative values whenever the initial boundaries are nonnegative, some delay-dependent sufficient conditions for the exponential stability of positive 2D switched systems with state delays are given.

Very recently, positive switched systems, whose states and outputs are nonnegative whenever the initial conditions and inputs are nonnegative, have been highlighted and investigated by many researchers due to their broad applications in communication systems [4, 5], viral mutation dynamics under drug treatment [6], formation flying [7] and system theory [8 12], to mention a few.

Definition 4.1 The numerical solution Y n ( n = 1, 2, … ) of Eq. (4) is said to be mean square stable if for every ε > 0, there exists δ = δ > 0 such that E | Y n | 2 ≤ ε, n = 1, 2, …, whenever the initial values satisfy E ( max n = − m, …, 0 | ξ ( t n ) | 2 ) < δ and, moreover, lim n → ∞ E | Y n | 2 = 0. Theorem 4.1 Assume the backward Euler method (33) is well defined.

A note was made whenever the initial INR value was unavailable.

It is expected that differences in amplification of single copy genes will reflect differences in copy number whenever the two initial DNA concentrations are kept equal and the genes are constitutively expressed.

The registration is based on notification forms that are completed by hospital departments (including departments of pathology and forensic medicine) and practicing physicians whenever a case of cancer is diagnosed or found at autopsy, and whenever changes to the initial diagnosis are made.

So whenever we know the initial state consisting of the system's position and momentum, and know all external forces acting on it, we also know what will be its later states.

Whenever blood is collected for a chemistry profile, a urine sample should be obtained whenever possible (especially on the initial sampling).

Under these circumstances, it is immediate that whenever the size of the initial buyer group is smaller than ( s^ ), its members are willing to admit new participants to increase size to ( s^ ), outsiders want to enter, and no side payments are needed since both insiders and outsiders are better off.