Three kinds of effective error bounds of the quadrature formulas with multiple nodes that generalize the well-known Micchelli-Rivlin quadrature formula, when the

integrand is an analytic function in the regions containing the confocal ellipses, were considered recently in [13, 15].

Furthermore, using the Gaussian mixture model representation from (26) and (27) and assuming Gaussian measurement noise, the

integrand can be expressed as a sum of weighted Gaussian PDFs; namely,

Therefore, the

integrand in (37) contains a constant component and [mathematical expression not reproducible].

The existence of optimal control [u.sup.*] can be established since the

integrand in (19) and the right hand side of system (18), denoted by f = ([f.sub.1], [f.sub.2], [f.sub.3], [f.sub.4]), are continuously differentiable functions and concave in both (S, I, C, T) and u = ([omega], [sigma]).

When regarding the

integrand in (24) as the Lagrangian, we can define the "Hamiltonian" H as

Then the

integrand is approximated at the saddlepoint using an integration path passing through the saddlepoint.

Next, proceeding as in [17,24] makes it possible to deform the integration contour so that it is wrapped around the singularities of the

integrand located in the upper-half of the complex plane, as shown in Fig.

Proof: Noting that both [mathematical expression not reproducible] are totally independent of [r.sub.0], the partial derivative w.r.t [r.sub.0] of the

integrand in the definition of [??] is

(iv) The

integrand of the objective functional is convex.

By the definition of [u.sub.n+1] and the fact that the Ito integral coincides with the Skorohod integral if the

integrand is predictable, it follows that, for any (t, x) [member of] [0, T] x R,

The neural network completed the mapping relationship between the input variable X and the

integrand y.