This is an addendum to [1]. In this report, we initially present the derivation of the equilibrium strategies from Isaac’s conditions. Then we derive the final values of the adjoint variables on the UP. Once we have the final values of the adjoint variables, we solve the adjoint equation backward in time to construct a value function in the state-space. From those value functions, we derive the...

Some oscillation criteria for the oscillatory behavior of fourth order superlinear dynamic equations on time scales are established. Criteria are proved that ensure that all solutions of superlinear and linear equations are oscillatory. Many of our results are new for corresponding fourth order superlinear di¤erential equations and fourth order superlinear di¤erence equations.

In this work, a strategy capable of simulating polydisperse flows in complex geometries is employed where the fluid transport equations are solved in an Eulerian framework and the dispersed phase is represented as Lagrangian particles. Volume filtered equations for the carrier phase are derived in detail for variable density flows, and all unclosed terms are discussed. Special care is given to ...

We consider the pair of second-order dynamic equations, (r(t)(xΔ)γ)Δ + p(t)xγ(t) = 0 and (r(t)(xΔ)γ)Δ + p(t)xγσ(t) = 0, on a time scale T, where γ > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when T = R involve the wellknownHille-Kneser-type criteria of second-order line...

In this work, we investigate some new Gronwall-Bellman type dynamic inequalities on time scales in two independent variables, which provide a handy tool in deriving explicit bounds on unknown functions in certain dynamic equations on time scales. The established results generalize the main results on integral inequalities for continuous functions in [1] and their corresponding discrete analysis...

This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation r t xΔ t γ Δ p t x g t 0 on an arbitrary time scale T with sup T ∞, where g t ≥ t and ∫∞ to Δs/ r1/γ s < ∞. Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify th...

Abstraction refinement is a key technique for applying model checking to the verification of real-world digital systems. In previous work, the abstraction granularity is often limited at the state variable level, which is too coarse for verifying industrial-scale designs. In this paper, we propose a finer grain abstraction in which intermediate variables are selectively inserted to partition la...