Explicit solution formulas are presented for systems of the form Exk+1 = Axk + fk with k ∈ K, where K ⊂ Z is a discrete interval and the pencil λE − A is regular. Different results are obtained when one starts with an initial condition at the point k = 0 and calculates into the future (i.e., Exk+1 = Axk + fk with k ∈ N) and when one wants to get a complete solution (i.e., Exk+1 = Axk + fk with ...

We provide a complete analysis of the asymptotics for the semi-infinite Schur flow: αj(t) = (1 − |αj(t)|)(αj+1(t) − αj−1(t)) for α−1(t) = 1 boundary conditions and n = 0, 1, 2, . . . , with initial condition αj(0) ∈ (−1, 1). We also provide examples with αj(0) ∈ D for which α0(t) does not have a limit. The proofs depend on the solution via a direct/inverse spectral transform.

This paper proposes and evaluates a model of supply chains as networks of auctions. In this model, companies are represented according to the first level of the Supply Chain Council’s SCOR model and the trading strategy of the agents is adapted from a model proposed by Steiglitz and colleagues. Specifically, the highest level of SCOR treats a company as comprising three functions, namely Source...

and Applied Analysis 3 4. Solution by Homotopy Perturbation Sumudu Transform Method (HPSTM) 4.1. Basic Idea of HPSTM. To illustrate the basic idea of this method, we consider a general fractional nonlinear nonhomogeneous partial differential equationwith the initial condition of the form D α t U (x, t) + RU (x, t) + NU (x, t) = g (x, t) , (11) U (x, 0) = f (x) , (12) where Dα t U(x, t) is the C...

We present a model of slow-roll inflation in the context of effective Supergravities arising from string theories. The uplifting of the potential (to generate dS or Minkowski vacua) is provided by the D-term associated to an anomalous U(1), in a fully consistent and gauge invariant formulation. We develop a minimal working model which incorporates eternal topological inflation and complies with...

We point out that for a class of ‘new inflation’ models in supergravity the required initial value of the inflaton field is dynamically set if there is another inflation (‘pre-inflation’) before the ‘new inflation’. We study the dynamics of both inflatons by taking a hybrid inflation model for the ‘pre-inflation’ as an example. We find out that our ‘new inflation’ model provides reheating tempe...

In this paper, the control problem of the forced Burgers equation: ∂u ∂t = ν ∂2u ∂x2 − u ∂u ∂x +mu+ f(x), 0 < x < 2π, t > 0 subject to Neumann boundary conditions: ∂u ∂x (0, t) = ũ1(t), ∂u ∂x (2π, t) = ũ2(t) t > 0. and the initial condition: u(x, 0) = u0(x), x ∈ (0, 2π) t > 0, where ν is a positive constant, the parameter m ∈ IR, and ũ1(t), ũ2(t) are two control inputs is considered. We show th...

This problem models heat propagation in a rod where the left end is kept at constant temperature T1, the right end is kept at temperature T2, the initial temperature is f(x) and at each point x, there is heat radiating at the rate F (x, t) at time t. Note that the PDE, the boundary conditions, and the initial condition, are nonhomogeneous. We can use the principle of superposition to decompose ...

If our universe underwent inflation, its entropy during the inflationary phase was substantially lower than it is today. Because a low-entropy state is less likely to be chosen randomly than a high-entropy one, inflation is unlikely to arise through randomlychosen initial conditions. To resolve this puzzle, we examine the notion of a natural state for the universe, and argue that it is a nearly...

Analytic solutions for Burgers equations with source terms, possibly stiff, represent an important element to assess numerical schemes. Here we present a procedure, based on the characteristic technique to obtain analytic solutions for these equations with smooth initial conditions.