Existence and Uniqueness of Endemic States for the Age-structured MSEIR Epidemic Model

The existence and uniqueness of positive steady states for the age-structured MSEIR epidemic model with age-dependent transmission coefficient is considered. Threshold results for the existence of endemic states are established; under certain conditions, uniqueness is also shown.     

A different construction of Gaussian fields from Markov chains: Dirichlet covariances

We study a class of Gaussian random fields with negative correlations. These fields are easy to simulate. They are defined in a natural way from a Markov chain that has the index space of the Gaussian field as its state space. In parallel with Dynkin's investigation of Gaussian fields having covariance given by the Green's function of a Markov process, we develop connections between the occupation times of the Markov chain and the prediction properties of the Gaussian field. Our interest in such fields was initiated by their appearance in random matrix theory.     

One-Dimensional Poisson Growth Models With Random and Asymmetric Growth

Suppose there is a Poisson process of points X _i on the line. Starting at time zero, a grain begins to grow from each point X _i , growing at rate A _i to the left and rate B _i to the right, with the pairs (A _i , B _i ) being i.i.d. A grain stops growing as soon as it touches another grain. When all growth stops, the line consists of covered intervals (made up of contiguous grains) separated by gaps. We show (i) a fraction 1/e of the line remains uncovered, (ii) the fraction of covered intervals which contain exactly k grains is (k–1)/k!, (iii) the length of a covered interval containing k grains has a gamma(k–1) distribution, (iv) the distribution of the grain sizes depends only on the distribution of the total growth rate A _i +B _i , and other results. Similar theorems are obtained for growth processes on a circle; in this case we need only assume the pairs (A _i , B _i ) are exchangeable. These results extend those of Daley, et al. (2000) who studied the case where A _i =B _i =1. Simulation results are given to illustrate the various theorems.     

Redefining Event Variables for Efficient Modeling of Continuous-Time Batch Processing

We define events so as to reduce the number of events and decision variables needed for modeling batch-scheduling problems such as described in [15]. We propose a new MILP formulation based on this concept, defining non-uniform time periods as needed and decision variables that are not time-indexed. It can handle complicated multi-product/multi-stage machine processes, with production lines merging and diverging, and with minimum and maximum batch sizes. We compare it with earlier models and show that it can solve problems with small to medium demands relative to batch sizes in reasonable computer times.     

Nonsensitiveness Regions for Threshold Ellipsoids

The problem is to determine nonsensitiveness regions for threshold ellipsoids within a regular mixed linear model.     

On the Additive Group Structure of the Nonstandard Models of the Theory of Integers

Let $\hat \mathbb{Z}$ denote the inverse limit of all finite cyclic groups. Let F, G and H be abelian groups with H ≤ G. Let FβH denote the abelian group (F × H, +^β), where +^βis defined by (a, x) +^β (b, y) = (a + b, x + y + β(a) + β(b) — β(a + b)) for a certain β : F → G linear mod H meaning that β(0) = 0 and β(a) + β(b) — β(a + b) ∈ H for all a, b in F. In this paper we show that the following hold: (1) The additive group of any nonstandard model ℤ* of the ring ℤ is isomorphic to (ℤ*₊/H)βH for a certain β : ℤ*₊/H → $\hat \mathbb{Z}$ linear mod H. (2) $\hat \mathbb{Z}$ is isomorphic to (ℤ₊/H )βH for some β : $\hat \mathbb{Z}$/H →ℚ linear mod H, though $\hat \mathbb{Z}$ is not the additive group of any model of Th(ℤ, +, ×) and the exact sequence H → $\hat \mathbb{Z}$ → $\hat \mathbb{Z}$/H is not splitting.     

Asymptotic and Spectral Analysis of the Spatially Nonhomogeneous Timoshenko Beam Model

We develop spectral and asymptotic analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the spatially nonhomogeneous Timoshenko beam model with a 2–parameter family of dissipative boundary conditions. Our results split into two groups. We prove asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues and each branch has only two points of accumulation: +∞ and —∞), and for their generalized eigenvectors. Our second main result is the fact that these operators are Riesz spectral. To obtain this result, we prove that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces. We also obtain the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators. The pencil asymptotics are essential for the proofs of the spectral results for the aforementioned dynamics generators.     

Stability for a random evolution equation with Gaussian perturbation

In this paper we consider a random evolution equation which is perturbed by Gaussian type noise, and show how to construct its coupled solutions. On the basis of the coupling results, we discuss the asymptotic flatness and the stability in total variation norm for the solutions of the equation. In addition, we also prove the Feller continuity, and the existence and uniqueness of invariant measures of the solutions.     

Effects of a Degeneracy in the Competition Model: Part II. Perturbation and Dynamical Behaviour

This is the second part of our study on the competition model where the coefficient functions are strictly positive over the underlying spatial region Ω except b(x), which vanishes in a nontrivial subdomain of Ω, and is positive in the rest of Ω. In part I, we mainly discussed the existence of two kinds of steady-state solutions of this system, namely, the classical steady-states and the generalized steady-states. Here we use these solutions to determine the dynamics of the model. We do this with the help of the perturbed model where b(x) is replaced by b(x)+ε, which itself is a classical competition model. This approach also reveals the interesting relationship between the steady-state solutions (both classical and generalized) of the above system and that of the perturbed system.     

On the Dynamics of the Classical Transverse Field Ising Chain

We investigate stationary and travelling wave solutions of a special lattice differential equation in one space dimension. Depending on a parameter λ, results are given on the existence, shape and stability for these kind of solutions. The analysis of travelling wave solutions leads us to a functional differential equation with both forward and backward shifts. The existence of solutions of this equation will be proved by use of the implicit function theorem. In particular, we consider kink solutions and periodic solutions.