The analysis and application of an HBV model

A mathematical model is formulated to describe the spread of hepatitis B. The stability of equilibria and persistence of disease are analyzed. The results shows that the dynamics of the model is completely determined by the basic reproductive number ρ₀. If ρ₀ < 1, the disease-free equilibrium is globally stable. When ρ₀ > 1, the disease-free equilibrium is unstable and the disease is uniformly persistent. Furthermore, under certain conditions, it is proved that the endemic equilibrium is globally attractive. Numerical simulations are conducted to demonstrate our theoretical results. The model is applied to HBV transmission in China. The parameter values of the model are estimated based on available HBV epidemic data in China. The simulation results matches the HBV epidemic data in China approximately.     

Monotonicity of the search number in the Golovach problem

The Golovach problem, also known as the ɛ-search problem, is as follows. A team of pursuers pursues an evader on a topological graph. The objective of the pursuers is to catch the evader, that is, approach the evader to a distance not exceeding a given nonnegative number ɛ. It is assumed that the evader is invisible to the pursuers and is fully informed beforehand about the search program of the pursuers. The problem is to find the ɛ-search number, i.e., the least number of pursuers sufficient for capturing the evader. Graphs with monotone ɛ-search number are studied; the ɛ-search number of a graph G is said to be monotone if it is not exceeded by the ɛ-search numbers of all connected subgraphs H of G. It is known that the ɛ-search number of any tree is monotone for all nonnegative ɛ. The edgesearch number, which is equal to the 0-search number, is monotone for all connected subgraphs of an arbitrary graph. A sufficient monotonicity condition for the ɛ-search number of any graph is obtained. This result is improved in the case of complete subgraphs. The Golovach function is constructed for graphs obtained by removing one edge from complete graphs with unit edges.     

Lifshitz tails for Schrödinger operators with random breather potential

We prove a Lifshitz tail bound on the integrated density of states of random breather Schrödinger operators. The potential is composed of translated single-site potentials. The single-site potential is an indicator function of the set tA where t is from the unit interval and A is a measurable set contained in the unit cell. The challenges of this model are that, since A is not assumed to be star-shaped, the dependence of the potential on the parameter t is not monotone. It is also non-linear and not differentiable.     

A priori reduction method for solving the two-dimensional Burgers’ equations

The two-dimensional Burgers’ equations are solved here using the A Priori Reduction method. This method is based on an iterative procedure which consists in building a basis for the solution where at each iteration the basis is improved. The method is called a priori because it does not need any prior knowledge of the solution, which is not the case if the standard Karhunen-Loéve decomposition is used. The accuracy of the APR method is compared with the standard Newton-Raphson scheme and with results from the literature. The APR basis is also compared with the Karhunen-Loéve basis.     

Equivalence between modified symplectic Gram-Schmidt and Householder SR algorithms

The SR factorization for a given matrix A is a QR-like factorization A=SR, where the matrix S is symplectic and R is J-upper triangular. This factorization is fundamental for some important structure-preserving methods in linear algebra and is usually implemented via the symplectic Gram-Schmidt algorithm (SGS). There exist two versions of SGS, the classical (CSGS) and the modified (MSGS). Both are equivalent in exact arithmetic, but have very different numerical behaviors. The MSGS is more stable. Recently, the symplectic Householder SR algorithm has been introduced, for computing efficiently the SR factorization. In this paper, we show two new and important results. The first is that the SR factorization of a matrix A via the MSGS is mathematically equivalent to the SR factorization via Householder SR algorithm of an embedded matrix. The later is obtained from A by adding two blocks of zeros in the top of the first half and in the top of the second half of the matrix A. The second result is that MSGS is also numerically equivalent to Householder SR algorithm applied to the mentioned embedded matrix.     

Conditional ɛ-uniform convergence of adaptive algorithms in the finite element method as applied to singularly perturbed problems

The boundary value problem for the ordinary differential equation of reaction-diffusion on the interval [−1, 1] is examined. The highest derivative in this equation appears with a small parameter ɛ² (ɛ ∈ (0, 1]). As the small parameter approaches zero, boundary layers arise in the neighborhood of the interval endpoints. An algorithm for the construction of a posteriori adaptive piecewise uniform grids is proposed. In the adaptation process, the edges of the boundary layers are located more accurately and the grid on the boundary layers is repeatedly refined. To find an approximate solution, the finite element method is used. The sequence of grids constructed by the algorithm is shown to converge “conditionally ɛ-uniformly” to some limit partition for which the error estimate O(N ⁻²ln³ N) is proved. The main results are obtained under the assumption that ɛ ≪ N ⁻¹, where N is number of grid nodes; thus, conditional ɛ-uniform convergence is dealt with. The proofs use the Galerkin projector and its property to be quasi-optimal.     

Initial-boundary value problems for incomplete singular perturbations of hyperbolic systems

The general problem studied has as a prototype the full non-linear Navier-Stokes equations for a slightly viscous compressible fluid including the heat transfer. The boundaries are of inflow-outflow type, i.e. non-characteristic, and the boundary conditions are the most general ones with any order of derivatives. It is assumed that the uniform Lopatinsky condition is satisfied. The goal is to prove uniform existence and boundedness of solution as the viscosity tends to zero and to justify the boundary layer asymptotics. The paper consists of two parts. In Part I the linear problem is studied. Here, uniform lower and higher order tangential estimates are derived and the existence of a solution is proved. The higher order estimates depend on the smoothness of coefficients; however this smoothness does not exceed the smoothness of the solution. In Part II the quasilinear problem is studied. It is assumed that for zero viscosity the overall initial-boundary value problem has a smooth solutionu ₀ in a time interval 0≦t≦T ₀. As a result the boundary laye, is weak and is uniformlyC ¹ bounded. This makes the linear theory applicable. an iteration scheme is set and proved to converge to the viscous solution. The convergence takes place for small viscosity and over the original time interval 0≦t≦T ₀.     

A sequential sampling rule for selecting the most probable multinomial event

Summary A sequential sampling rule is given for selecting the most probable event from a multinomial distribution withk cells. A random number of observations is taken from the given multinomial distribution at each stage of sampling, where the number is distributed according to a Poisson distribution with mean λ. The sampling is stopped when the count in any cell is greater than or equal to a given positive integerN. The cell with the highest count is selected for the most probable event. The mathematical analysis of the problem is simplified as a result of the statistical independence of the cell frequencies due to the randomization of the sample number. The expected value of the stage when the sampling terminates is decreasing in λ. The sequential sampling scheme in which one observation is taken at a time until the highest cell count is equal toN, corresponds to λ→0. A table is given showing some properties of the given selection procedure.     

Dynamics of Bianchi Type I Solutions of the Einstein Equations with Anisotropic Matter

We analyze the global dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter. The matter model is not specified explicitly but only through a set of mild and physically motivated assumptions; thereby our analysis covers matter models as different from each other as, e.g., collisionless matter, elastic matter and magnetic fields. The main result we prove is the existence of an ‘anisotropy classification’ for the asymptotic behaviour of Bianchi type I cosmologies. The type of asymptotic behaviour of generic solutions is determined by one single parameter that describes certain properties of the anisotropic matter model under extreme conditions. The anisotropy classification comprises the following types. The convergent type A₊: Each solution converges to a Kasner solution as the singularity is approached and each Kasner solution is a possible past asymptotic state. The convergent types B₊ and C₊: Each solution converges to a Kasner solution as the singularity is approached; however, the set of Kasner solutions that are possible past asymptotic states is restricted. The oscillatory type D₊: Each solution oscillates between different Kasner solutions as the singularity is approached. Furthermore, we investigate non-generic asymptotic behaviour and the future asymptotic behaviour of solutions. Submitted: October 28, 2008.; Accepted: January 26, 2009.     

Distribution dependent SDEs for Landau type equations

The distribution dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a nonlinear PDE. Due to the distribution dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a nonlinear semigroup $P_t^1$ on the space of probability measures. The exponential contraction as well as Harnack inequalities and applications are investigated for the nonlinear semigroup $P_t^1$ using coupling by change of measures. The main results are illustrated by homogeneous Landau equations.     

Recurrence set-relations in stochastic multiobjective dynamic decision processes

Finite horizon stochastic dynamic decision processes with R^p valued additive returns are considered. The optimization criterion is a partial-order preference relation induced from a convex cone in R^p . The state space is a countable set, and the action space is a compact metric spaces. The optimal value function, which is of a set-valued mapping, is defined. Under certain assumptions on the continuity of the reward vector and the transition probability, a system of a recurrence set-relations concerning the optimal value functions is given.     

Economic and economic-statistical design of a chi-square chart for CBM

In this paper, the economic and economic-statistical design of a χ² chart for a maintenance application is considered. The machine deterioration process is described by a three-state continuous time Markov chain. The machine state is unobservable, except for the failure state. To avoid costly failures, the system is monitored by a χ² chart. The observation process stochastically related to the machine condition is assumed to be multivariate, normally distributed. When the chart signals, full inspection is performed to determine the actual machine condition. The system can be preventively replaced at a sampling epoch and must be replaced upon failure; preventive replacement costs less than failure replacement. The objective is to find the optimal control chart parameters that minimize the long-run average maintenance cost per unit time. For the economic-statistical design, an additional constraint guaranteeing the occurrence of the true alarm signal on the chart before failure with given probability is considered. For both designs, the objective function is derived using renewal theory.     

Maximum likelihood estimator for the drift of a Brownian flow

The maximum likelihood estimator for the drift of a Brownian flow on ℝ^d, d ⩾ 2, is found with the assumption that the covariance is known. By approximation of the drift with known functions, the statistical model is reduced to a parametric one that is a curved exponential family. The data is the n‐point motion of the Brownian flow throughout the time interval [0, T]. The asymptotic properties of the MLE are also investigated. Copyright © 2000 John Wiley & Sons, Ltd.     

Analysis of the multistrain asymmetric SI model for arbitrary strain diversity

The asymmetric multistrain $SI$ model is studied within a history-based framework. The governing differential equation in set notation is solved by making use of the powerful set manipulation functions of the Mathematica programming language. The algorithm allows, for the first time, the solution of both the temporal and the equilibrium equations for arbitrary strain diversity. Since Mathematica is an algebraic manipulator, analytical expressions are presented for the equilibrium population variables in terms of the forces of infection for arbitrary number of strains, $n$. Since there are no recoveries allowed in this model, it is found that coinfection always dominates the system if the basic reproductive number of both strains is greater than 1. The danger of coinfection is already evident for the relatively simple case of $n=2$ and becomes more drastic as $n$ increases. Strains which are not sustainable on their own are prevalent in the host population due to coinfection. The notion of a prevalence distribution function is introduced, which shows how the total prevalence is distributed amongst the different levels of infection. Results indicate that higher values of $n$ lead to a faster increase in coinfection prevalences since there are no recoveries in the model.     

Magnetic field in a turbulent conducting fluid

The problem of magnetic field in conducting turbulent, incompressible fluid is considered. The velocity of the fluid is taken to be independent of the magnetic field and is described by a Gaussian field, ‘white noise’ in time with smooth space correlation. The main result is that no fast dynamo (by which is meant almost sure exponential growth of magnetic field) can exist for an incompressible fluid when the magnetic viscosity is positive. For d = 2, sharper results are obtained; the magnetic field dies out when the magnetic viscosity is strictly positive. Furthermore, when d = 2, existence and characterization of invariant measure are given for d = 2 when the magnetic viscosity is zero. The results are compared to those discussed by Baxendale and Rosovskii in [2]     

The solvability conditions for inverse eigenproblem of symmetric and anti‐persymmetric matrices and its approximation

The problem of generating a matrix A with specified eigen‐pair, where A is a symmetric and anti‐persymmetric matrix, is presented. An existence theorem is given and proved. A general expression of such a matrix is provided. We denote the set of such matrices by ??????_Eⁿ. The optimal approximation problem associated with ??????_Eⁿ is discussed, that is: to find the nearest matrix to a given matrix A* by A∈??????_Eⁿ. The existence and uniqueness of the optimal approximation problem is proved and the expression is provided for this nearest matrix. Copyright © 2002 John Wiley & Sons, Ltd.     

Continuous time control of the arrival process in an m/g/1 queue

The problem of continuously controlling the arrival process in an M/G/1 queue is studied. The control is exercised by keeping the facility open or closed for potential arrivals, and is based on the residual workload process. The reward structure includes a reward rate R when the server is busy, and a holding cost rate cx when the residual workload is x. The economic criterion used is long run average return. A control limit policy is shown to be optimal. An iterative method for calculating this control limit policy is suggested.     

k-eccentricity and absolute k-centrum of a probabilistic tree

The k-eccentricity evaluated at a point x of a graph G is the sum of the (weighted) distances from x to the k vertices farthest from it. The k-centrum is the set of vertices for which the k-eccentricity is a minimum. The concept of k-centrum includes, as a particular case, that of center and that of centroid (or median) of a graph. The absolute k-centrum is the set of points (not necessarily vertices) for which the k-eccentricity is a minimum. In this paper it will be proven that, for a weighted tree, both deterministic and probabilistic, the k-eccentricity is a convex function and that the absolute k-centrum is a connected set and is contained in an elementary path. Hints will be given for the construction of an algorithm to find the k-centrum and the absolute k-centrum.