On global stability of an HIV pathogenesis model with cure rate

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4⁺ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12 : 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4⁺ T cells. Copyright © 2014 John Wiley & Sons, Ltd.     

Global properties of a class of HIV infection models with Beddington–DeAngelis functional response

In this paper, the global properties of a class of human immunodeficiency virus (HIV) models with Beddington–DeAngelis functional response are investigated. Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of three HIV infection models. The first model considers the interaction process of the HIV and the CD4 ⁺ T cells and takes into account the latently and actively infected cells. The second model describes two co‐circulation populations of target cells, representing CD4 ⁺ T cells and macrophages. The third model is a two‐target‐cell model taking into account the latently and actively infected cells. We have proven that if the basic reproduction number R₀ is less than unity, then the uninfected steady state is globally asymptotically stable, and if R₀ > 1, then the infected steady state is globally asymptotically stable. Copyright © 2012 John Wiley & Sons, Ltd.     

On the solutions of the Moisil–Théodoresco system

A structure theorem is proved for the solutions to the Moisil–Théodoresco system in open subsets of ?³. Furthermore, it is shown that the Cauchy transform maps L₂(?², ?_{0, 2}⁺) isomorphically onto H²(?₊³, ?_{0, 3}⁺), thus proving an elegant generalization to ?² of the classical notion of an analytic signal on the real line. Copyright © 2008 John Wiley & Sons, Ltd.     

Implementation of a Preconditioning–Technique in the Hybrid Flow–Solver TAU+

The present paper describes the implementation of a preconditioning method in the hybrid DLR–TAU⁺–code and its application to nearly incompressible flows. The method is designed in order to get an efficient and accurate solution even for very low Mach numbers using a time stepping scheme for the solution of the compressible Navier–Stokes equations. The algorithm is based on the work of Choi and Merkle. The numerical results obtained for inviscid and viscous flows indicate, that for Mach numbers lower than 0.1 the accuracy as well as the convergence properties are almost independent of the fluid speed, like for incompressible codes.     

Bifurcation analysis of the HIV‐1 within host model

In this paper, a bifurcation solution's analysis is proposed for an HIV‐1 within the host model around its chronic equilibrium point, this is carried out based on Lyapunov–Schmidt approach. It is shown that the coefficient b, which represents the healthy CD4⁺ T‐cells growth rate, is a bifurcation parameter; this means that the rate of multiplication of healthy cells can have serious effects on the qualitative dynamical properties and structural stability of the infection evolution dynamics. Copyright © 2015 John Wiley & Sons, Ltd.     

Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation ?u_t+Δu+u^p=0 in ?ⁿ×[0, ∞), for exponents p?3 and space dimensions n?3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright © 2009 John Wiley & Sons, Ltd.     

An artificial boundary condition for an advection–diffusion equation

Consider the advection–diffusion equation: u₁ + au_x1 ? vδu = 0 in ?ⁿ × ?⁺ with initial data u⁰; the Support of u⁰ is contained in ?(x₁ < 0) and a: ?ⁿ → ? is positive. In order to approximate the full space solution by the solution of a problem in ? × ?⁺, we propose the artificial boundary condition: u₁ + au_x1 = 0 on ∑. We study this by means of a transmission problem: the error is an O(v²) for small values of the viscosity v.     

A global existence theorem for the general coagulation–fragmentation equation with unbounded kernels

In this article an existence theorem is proved for the coagulation–fragmentation equation with unbounded kernel rates. Solutions are shown to be in the space X⁺ = {c∈L¹: ∫ (1 + x)∣c(x)∣dx < ∞} whenever the kernels satisfy certain growth properties and the non-negative initial data belong to X⁺. The proof is based on weak L¹ compactness methods applied to suitably chosen approximating equations.     

A non‐existence result on Cameron–Liebler line classes

Cameron–Liebler line classes are sets of lines in PG(3, q) that contain a fixed number x of lines of every spread. Cameron and Liebler classified Cameron–Liebler line classes for x ∈ {0, 1, 2, q² ? 1, q², q² + 1} and conjectured that no others exist. This conjecture was disproven by Drudge for q = 3 [8] and his counterexample was generalized to a counterexample for any odd q by Bruen and Drudge [4]. A counterexample for q even was found by Govaerts and Penttila [9]. Non‐existence results on Cameron–Liebler line classes were found for different values of x. In this article, we improve the non‐existence results on Cameron–Liebler line classes of Govaerts and Storme [11], for q not a prime. We prove the non‐existence of Cameron–Liebler line classes for 3 ≤ x < q/2. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 342–349, 2008     

A note on bound for norms of Cauchy–Hankel matrices

We determine bounds for the spectral and ??_p norm of Cauchy–Hankel matrices of the form H_n=[1/(g+h(i+j))]ⁿ_i,j=1≡ ([1/(g+kh)]ⁿ_i,j=1), k=0, 1,…, n –1, where k is defined by i+j=k (mod n). Copyright © 2002 John Wiley & Sons, Ltd.     

A Description of the Global Attractor for a Class of Reaction – Diffusion Systems with Periodic Solutions

We examine the autonomous reaction–diffusion system with Dirichlet boundary conditions on (0, 1), where α, β are real, α > 0, and g is C¹ and satisfies some conditions which we need in order to prove the existence of solutions. We construct a solution of (RD) for every initial value in L²((0, 1)) × L²((0, 1)), we show that this solution is uniquely determined and that the solution has C^∞–smooth representatives for all positive t. We determine the long time behaviour of each solution. In particular, we show that each solution of (RD) tends either to the zero solution or to a periodic orbit. We construct all periodic orbits and show that their number is always finite. It turns out that the global attractor is a finite union of subsets of L² × L², which are finite–dimensional manifolds, and the dynamics in these sets can be described completely.     

Perfectness of Conelike *–Semigroups in ℚk

Let S be an abelian *–semigroup in ℚ^k. We give a sufficient condition for every positive definite function on S to have a unique representing measure on the dual semigroup of S (i.e. S is perfect). To characterize perfectness for any abelian *–semigroupis a challenging, but not yet generally solved problem. In this paper, we characterize the structure of involutions on an abelian *–semigroup which is a subset of ℚ^k, and show that any conelike *–semigroups in ℚ^k are perfect.     

Approximate string matching on Ziv–Lempel compressed text

We present the first nontrivial algorithm for approximate pattern matching on compressed text. The format we choose is the Ziv–Lempel family. Given a text of length u compressed into length n, and a pattern of length m, we report all the R occurrences of the pattern in the text allowing up to kinsertions, deletions and substitutions. On LZ78/LZW we need O(mkn+R) time in the worst case and O(k²n+mkmin(n,(mσ)^k)+R) on average where σ is the alphabet size. The experimental results show a practical speedup over the basic approach of up to 2X for moderate m and small k. We extend the algorithms to more general compression formats and approximate matching models.     

Global sharp interface limit of the Hele–Shaw–Cahn–Hilliard system

In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two‐phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by ? , and the mobility coefficient (the diffusional Peclet number) is ? ^α . We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as ? →0 in the case of 0≤α  < 1. Copyright © 2016 John Wiley & Sons, Ltd.     

A new difference scheme with high accuracy and absolute stability for solving convection–diffusion equations

In this paper, we use a semi-discrete and a padé approximation method to propose a new difference scheme for solving convection–diffusion problems. The truncation error of the difference scheme is O(h⁴+τ⁵). It is shown through analysis that the scheme is unconditionally stable. Numerical experiments are conducted to test its high accuracy and to compare it with Crank–Nicolson method.     

Stability of the Crank–Nicolson–Adams–Bashforth scheme for the 2D Leray‐alpha model

We consider the stability of an efficient Crank–Nicolson–Adams–Bashforth method in time, finite element in space, discretization of the Leray‐α model. We prove finite‐time stability of the scheme in L², H¹, and H², as well as the long‐time L‐stability of the scheme under a Courant‐Freidrichs‐Lewy (CFL)‐type condition. Numerical experiments are given that are in agreement with the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1155–1183, 2016     

Full asymptotic expansion of the heat trace for non–self–adjoint elliptic cone operators

The operator e^–tA and its trace Tr e^–tA, for t > 0, are investigated in the case when A is an elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter–ellipticity) we obtain a full asymptotic expansion in t of the heat trace as t → 0⁺. As in the smooth compact case, the problem is reduced to the investigation of the resolvent (A–λ)^–1. The main step consists in approximating this family by a parametrix of A – λ constructed within a suitable parameter–dependent calculus.     

Existence and asymptotic behaviour for the time‐fractional Keller–Segel model for chemotaxis

One of the most important systems for understanding chemotactic aggregation is the Keller–Segel system. We consider the time‐fractional Keller–Segel system of order . We prove an existence result with small initial data in a class of Besov–Morrey spaces. Self‐similar solutions are obtained and we also show an asymptotic behaviour result.     

A conservative compact difference scheme for the coupled Klein–Gordon–Schrödinger equation

In this article, a conservative compact difference scheme is presented for the periodic initial‐value problem of Klein–Gordon–Schrödinger equation. On the basis of some inequalities about norms and the priori estimates, convergence of the difference solution is proved with order O(h⁴ +τ ²) in maximum norm. Numerical experiments demonstrate the accuracy and efficiency of the compact scheme. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013