Global dynamics of an SEI model with acute and chronic stages

A model with acute and chronic stages in a population with exponentially varying size is proposed. An equivalent system is obtained, which has two equilibriums: a disease-free equilibrium and an endemic equilibrium. The stability of these two equilibriums is controlled by the basic reproduction number _R0$R_0$. When _R0<1$R_0<1$, the disease-free equilibrium is globally stable. When _R0>1$R_0>1$, the disease-free equilibrium is unstable and the unique endemic equilibrium is locally stable. When _R0>1$R_0>1$ and γ=0,α=0$\gamma =0,\alpha =0$, the endemic equilibrium is globally stable in Γ⁰${\Gamma }^0$.     

Stability of the virus dynamics model with Beddington–DeAngelis functional response and delays

A virus dynamics model with Beddington–DeAngelis functional response and delays is introduced. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and the LaSalle invariance principle, we show that the infection-free equilibrium is globally asymptotically stable if _R0?1$R_0?1$ and the chronic-infection equilibrium is globally asymptotically stable if _R0>1$R_0>1$. Numerical simulations are also given to explain our results.     

Globally generated vector bundles of rank 2 on a smooth quadric threefold

We investigate the existence of globally generated vector bundles of rank 2 with _c1≤3$c_1\le 3$ on a smooth quadric threefold and determine their Chern classes. As an automatic consequence, every rank 2 globally generated vector bundle on Q$Q$ with _c1=3$c_1=3$ is an odd instanton up to twist.     

Complete global analysis of an SIRS epidemic model with graded cure and incomplete recovery rates

In this paper, applying two types of Lyapunov functional techniques to an SIRS epidemic model with graded cure and incomplete recovery rates, we establish complete global dynamics of the model whose threshold parameter is the basic reproduction number _R0$R_0$ such that the disease-free equilibrium is globally asymptotically stable when _R0?1$R_0?1$, and the endemic equilibrium is globally asymptotically stable when _R0>1$R_0>1$.     

On the general excess bound for binary codes with covering radius one

Let K(n,1)$K(n,1)$ denote the minimal cardinality of a binary code of length n$n$ and covering radius one. Fundamental for the theory of lower bounds for K(n,1)$K(n,1)$ is the covering excess method introduced by Johnson and van Wee. Let _δi${\delta }_i$ denote the covering excess on a sphere of radius i$i$, 0≤i≤n$0\le i\le n$. Generalizing an earlier result of van Wee, Habsieger and Honkala showed _δp−1≥p−1${\delta }_{p-1}\ge p-1$ whenever n≡−1$n\equiv -1$ (mod p$p$) for an odd prime p$p$ and _δ0=_δ1=?=_δp−2=0${\delta }_0={\delta }_1=?={\delta }_{p-2}=0$ holds. In the present paper we give the new estimation _δp−1≥(p−2)p−1${\delta }_{p-1}\ge (p-2)p-1$ instead. This answers a question of Habsieger and yields a “general improvement of the general excess bound” for binary codes with covering radius one. The proof uses a classification theorem for certain subset systems as well as new congruence properties for the δ$\delta$-function, which were conjectured by Habsieger.     

Stability analysis of an HIV/AIDS epidemic model with treatment

An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number _ℜ0${\Re }_0$. If _ℜ0≤1${\Re }_0\le 1$, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if _ℜ0>1${\Re }_0>1$. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.     

Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response

In this paper, we investigate the dynamical behavior of a virus infection model with delayed humoral immunity. By using suitable Lyapunov functional and the LaSalle?s invariance principle, we establish the global stabilities of the two boundary equilibria. If _R0<1$R_0<1$, the uninfected equilibrium _E0$E_0$ is globally asymptotically stable; if _R1<1<_R0$R_1<1<R_0$, the infected equilibrium without immunity _E1$E_1$ is globally asymptotically stable. When _R1>1$R_1>1$, we obtain the sufficient conditions to the local stability of the infected equilibrium with immunity _E2$E_2$. The time delay can change the stability of _E2$E_2$ and lead to the existence of Hopf bifurcations. The stabilities of bifurcating periodic solutions is also studied. We check our theorems with numerical simulations in the end.     

Joubert's theorem fails in characteristic 2

Let L/K$L/K$ be a separable field extension of degree 6. A 1867 theorem of P. Joubert asserts that if char(K)≠2$\mathrm{char}(K)\ne 2$, then L is generated over K   by an element whose minimal polynomial is of the form t⁶+at⁴+bt²+ct+d$t^6+a{t}^4+b{t}^2+ct+d$ for some a,b,c,d∈K$a,b,c,d\in K$. We show that this theorem fails in characteristic 2.     

Analysis of a SEIV epidemic model with a nonlinear incidence rate

In this paper, a SEIV epidemic model with a nonlinear incidence rate is investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is shown that if the basic reproduction number _R0<1${\mathfrak{R}}_0<1$, the disease-free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist. Moreover, we show that if the basic reproduction number _R0>1${\mathfrak{R}}_0>1$, the disease is uniformly persistent and the unique endemic equilibrium of the system with saturation incidence is globally asymptotically stable under certain conditions.     

Fast Lagrange–Newton transformations

In this paper we present explicit vector formulae for the Lagrange–Newton transformation L:Kⁿ→Kⁿ$L:{\mathrm{K}}^n\to {\mathrm{K}}^n$ and its inverse L^-1$L^{-1}$ with respect to interpolating knots _xi=α_xi-1+β(i=1,2,…,n-1;_x0=γ)$x_i=\alpha x_{i-1}+\beta \left(i=1,2,\dots ,n-1;x_0=\gamma \right)$, where α≠0,β,γ$\alpha \ne 0,\beta ,\gamma$ belong to a field K. These formulae depend on the wrapped convolution, Horner transformation, iterative product and coordinatewise vector operations. All these transformations and operations, except of O(nlogn)$O\left(n\log n\right)$—wrapped convolution, have running time of O(n)$O\left(n\right)$ base operations from the field K. Moreover, we give an application of these fast interpolating transformations to threshold secret sharing schemes in cryptography.     

A Sobolev inequality with reciprocal weights

We give a Sobolev inequality with the weight K(x)$K\left(x\right)$ belonging to the class _A2∩_Gn$A_2\cap G_n$ for the function |u|^t${\left|u\right|}^t$ and the weight K(x)⁻¹$K{\left(x\right)}^{-1}$ for |∇u|²${\left|\nabla u\right|}^2$. The constant in the relevant inequality is seen to depend on the _Gn$G_n$ and _A2$A_2$ constants of the weight.     

On convergence of the inexact Rayleigh quotient iteration with MINRES

For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow the residual norm _ξk≥1${\xi }_k\ge 1$ of the inner linear system at outer iteration k+1$k+1$ and can be considerably weaker than the condition _ξk≤ξ<1${\xi }_k\le \xi <1$ with ξ$\xi$ a constant not near one commonly used in the literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditioned MINRES methods for the linear systems. Some attractive properties are derived for the residuals obtained by MINRES. Based on them and the new general theory, we make a refined analysis and establish a number of new convergence results. Let ‖_rk‖$\Vert r_k\Vert$ be the residual norm of approximating eigenpair at outer iteration k$k$. Then all the available cubic and quadratic convergence results require _ξk=O(‖_rk‖)${\xi }_k=O\left(\Vert r_k\Vert \right)$ and _ξk≤ξ${\xi }_k\le \xi$ with a fixed ξ$\xi$ not near one, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that _ξk≤ξ${\xi }_k\le \xi$ with a constant ξ<1$\xi <1$ not near one, _ξk=1−O(‖_rk‖)${\xi }_k=1-O\left(\Vert r_k\Vert \right)$ and _ξk=1−O(‖_rk‖²)${\xi }_k=1-O\left({\Vert r_k\Vert }^2\right)$, respectively. The new convergence conditions are much more relaxed than ever before. The theory can be used to design practical stopping criteria to implement the method more effectively. Numerical experiments confirm our results.     

On the local representation of piecewise smooth equations as a Lipschitz manifold

We study systems of equations, F(x)=0$F(x)=0$, given by piecewise differentiable functions F:Rⁿ→R^k$F:{\mathbb{R}}^n\to {\mathbb{R}}^k$, k?n$k?n$. The focus is on the representability of the solution set locally as an (n−k)$(n-k)$-dimensional Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we introduce a strong full-rank assumption (SFRA) in terms of Clarke?s generalized Jacobians. The SFRA claims the existence of a basis in which Clarke?s inverse function theorem can be applied. Aiming at a characterization of SFRA, we consider also a full-rank assumption (FRA). The FRA insures the full rank of all matrices from the Clarke?s generalized Jacobian. The article is devoted to the conjectured equivalence of SFRA and FRA. For min-type functions, we give reformulations of SFRA and FRA using orthogonal projections, basis enlargements, cross products, dual variables, as well as via exponentially many convex cones. The equivalence of SFRA and FRA is shown to be true for min-type functions in the new case k=3$k=3$.     

Blow-up and propagation of disturbances for fast diffusion equations

This paper is concerned with the Cauchy problem for the fast diffusion equation _ut−Δu^m=αu^_p1$u_t-\Delta u^m=\alpha u^{p_1}$ in R^N${\mathbb{R}}^N$ (N≥1$N\ge 1$), where m∈(0,1)$m\in (0,1)$, _p1>1$p_1>1$ and α>0$\alpha >0$. The initial condition _u0$u_0$ is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value _u0$u_0$ so that u(t,x)$u(t,x)$ blows up in finite time, and we show how to get estimates on the profile of u(t,x)$u(t,x)$ for small enough values of t>0$t>0$.     

Remark on the stability of the large stationary solutions to the Navier–Stokes equations under the general flux condition

Consider stationary weak solutions of the Navier–Stokes equations in a bounded domain in R³${\mathbb{R}}^3$ under the nonhomogeneous boundary condition. We give a new approach for the stability of the stationary flow in the L²$L^2$-framework. Furthermore, we give some examples of stable solutions which may be large in L³(Ω)$L^3\left(\Omega \right)$ or W^1,3/2(Ω)$W^{1,3/2}\left(\Omega \right)$.     

Rate of escape and central limit theorem for the supercritical Lamperti problem

The study of discrete-time stochastic processes on the half-line with mean drift at x$x$ given by _μ1(x)→0${\mu }_1\left(x\right)\to 0$ as x→∞$x\to \infty$ is known as Lamperti’s problem  . We give sharp almost-sure bounds for processes of this type in the case where _μ1(x)${\mu }_1\left(x\right)$ is of order x^−β$x^{-\beta }$ for some β∈(0,1)$\beta \in (0,1)$. The bounds are of order t^1/(1+β)$t^{1/(1+\beta )}$, so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)$(2+2\beta +\epsilon )$-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where x^β_μ1(x)$x^{\beta }{\mu }_1\left(x\right)$ has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0$\beta =0$. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.     

On permanent and breaking waves in hyperelastic rods and rings

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point (_t0,_x0)∈R⁺×S¹$(t_0,x_0)\in {\mathbb{R}}^{+}\times {\mathbb{S}}^1$ is the identically zero solution. Such conclusion holds provided the physical parameter γ   of the model (related to the Finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa–Holm equation, corresponding to γ=1$\gamma =1$. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincaré inequalities.     

A structure theorem for small sumsets in nonabelian groups

Let G$G$ be an arbitrary finite group and let S$S$ and T$T$ be two subsets such that |S|≥2$\left|S\right|\ge 2$, |T|≥2$\left|T\right|\ge 2$, and |TS|≤|T|+|S|−1≤|G|−2$\left|TS\right|\le \left|T\right|+\left|S\right|-1\le \left|G\right|-2$. We show that if |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ then either S$S$ is a geometric progression or there exists a non-trivial subgroup H$H$ such that either |HS|≤|S|+|H|−1$\left|HS\right|\le \left|S\right|+\left|H\right|-1$ or |SH|≤|S|+|H|−1$\left|SH\right|\le \left|S\right|+\left|H\right|-1$. This extends to the nonabelian case classical results for abelian groups. When we remove the hypothesis |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ we show the existence of counterexamples to the above characterization whose structure is described precisely.