香港艾滋病近期传播特征与动态预测

首次研究出香港艾滋病目前传播势态为一元线性回归方程,并由方程的预测值和斜率的变化动态监测香港艾滋病近期发病趋势,艾滋病病毒感染者年平均发病率为255至271个,艾滋病病患者年平均发病率为56个左右.本文研究方法简单易行,对香港政府近期预防艾滋病有重要的理论和实用价值.     

The velocity and shear stress functions of the Falkner-Skan equation arising in boundary layer theory

New compactness results on the velocity functions and shear stress functions of the well-known Falkner-Skan equation are obtained. The methodology is to utilize the equivalence between the Falkner-Skan equation and a singular integral equation established recently by Lan and Yang.     

Conservative finite difference schemes for the Degasperis–Procesi equation

We consider the numerical integration of the Degasperis–Procesi equation, which was recently introduced as a completely integrable shallow water equation. For the equation, we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi-Hamiltonian form of the equation at the same time. We also prove the unique solvability of the schemes, and show some numerical examples.     

Nonoscillation and oscillation of second-order impulsive differential equations with periodic coefficients

In this paper, we give a nonoscillation criterion for half-linear equations with periodic coefficients under fixed moments of impulse actions. The method is based on the existence of positive solutions of the related Riccati equation and a recently obtained comparison principle. In the special case when the equation becomes impulsive Hill equation new oscillation criteria are also obtained.     

Elliptic traveling waves of the Olver equation

Nonlinear waves on water are studied. The method recently developed by Demina and Kudryashov is applied to the Olver water wave equation. New solutions of this equation are found. These solutions are expressed in terms of the Weierstrass elliptic function.     

Blow-up phenomenon for the integrable Novikov equation

In this paper we investigate a new integrable equation derived recently by V.S. Novikov [Generalizations of the Camassa–Holm equation, J. Phys. A 42 (34) (2009) 342002, 14 pp.]. Analogous to the Camassa–Holm equation and the Degasperis–Procesi equation, this new equation possesses the blow-up phenomenon. Under the special structure of this equation, we establish sufficient conditions on the initial data to guarantee the formulation of singularities in finite time. A global existence result is also found.     

The Beverton–Holt dynamic equation

The Cushing–Henson conjectures on time scales are presented and verified. The central part of these conjectures asserts that based on a model using the dynamic Beverton–Holt equation, a periodic environment is deleterious for the population. The proof technique is as follows. First, the Beverton–Holt equation is identified as a logistic dynamic equation. The usual substitution transforms this equation into a linear equation. Then the proof is completed using a recently established dynamic version of the generalized Jensen inequality.     

A boundary integral equation for the transmission eigenvalue problem for Maxwell equation

We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two‐by‐two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric‐to‐magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation.     

A boundary integral equation method for the transmission eigenvalue problem

We propose a new integral equation formulation to characterize and compute transmission eigenvalues for constant refractive index that play an important role in inverse scattering problems for penetrable media. As opposed to the recently developed approach by Cossonnière and Haddar [1,2] which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators which results in a noticeable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further we employ the numerical algorithm for analytic non-linear eigenvalue problems that was recently proposed by Beyn [3] for the numerical computation of transmission eigenvalues via this new integral equation.     

Nonlinear self-adjointness of a 2D generalized second order evolution equation

We study the nonlinear self-adjointness of a general class of quasilinear 2D second order evolution equations which do not possess variational structure. For this purpose, we use the method of Ibragimov, devised and developed recently. This approach enables one to establish the conservation laws for any differential equation. We first obtain conditions determining the self-adjoint sub-class in the general case. Then, we establish the conservation laws for important particular cases: the Ricci Flow equation, the modified Ricci Flow equation and the nonlinear heat equation.     

Comments on “Similarity variables and reduction of the heat equation on torus” [Commun Nonlinear Sci Numer Simulat 2012;17:1251–57]

A brief account of symmetry analysis of heat equation on torus is presented, correcting errors in a recently published paper.     

Discrete equation on a square lattice with a nonstandard structure of generalized symmetries

We clarify the integrability nature of a recently found discrete equation on the square lattice with a nonstandard symmetry structure. We find its L-A pair and show that it is also nonstandard. For this discrete equation, we construct the hierarchies of both generalized symmetries and conservation laws. This equation yields two integrable systems of hyperbolic type. The hierarchies of generalized symmetries and conservation laws are also nonstandard compared with known equations in this class.     

Rational approximation to the Thomas-Fermi equations

We show that a simple and straightforward rational approximation to the Thomas-Fermi equation provides the slope at origin with unprecedented accuracy and that Padé approximants of relatively low order are far more accurate than more elaborate approaches proposed recently by other authors. We consider both the Thomas-Fermi equation for isolated atoms and for atoms in strong magnetic fields.     

Integrability properties of a generalized reduction of the KdV6 equation

We consider the integrability properties of a generalized version of a similarity reduction of the so-called KdV6 equation, an equation that has recently generated much interest. We give a linear problem for this generalized reduction and show that it satisfies the requirements of the Ablowitz-Ramani-Segur algorithm. In addition we give a Bäcklund transformation to a related equation, giving also an auto-Bäcklund transformation for this last. Our results mirror those for the Korteweg-de Vries equation itself, which has a similarity reduction to an ordinary differential equation which is related by a Bäcklund transformation to the second Painlevé equation, this last having an auto-Bäcklund transformation.     

Supersymmetric integrable systems from geodesic flows on superconformal groups

We discuss the possible relation between geodesic flow, integrability, and supersymmetry, using fermionic extensions of teh KdV equation and the recently introduced supersymmetrization of the Camassa-Holm equation as illustrative examples. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 182–188, May, 2000.     

EQUIVALENCE BETWEEN EXACT INTERNAL CONTROLLABILITY OF THE KIRCHHOFF PLATE-LIKE EQUATION AND THEWAVE EQUATION

1.IntroductionConsiderthefollowingillitial-boundaryvalueproblemLetD(A)={vEHI(fl)ItvEL'(fl)},V~Ha(~~),H~L'(fl).Thenthefiniteenergystatespacesofthesystems(1.1),(1.2)are,respectively,uk~D(A)xV,U.~VxH.ConsiderthegeneralizedfunctionspaceH-1~V'DHDV.ForbEC'(fl)anduEH-',wedefinebaEH-1by(bu)(()~u(bo,V(6HJ(n).(1.3)WehaveusedtheSobolevspaces.Wereferthereadersto[1]fortheknowledgeofthosespaces.Definition1.1.Thesystem(1.1)issaidtobeexactlycontrollableinukbyH-1-controlsifforevery(yo,yi)6uk,th…     

A New Integrable Equation with Peakon Solutions

We consider a new partial differential equation recently obtained by Degasperis and Procesi using the method of asymptotic integrability; this equation has a form similar to the Camassa–Holm shallow water wave equation. We prove the exact integrability of the new equation by constructing its Lax pair and explain its relation to a negative flow in the Kaup–Kupershmidt hierarchy via a reciprocal transformation. The infinite sequence of conserved quantities is derived together with a proposed bi-Hamiltonian structure. The equation admits exact solutions as a superposition of multipeakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa–Holm peakons.     

An Integrable Discretization of a 2+1-dimensional Sine-Gordon Equation

We show that the complex discrete BKP equation that has been recently identified as an integrable discretization of the 2+1-dimensional sine-Gordon system introduced by Konopelchenko and Rogers admits a natural reduction to a discrete 2+1-dimensional sine-Gordon equation. We discuss three important properties of this equation. First, it may be interpreted as a superposition principle associated with a constrained Moutard transformation. Second, the complexified discrete sine-Gordon equation constitutes an eigenfunction equation for the discrete sine-Gordon system. Third, we derive a form of the equation in terms of trigonometric functions that has been studied by Konopelchenko and Schief in a discrete geometric context. A discrete Moutard transformation for the discrete sine-Gordon equation and the corresponding Bäcklund equations are also recorded.     

On the Inverse Scattering Transform for the Kadomtsev-Petviashvili Equation

The initial value problem of the Kadomtsev-Petviashvili equation for one choice of sign in the equation has been recently investigated in the literature. Here we consider the other choice of sign. We introduce suitable eigenfunctions which though bounded are not analytic in the spectral parameter. This, in contrast to the known case, prevents us from formulating the inverse problem as a nonlocal Riemann-Hilbert boundary value problem. Nevertheless a suitable formulation is given and a formal solution is constructed via a linear integral equation.     

Universal Models of Soliton Hierarchies

We consider commutativity equations and a related new model similar to the Kadomtsev–Petviashvili equation in the Sato theory. Integration of this model equation with three independent variables is based on a generalization of the Dubrovin equations and the recently developed theory of transformations of spectral problems. We give examples of equations with a fractional-power dispersion law that can be linearized in this theory.