Structure of the spectrum of infinite dimensional Hamiltonian operators

This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.     

Exact solutions of the generalized Dugdale model of two-dimensional decagonal quasicrystals

The basic properties of the generalized Dugdale model of quasicrystals have significance in the theory of fracture mechanics of this solids. This study suggests a conformal mapping which transforms the complicated region in physical plane onto the exterior of the unit circle in the ζ plane, then in terms of Cauchy integral theory, the complicated boundary value problem is solved, and the solutions are expressed in close form exactly. The crack tip opening displacement (CTOD), cohesive force zone and the most important physical quantity, stress intensity factor, are obtained with exact analytic form, which are useful in materials science and engineering.     

对角无穷维Hamilton算子点谱关于实轴的对称性

在不同条件下得到对角无穷维Hamilton算子点谱的两个组成部分σp（A）与σp（-A^＊）关于实轴对称的充分必要条件.以此为基础,完全刻画了对角无穷维Hamilton算子点谱关于实轴的对称性.     

Exact and analytical solutions for a nonlinear sigma model

We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error‐minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.     

Exact solutions for two dimensional flows of couple stress fluids

Exact solutions are derived for the class of two dimensional couple stress flows. This class consists of flows for which the vorticity distribution is proportional to the stream function perturbed by a uniform stream. The solutions are obtained by applying the so-called inverse method which makes certain hypothesis a priori on the form of the velocity field and pressure without making any on the boundaries of the domain occupied by the fluid. Exact solutions are obtained for both steady and unsteady cases.     

Exact solutions for two dimensional flows of couple stress fluids

Exact solutions are derived for the class of two dimensional couple stress flows. This class consists of flows for which the vorticity distribution is proportional to the stream function perturbed by a uniform stream. The solutions are obtained by applying the so-called inverse method which makes certain hypothesis a priori on the form of the velocity field and pressure without making any on the boundaries of the domain occupied by the fluid. Exact solutions are obtained for both steady and unsteady cases.     

Poisson limits for a hard-core clustering model

Let X_1n,…,X_>nn denote the locations of n points in a bounded, γ-dimensional, Euclidean region D_n which has positive γ-dimensional Lebesgue measure μ(D_n). Let {Y_n(r): r > 0} be the interpoint distance process for these points where Y_n(r) is the number of pairs of points(X_in, X_in) which with i < j have Euclidean distance 6X_in ? X_>in6 < r. In this article we study the limiting distribution of Y_n(r) when n → ∞ and μ(D_n) → ∞, and the joint density of X_1n,…,X_nnis of the form

where r₀ is a positive constant and C_n is a normalizing constant. These joint densities modify the Strauss [11] clustering model densities by introducing a hard-core component (no two points can have 6X_in ? X_in6 < r₀) found in the Matérn [4] models. In our main result we show that the interpoint distance process converges to a non-homogeneous Poisson process for r values in a bounded interval 0 < r₀ < r < r₀₀ provided sparseness conditions discussed by Saunders and Funk [9] hold. The sparseness conditions which require $\text{μ(D}{}_{\mathrm{n}}\text{)}\text{n}{}^2$ converges to a positive constant and the boundary of D_n is negligible are essentially equivalent to requiring that although the number of points n is large the region is large enough so that the points are sparse in this region. That is, it is rare for a point to have another point close to it. These results extend results for v ? 0 given by Saunders and Funk [9] where it is shown that without the hard core component such results do not hold for v > 0. Statistical applications are discussed.     

高维Wiener sausage的强逼近

本文研究了四维及四维以上的Wiener sausage 的体积, 得到它们可以由一维Brown 运动强逼近. 作为应用, 推出了弱收敛和重对数率.     

Global solutions of a two‐dimensional chemotaxis system with attraction and repulsion rotational flux terms

We consider the attraction–repulsion chemotaxis system with rotational flux terms where is a bounded domain with smooth boundary. Here, S₁ and S₂ are given parameter functions on [0,∞)²×Ω with values in . It is shown that for any choice of suitably regular initial data (u₀,v₀,w₀) fulfilling a smallness condition on the norm of v₀,w₀ in L^∞(Ω), the corresponding initial‐boundary value problem possesses a global bounded classical solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Study of the Hubbard model at half filling

Under the conditions of a strong Coulomb repulsion between electrons at a lattice site, we now study the equations for the self-energy part of the electron Green’s function, which we previously obtained using the generating functional method. These equations have a form close to that corresponding to the self-consistent Born approximation in the weak-coupling theory. In these equations, we omit the dependence of the self-energy on the momentum, which corresponds to the infinite-dimensional space limit. We then numerically solve the integral equations, where all the variables depend only on the frequency, and obtain results consistent with the dynamical mean field theory. In particular, we show that as the Coulomb repulsion increases, the three-peak structure of the quasiparticle spectrum changes into a two-peak structure and the metal–insulator phase transition occurs. The proposed method can be used to study other models of the theory of strongly correlated systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 63–76, January, 2008.     

Exact solutions to the equations of a dynamic asymmetric pseudoelasticity model

Using a group foliation of the equations of the dynamic asymmetric model of pseudoelasticity effectively used in studying the elastic materials made of polymers, we obtain a system that, after renaming the functions, becomes equivalent to these equations and contains fewer additional functions than the union of the resolving and automorphic systems of the accomplished group foliation. Among the first-order systems equivalent to these equations, it contains the least number of additional functions and is the only such a system up to a nondegenerate linear transformation of the additional functions. For this system, we find the main Lie group of transformation, an optimal system of its subgroups, and their universal invariants. Some invariant and partially invariant exact solutions are obtained, and their physical meaning is explained.     

Asymptotic dynamics of the one‐dimensional attraction–repulsion Keller–Segel model

The asymptotic behavior of the attraction–repulsion Keller–Segel model in one dimension is studied in this paper. The global existence of classical solutions and nonconstant stationary solutions of the attraction–repulsion Keller–Segel model in one dimension were previously established by Liu and Wang (2012), which, however, only provided a time‐dependent bound for solutions. In this paper, we improve the results of Liu and Wang (2012) by deriving a uniform‐in‐time bound for solutions and furthermore prove that the model possesses a global attractor. For a special case where the attractive and repulsive chemical signals have the same degradation rate, we show that the solution converges to a stationary solution algebraically as time tends to infinity if the attraction dominates. Copyright © 2014 John Wiley & Sons, Ltd.     

High dimensional model representation for stochastic finite element analysis

This paper presents a generic high dimensional model representation (HDMR) method for approximating the system response in terms of functions of lower dimensions. The proposed approach, which has been previously applied for problems dealing only with random variables, is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modelled as random fields. The formulation of the extended HDMR is similar to the spectral stochastic finite element method in the sense that both of them utilize Karhunen–Loève expansion to represent the input, and lower-order expansion to represent the output. The method involves lower dimensional HDMR approximation of the system response, response surface generation of HDMR component functions, and Monte Carlo simulation. Each of the low order terms in HDMR is sub-dimensional, but they are not necessarily translating to low degree polynomials. It is an efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the approximate form of the system response is defined, the failure probability can be obtained by statistical simulation. The proposed approach decouples the finite element computations and stochastic computations, and consecutively the finite element code can be treated as a black box, as in the case of a commercial software. Numerical examples are used to illustrate the features of the extended HDMR and to compare its performance with full scale simulation.     

Explicit solutions of the one‐dimensional Vlasov–Poisson system with infinite mass

A collisionless plasma is modelled by the Vlasov–Poisson system in one dimension. A fixed background of positive charge, dependent only upon velocity, is assumed and the situation in which the mobile negative ions balance the positive charge as |x| → ∞ is considered. Thus, the total positive charge and the total negative charge are infinite. In this paper, the charge density of the system is shown to be compactly supported. More importantly, both the electric field and the number density are determined explicitly for large values of |x|. Copyright © 2007 John Wiley & Sons, Ltd.     

The asymptotic behaviour of global smooth solutions to the multi‐dimensional hydrodynamic model for semiconductors

We establish the global existence of smooth solutions to the Cauchy problem for the multi‐dimensional hydrodynamic model for semiconductors, provided that the initial data are perturbations of a given stationary solutions, and prove that the resulting evolutionary solution converges asymptotically in time to the stationary solution exponentially fast. Copyright © 2003 John Wiley & Sons, Ltd.     

Exact solutions of the phase transition problem in a one-dimensional model case

Solutions of the phase transition problem are obtained explicitly in a one-dimensional model case for both positive and zero surface tension coefficients. The dependence of the equilibrium states on the parameters of the problem is investigated.     

Exact invariant solutions for a class of energy-transport models of semiconductors in the two dimensional stationary case

The symmetry classification of a class of energy-transport models for semiconductors is performed in the two dimensional stationary case. Reduced systems and examples of exact invariant solutions are shown.     

Symmetry of the point spectrum of infinite dimensional hamiltonian operators and its applications

This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp(H) = σp(A) ∪σp1(-A*). Using the characteristic of the set σp1(-A*), we divide the point spectrum σp(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1(-A*) and one part of σp(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.