关于Hurwitz Zeta－函数

本文的主要目的是利用解析方法及三角和的估计给出ＨｕｒｗｉｔｚＺｅｔａ－函数的二次积分均值的一个很强的渐近公式。     

Rolling-ball method for estimating the boundary of the support of a point-process intensity

We suggest a generalisation of the convex-hull method, or ‘DEA’ approach, for estimating the boundary or frontier of the support of a point cloud. Figuratively, our method involves rolling a ball around the cloud, and using the equilibrium positions of the ball to define an estimator of the envelope of the point cloud. Constructively, we use these ideas to remove lines from a triangulation of the points, and thereby compute a generalised form of a convex hull. The radius of the ball acts as a smoothing parameter, with the convex-hull estimator being obtained by taking the radius to be infinite. Unlike the convex-hull approach, however, our method applies to quite general frontiers, which may be neither convex nor concave. It brings to these contexts the attractive features of the convex hull: simplicity of concept, rotation-invariance, and ready extension to higher dimensions. It admits bias corrections, which we describe and illustrate through implementation.     

Integrating partial optimization with scatter search for solving bi-criteria {0, 1}-knapsack problems

This paper deals with the problem of inaccuracy of the solutions generated by metaheuristic approaches for combinatorial optimization bi-criteria {0, 1}-knapsack problems. A hybrid approach which combines systematic and heuristic searches is proposed to reduce that inaccuracy in the context of a scatter search method. The components of this method are used to determine regions in the decision space to be systematically searched.     

LSQR iterative common symmetric solutions to matrix equations AXB = E and CXD = F

A new matrix based iterative method is presented to compute common symmetric solution or common symmetric least-squares solution of the pair of matrix equations AXB = E and CXD = F. By this iterative method, for any initial matrix X₀, a solution X^∗ can be obtained within finite iteration steps if exact arithmetic was used, and the solution X^∗ with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. In addition, the unique nearest common symmetric solution or common symmetric least-squares solution to given matrix in Frobenius norm can be obtained by first finding the minimum Frobenius norm common symmetric solution or common symmetric least-squares solution of the new pair of matrix equations. The given numerical examples show that the matrix based iterative method proposed in this paper has faster convergence than the iterative methods proposed in [1] and [2] to solve the same problems.     

Positive definite solutions of the nonlinear matrix equation X + AXA = Q (q > 0)

In this paper, the nonlinear matrix equation X + A^∗X^qA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods.     

Symmetry reductions and explicit solutions of a (3 + 1)-dimensional PDE

The symmetry of the (3 + 1)-dimensional partial differential equation has been derived via a direct symmetry method and proved to be infinite dimensional non-Virasoro type symmetry algebra. Many kinds of symmetry reductions have been obtained, including the (2 + 1)-dimensional ANNV equation and breaking soliton equation. And some new soliton solutions and complex solutions are obtained due to the Riccati equation method and symbolic computation.     

Explicit solutions of (2 + 1)-dimensional nonlinear KP-BBM equation by using Exp-function method

This paper is devoted to studying the (2 + 1)-dimensional KP-BBM wave equation. Exp-function method is used to conduct the analysis. The generalized solitary solutions, periodic solutions and other exact solutions for the (2 + 1)-dimensional KP-BBM wave equation are obtained via this method with the aid of symbolic computational system. It is also shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

AC 0-collocation-like method for two-point boundary value problems

Summary A method of a collocation type based onC ⁰-piecewise polynomial spaces is presented for a two-point boundary value problem of the second order. The method has an optimal order of convergence under smoothness requirements on the exact solution which are weaker than forC ¹-collocation methods. If the differential operator is symmetric, a modification of this method leads to a symmetric system of linear equations. It is shown that if the collocation solution is a piecewise polynomial of degree not greater thanr, the method is stable and convergent with orderh ^r inH ¹-norm. A similar symmetric modification forC ⁰-colloction-finite element method [7] is also obtained. Superconvergence at the nodes is established.     

A compact C0 discontinuous Galerkin method for Kirchhoff plates

A compact C⁰ discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C⁰ finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C⁰ interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H¹‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015     

Elements of reduced trace 0

Every elementr of reduced trace 0 in a simple finite dimensional algebraR is a sum of at most 2 commutators. IfR innot a division ring thenr is a commutator, unlessr is a scalar (in which case char(R)≠0). The method of proof provides a generic division algebra of transcendence degreen ²−1.     

Alternating projections in CAT(0) spaces

By using recently developed theory which extends the idea of weak convergence into CAT(0) space we prove the convergence of the alternating projection method for convex closed subsets of a CAT(0) space. Given the right notion of weak convergence it turns out that the generalization of the well-known results in Hilbert spaces is straightforward and allows the use of the method in a nonlinear setting. As an application, we use the alternating projection method to minimize convex functionals on a CAT(0) space.     

A variety of exact travelling wave solutions for the (2 + 1)-dimensional Boiti-Leon-Pempinelli equation

In this paper the (2 + 1)-dimensional Boiti-Leon-Pempinelli (BLP) equation will be studied. The tanh-coth method will be used to obtain exact travelling wave solutions for this equation. The Exp-function method will also be applied to the BLP equation to derive a new variety of travelling wave solutions with distinct physical structures.     

Decomposition and Linearization for 0-1 Quadratic Programming

This paper presents a general decomposition method to compute bounds for constrained 0-1 quadratic programming. The best decomposition is found by using a Lagrangian decomposition of the problem. Moreover, in its simplest version this method is proved to give at least the bound obtained by the LP-relaxation of a non-trivial linearization. To illustrate this point, some computational results are given for the 0-1 quadratic knapsack problem.     

Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter

In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation σ field and the tail σ field of the signal, in the stationary filtering problem. This property can be viewed as the permissibility of the interchange of the order of the operations of maximum and countable intersection for certain σ-fields. Under suitable conditions, it is shown that the above property is equivalent to various desirable properties of the filter such as

(a) uniqueness of invariant measure for the signal,

(b) uniqueness of invariant measure for the pair (signal, filter),

(c) a finite memory property of the filter,

(d) a property of finite time dependence between the signal and observation σ fields and

(e) asymptotic stability of the filter.

Previous works on the asymptotic stability of the filter for a variety of filtering models then identify a rich class of filtering problems for which the above equivalent properties hold.     

A continuation method for linear complementarity problems with P 0 matrix

In this article, we propose a new continuation method for solving the linear complementarity problem (LCP). The method solves one system of linear equations and carries out only a one-line search at each iteration. The continuation method is based on a modified smoothing function. The existence and continuity of a smooth path for solving the LCP with a P ₀ matrix are discussed. We investigate the boundedness of the iteration sequence generated by our continuation method under the assumption that the solution set of the LCP is nonempty and bounded. It is shown to converge to an LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution under suitable assumption. In addition, some numerical results are also reported in this article.     

A reduced local C0 discontinuous galerkin method for Kirchhoff plates

We propose and analyze a reduced local C⁰ discontinuous Galerkin (reduced LCDG) method with minimal penalization for Kirchhoff plate bending problems. The resulting linear system of the method can be solved efficiently by Gaussian elimination. Based on the key observation that the reduced LCDG method can be viewed as the localization of Hellan–Herrmann–Johnson method, we are inspired to use the techniques for dealing with the latter method to derive the well‐posedness and a priori error estimates of the reduced LCDG method. With the help of Zienkiewicz–Guzmán–Neilan element space, the a posteriori error analysis is also developed. Some numerical results are provided to demonstrate the theoretical estimates.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1902–1930, 2014     

Epsilon substitution method for [Π0 1, Π0 1]-FIX

We formulate epsilon substitution method for a theory [Π⁰₁, Π⁰₁]-FIX for two steps non-monotonic Π⁰₁ inductive definitions. Then we give a termination proof of the H-processes based on Ackermann [1].     

N-soliton solutions for shallow water waves equations in (1 + 1) and (2 + 1) dimensions

A variety of shallow water waves equations in (1 + 1) and (2 + 1) dimensions are investigated. We first show that these models are completely integrable. We next determine multiple-soliton solutions for each equation. The simplified Hirota’s bilinear method developed by Hereman will be employed to achieve this goal. A comparison between dispersion relations and the phase shifts will be conducted. (But possess the same coefficients for the polynomials of exponentials.)     

The interface between regions where u<0 andu > 0in the porous medium equation

We consider the porous medium equation with sign changes. In particular this equation describes the mixing of fresh and salt groundwater due to mechanical dispersion. The unknown function u, which denotes the velocity of the fluids, may take positive as well as negative values. Our main result is the following : under certain monotonicity hypotheses on the initial function, there exists a time T> 0 after which the regions where u < 0 and u > 0 are separated by an interface x = ζ(t) such that ζ is continuously differentiable on [T,∞]. The method of proof is based on a priori estimates for solutions of regularized problems and for their level lines     

On orthogonal collocation by the zeros of orthogonal 0-interpolants

The orthogonal collocation method is quite prominent among the collocation methods that determine approximate solutions of boundary value problems. This method involves Gaussian knots as collocation points and, in general, has fourth order convergence. Here, we discuss a similar method in which the knots are based on pre-assigned and free zeros of polynomials that arise in orthogonal 0-interpolants. Some numerical results demonstrate an advantage of the proposed collocation points over the Gaussian knots. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)