广义Reed-Solomon 码的深洞

深洞在广义Reed-Solomon 码的译码中发挥重要的作用. 最近, Wu 和Hong 通过循环码对于标准Reed-Solomon 码发现了一类新的深洞. 本文给出一个简洁的方法, 对于一般广义Reed-Solomon 码给出新的一类深洞. 特别地, 对于标准Reed-Solomon 码, 我们得到了Wu 和Hong 给出的深洞. 对于广义Reed-Solomon 码GRS_k(F_q,D), Li 和Wan 研究和刻画了k+1 次多项式定义的深洞, 并且指出这个问题归结为在有限域中的子集和问题. 在偶特征的情形下, 利用他们的方法, 我们对于一些特殊的Reed-Solomon 码得到了更多一类新的深洞. 此外, 我们研究扩展Reed-Solomon 码(即赋值集合为D=F_q) k+2 次多项式定义的深洞, 并且证明没有k+2次多项式定义的深洞.     

Numerical solution of fuzzy differential equations under generalized differentiability

In this paper, we interpret a fuzzy differential equation by using the strongly generalized differentiability concept. Utilizing the Generalized Characterization Theorem, we investigate the problem of finding a numerical approximation of solutions. Then we show that any suitable numerical method for ODEs can be applied to solve numerically fuzzy differential equations under generalized differentiability. The generalized Euler approximation method is implemented and its error analysis, which guarantees pointwise convergence, is given. The method’s applicability is illustrated by solving a linear first-order fuzzy differential equation.     

Moments of the generalized hyperbolic distribution

In this paper we demonstrate a recursive method for obtaining the moments of the generalized hyperbolic distribution. The method is readily programmable for numerical evaluation of moments. For low order moments we also give an alternative derivation of the moments of the generalized hyperbolic distribution. The expressions given for these moments may be used to obtain moments for special cases such as the hyperbolic and normal inverse Gaussian distributions. Moments for limiting cases such as the skew hyperbolic t and variance gamma distributions can be found using the same approach.     

A geometrical treatment of singular trajectories

The nonlinear field equations often arising in geometrodynamical theories of matter generally exhibit nonremovable singularities. Assuming that the field equations are either (1) analytic, or (2) structurally stable, we show that the Christoffel symbols of the second kind have certain properties (4), (9). The singularities are such that wedge-shaped sets can be found containing n-parameter families of trajectories emanating from a given point on a singularity. In particular cases where the singularity is an isolated point, entire neighborhoods have been found, composed of trajectories. The latter situation is especially convenient in that a generalized tangent space can be defined, in which various manipulations of other field equations can be done (separation of variables, potential theory) and for which an exponential map can be set up. We show that under (4) the geodesies (trajectories) vary continuously with respect to limit tangent vectors at the singularity. Under a slightly stronger condition (23), trajectories vary differentiably with respect to limit tangent vectors. The limit tangent vectors are the elements of the generalized tangent space.     

The Tits Alternative for Groups Defined by Periodic Paired Relations

The class of groups defined by periodic paired relations, as introduced by Vinberg, includes the generalized triangle groups, the generalized tetrahedron groups, and the generalized Coxeter groups. We observe that any group defined by periodic paired relations Γ can be realized as a so-called “Pride group”. Using results of Howie and Kopteva we give necessary and sufficient conditions for this Pride group to be non-spherical. Under such conditions, we show that Γ satisfies the Tits alternative.

Communicated by A. Olshanskiy     

Automorphism groups of some algebras

The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra A _m,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n). This work was supported by 2007 Research fund of Hanyang University     

Adaptive Smoothing Method,Deterministically Computable Generalized Jacobians,and the Newton Method

In this note, we show that a well-known integral method, which was used by Mayne and Polak to compute an -subgradient, can be exploited to compute deterministically an element of the plenary hull of the Clarke generalized Jacobian of a locally Lipschitz mapping regardless of its structure. In particular, we show that, when a locally Lipschitz mapping is piecewise smooth, we are able to compute deterministically an element of the Clarke generalized Jacobian by the adaptive smoothing method. Consequently, we show that the Newton method based on the plenary hull of the Clarke generalized Jacobian can be implemented in a deterministic way for solving Lipschitz nonsmooth equations.     

Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

Existence Problem of a Generalized Universal Covering Space

In this paper we study the existence problem of a generalized universal covering space of a given digital space, which can be essentially used in classifying digital spaces. To be specific, we show a method to establish a generalized universal covering space of a simple closed k-curve and prove that a digital wedge need not have a generalized universal covering space.     

广义数及其应用(Ⅰ)

本文在文献[2]的基础上引进广义数系统,定义了以广义数为基础的广义函数(本质不同于L.Schwartz的分布),研究了勒贝格积分的推广,将这理论应用于分布,便得到对σ函数等的自然理解,对广义数应用于量子场论中,也作了一些尝试性的工作。     

Exact peakon solutions given by the generalized hyperbolic functions for some nonlinear wave equations

In 1993, Camassa and Holm drived a shallow water equation and found that this equation has a peakon solution with the form $\phi(\xi)=ce^{-|\xi|}$. In this paper, we show that three nonlinear wave systems have peakon solutions which needs to be represented as generalized hyperbolic functions. For the existence of these solutions, some constraint parameter conditions are derived.     

Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13–29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144:13–29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green’s theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.     

Finite termination of a Newton-type algorithm for a class of affine variational inequality problems

Many optimization problems can be reformulated as a system of equations. One may use the generalized Newton method or the smoothing Newton method to solve the reformulated equations so that a solution of the original problem can be found. Such methods have been powerful tools to solve many optimization problems in the literature. In this paper, we propose a Newton-type algorithm for solving a class of monotone affine variational inequality problems (AVIPs for short). In the proposed algorithm, the techniques based on both the generalized Newton method and the smoothing Newton method are used. In particular, we show that the algorithm can find an exact solution of the AVIP in a finite number of iterations under an assumption that the solution set of the AVIP is nonempty. Preliminary numerical results are reported.     

Generalized Thrackle Drawings of Non-bipartite Graphs

A graph drawing is called a generalized thrackle if every pair of edges meets an odd number of times. In a previous paper, we showed that a bipartite graph G can be drawn as a generalized thrackle on an oriented closed surface M if and only if G can be embedded in M. In this paper, we use Lins’ notion of a parity embedding and show that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.     

On Several Types of Basic Constraint Qualifications via Coderivatives for Generalized Equations

In this paper, we study several types of basic constraint qualifications in terms of Clarke/Fréchet coderivatives for generalized equations. Several necessary and/or sufficient conditions are given to ensure these constraint qualifications. It is proved that basic constraint qualification and strong basic constraint qualification for convex generalized equations can be obtained by these constraint qualifications, and the existing results on constraint qualifications for the inequality system can be deduced from the given conditions in this paper. The main work of this paper is an extension of the study on constraint qualifications from inequality systems to generalized equations.     

Riemann problem with delta initial data for the isentropic relativistic Chaplygin Euler equations

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the isentropic relativistic Chaplygin Euler equations. Under suitably generalized Rankine–Hugoniot relation and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, it can be found that the solutions constructed here are stable for the perturbation of the initial data.     

Explicit analytical solutions of the generalized Burger and Burger–Fisher equations by homotopy perturbation method

In this article, we apply the homotopy perturbation method (HPM) to obtain approximate analytical solutions of the generalized Burger and Burger‐Fisher (B–F) equations. Several numerical examples are given to illustrate the efficiency of the HPM. Comparison of the result obtained by the present method with exact solution reveals that the accuracy and fast convergence of the new method. It is predicted that the HPM can be found wide application in engineering problems. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method

The generalized Taylor theorem is the foundation of the homotopy analysis method proposed by Liao. This theorem is interesting but hard to understand from the mathematical point of view. Especially, there is a key parameter h whose meaning is still unknown. In the paper, we derive the generalized Taylor theorem from a usual way, that is, we prove that the generalized Taylor expansion is equivalent to a different representation of the usual Taylor expansion at different points. Therefore the meaning of the auxiliary parameter h is clarified. These results give a reasonable explanation of the parameter h and uncover the essence of the generalized Taylor theorem from which we can deeply understand the homotopy analysis method. Through the detailed analysis of some examples, we compare the series solution at the different points with the generalized Taylor series solution obtained by the homotopy analysis method.     

Fast inexact subspace iteration for generalized eigenvalue problems with spectral transformation

We study inexact subspace iteration for solving generalized non-Hermitian eigenvalue problems with spectral transformation, with focus on a few strategies that help accelerate preconditioned iterative solution of the linear systems of equations arising in this context. We provide new insights into a special type of preconditioner with “tuning” that has been studied for this algorithm applied to standard eigenvalue problems. Specifically, we propose an alternative way to use the tuned preconditioner to achieve similar performance for generalized problems, and we show that these performance improvements can also be obtained by solving an inexpensive least squares problem. In addition, we show that the cost of iterative solution of the linear systems can be further reduced by using deflation of converged Schur vectors, special starting vectors constructed from previously solved linear systems, and iterative linear solvers with subspace recycling. The effectiveness of these techniques is demonstrated by numerical experiments.     

Symmetric Solutions of Nonlinear Fractional Integral Equations via a New Fixed Point Theorem under FG-Contractive Condition

We set up the existence of a symmetric outcome of a system of simultaneous nonlinear fractional integral equations, that arises in motion of water wave on smooth surface, with the help of a common fixed point theorem satisfying a generalized FG-contractive condition. To accomplish this, we introduce first the concept of generalized FG-contractive condition for two pairs of self-mappings in a complete metric space and then we establish requisites for common fixed point results for weakly compatible mappings followed by a suitable example.