一个与Toda格相关的微分差分方程族及其反散射变换

By combined power evolution laws of the spectral parameter and the initial constants of integration,a new differential-difference hierarchy is presented from the Toda spectral problem.The hierarchy contains the classic Toda lattice equation,the nonisospectral Toda lattice equation and the mixed Toda lattice equation as reduced cases.The evolution of the scattering data in the inverse scattering transform is analyzed in detail and exact soliton solutions are computed through the corresponding inverse scattering transform.     

幂格与商格的关系

该文在文［1］中幂格概念的基础上,得到了幂格的一个充要条件,给出了正则幂格、相对幂格的概念,并讨论了商格与正则幂格、相对幂格的关系.〖HT5”H〗关键词:〖HT5”SS〗格;幂格;商格;正则幂格;相对幂格.     

The rapidly decreasing solution of the Cauchy problem for the Toda lattice

We prove the existence of a rapidly decreasing solution of the Cauchy problem for the Toda lattice. We find a class of initial data guaranteeing the existence of such a solution.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 5–12, January, 2005.     

Delaunay Transformations of a Delaunay Polytope

Let P be a Delaunay polytope in ⁿ . Let T(P) denote the set of affine bijections f of ⁿ for which f (P) is again a Delaunay polytope. The relation: fg if f, g differ by an orthogonal transformation and/or a translation is an equivalence relation on T(P). We show that the dimension (in the topological sense) of the quotient set T(P)/ coincides with another parameter of P, namely, with its rank.Let V denote the set of vertices of P and let dp denote the distance on V defined by dp(u, v)=u–v ² for u, vV. Assouad [1] shows that dp belongs to the cone |V|:={d | _u,vV b _u b _v d(u,v) 0 for b ^V with _uV b _u = 1}. Then, the rank of P is defined as the dimension of the smallest face of the cone |V| that contains dp. AMS Subject Classification (1991): 11H06, 52C07.     

Gorenstein simplices with a given δ-polynomial

To classify the lattice polytopes with a given $\delta$-polynomial is an important open problem in Ehrhart theory. A complete classification of the Gorenstein simplices whose normalized volumes are prime integers is known. In particular, their $\delta$-polynomials are of the form $1+t^k+?+t^{(v?1)k}$, where $k$ and $v$ are positive integers. In the present paper, a complete classification of the Gorenstein simplices with the above $\delta$-polynomials will be performed, when $v$ is either $p^2$ or $pq$, where $p$ and $q$ are prime integers with $p\ne q$. Moreover, we consider the number of Gorenstein simplices, up to unimodular equivalence, with the expected $\delta$-polynomial.     

Reduction and realization in toda and volterra

We construct a new symplectic, bi-Hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-Hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems.      

Towards an Effectivisation of the Riemann Theorem

Let Q be a connected and simply connected domain on the Riemann sphere, not coinciding with the Riemann sphere and with the whole complex plane ℂ. Then, according to the Riemann Theorem, there exists a conformal bijection between Q and the exterior of the unit disk. In this paper, we find an explicit form of this map for a broad class of domains with analytic boundaries. Communicated by M. A. Shubin (Moscow) Mathematics Subject Classifications (2000): 30Cxx, 37Kxx.     

A Kind of Discretization of the KdV Hierarchy

We propose a method for introducing higher-order terms in the potential expansion in order to study the continuum limits of the Toda hierarchy. These higher-order terms are differential polynomials in the lower-order terms. This type of potential expansion allows using fewer equations in the Toda hierarchy to recover the KdV hierarchy by the so-called recombination method. We show that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend toward the corresponding objects of the KdV hierarchy in the continuum limit. This method gives a kind of discretization of the KdV hierarchy.     

Generalized dressing method for the extended two-dimensional Toda lattice hierarchy and its reductions

In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed in Liu-Lin-Jin-Zeng (2009). General Casoratian determinant solutions for this hierarchy are obtained. In particular, explicit solutions of soliton-type are formulated by using the τ-function in the form of exponential functions. The periodic reduction and one-dimensional reduction of ex2DTLH are studied by finding the constraints. Many reduced systems are shown, including the pe...     

Weighted lattice polynomials of independent random variables

We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems.     

幂格

本文引入了幂格的概念并讨论了其相关性质。