Difference discrete connection and curvature on cubic lattice

In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice. One of the definitions can be extended to the case over the random lattice. We also discuss the relation between our approach and the lattice gauge theory and apply to the discrete integrable systems. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

离散外微分与计算电磁学

计算电磁学的核心之一是数值求解Maxwell方程组.适当的离散方式是保证结果能真实反映物理现象的关键.为了在离散的过程中保持该方程组的几何性质,我们建立了基于棱柱网格的系数为R的格点规范理论,其离散曲率满足相应的Bianchi恒等式.通过适当定义离散微分形式之间的内积和棱柱网格上的Hodge星算子,我们由离散变分导出源方程和连续性方程,和Bianchi恒等式一起称为真空中的离散Maxwell方程组.这组方程是内蕴的,并具有规范不变性.     

Dependence on parameters for discrete second-order boundary value problems

We investigate the dependence on parameters for second-order difference equations with two-point boundary value conditions using a variational method in case when the corresponding Euler action functional is coercive. Some applications for discrete Emden–Fowler equation are also given.     

Cauchy problem for integrable discrete equations on quad-graphs

Initial value problems for the integrable discrete equations on quad-graphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete KdV equation an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice are discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.Dedicated to S. P. Novikov on his 65 birthdayOn leave from Landau Institute for Theoretical Physics, Chernogolovka, Russia.     

On a discrete version of alexandrov’s projection theorem

In this paper, we consider a discrete version of Aleksandrov＇s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice＇s y-coordinates＇ absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov＇s projection theorem which was proposed by Gardner, Gronchi and Zong in 2005.     

Zd中离散分形指标的线性不变性质

研究欧几里得格 Zd 内离散分形指标的线性不变性质 ,即证明了上、下离散质量维数的线性不变性质 ,离散 Hausdorff维数的线性不变性质以及离散填充维数的线性不变性质 .     

Discrete linear evolution equations in a finite lattice

We present a general and effective method, known as the Fokas method, to solve an arbitrary discrete linear evolution equation posed in a finite lattice. The method is based on the simultaneous analysis of both parts of Lax pair, as well as the global relation that involves initial and boundary values. We show that, as in the continuum problems, the method can be applied effectively to solve general linear differential-difference equations in a finite lattice. In particular, we demonstrate the method by addressing a number of significant examples and we discuss the continuum limits of the solution and the boundary values.     

Arithmetic on superelliptic curves

This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form which has only one point at infinity. Divisors are represented as ideals, and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time. An algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field.

     

Orthogonal Polynomials on a Bi-lattice

We investigate generalizations of the Charlier and the Meixner polynomials on the lattice ? and on the shifted lattice ?+1???. We combine both lattices to obtain the bi-lattice ???(?+1???) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients that satisfy a nonlinear system of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlevé equation.     

Compactness properties of operators dominated by AM-compact operators

We study several properties about the problem of domination in the class of positive AM-compact operators, and we obtain some interesting consequences on positive compact operators. Also, we give a sufficient condition under which a Banach lattice is discrete.

     

On Discrete Painlevé Equations Associated with the Lattice KdV Systems and the Painlevé VI Equation

A new integrable nonautonomous nonlinear ordinary difference equation is presented that can be considered to be a discrete analogue of the Painlevé V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial differential equations of Korteweg–de Vries (KdV) type. The new equation, which is referred to as generalized discrete Painlevé equation (GDP), contains various "discrete Painlevé equations" as subcases for special values/limits of the parameters, some of which have already been given in the literature. The general solution of the GDP can be expressed in terms of Painlevé VI (PVI) transcendents. In fact, continuous PVI emerges as the equation obeyed by the solutions of the discrete equation in terms of the lattice parameters rather than the lattice variables that label the lattice sites. We show that the bilinear form of PVI is embedded naturally in the lattice systems leading to the GDP. Further results include the establishment of Bäcklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.     

Spectral properties of singular discrete linear Hamiltonian systems

This paper is concerned with spectral properties of singular discrete linear Hamiltonian systems. It is shown that properties of the essential spectrum of each self-adjoint subspace extension (SSE) of the corresponding minimal subspace are independent of the values of the coefficients of the system on any finite subinterval. The analyticity of the Weyl function is studied by employing the Schwarz reflection principle for the system in the limit point case. Based on the above work, several sufficient conditions are obtained for each SSE to have no essential spectrum points in an interval of the real line in the strong limit point case, and then a sufficient condition for the essential spectrum to be bounded from below (above) and a sufficient condition for the pure discrete spectrum are presented, respectively. As a direct consequence, the related spectral properties of singular higher order symmetric vector difference expressions are given.     

Constructing exact solutions to discrete systems with the trial function method

Based on the homogenous balance method and the trial function method, several trial function methods composed of exponential functions are proposed and applied to nonlinear discrete systems. With the.help of symbolic computation system, the new exact solitary wave solutions to discrete nonlinear mKdV lattice equation, discrete nonlinear （2 ＋ 1） dimensional Toda lattice equation, Ablowitz-Ladik-lattice system are constructed.The method is of significance to seek exact solitary wave solutions to other nonlinear discrete systems.     

Helmholtz's inverse problem of the discrete calculus of variations

We derive the discrete version of the classical Helmholtz's condition. Precisely, we state a theorem characterizing second-order finite difference equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.     

On the moments and distribution of discrete Choquet integrals from continuous distributions

We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means, ordered weighted averaging functions, and lattice polynomial functions as particular cases, our results encompass the corresponding results for these aggregation functions. After detailing the results obtained in [J.-L. Marichal, I. Kojadinovic, Distribution functions of linear combinations of lattice polynomials from the uniform distribution, Statistics & Probability Letters 78 (2008) 985–991] in the uniform case, we present results for the standard exponential case, show how approximations of the moments can be obtained for other continuous distributions such as the standard normal, and elaborate on the asymptotic distribution of the Choquet integral. The results presented in this work can be used to improve the interpretation of discrete Choquet integrals when employed as aggregation functions.     

A discrete “three-particle” Schrödinger operator in the Hubbard model

In the space L ₂(T ^ν ×T ^ν ), where T ^ν is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrödinger operator ? = H₀ + H₁ + H₂, where H₀ is the operator of multiplication by a function and H₁ and H₂ are partial integral operators. We prove several theorems concerning the essential spectrum of ?. We study the discrete and essential spectra of the Hamiltonians H^t and h arising in the Hubbard model on the three-dimensional lattice.     

A perturbation theory for the discrete harmonic oscillator equation

This work deals with the existence, uniqueness and stability of solutions for the semilinear discrete harmonic oscillator equation on Banach spaces by using recent characterization of maximal regularity for a best difference approximation of the discrete harmonic oscillator equation.     

Integrable Seven-Point Discrete Equations and Second-Order Evolution Chains

We consider differential–difference equations defining continuous symmetries for discrete equations on a triangular lattice. We show that a certain combination of continuous flows can be represented as a secondorder scalar evolution chain. We illustrate the general construction with a set of examples including an analogue of the elliptic Yamilov chain.     

A NEW INTEGRABLE SYMPLECTIC MAP ASSOCIATED WITH A DISCRETE MATRIX SPECTRAL PROBLEM

A hierarchy of lattice soliton equations is derived from a discrete matrix spectral problem. It is shown that the resulting lattice soliton equations are all discrete Liouville integrable systems. A new integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonli-nearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resulting lattice soliton equations.     

Explicit solution and Darboux transformation for a new discrete integrable soliton hierarchy with 4×4 Lax pairs

The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd.