A reciprocity relation for WP-Bailey pairs

We derive a new general transformation for WP-Bailey pairs by considering a certain limiting case of a WP-Bailey chain previously found by the authors, and examine several consequences of this new transformation. These consequences include new summation formulae involving WP-Bailey pairs. Other consequences include new proofs of some classical identities due to Jacobi, Ramanujan and others, and indeed extend these identities to identities involving particular specializations of arbitrary WP-Bailey pairs.     

A pair of difference differential equations of Euler-Cauchy type

We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.

     

Some new transformations for Bailey pairs and WP-Bailey pairs

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.     

Algebraic properties of classes of path ideals of posets

We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen–Macaulay in terms of the underlying poset. Moreover, monomial ideals, which arise in algebraic statistics from the Luce-decomposable model and the ascending model, can be viewed as path ideals of certain posets. We study invariants of these so-called Luce-decomposable monomial ideals and ascending ideals for diamond posets and products of chains. In particular, for these classes of posets, we explicitly compute their Krull dimension, their projective dimension, their Castelnuovo–Mumford regularity and their Betti numbers.     

一类特殊非齐次树上马氏链的若干强大数定律

首先给出了一类特殊非齐次树上可数状态马氏链的局部收敛定理,作为推论,得到了此类树上可数状态马氏链关于状态与状态序偶出现频率的若干极限性质,最后得到了这类特殊非齐次树上有限状态马氏链关于状态与状态序偶出现频率的强大数定律.     

The weight distributions of a class of cyclic codes

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ma et al. (2011) [14], Ding et al. (2011) [5], Wang et al. (2011) [20]. In this paper we provide a slightly different approach toward the general problem and use it to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums to transform the problem of finding the weight distribution into a problem of evaluating certain character sums over finite fields, which on the special case is related with counting the number of points on some elliptic curves over finite fields. Other cases are also possible by this method.     

Some identities between basic hypergeometric series deriving from a new Bailey-type transformation

We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.     

Projection iterative methods for extended general variational inequalities

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems.     

On the convex hull of the union of certain polyhedra

We consider a finite collection of polyhedra whose defining linear systems differ only in their right hand sides. Jeroslow [5] and Blair [4] specified conditions under which the convex hull of the union of these polyhedra is defined by a system whose left hand side is the common lefthand side of the individual systems, and whose right hand side is a convex combination of the individual right hand sides. We give a new sufficient condition for this property to hold, which is often easier to recognize. In particular, we show that the condition is satisfied for polyhedra whose defining systems involve the node-arc incidence matrices of directed graphs, with certain righ hand sides. We also derive as a special case the compact linear characterization of the two terminal Steiner tree polytope given in Ball, Liu and Pulleyblank [3].     

A General Law of Precise Asymptotics for the Complete Moment Convergence

The authors achieve a general law of precise asymptotics for a new kind of complete moment convergence of i.i.d, random variables, which includes complete convergence as a special case. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in studies of complete convergence. This extends and generalizes the corresponding results of Liu and Lin in 2006.     

Continuous refinement equations and subdivision

This paper is concerned with the study of a general class of functional equations covering as special cases the relation which defines theup-function as well as equations which arise in multiresolution analysis for wavelet construction. We discuss various basic properties of solutions to these functional equations such as regularity, polynomial containment within the space spanned by their integer shifts and their computability by subdivision algorithms.     

On the structure of random plane-oriented recursive trees and their branches

This paper is an investigation of the structural properties of random plane-oriented recursive trees and their branches. We begin by an enumeration of these trees and some general properties related to the outdegrees of nodes. Using generalized Pólya urn models we study the exact and limiting distributions of the size and the number of leaves in the branches of the tree. The exact distribution for the leaves in the branches is given by formulas involving second-order Eulerian numbers. A martingale central limit theorem for a linear combination of the number of leaves and the number of internal nodes is derived. The distribution of that linear combination is a mixture of normals with a beta distribution as its mixing density. The martingale central limit theorem allows easy determination of the limit laws governing the leaves in the branches. Furthermore, the asymptotic joint distribution of the number of nodes of outdegree 0, 1 and 2 is shown to be trivariate normal. © 1993 John Wiley & Sons, Inc.     

On the solution of a class of integral equations

In this paper we consider a class of Fredholm integral equations of the first kind which arise in a large number of problems in applied mathematics. Although only certain special cases of the equations can be solved exactly, it is shown that a constructive method can be developed for reformulating the equations as Fredholm integral equations of the second kind. This approach will be seen to cover and bring together the large number of isolated cases of the equations which have appeared in the literature. Several examples are given to illustrate the general method.     

Algebras in the Positive Cone of po-Groups

We systemize a number of algebras that are especially known in the field of quantum structures and that in particular arise from the positive cones of partially ordered groups. Generalized effect algebras, generalized difference posets, cone algebras, commutative BCK-algebras with the relative cancellation property, and positive minimal clans are included in the text.All these structures are conveniently characterizable as special cases of generalized pseudoeffect algebras, which we introduced in a previous paper. We establish the exact relations between all mentioned structures, thereby adding new structures whenever necessary to make the scheme of order complete.Generalized pseudoeffect algebras were under certain conditions proved to be representable by means of a po-group. From this fact, we will easily establish representation theorems for all of the structures included in discussion.     

Steiner minimal trees for bar waves

A Steiner minimal tree (SMT) for a set of pointsP in the plane is a shortest network interconnectingP. The construction of a SMT for a general setP is known to be anNP-complete problem. Recently, SMTs have been constructed for special setsP such as ladders, splitting trees, zigzag lines and co-circular points. In this paper we study SMTs for a wide class of point-sets called mild bar wave. We show that a SMT for a mild bar wave must assume a special form, thus the number of trees needed to be inspected is greatly reduced. Furthermore if a mild bar wave is also a mild rectangular wave, then we produce a Steiner tree constructible in linear time whose length can exceed that of a SMT by an amount bounded by the difference in heights of the two endpoints of the rectangular wave, thus independent of the number of points. When a rectangular wave satisfies some other conditions (including ladders as special cases), then the Steiner tree we produced is indeed a SMT.     

Stationary focusing mean-field games

We consider stationary viscous mean-field games (MFG) systems in the case of local, decreasing and unbounded coupling. These systems arise in ergodic MFG theory and describe Nash equilibria of games with a large number of agents aiming at aggregation. We show how the dimension of the state space, the behavior of the coupling, and the Hamiltonian at infinity affect the existence and nonexistence of regular solutions. Our approach relies on the study of Sobolev regularity of the invariant measure and a blow-up procedure that is calibrated on the scaling properties of the system. In very special cases, we observe uniqueness of solutions. Finally, we apply our methods to obtain new existence results for MFG systems with competition, namely, when the coupling is local and increasing.     

Double convolution integral equations involving a general class of multivariable polynomials and the multivariableH-functions

In this paper we have solved a double convolution integral equation whose kernel involves the product of theH-functions of several variables and a general class of multivariable polynomials. Due to general nature of the kernel, we can obtain from it, solutions of a large number of double and single convolution integral equations involving products of several classical orthogonal polynomials and simpler functions. We have also obtained here solutions of two double convolution integral equations as special cases of our main result. Exact reference of three known results, which are obtainable as particular cases of one of these special cases, have also been included.     

On composition of some general fractional integral operators

In the present paper we derive three interesting expressions for the composition of two most general fractional integral oprators whose kernels involve the product of a general class of polynomials and a multivariableH-function. By suitably specializing the coefficients and the parameters in these functions we can get a large number of (new and known) interesting expressions for the composition of fractional integral operators involving classical orthogonal polynomials and simpler special functions (involving one or more variables) which occur rather frequently in problems of mathematical physics. We have mentioned here two special cases of the first composition formula. The first involves product of a general class of polynomials and the Fox’sH-functions and is of interest in itself. The findings of Buschman [1] and Erdélyi [4] follow as simple special cases of this composition formula. The second special case involves product of the Jacobi polynomials, the Hermite polynomials and the product of two multivariableH-functions. The present study unifies and extends a large number of results lying scattered in the lierature. Its findings are general and deep.     

A class of nonlinear nonseparable continuous knapsack and multiple-choice knapsack problems

This paper considers a general class of continuous, nonlinear, and nonseparable knapsack problems, special cases of which arise in numerous operations and financial contexts. We develop important properties of optimal solutions for this problem class, based on the properties of a closely related class of linear programs. Using these properties, we provide a solution method that runs in polynomial time in the number of decision variables, while also depending on the time required to solve a particular one-dimensional optimization problem. Thus, for the many applications in which this one-dimensional function is reasonably well behaved (e.g., unimodal), the resulting algorithm runs in polynomial time. We next develop a related solution approach to a class of continuous, nonlinear, and nonseparable multiple-choice knapsack problems. This algorithm runs in polynomial time in both the number of variables and the number of variants per item, while again dependent on the complexity of the same one-dimensional optimization problem as for the knapsack problem. Computational testing demonstrates the power of the proposed algorithms over a commercial global optimization software package.