Abundant periodic and aperiodic solutions of a discontinuous three-term recurrence relation

In this note, we study a discontinuous three-term recurrence relation which arises from seeking the steady states of a cellular neural network with step control function. Several collections of periodic solutions are found. A necessary and sufficient condition for a solution to be periodic is stated and aperiodic solutions are found as consequences. We also show that any periodic solution can be derived from a primary periodic solution with least period not divisible by 5. Although the periodic or aperiodic solutions we found are not exhaustive, they are quite abundant and may reflect some of the rich physical phenomena in true biological systems. Our method in this note may also provide a general approach to analyze the periodicity of solutions of similar recurrence relations.     

A multidimensional continued fraction based on a high-order recurrence relation

The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers should not become too large. The problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one-parameter family of lattices; the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero. The paper discusses two rounding procedures to solve the underlying inhomogeneous Diophantine approximation problem: the first, which we call ``naive rounding' leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties; in particular, when it is applied to the familiar problem of simultaneous rational approximation, the algorithm reduces to the classical Jacobi-Perron algorithm. The second rounding procedure is Babai's nearest-plane procedure. We compare the two rounding procedures numerically; our experiments suggest that the multidimensional continued fraction corresponding to nearest-plane rounding converges at an optimal asymptotic rate.

     

Open covers and the square bracket partition relation

An open cover of an infinite separable metric space is an -cover of if and for every finite subset of there is a such that . Let be the collection of -covers of . We show that the partition relation holds if, and only if, the partition relation holds.

     

Generating functions and duality for integer programs

We consider the integer program P→max c^′x|Ax=y;xNⁿ . Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables λR^m have discrete analogues zC^m, related in a simple manner. Moreover, both optimal values of P and the LP obey the same formula, using z for P and |z| for the LP. One retrieves (and refines) the so-called group-relaxations of Gomory which, in this dual approach, arise naturally from a detailed analysis of a generalized residue formula of Brion and Vergne. Finally, we also provide an explicit formulation of a dual problem P^*, the analogue of the dual LP in linear programming.     

Some set partition statistics in non-crossing partitions and generating functions

In this paper we shall give the generating functions for the enumeration of non-crossing partitions according to some set partition statistics explicitly, which are based on whether a block is singleton or not and is inner or outer. Using weighted Motzkin paths, we find the continued fraction form of the generating functions. There are bijections between non-crossing partitions, Dyck paths and non-nesting partitions, hence we can find applications in the enumeration of Dyck paths and non-nesting partitions. We shall also study the integral representation of the enumerating polynomials for our statistics. As an application of integral representation, we shall give some remarks on the enumeration of inner singletons in non-crossing partitions, which is equivalent to one of udu's at high level in Dyck paths investigated in [Y. Sun, The statistic “number of udu's” in Dyck paths, Discrete Math. 284 (2004) 177-186].     

Residue formulae, vector partition functions and lattice points in rational polytopes

We obtain residue formulae for certain functions of several variables. As an application, we obtain closed formulae for vector partition functions and for their continuous analogs. They imply an Euler-MacLaurin summation formula for vector partition functions, and for rational convex polytopes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope.

     

Generating functions, Fibonacci numbers and rational knots

We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots. Then we show how to enumerate rational knots of given crossing number depending on genus and/or signature. This allows to determine the asymptotical average value of these invariants among rational knots. We give also an application to the enumeration of lens spaces.     

On optimal message vector length for block single parallel partition algorithm in a three-dimensional ADI solver

It has long been recognized that many direct parallel tridiagonal solvers are only efficient for solving a single tridiagonal equation of large sizes, and they become inefficient when naively used in a three-dimensional ADI solver. In order to improve the parallel efficiency of an ADI solver using a direct parallel solver, we implement the single parallel partition (SPP) algorithm in conjunction with message vectorization, which aggregates several communication messages into one to reduce the communication costs. The measured performances show that the longest allowable message vector length (MVL) is not necessarily the best choice. To understand this observation and optimize the performance, we propose an improved model that takes the cache effect into consideration. The optimal MVL for achieving the best performance is shown to depend on number of processors and grid sizes. Similar dependence of the optimal MVL is also found for the popular block pipelined method.     

A polarized partition relation for cardinals of countable cofinality

We prove that if and , then

for all . This polarized partition relation holds if for every partition either there are and with or there are and with .

     

Positive Harris recurrence and exponential ergodicity of the basic affine jump-diffusion

In this article, we find the transition densities of the basic affine jump-diffusion (BAJD), which has been introduced by Duffie and Gârleanu as an extension of the CIR model with jumps. We prove the positive Harris recurrence and exponential ergodicity of the BAJD. Furthermore, we prove that the unique invariant probability measure π of the BAJD is absolutely continuous with respect to the Lebesgue measure and we also derive a closed-form formula for the density function of π.     

Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings

In this paper, some gap functions for three classes of a system of generalized vector quasi-equilibrium problems with set-valued mappings (for short, SGVQEP) are investigated by virtue of the nonlinear scalarization function of Chen, Yang and Yu. Three examples are then provided to demonstrate these gap functions. Also, some gap functions for three classes of generalized finite dimensional vector equilibrium problems (GFVEP) are derived without using the nonlinear scalarization function method. Furthermore, a set-valued function is obtained as a gap function for one of (GFVEP) under certain assumptions.      

Application of a composition of generating functions for obtaining explicit formulas of polynomials

Using notions of composita and composition of generating functions, we obtain explicit formulas for the Chebyshev polynomials, the Legendre polynomials, the Gegenbauer polynomials, the Associated Laguerre polynomials, the Stirling polynomials, the Abel polynomials, the Bernoulli Polynomials of the Second Kind, the Generalized Bernoulli polynomials, the Euler Polynomials, the Peters polynomials, and the Narumi polynomials.     

Differential and sensitivity properties of gap functions for Minty vector variational inequalities

The purpose of this paper is to investigate differential properties of a class of set-valued maps and gap functions involving Minty vector variational inequalities. Relationships between their contingent derivatives are discussed. An explicit expression of the contingent derivative for the class of set-valued maps is established. Optimality conditions of solutions for Minty vector variational inequalities are obtained.     

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix‐like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes‐Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general q‐polynomials.     

Complete set of periodic solutions of a discontinuous recurrence equation

A nonlinear three-term recurrence relation arising from seeking the steady states of a cellular neural network with bang‐bang control is studied. We show that each solution is periodic and its prime period can be determined by three of its consecutive terms. By means of this periodicity property, we may then solve the steady state problem which to our knowledge is not solved by other means.     

Gap functions for constrained vector variational inequalities with applications

In this paper, the image space analysis is applied to investigate scalar-valued gap functions and their applications for a (parametric)-constrained vector variational inequality. Firstly, using a non-linear regular weak separation function in image space, a gap function of a constrained vector variational inequality is obtained without any assumptions. Then, as an application of the gap function, two error bounds for the constrained vector variational inequality are derived by means of the gap function under some mild assumptions. Further, a parametric gap function of a parametric constrained vector variational inequality is presented. As an application of the parametric gap function, a sufficient condition for the continuity of the solution map of the parametric constrained vector variational inequality is established within the continuity and strict convexity of the parametric gap function. These assumptions do not include any information on the solution set of the parametric constrained vector variational inequality.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.