Strictly positive definiteness of Hermite interpolation on spheres

In this paper, we obtain some sufficient conditions for positive definite kernels to be strictly positive definite and hence well‐posed for Hermite scattered data interpolation on Euclidean unit spheres. This revised version was published online in June 2006 with corrections to the Cover Date.     

Strict positive definiteness on products of compact two-point homogeneous spaces

We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, covering almost all possible choices for the spaces. The result complements similar characterizations previously obtained for products of high-dimensional spheres.     

Controlled rounding and cell perturbation: statistical disclosure limitation methods for tabular data

Rounding methods are common techniques in many statistical offices to protect sensitive information when publishing data in tabular form. Classical versions of these methods do not consider protection levels while searching patterns with minimum information loss, and therefore typically the so-called auditing phase is required to check the protection of the proposed patterns. This paper presents a mathematical model for the whole problem of finding a protected pattern with minimum loss of information, and proposes a branch-and-cut algorithm to solve it. It also describes a new methodology closely related to the classical Controlled Rounding methods but with several advantages. The new methodology is named Cell Perturbation and leads to a different optimization problem which is simpler to solve than the previous problem. This paper presents a cutting-plane algorithm for finding an exact solution of the new problem, which is a pattern guaranteeing the same protection level requirements but with smaller loss of information when compared with the classical Controlled Rounding optimal patterns. The auditing phase is unnecessary on the solutions generated by the two algorithms. The paper concludes with computational results on real-world instances and discusses a modification in the objective function to guarantee statistical properties in the solutions. Received: April, 2004     

Lifting,superadditivity, mixed integer rounding and single node flow sets revisited

In this survey we attempt to give a unified presentation of a variety of results on the lifting of valid inequalities, as well as a standard procedure combining mixed integer rounding with lifting for the development of strong valid inequalities for knapsack and single node flow sets. Our hope is that the latter can be used in practice to generate cutting planes for mixed integer programs. The survey contains essentially two parts. In the first we present lifting in a very general way, emphasizing superadditive lifting which allows one to lift simultaneously different sets of variables. In the second, our procedure for generating strong valid inequalities consists of reduction to a knapsack set with a single continuous variable, construction of a mixed integer rounding inequality, and superadditive lifting. It is applied to several generalizations of the 0–1 single node flow set. This paper appeared in 4OR, 1, 173–208 (2003). The first author is supported by the FNRS as a chercheur qualifié. This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors.     

Lifting,superadditivity, mixed integer rounding and single node flow sets revisited

In this survey we attempt to give a unified presentation of a variety of results on the lifting of valid inequalities, as well as a standard procedure combining mixed integer rounding with lifting for the development of strong valid inequalities for knapsack and single node flow sets. Our hope is that the latter can be used in practice to generate cutting planes for mixed integer programs. The survey contains essentially two parts. In the first we present lifting in a very general way, emphasizing superadditive lifting which allows one to lift simultaneously different sets of variables. In the second, our procedure for generating strong valid inequalities consists of reduction to a knapsack set with a single continuous variable, construction of a mixed integer rounding inequality, and superadditive lifting. It is applied to several generalizations of the 0-1 single node flow set.Received: December 2002, Revised: April 2003, AMS classification: 90C11, 90C27Laurence A. Wolsey: Corresponding author: CORE, Voie du Roman Pays 34, 1348 Louvain-la-Neuve, Belgium. The first author is supported by the FNRS as a research fellow. This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Ministers Office, Science Policy Programming. The scientific responsibility is assumed by the authors.Laurence A. Wolsey: This research was also supported by the European Commission GROWTH Programme, Research Project LISCOS, Large Scale Integrated Supply Chain Optimization Software Based on Branch-and-Cut and Constraint Programming Methods, Contract No. GRDI-1999-10056, and the project TMR-DONET nr. ERB FMRX-CT98-0202.     

q-positive definiteness and related operators

We are in progress of extending the family of ‘q-deformed operators’ considered in the previous papers by joining to them q-subnormal as well as q-formally subnormal ones. It turns out that q-positive definiteness, a notion generalizing Halmos' standard positive definiteness of bounded subnormal operators, is likewise central for our new scheme.     

Krein duality,positive 2-algebras,and dilation of comultiplications

The Krein-Tannaka duality for compact groups was a generalization of the Pontryagin-van Kampen duality for locally compact Abelian groups and a remote predecessor of the theory of tensor categories. It is less known that it found applications in algebraic combinatorics (“Krein algebras”). Later, this duality was substantially extended: in [29], the notion of involutive algebras in positive vector duality was introduced. In this paper, we reformulate the notions of this theory using the language of bialgebras (and Hopf algebras) and introduce the class of involutive bialgebras and positive 2-algebras. The main goal of the paper is to give a precise statement of a new problem, which we consider as one of the main problems in this field, concerning the existence of dilations (embeddings) of positive 2-algebras in involutive bialgebras, or, in other words, the problem of describing subobjects of involutive bialgebras; we define two types of subobjects of bialgebras, strict and nonstrict ones. The dilation problem is illustrated by the example of the Hecke algebra, which is viewed as a positive involutive 2-algebra. We consider in detail only the simplest situation and classify two-dimensional Hecke algebras for various values of the parameter q, demonstrating the difference between the two types of dilations. We also prove that the class of finite-dimensional involutive semisimple bialgebras coincides with the class of semigroup algebras of finite inverse semigroups.     

Piecewise polynomial,positive definite and compactly supported radial functions of minimal degree

We construct a new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support. For given smoothness and space dimension it is proved that they are of minimal degree and unique up to a constant factor. Finally, we establish connections between already known functions of this kind.     

Existence,uniqueness and stability of positive steady states to a prey-predator diffusion system

In the paper, we study the positive solutions of an elliptic system coming from a preypredator model with modified Leslie-Gower and Holling-Type II schemes. We study the existence, non-existence, bifurcation, uniqueness and stability of positive solutions. In particular, we obtain a continuum of positive solutions connecting a semi-trivial solution to the unique positive solution of the limiting system. This work was supported by National Natural Science Foundation of China (Grant Nos. 10471022, 10771032) and Natural Science Foundation of Jiangsu Province (Grant No. BK2006088)     

Densities of lattices corresponding to spaces of positive, negative, and variational dimension, and their application to time series

We prove a general theorem concerning a distribution of Bose-Einstein type. Using this theorem, we apply the notions of lattice dimension and lattice density to oscillatory time series.