Constructions of External Difference Families and Disjoint Difference Families

External difference families (EDFs) are a type of new combinatorial designs originated from cryptography. In this paper, some earlier ideas of recursive and cyclotomic constructions of combinatorial designs are extended, and a number of classes of EDFs and disjoint difference families are presented. A link between a subclass of EDFs and a special type of (almost) difference sets is set up.     

Semiconvergence in distribution of random closed sets with application to random optimization problems

The paper considers upper semicontinuous behavior in distribution of sequences of random closed sets. Semiconvergence in distribution will be described via convergence in distribution of random variables with values in a suitable topological space. Convergence statements for suitable functions of random sets are proved and the results are employed to derive stability statements for random optimization problems where the objective function and the constraint set are approximated simultaneously. The author is grateful to two anonymous referees for helpful suggestions.     

Estimates for transition probabilities on a compact manifold

The purpose of this note is to describe a procedure for transferring familiar estimates for transition probabilities on RN to transition probabilities on compact manifolds.     

Some results on quasipolyhedral convexity

The objective of this paper is to analyze under what well-known operations the class of quasipolyhedral convex functions, which can be regarded as an extension of that of polyhedral convex functions, is closed. The operations that will be considered are those that preserve polyhedral convexity, such that the image and the inverse image under linear transformations, right scalar multiplication (including the case where λ=0⁺) and pointwise addition.      

Compact directed acyclic word graphs for a sliding window

Suffix trees are a well-known and widely-studied data structure highly useful for string matching. The suffix tree of a string w can be constructed in O(n) time and space, where n denotes the length of w. Larsson achieved an efficient algorithm to maintain suffix trees for a sliding window. It contributes to prediction by partial matching (PPM) style statistical data compression scheme. Compact directed acyclic word graphs (CDAWGs) are a more space-economical data structure for indexing strings. In this paper we propose a linear-time algorithm to maintain CDAWGs for a sliding window.     

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.     

Second Order Nonlinear Evolution Inclusions Ⅰ： Existence and Relaxation Results

This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.     

Proximality and Chebyshev sets

This paper is a companion to a lecture given at the Prague Spring School in Analysis in April 2006. It highlights four distinct variational methods of proving that a finite dimensional Chebyshev set is convex and hopes to inspire renewed work on the open question of whether every Chebyshev set in Hilbert space is convex.     

Barycentric coordinates for convex sets

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.      

Dual Problems and Separation of Sets: Lagrange and Fenchel Duality

An extremum problem is embedded in a parametric scheme which contains, as a particular case, the classic perturbation function. The introduction of the image of the embedded problem allows one to derive a generalized duality and, in particular, Lagrangian and Fenchel duality.