On the structure of the set bases of a degenerate point

Consider an extreme point (EP)x ⁰ of a convex polyhedron defined by a set of linear inequalities. If the basic solution corresponding tox ⁰ is degenerate,x ⁰ is called a degenerate EP. Corresponding tox ⁰, there are several bases. We will characterize the set of all bases associated withx ⁰, denoted byB ⁰. The setB ⁰ can be divided into two classes, (i) boundary bases and (ii) interior bases. For eachB ⁰, there is a corresponding undirected graphG ⁰, in which there exists a tree which connects all the boundary bases. Some other properties are investigated, and open questions for further research are listed, such as the connection between the structure ofG ⁰ and cycling (e.g., in linear programs).     

On set systems determined by intersections

The set systems determined by intersections are studied and a sufficient condotion for this property is given. For case of graphs a necessary and sufficient condition is established. Some connections to other results are discussed.     

On the set of directions at a point of strict tangency in which one can draw shortest geodesics

Translated from Sibirskii Matematicheskii, Vol. 36, No. 1, pp. 149–155, January–February, 1995.     

On the number of directions determined by a pair of functions over a prime field

A three-dimensional analogue of the classical direction problem is proposed and an asymptotically sharp bound for the number of directions determined by a non-planar set in AG(3,p), p prime, is proved. Using the terminology of permutation polynomials the main result states that if there are more than pairs with the property that f(x)+ag(x)+bx is a permutation polynomial, then there exist elements c,d,e∈Fp with the property that f(x)=cg(x)+dx+e.     

On convex decompositions of a planar point set

Let P be a planar point set in general position. Neumann-Lara et al. showed that there is a convex decomposition of P with at most elements. In this paper, we improve this upper bound to .     

On large families of subsets of the set of the integers not exceeding N

In an earlier paper Dartyge and Sárközy defined the measures of pseudo-randomness of subsets of {1,2,…,N}, and they presented several examples for subsets with strong pseudo-random properties. However, in applications one usually needs large families of subsets with strong pseudo-random properties. Here two constructions of this type are given. The notion of complexity of families of subsets of {1,…,N} is also introduced and studied.     

On the number of rectangulations of a planar point set

We investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (nonintersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable permutation, the number of rectangulations is exactly the (n+1)st Baxter number. We also show that no matter what the order of the points is, the number of guillotine rectangulations is always the nth Schröder number, and the total number of rectangulations is O(n²⁰/n⁴).     

On the structure of a random sum-free set

Summary There is a natural probability measure on the set of all sum-free sets of natural numbers. If we represent such a set by its characteristic functions, then the zero-one random variabless(i) are far from independent, and we cannot expect a law of large numbers to hold for them. In this paper I conjecture a decomposition of into countably many more tractible pieces (up to a null set). I prove that each piece has positive measure, and show that, within each piece, a random set almost surely has a density which is a fixed rational number depending only on the piece. For example, the first such piece is made up of sets consisting entirely of odd numbers; it has probability 0.218 ..., and its members almost surely have density 1/4.     

On empty convex polygons in a planar point set

Let P be a set of n points in general position in the plane. Let X_k(P) denote the number of empty convex k-gons determined by P. We derive, using elementary proof techniques, several equalities and inequalities involving the quantities X_k(P) and several related quantities. Most of these equalities and inequalities are new, except for a few that have been proved earlier using a considerably more complex machinery from matroid and polytope theory, and algebraic topology. Some of these relationships are also extended to higher dimensions. We present several implications of these relationships, and discuss their connection with several long-standing open problems, the most notorious of which is the existence of an empty convex hexagon in any point set with sufficiently many points.     

On embedding an outer-planar graph in a point set

Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(nlog³n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [8,12] that requires O(n²) time. Our algorithm is near-optimal as there is an Ω(nlogn) lower bound for the problem [4]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where lognd2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(nlogn) and O(n²) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ(nlogn) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.     

On the structure of commutative affine group schemes over a non-perfect field

Let k be a non-perfect field of characteristic p>O with a p-basisB and k_s the algebraic separable closure of k. Starting from the ring of Schoeller D ^B [3] and the topological Galois group II of k_s over k, we construct a new ring such that the category of commutative affine k-group schemes is anti-equivalent to the category ofeffaceable left -modules. (The effaceability is defined in the text).