An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least [Formula: see text] ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We...

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent “almost all” pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular ...

A notion of random walks for circle packings is introduced. The geometry behind this notion is discussed, together with some applications. In particular, we obtain a short proof of a result regarding the type problem for circle packings, which shows that the type of a circle packing is closely related to the type of its tangency graph.

In this paper we describe various applications of the RiemannHilbert method to the theory of orthogonal polynomials on the line and on the circle.

The cyclohedron Wn, known also as the Bott-Taubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x1x2 . . . xn and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S1. The “polygonal pegs problem” asks whether every simple, closed curve in the plane or in...

The most important property of humankind is the capacity to form and maintain relationships. These relationships are absolutely necessary for any of us to survive, learn, work, love and procreate. Human relationships take many forms but the most intense, most pleasurable and most painful are those relationships with family, friends and loved ones. Within this inner circle of intimate relationsh...

Foam flooding (or injection of foam) is a common technology to enhance oil recovery. Although the effects of permeability on foam flooding were well studied in many laboratory experiments, little research has been focused on the specificity of low permeability. In this paper, a series of constant-quality nitrogen foam flow experiments were conducted to investigate the effects of permeability on...

Study of permeability of Fibrous Composites is important in several natural and industrial processes in mechanical engineering. In this study, a comprehensive mathematical model is presented for calculation of normal permeability of ordered elliptical fibrous media. An innovative scale-analysis technique is employed for determining the normal permeability of elliptical fibrous media. In this te...