1. INTRODUCTION. Although it is well known that every tetrahedron circum-scribes a sphere, the following two simple consequences apparently have not been previously recorded. First, any plane through the center of the inscribed sphere divides the tetrahedron into two smaller solids whose surface areas are equal if and only if their volumes are equal. Second, the centroid of the boundary surface...

A new IPM (Interior Point Method) for LPs has been discussed in [9, 10] based on a centering step that attempts to maximize the radius of the inscribed sphere with center on the current objective plane, and using descent directions derived without using matrix inversions. The method is a descent method and may be called the sphere method for LP. In contrast to all the existing IPMs which involv...

We derive an exact, analytic expression for the fourth virial coefficient of a system of polydisperse spheres under the constraint that the smallest sphere has a radius smaller than a given function of the radii of the three remaining particles.

The inscribed square conjecture, also known as Toeplitz’ conjecture or the square peg problem, asserts that every Jordan curve in the Euclidean plane admits an inscribed square. Although there exists no proof of the general conjecture, there are affirmative proofs of the conjecture subject to additional local smoothness conditions. The weakest of these local smoothness conditions include the sp...

In recent decades, computed tomography (CT) combined with suitable image processing software has become a valuable tool to enable the reading of texts written on scrolls, which are fragile, damaged, or fused together, without physically unwrapping them. X-ray based computer successfully been used scrolls made soft materials such as papyrus and parchment. Although in few cases inscriptions thin ...

Given a rectangle A and a set S of n points in A, the maximum empty rectangle problem is that of finding a largest-area rectangle which is contained in A. has its sides parallel to Ihose ofA. and does Dot contain any of the points in S. This Dote describes an efficient algorithm for solving this problem.

There are deep, but hidden, geometric structures within jammed systems, associated with hidden symmetries. These can be revealed by repeated transformations under which these structures lead to fixed points. These geometric structures can be found in the Voronoi tesselation of space defined by the packing. In this paper we examine two iterative processes: maximum inscribed sphere (MIS) inversio...