Via a computer search, Altshuler and Steinberg found that there are 1296 + 1 combinatorial 3-manifolds on nine vertices, of which only one is non-sphere. This exceptional 3-manifold K 9 triangulates the twisted S -bundle over S . It was first constructed by Walkup. In this paper, we present a computer-free proof of the uniqueness of this non-sphere combinatorial 3-manifold. As opposed to the co...

A combinatorial map is a connected topological graph cellularly embeddedin a surface. This monograph concentrates on the automorphism groupof a map, which is related to the automorphism groups of a Klein surfaceand a Smarandache manifold, also applied to the enumeration of unrootedmaps on orientable and non-orientable surfaces. A number of results for theautomorphism groups ...

Let M be an n-vertex combinatorial triangulation of a Z2-homology d-sphere. In this paper we prove that if n ≤ d+ 8 then M must be a combinatorial sphere. Further, if n = d+ 9 and M is not a combinatorial sphere then M can not admit any proper bistellar move. Existence of a 12-vertex triangulation of the lens space L(3, 1) shows that the first result is sharp in dimension three. In the course o...

Offer Shai received his Ph.D. (Summa Cum Laude) in Mechanical Engineering from Ben-Gurion University, Israel in 1997, and he is currently a faculty member in the Department of Mechanics, Materials and Systems at Tel-Aviv University. His scientific background is comprised of several diverse academic fields— mechanical engineering, computer science, advanced discrete mathematics, and operations r...

Generalizing the idea of the Lovász extension of a set function and the discrete Choquet integral, we introduce a combinatorial model that allows us to define and analyze matroid-type greedy algorithms. The model is based on a real-valued function v on a (finite) family of sets which yields the constraints of a combinatorial linear program. Moreover, v gives rise to a ranking and selection proc...

The affine general linear group acting on a vector space over prime field is well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use cryptography as way hide or detect weaknesses inside block ciphers. This paper focused building convenient representation of elements which suits better the purposes cry...

Distributing points on a (possibly high-dimensional) sphere with minimal energy is long-standing problem in and outside the field of mathematics. This paper considers novel function that arises naturally from statistics combinatorial optimization, studies its theoretical properties. Our result solves both exact optimal spherical point configurations certain cases asymptotics under general assum...

“Combinatorial number theory”, in very loose terms, can be described as an area of mathematics which is a cross between combinatorics and number theory. More precisely, the area concerns structures of integers (or similar sets), with some number theoretic properties, which can be studied mainly by combinatorial means, rather than, for example, by algebraic methods. This area really only truly e...

In this paper we study a notion of preorder that arises in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and its generalization to ultrafilters, which is related to the algebraical and topological structure of the Stone-Čech compactification of the discrete space of natural numbers. In particular, we prove that there exist ultrafilters maximal for...