We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in D(n, p) for nearly optimal p (up to a logc n factor). In particular, we show that given t = (1 − o(1))np Hamilton cycles C1, . . . , Ct, each of which is oriented arbitrarily, a digraph D ∼ D(n, p) w.h.p. contains edge disjoint copies of C1, . . . , Ct, provided p = ω(log 3 n/n). We also show that given an arbitr...

In a graph G = (V,E), a non-empty set S ⊆ V is said to be an open packing set if no two vertices of S have a common neighbour in G. An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The minimum and maximum cardinalities of a maximal open packing set are respectively called the lower open packing number and the open packing number and ...

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of well-rounded sublattices of Z2. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has the structure of a noncommutative infinitely generated monoid. We discuss the structure of a given simi...

The performance of Branch-and-Bound algorithms is severely impaired by the presence of symmetric optima in a given problem. We describe a method for the automatic detection of formulation symmetries in MINLP instances. A software implementation of this method is used to conjecture the group structure of the problem symmetries of packing equal circles in a square. We provide a proof of the conje...

The Lovász Local Lemma is known to have an extension for cases where independence is missing but negative dependencies are under control. We show that this is often the case for random injections, and we provide easy-to-check conditions for the non-trivial task of verifying a negative dependency graph for random injections. As an application, we prove existence results for hypergraph packing an...

injection molding is one of the most common processes which are used for manufacturing different plastic parts. this method includes 3 continuous steps: filling of the mold, cooling and driving the part out. a wide range of plastic and non-plastic materials can be used as raw material in this process. one of the most important advantages of this way of production is that it is very economic. in...