Related papers: Collatz mapping on $\mathbb{Z}/10\mathbb{Z}$
We prove the Strengthened Hanna Neumann Conjecture, in its common graph theoretic formulation. Our original approach to this conjecture used cohomology of sheaves on graphs, although here we give a short combinatorial proof that we found in…
If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form then, under suitable conditions on $p$ and $q$, almost all orbits of this map attain…
Consider a finite positive integer. If it is even, divide it by 2, and if it is odd, multiply it by 3 and add 1. This will give you a new integer. Following the procedure for the new integer, you will receive another integer. Repeat the…
Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways…
In this paper we present a proof of the BMZ Reduction Lemma with a motivational perspective, and state this lemma for maps to manifolds using the classical definition of cohomological dimension. The lemma, proved and utilized in [4], gives…
In this paper we deal with a classical problem in elementary number theory, namely repeating decimals. We show how the digits of the period of the decimal representation of any fraction $\frac{k}{m}$, where $k$ and $m$ are positive integers…
We define Collatz representations for a subset of rational numbers and prove that each real number \( x \notin (-1,1) \) can be approximated arbitrarily well by rational numbers which have only \( 2 \)'s and \( 1 \)'s in their Collatz…
A simple hierarchical structure is imposed on the set of Lipschitz functions on streams (i.e. sequences over a fixed alphabet set) under the standard metric. We prove that sets of non-expanding and contractive functions are closed under a…
In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…
This article is based upon previous work by Sousa Ramos and his collaborators. They first prove that the existence of only one orbit associated with the Collatz conjecture is equivalent to the determinant of each matrix of a certain…
This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. The main feature of our approach is…
A folklore result uses the Lovasz local lemma to analyze the discrepancy of hypergraphs with bounded degree and edge size. We generalize this result to the context of real matrices with bounded row and column sums.
A conjecture regarding the structure of expander graphs is discussed.
In 2009, G. Gr\"atzer and E. Knapp proved that every planar semimodular lattice has a rectangular extension. We prove that, under reasonable additional conditions, this extension is unique. This theorem naturally leads to a hierarchy of…
The Collatz conjecture, also known as the 3n+1 problem, is one of the most popular open problems in number theory. In this note, an algorithm for the verification of the Collatz conjecture is presented that works on a standard PC for…
In 20th century mathematics, the field of topology, which concerns the properties of geometric objects under continuous transformation, has proved surprisingly useful in application to the study of discrete mathematics, such as…
This is the completely revised version of math.AG/0101071.
We define an operation of jets on graphs inspired by the corresponding notion in commutative algebra and algebraic geometry. We examine a few graph theoretic properties and invariants of this construction, including chromatic numbers,…
The topic of this treatise is a combinatorial technique called Graph Pebbling. We investigate pebbling numbers, weight functions, flow networks, hypercubes, and the zero-sum conjecture of Erd\H{o}s and Lemke. This investigation is a…
Motivated by a fundamental geometrical object, the cut locus, we introduce and study a new combinatorial structure on graphs.