Set Of Positive Rational Numbers Research Articles

Further Results on a Curious Arithmetic Function

Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.

Free monoids and forests of rational numbers

The Calkin–Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each such number occurs in the tree exactly once and in the form a/b, where a and b are relatively prime positive integers. This tree is associated with the matrices L1=(1011) and R1=(1101), which freely generate the monoid SL2(N0) of 2×2 matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices Lu and Rv that freely generate submonoids of GL2(N0), there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.

Points at rational distances from the vertices of certain geometric objects

We consider various problems related to finding points in Q2 and in Q3 which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in Q2, and a cube or tetrahedron in Q3. In particular, as one of several results, we prove that the set of positive rational numbers a such that there exist infinitely many rational points in the plane which lie at rational distance from the four vertices of the rectangle with vertices (0,0), (0,1), (a,0), and (a,1), is dense in R+.

On the dovetail counting method

When the idea of dovetailing is introduced to the student, they can recognize that this method shows a way to count the set of positive rational numbers, but would typically have questions about th...

Analogs of the Stern Sequence

AbstractWe present two infinite families of sequences that are analogous to the Stern sequence. Sequences in the first family enumerate the set of positive rational numbers, while sequences in the second family enumerate the set of positive rational numbers with either an even numerator or an even denominator.

Resistor network-based algorithms for enumerating rational numbers

A class of simple resistor networks is used to enumerate the set of positive rational numbers. A one-to-one correspondence between the simple resistor networks and rational numbers is established, and efficient algorithms to find the rational for a given ordinal number, to find the ordinal number for a given rational, and to generate the rationals up to a given ordinal number are presented. The bijection and the algorithms answer some typical questions that are frequently asked by students in discrete mathematics or computing theory classes related to counting positive rational numbers.

The depression of a graph and the diameter of its line graph

An edge ordering of a graph G = ( V , E ) is an injection f : E → Q + where Q + is the set of positive rational numbers. A (simple) path λ for which f increases along its edge sequence is an f -ascent, and a maximal f -ascent if it is not contained in a longer f -ascent. The depression ε ( G ) of G is the least integer k such that every edge ordering of G has a maximal ascent of length at most k . It has been shown in [E.J. Cockayne, G. Geldenhuys, P.J.P. Grobler, C.M. Mynhardt, J. van Vuuren, The depression of a graph, Utilitas Math. 69 (2006) 143–160] that the difference diam ( L ( G ) ) − ε ( G ) may be made arbitrarily large. We prove that the difference ε ( G ) − diam ( L ( G ) ) can also be arbitrarily large, thus answering a question raised in [E.J. Cockayne, G. Geldenhuys, P.J.P. Grobler, C.M. Mynhardt, J. van Vuuren, The depression of a graph, Utilitas Math. 69 (2006) 143–160].

On some inequalities of probabilistic number theory

Let Q + denote the set of positive rational numbers. We define discrete probability measures ν x on the σ-algebra of subsets of Q +.We introduce additive functions ƒ: Q + → G and obtain a bound for νx(ƒ (r) ∉ X+X−X) using a probability related to some independent random variables. This inequality is an analogue to that proved by I. Ruzsa for additive arithmetical functions.

Classification of semisimple rank one monoids

Consider the classification problem for irreducible, normal, algebraic monoids with unit group $G$. We obtain complete results for the groups $\operatorname {Sl}_2(K) \times {K^\ast }$, $\operatorname {Gl}_2(K)$ and $\operatorname {PGl}_2(K) \times {K^\ast }$. If $G$ is one of these groups let $\mathcal {E}(G)$ denote the set of isomorphy types of normal, algebraic monoids with zero element and unit group $G$. Our main result establishes a canonical one-to-one correspondence $\mathcal {E}(G) \cong {{\mathbf {Q}}^ + }$, where ${{\mathbf {Q}}^ + }$ is the set of positive rational numbers. The classification is achieved in two steps. First, we construct a class of monoids from linear representations of $G$. That done, we show that any other $E$ must already be one of those constructed. To do this, we devise an extension principle analogous to the big cell construction of algebraic group theory. This yields a birational comparison morphism $\varphi :{E_r} \to E$, for some $r \in {{\mathbf {Q}}^ + }$, which is ultimately an isomorphism because the monoid ${E_r} \in \mathcal {E}(G)$ is regular. The relatively insignificant classification problem for normal monoids with group $G$ and no zero element is also solved. For each $G$ there is only one such $E$ with $G \subsetneqq E$.