Abstract:
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 are $a$ and $b$ are relatively prime positive integers. This tree is associated with the matrices $L_1 = \left( \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \right)$ and $R_1 = \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$, which freely generate the monoid $SL_2(\mathbf{N}_0)$ of $2 \times 2$ matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices $L_u$ and $R_v$ that freely generate submonoids of $GL_2(\mathbf{N}_0)$, there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.

Abstract:
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic progressions.

Abstract:
This study solves a problem on rational numbers distribution along number plane and number axis. It is shown, such distribution is linked to resonance phenomenon and also to stability of oscillating systems.

Abstract:
The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.

Abstract:
This short article is aimed at educators and teachers of mathematics.Its goal is simple and direct:to explore some of the basic/elementary properties of proper rational numbers.A proper rational number is a rational which is not an integer. A proper rational r can be written in standard form: r=c/b,where c and b are relatively prime integers; and with b greater than or equal to 2. There are seven theorems, one proposition, and one lemma; Lemma1, in this paper. Lemma1 is a very well known result, commonly known as Euclid's lemma.It is used repeatedly throughout this paper, and its proof can be found in reference[1]. Theorem4(i) gives precise conditions for the sum of two proper rationals to be an integer.Theorem5(a) gives exact conditions for the product to be an integer. Theorem7 states that there exist no two proper rationals both of whose sum and product are integers.This follows from Theorem6 which states that if two rational numbers have a sum being an integer; and a product being an integer;then these two rationals must both be in fact integers.

Abstract:
We prove that the necessary and sufficient condition for the state equation of a finite automaton M to have a rational solution is that the lexicographical G del numbers of the strings belonging to each of the end-sets of M form an ultimately periodic set. A method of determining the existence of a rational solution of the state equation is also given.

Abstract:
We solve the problem of computing characteristic numbers of rational space curves with a cusp, where there may or may not be a condition on the node. The solution is given in the form of effective recursions. We give explicit formulas when the dimension of the ambient projective space is at most 5. Many numerical examples are provided. A C++ program implementing most of the recursions is available upon request

Abstract:
We study real numbers $\beta$ with the curious property that the $\beta$-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let $\gamma(\beta)$ denote the supremum of the real numbers $c$ in $(0,1)$ such that all positive rational numbers less than $c$ have a purely periodic $\beta$-expansion. We prove that $\gamma(\beta)$ is irrational for a class of cubic Pisot units that contains the smallest Pisot number $\eta$. This result is motivated by the observation of Akiyama and Scheicher that $\gamma(\eta)=0.666 666 666 086 ...$ is surprisingly close to 2/3.

Abstract:
A binary representation of complex rational numbers and their arithmetic is described that is not based on qubits. It takes account of the fact that $0s$ in a qubit string do not contribute to the value of a number. They serve only as place holders. The representation is based on the distribution of four types of systems, corresponding to $+1,-1,+i,-i,$ along an integer lattice. Complex rational numbers correspond to arbitrary products of four types of creation operators acting on the vacuum state. An occupation number representation is given for both bosons and fermions.