Sieving the Rationals

Lithuanian Mathematical Journal - J. Kubilius developed the model for transferring the problems related to the value distributions of arithmetical functions to those for independent random...     

The Order on the Rationals has an Orthogonal Order with the Same Order Type

Two orders on the same set are orthogonal if the constant maps and the identity map are the only maps preserving both orders. We construct linear orders orthogonal to the order on the rationals.     

Differentiable Functions on the Rationals

We construct a smooth bijective function from the rationalsto the rationals whose inverse is nowhere continuous.     

Discrete Chebyshev Approximation by Interpolating Rationals

The problem of discrete Chebyshev approximation by a certainform of rational function is considered. The exchange algorithmis applied with a particular treatment of the levelling equations.Illustrative numerical examples are presented, including thedetection of noise levels.     

Arithmetic on Groups of Positive Rationals

In this paper we develop a theory of unique factorization for subgroups of the positive rationals. We show that this theory is strong enough to include arithmetic progressions and the theory of genera in algebraic number fields. We establish generalizations of both Dirichlet's theorem on primes in arithmetic progressions and the theory of genera for Abelian extensions of the rationals.     

How to Solve a Quadratic Equation In Rationals

The title alludes to a similar title of the paper [3] by Grunewaldand Segal, in which it is shown how to solve a quadratic equationin integers. This latter procedure seems to be quite difficult,and the algorithm outlined in [3] is rather involved, althoughit is completely effective in the logical sense. 1991 MathematicsSubject Classification 11D09.     

On the Approximation of Irrationals by Rationals

Let ξ be a real irrational number, and a ≧ 0, b ≧ 0, s > 1 be integers. A theorem of S. UCHIYAMA states that there are infinitely many pairs of integers u and v ≠ 0 such that OVBARξ?u/vOVBAR ≤ s²/4v² and u ? a, v ? b mod s, provided that it is not a ? 6 ? 0 mod s. It is shown that this result is best-possible for all integers s > 1.     

关于Williams的具有插值条件的有理切比雪夫逼近问题的一个注记

J.Williams在[3]中提出了对振荡衰减函数的具有插值条件的有理切比雪夫逼近问题。C.B.Dunham于1975年在[1]中指出了Williams的算法中在理论上存在一个漏洞。本文指出,如[2]中所说,则[1]中C.B.Dunham的论证事实上不可能发生。     

Algebras of p-Groups of Class 2 Over the Rationals

In this work we analyze p-groups of class 2 G and H, with same rational group algebras. We prove that if QG = QH, then their commutators are equal and the centers, 𝒵(G) and 𝒵(H), have their orders preserved. We apply our results to Frattini Central p-groups, and we present an example of two groups of order p ⁷, with no isomorphic centers and different central cyclic components intersecting the cyclic components of the respective commutators groups.     

On Dominions of the Rationals in Nilpotent Groups

The dominion of a subgroup H of a group G in a class M is the set of all a ∈ G that have the same images under every pair of homomorphisms, coinciding on H from G to a group in M. A group H is n-closed in M if for every group G = gr(H, a₁,..., a_n) in M that includes H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rationals is 2-closed in every quasivariety of torsion-free nilpotent groups of class at most 3.     

Rationals and decimals as required in the school curriculum: Part 1: Rationals as measurement

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment, carried out jointly with his wife, Nadine, in her classroom at the École Jules Michelet, was to teach all of the material on rational and decimal numbers required by the national program with a carefully structured, tightly woven and interdependent sequence of “situations.” This article describes and discusses the first portion of that experiment.     

Interpreting the Action of the Endomorphism Monoid of the Rationals

Algebra and Logic - In a first order language we interpret the action of the monoid M of embeddings of (?,<) on the set ? inside (M,°). A similar result is proved for the...     

Rationals and decimals as required in the school curriculum: Part 3. Rationals and decimals as linear functions

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment, carried out by a team of researchers and teachers that included his wife, Nadine, in classrooms at the École Jules Michelet, was to teach all of the material on rational and decimal numbers required by the national programme with a carefully structured, tightly woven and interdependent sequence of “situations.” This article describes and discusses the third portion of that experiment.     

Central Factorial Numbers and Values of Bernoulli and Euler Polynomials at Rationals

The nth order derivatives of tan x and sec x may be represented by polynomials P _n (u) and Q _n (u) in u = tan x, which are known as the derivative polynomials for the tangent and secant and have occurred in diverse contexts. In this paper, explicit representations of P _n (u) and Q _n (u) are derived in terms of the central factorial numbers of the second kind, and the values of the Bernoulli and Euler polynomials at rationals are expressed by means of these polynomials.     

Does a cube have an equation?

This article looks at the question of whether and how a geometricalcube may be determined as the solution set of a single equation.Beyond this being an interesting and surprising result for upperhigh school and first year college students, we argue that,as an investigative activity, it has value in refining students’images of what solutions of equations can be and, with that,refining their ideas of what an equation is.     

几类可升分解的图

Ａｌａｖｉ等人在文献［１］中定义了图的一种新分解，即“升分解”，并且猜想：任意有正数条边的图都可升分解。本文证明了下面三类图可升分解，并得到了一些有意义的推论。１设Ｒｎ是一个至多含有ｎ个顶点和至多含有ｎ条边的图，Ｋｎ－Ｒｎ可升分解（ｎ≥５）；２对称图可升分解；３对称图Ｇ的混合积（Ｇ；ｋ）可升分解。     

Cycles of Polynomial Mappings in Several Variables over Rings of Integers in Finite Extensions of the Rationals,II

We prove (Theorem 1.1) an asymptotic for maximal cycle lengths of polynomial mappings in several variables over rings of integers in algebraic number fields. In addition (Proposition 3.1), we strengthen the shape of possible lengths of polynomial cycles over discrete valuation domains.