Quadrature of the circle in ancient Egypt

The mathematicians of ancient Egypt approximated the area of a circle by a square with astonishing accuracy. The way to find this approximation is not handed down. In this paper a conjecture is given which seems to be much more simple than earlier attempts.     

Tasoev's continued fractions and Rogers-Ramanujan continued fractions

We show some new variations on Tasoev's continued fractions , where the periodic parts include the exponentials in k instead of the polynomials in k. We also mention some relations with other kinds of continued fractions, in particular, with Rogers-Ramanujan continued fractions.     

Relation between interpolating integral continued fractions and interpolating branched continued fractions

We construct and investigate an interpolating integral continued fraction, which represents a natural generalization of interpolating continued fractions. We also point out the optimal choice of the sequence of interpolating knots.     

Circle measurements in ancient China

This paper discusses the method of Liu Hui (3rd century) for evaluating the ratio of the circumference of a circle to its diameter, now known as π. A translation of Liu's method is given in the Appendix. Also examined are the values for π given by Zu Chongzhi (429–500) and unsurpassed for a millenium. Although the method used by Zu is not extant, it is almost certain that the applied Liu's method. With the help of an electronic computer, a table of computations adhering to Liu's method is given to show the derivation of Zu's results. The paper concludes with a survey of circle measurements in China.     

Modules of fractions

Siberian Mathematical Journal -     

A survey of OR utilization in Greece

A recent survey concerning OR utilization in firms and public organizations in Greece showed a rather poor profile. The survey covered 20 big organizations including the National Defense General Staff, national enterprises of public utilities, banks and firms from the textile, cement, oil, construction and other industries. Only a few of these organizations are using OR to a significant degree. The rest reported ‘poor’ or ‘no’ use of OR.Research/implementation problems most frequently appearing in public sector organizations include lack of data, insufficient qualified personnel and unfavorable attitude towards cooperation by affected parties. In private firms, the need for quick solutions and the use of OR analysts in line functions quite often imply the use of a ‘rule of thumb’ rather than a mathematical OR technique. In addition, OR study results quite often do not get implemented. One of the reasons is resistance by the affected parties, considering any proposal for reorganization put forward by other parties as a threat.The use of OR has proven beneficial for the using firm or organization. These benefits, however, are not as great as one might expect, due to the problems mentioned above. The future growth of OR utilization in Greece will depend on developments concerning the education of managers in OR methods, creation of OR groups in firms that will seek solutions to real-world, worth-while problems, publicizing successful OR applications, active support by top management, reforms in the educational system etc. The real push for OR utilization in Greece will come from drastic changes in administrative practices, which are directly related to the socioeconomic developments in the country.     

Length and denominators of Egyptian fractions

We say that an algorithm which could yield a short unit fraction expansion in which the denominators do not get too large is an ideal expansion. It is shown that Bleicher and Erdös algorithm can be modified to be an ideal algorithm.     

Generalized orderings and rings of fractions

Every total ordering of a commutative domain can be extended uniquely to its field of fractions. This result is extended in two directions. Firstly, the notion of a total ordering is generalized so that a nonzero element can have more than two signs (in fact, these signs form a group). Secondly, commutative domains are replaced by noncommutative ones and we consider the following types of rings of fractions: Ore extensions, maximal (right or two-sided) rings of fractions, division hulls of free algebras and epic fields. Throughout the paper several examples are given to illustrate the theory. Received January 8, 2005; accepted in final form November 1, 2005.     

Operational and economic efficiency analysis of public hospitals in Greece

The continuous growth of hospital costs has driven governments in many countries to seek ways to improve their efficiency. In Greece, this has consistently been a major issue for almost two decades, as efficiency assessment and monitoring systems are lacking. In response to this need, the evaluation of the National Health System hospitals’ efficiency level is a precondition for planning, implementing and monitoring any promising reform. In this paper, a non-parametric modeling approach is employed to assess and analyze the efficiency of 87 Greek public hospitals over the period 2005–2009, using data envelopment analysis. The operational and economic aspects of the hospitals’ operation are considered on the basis of their service/case mix and cost structure. We also investigate the efficiency trends over time with the Malmquist index and a second stage regression analysis is performed to explain the operational and economic efficiency results in terms of the hospitals’ operating characteristics and the environment in which they operate.     

Snake graphs and continued fractions

This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory.     

Hypergeometric series and continued fractions

Ramanujan’s results on continued fractions are simple consequences of three-term relations between hypergeometric series. Theirq-analogues lead to many of the continued fractions given in the ‘Lost’ notebook in particular the famous one considered by Andrews and others.     

Continued fractions and irrational rotations

Let \(\alpha \in (0, 1)\) be an irrational number with continued fraction expansion \(\alpha =[0; a_1, a_2, \ldots ]\) and let \(p_n/q_n= [0; a_1, \ldots , a_n]\) be the nth convergent to \(\alpha \). We prove a formula for \(p_nq_k-q_np_k\) \((k<n)\) in terms of a Fibonacci type sequence \(Q_n\) defined in terms of the \(a_n\) and use it to provide an exact formula for \(\{n\alpha \}\) for all n.     

Geodesic continued fractions and LLL

We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to d$d$ real numbers _α1,…,_αd${\alpha }_1,\dots ,{\alpha }_d$. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter t$t$ as t↓0$t\downarrow 0$. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in t$t$.     

Frequency analysis and continued fractions

The frequency analysis problem is solving by the positive Perron-Carathéodory continued fraction, associated with monic Szegö polynomials.     

Pythagorean approximations and continued fractions

In this article, we will show that the Pythagorean approximationsof coincide with those achieved in the 16th century by means of continued fractions. Assumingthis fact and the known relation that connects the Fibonaccisequence with the golden section, we shall establish a procedureto obtain sequences of rational numbers converging to differentalgebraic irrationals. We will see how approximations to someirrational numbers, using known facts from the history of mathematics,may perhaps help to acquire a better comprehension of the realnumbers and their properties at further mathematics level.     

S-units and periodicity of continued fractions in hyperelliptic fields

Let L be the function field of a hyperelliptic curve defined over any field of characteristic different from 2, and let S be a set consisting of an infinite and a finite valuation of L. A relationship between the problem of the existence of nontrivial S-units in the field L and the periodicity of the continued fraction expansion of certain key elements of L is discovered for the first time for finite valuations.