bluuebunny/crossref_metadata_2025_split huggingface dataset:
A Net to Hold a Sphere
Authors: Besicovitch A. S. 
Published: January 2007

Problem. A net is made of string round a unit sphere, so that the sphere cannot slip out of it. How long need the string be?This problem was offered by me as a “Contest Problem” to undergraduates of Cambridge last Lent Term and a correct solution was given by Mr. R. Schwarzenberger of Trinity College. I am publishing my own solution, as his solution is considerably more complicated.
On Formulae for the Nth Prime Number
Authors: Willans C. P. 
Published: January 2007

Let pn denote the nth prime number. (p1=2, p2 = 3, etc.)Let [x] denote the greatest integer which is not greater than x.From Wilson’s theorem, is an integer for x = 1 and for all prime values of x; but is fractional for all composite values of x.
Mathematical Education in Russian Schools|A Report on a Visit to Russia in August 1966
Authors: Thwaites Bryan 
Published: January 2007

The National Council of Teachers of Mathematics in the USA took the occasion of the 1966 International Congress of Mathematicians as the opportunity for arranging with the Russian authorities a tour of Soviet mathematical centres by a small party of selected American mathematicians with special experience and interests in various aspects of education. The tour covered Kiev, Moscow and Leningrad and it was hoped that, in each city, a full range of mathematical education would be seen, from the department of mathematics in the University to the famous special boarding schools for pupils of outstanding mathematical ability and finally to the elementary level.I was honoured to be invited to join the party as the only non-American and am very grateful to the Schools Council for enabling me to go by offering to meet the costs of travel and subsistence.
On Scoring in Games
Authors: Watson Donald 
Published: January 2007

The purpose of this paper is to consider the question: if p is the probability that a player P will win a given rally against a player Q, what is the probability that P will win a given game? The answer, of course, depends upon the method of scoring; and the methods considered here will be those used in squash and, for comparison table tennis.In table tennis P earns a point each time that he wins a rally. But in squash he must serve the rally that he wins in order to gain a point; and he relinquishes the service when he loses a rally that he serves.
Some sequences by inverse iteration
Authors: Clegg David , Bryant Victor 
Published: January 2007

This article was inspired by the following problem in a recent addition of the Bulletin of the IMA:Problem. The sequence a0, a1, a2, a3,… is defined by a0 = 3 and, for n ⩾ 1, an is the real root of the equationProve that the sequence converges to 5.
61.9 A permutation algorithm
Authors: Owen G. I. 
Published: January 2007

A computing colleague asked me recently if I could devise an algorithm for obtaining the ith permutation in an alphabetical list.Suppose that the n! permutations of n distinct letters A, B, C,..., N are listed in alphabetical order. The problem is: Given i (where i = 1,2,...,n!), find the ith permutation.
An improbable game of darts
Authors: Crilly Tony , Suen Stephen 
Published: January 2007

The general utility of the ‘pigeon hole principle’ (that if n + 1 pigeons are placed in n boxes one of the boxes contains more than one pigeon!) is neatly illustrated by the problem of showing that:
If seven darts land on a dartboard in any configuration whatsoever, there will always be two darts which are not more than a distance of one radius apart.
Great Circles
Authors: Cook Ian 
Published: January 2007

Spherical trigonometry has fallen somewhat out of fashion and so it was a salutary experience to be consulted recently by a would-be circumnavigator of the world.A great circle route can be analysed graphically by first using a gnomonic chart. Such a chart is obtained by projection from the centre of the earth on to any plane. The distance of the plane from the centre determines the scale of the chart. On a gnomonic chart every great circle appears as a straight line and every straight line on the chart represents a great circle. Of course chart can only represent one hemisphere and even then the whole hemisphere would need an infinite chart.
The importance of being 259
Authors: Goodstein R. L. 
Published: January 2007

The number of the Mathematical Association’s new headquarters in London Road, Leicester is 259.This number has some quite interesting features. It is for instance the sum of the three squares32 + 52 + 32 × 52,and the sum of four cubes23 + 23 + 33 + 23 × 33 and 13 + 23 + 53 + 53.
The pigeonhole principle: “Three into two won’t go”
Authors: Walker Richard 
Published: January 2007

When I began teaching secondary mathematics after several years in industry, I hadn’t been trained as a teacher. Having been away from school mathematics for so long, I was sometimes nonplussed by pupils’ questions.One day I was showing a mixed-ability first-year class how to obtain the decimal expansion of a vulgar fraction. After explaining the how and, I hoped, the why of the method I did several examples on the blackboard and concluded by telling them that the expansion would either terminate or recur (or words to that effect). At once a small boy demanded to be told how I knew this. This floored me: I hadn’t got a ready answer and I had to promise to think about it.
Re-designing the dartboard
Authors: Selkirk Keith 
Published: January 2007

Have you ever looked at a dartboard and wondered why it is numbered in its distinctive order?If we ignore the bull, the inner, and the double and treble rings, the standard dartboard takes the form of a circle divided into 20 equal sectors numbered as in Fig. 1. (There are boards to be found in the Manchester area—and perhaps elsewhere—with a different arrangement, but the one shown is by far the commonest.)
Changes in the teaching of mathematics in France
Authors: Revuz A. 
Published: January 2007

The movement for reform of mathematical education in France, of which the first signs appeared about 1955, had two main objectives:
(a)to reduce the mis-match which had grown up between contemporary mathematics and mathematical education at all levels;(b)to equip all pupils, as far as possible, with a view of mathematics which would enable them to understand and to master the increasing involvement of mathematics across an ever-widening spectrum.
Three progressions in a foursome
Authors: Cundy H. Martyn 
Published: January 2007

In their valuable book Board games around the world. Bell and Cornelius discuss possible investigations arising from the rather complicated medieval game of Rithmomachia. One such investigation gives rise to the following problem:Find all sets of four positive integers in which three of the four are in arithmetic progression, another three are in geometric progression, and another three are in harmonic progression (that is to say, their reciprocals are in arithmetic progression).The authors give two such sets, and state without proof that they are the only ones, apart from multiplication throughout by a common factor. The proof that this is so is an interesting investigation in itself, and opens up some further mathematics.
A circular tour of some circle theorems
Authors: Lord Nick , Wells David 
Published: January 2007

The following theorem about circles (which can be traced back at least as far as Archimedes) does not seem to be widely known and may be used to “pep-up” a discussion of circle theorems.THEOREM. Let two chords of a circle meet at P and make a fixed angle θ (radians, say) with each other. Then the sum of the (opposite) arc lengths, L + L′, is independent of the position of P inside the circle (and thus, taking P at the centre, this sum equals 2rθ, where r is the radius of the circle).
Polynomials with equally spaced zeros
Authors: McLean K. Robin 
Published: January 2007

Polynomials like (x + l)x(x - 1) or 5 (x - 2)(x - 4)(x - 6)(x - 8) are particularly easy to sketch because their zeros are real, distinct and equally spaced along the x-axis. As a simple exercise, please picture in your mind the graph ofy=(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)
Lagrange’s inversion formula and function matrices
Authors: Smith Cedric A. B. 
Published: January 2007

Lagrange's inversion formula is usually presented in the following form. Let f be a regular (= analytic or holomorphic) complex function with the propertiesThen it is a standard theorem that it has a regular inverse function g, such that g (f (z)) = z, with similar properties. (I assume here standard results to be found in appropriate text books.
Mathematical people: Roger Heath-Brown
Authors: Heath-Brown Roger 
Published: January 2007

Roger Heath-Brown (37) is a Reader in Pure Mathematics at Oxford University, and a Tutorial Fellow at Magdalen College. He works in analytic number theory, with interests covering the distribution of prime numbers, Waring's problem and Fermat's Last TheoremThis term I'm giving a first year lecture course—Groups, Rings and Fields. I'm not at my best at 9 am and despite my efforts the undergraduates find me rather dull. The weaker ones have to have everything spelled out in full, with the result that the interesting things are too sparsely scattered for the stronger students. I'm back in college by 10.30 am having collected a pile of mail. There's a request from the Mathematical Gazette for a “day in the life of” style article, and I toy with the idea of describing an occasion on which a journal asks me to describe a typical day's work …. I also receive a referee's report on a paper submitted to me as an editorial advisor to the London Mathematical Society. The report is not encouraging and I reject the paper—a frequent enough duty, but not one I relish. The associated paperwork fills the next three-quarters of an hour. This particular manuscript is rather too similar to something published in 1963. While double-checking this in “Reviews in Number Theory” I notice an abstract of a quite unrelated paper, by a Finnish mathematician.
The mathematics of bowls
Authors: Roper Tom 
Published: January 2007

A little altering of the one side maketh the bowl to run biasse waiesRobert Recorde, The Castle of Knowledge, 1556John Branfield’s request [1] for the mathematics of bowls was too tempting to resist. What follows comprises a simple model for the curve upon which the bowls move, an explanation of how and why they move as they do and an attempt to refine the model of the curve. The search for a model proved to be addictive and effectively displaced many other activities. One of the reasons for this was the way in which the investigation involved so many people with interesting contributions to make. In order to give a flavour of the investigation, I have taken the liberty of giving a somewhat personal rather than strictly mathematical account, the mathematics being no less important.
The enhancement of Machin’s formula by Todd’s process
Authors: Wetherfield Michael 
Published: January 2007

1. BackgroundFrom the end of the 17th century until the 1980s, most approximations to the value of π made use of Gregory’s series:which converges for values of tanθ between -1 and +1. Since , the simplest, and most slowly convergent, series of this kind isWriting x = cotθ, the series becomesExpressions for π/4 have been devised which involve the inverse cotangents of larger numbers, causing the series to converge much faster; of such expressions (referred to herein as ‘identities’) probably the most celebrated is ‘Machin’s Formula’:
The main body of some irrational complex functions
Authors: Kosi-Ulbl Irena 
Published: January 2007

We suggest a way to make traditional geometry more interesting for students.
Investigating the connection between classical geometry and modern chaos theory
seems to be a suitable way of motivating them. Excursions into complex number
iterations and interesting Mandelbrot sets offer many methods of discovering and
exploring.
In H. Zeitler considers the complex function The classic case
n = 2 was investigated by Mandelbrot. We are
discussing a more general case where the exponent n is a rational number.
Regular polygons on sides of special triangles
Authors: Čerin Zvonko 
Published: January 2007

For a natural number n ≥ 3 and a positive real number ξ let Pn (ξ) denote the area of the regular n-gon with side of length ξ and let P(ABC … Z) be the area of a convex polygon ABC … Z.
In 1996 Josip Kovačević observed [1] that in every triangle ABC with sides a, b and c and with the angle A equal to π/3 radians the following relation holds:

Similarly, if the angle A is equal to 2π/3 radians then

On the other hand, when the angle A is π/2 radians, then Pythagoras’ theorem implies which could be rewritten as
so

because P(ABC) = bc/2. The last observation is the starting point for Veljan’s results in [2] and [3] that

when the angle A is π - 2π/n radians and that

when the angle A is 2π/n radians.
Excavations at Hacılar, First Preliminary Report
Authors: Mellaart James 
Published: January 2007

The village of Hacilar is situated in the Vilayet of Burdur in South-west Anatolia, about 25 km. west of Burdur itself on the main road to Yeşilova and Denizli. The chalcolithic site lies about 1·5 km. west of the village and just beyond the orchards, which are irrigated by a plentiful spring at the foot of a great limestone crag which overlooks the village. It is this spring which since neolithic times has been the main reason for more or less continuous occupation in this region. Apart from the neolithic and early chalcolithic site at Hacılar there is a large Early Bronze Age mound on the northern outskirts and a classical site to the south-west of the village.The prehistoric site is an inconspicuous mound, about 150 metres in diameter, rising to a height of not more than 1·50 m. above the level of the surrounding fields (Fig. 1 and Pl. XXIXa). The entire surface of the mound is under cultivation and a series of depressions show the holes made by a local antique-dealer in search of painted pots and small objects. About 1 km. west of the site runs the Koca Çay, the ancient Lysis, and on the eastern scarp of this river valley lies the cemetery of the Early Bronze Age settlement. Not a single burial has yet been found in the chalcolithic or neolithic levels of our site and it is therefore not unreasonable to suggest that its cemetery also may eventually be located there.
Excavations at Can Hasan: Third Preliminary Report, 1963
Authors: French D. H. 
Published: January 2007

The third season of work at Can Hasan took place between 11th and 25th September, with additional work on the site between 26th and 28th September. This year we employed, as usual, Veli Karaaslan as foreman, Rifat Çelimli and Mustafa Duman as ustas, seven other men from Menteş and two local men from Canasun village. At the end of the season the site was completely filled in with earth from our dumps, except for areas where we hope to continue work in future seasons. Finally we fenced in, with posts and wire, an enclosure approximately 110 by 90 m. around the excavated area.This season's staff included Messrs. N. H. S. Kindersley and D. J. Blackman, field assistants; Mrs. D. J. Blackman, Messrs. J. E. Reade and M. C. C. Davie, pottery assistants. Bay Behçet Erdal of the İstanbul Archaeological Museum represented the Turkish Government.
Excavations at Can Hasan, Second Preliminary Report, 1962
Authors: French D. H. 
Published: January 2007

A second season of work at Can Hasan took place between 10th September and 10th October with nine days at the end to complete the final cleaning of the site and the registering of the finds. As last year, we employed three trained men from Beycesultan and ten unskilled men from the village.In addition to myself and my wife, the staff included Mr. N. H. S. Kindersley, field assistant; Messrs. R. Fonseca and I. Walls, architects; Mr. R. E. Oakley, field assistant; and Misses A. C. Cruikshank, C. MacLucas and A. Searight, pottery assistants. Bay Hayrettin Solmaz of the Konya Museum represented the Turkish Government.
Three Cilician Castles
Authors: Youngs G. R. 
Published: January 2007

The 1962 Cambridge Expedition to South-East Turkey spent five weeks in Cilicia studying the castles of Tumlu, Gökvelioğlu, and Yılan. No member of the expedition had any previous experience of this work and this report is therefore confined to facts: descriptions, measurements, plans and photographs. It is hoped that this material may help others, more experienced than ourselves, to make wider generalisations and perhaps to write the architectural history of the buildings.The castles probably date from the period of the Armenian occupation of Cilicia (1080–1375) during which time numerous castles were constructed and existing Byzantine foundations were strengthened. The castles were frequently built upon precipitous mountain sites and conformed with the sinuosities of the rock, thus making the best of the natural strength of the site. The kingdom of Lesser Armenia was centred chiefly on the Cilician plain and surrounding mountains (Fig. 1).
Aşvan Kale: the Third Millennium Pottery
Authors: French David , Helms Svend 
Published: January 2007

The pottery illustrated here is only a small proportion of the total recovered; the intention is to present a brief survey of the commoner fabrics and surface treatments.The stratigraphic connections from trench to trench were, as discovered, not entirely satisfactory. Though the lowest two of the four trenches which were dug (1969) into settlement debris on the north scarp, viz. G1d and G2b, produced no substantial structural features, the sequence of surfaces, tip lines and aggradations was reasonably clear (Fig. 1). In the trenches above G2b, i.e. G2d and G3b, stratigraphic observations were limited. No lines of aggradation could be observed, the trenches apparently being taken up by the remains or debris of a thick mud brick wall or walls. Into this mass of mud-brick were dug pits. Above both features (mud-brick and pits) began a series of structures and associated deposits, datable (after the 1972 season) to the middle or second half of the first millennium B.C.The pottery presented here comes from the lowest two trenches, G2b and G1d.
The Aşvan Hoard: Coins of two Cappadocian Monarchs
Authors: McNicoll Anthony 
Published: January 2007

During the second season of excavations on Aşvan Kale (June – September 1969) an area of over two hundred square metres was cleared to two related late Hellenistic phases (“Hellenistic lla and b” –Mitchell, above, p. 123). On 28th September, during the removal of a pisé wall of the earlier phase, a crudely made pot with two pierced lug handles was found set in the pisé. In it was a hoard of forty-eight silver coins (see Figs. 2–3).The coins are catalogued and registered as Aşv./69/326 nos.l –48. They are deposited in Elâziğ, and will be displayed in the new museum which is being built to house the finds from the various digs in the Keban region. All forty-eight coins are drachms; forty-seven are of Ariobarzanes 1 Philoromaios (96–63 B.C.) and one is of his rival Ariarathes IX Eusebes (101–87 B.C.). The coins had a green incrustation probably similar to that of the Cappadocian hoard discussed by Mørkholm, who thought it was caused by burial in a copper or bronze container. This is not so for the Aşvan hoard, where the incrustation must be caused by the corrosion of the copper content in the coins. It appears that the late silver coinage of Ariobarzanes I was far from pure.
Aşvan Project
Authors: French David 
Published: January 2007

The Aşvan Project grew out of the second aspect of the Aşvan excavations, viz. the recovery of archaeologically relevant information from the modern situation, and began (in 1969) with the botanical work of Gordon Hillman.Very quickly, particularly in 1970, with the help of those who undertook to carry out area-studies, the outline of a scheme for collecting data was sketched and, within a general framework, a number of problems were suggested; of all the problems, those of subsistence and subsistence-patterns were selected as offering the widest base for developing a co-ordinated programme of investigation. The choice was dictated, firstly, by the nature of excavated materials (structures, tools, refuse of stored food remains, both plant and animal) and, secondly, by the availability of a defined geographical area (the concession area i.e. the Aşvan bölgesi) in which was found a number of ancient settlement sites. All these sites could be excavated, together or separately, not only as part of a larger programme, but also by the same methods, standards and techniques and to the same ends; together, the sites covered a time-span of c. 7000–8000 years. In other words the Aşvan region was seen as a unit with clear, recognizable potential for area-studies while at the same time a good, ethical reason for undertaking a regional survey was apparent — any information rescued would be valuable.
Crop Husbandry and Food Production: Modern Basis for the Interpretation of Plant remains
Authors: Hillman Gordon 
Published: January 2007

The full analysis of the initial and most exhaustive of our studies of present-day agriculture at Aşvan could not be completed in time for the present volume. It seemed relevant, however, to present a brief outline of some of the principles involved and the data collected.Samples of vegetable remains represent a body of information which is concerned (in part, at least) with human manipulations of plant resources. These plants may have functioned as foods, fuels, building-timber, tools, dyes, drugs, cosmetics or as decorations. The compounded information may, further, suggest certain forms of ancient economy, though quantification of the data at a level representative of the settlement as a whole is rarely possible. Any such inferences can, however, only exist within the realms of one's own familiarity with equivalent modern situations (or, less consistently, historical situations). If our deductions here are to be repeatable, our present-day (or historical) models must be defined: it is not sufficient to have hazy analogues lurking in the data banks of our subconscious.
Urartian Hieroglyphic Inscriptions from Altintepe
Authors: Klein Jeffrey J. 
Published: January 2007

“Hieroglyphic inscriptions” of various types are common at Urartian sites, but even without a systematic collection and study of them as a whole it is evident that the great majority are at best only pictographic in nature. The relatively small number of such texts, their brevity, the isolated occurrence of many of the signs and the lack of repetition from site to site make it unlikely that they represent any phonetically based writing system. Indeed, the claim of some authors that there existed a local, specifically Urartian hieroglyphic script has yet to be demonstrated.The hieroglyphic inscriptions from Altıntepe, on the other hand, clearly belong to a different category. The length and consistency of the texts strongly suggest the employment of a regular script, while the close correspondence of the signs to those of the Hittite Hieroglyphic syllabary has led a number of scholars to contend that these are purely Hittite documents. The following study will show, however, not only that the language of the inscriptions is in fact Urartian, but also that they present some rather important new evidence for the reading of Hieroglyphic Hittite itself.