Maximum distance between consecutive primes and other related questions

The paper considers the asymptotic of the ratio of the number of primes not exceeding the primorial and the number of residues in the reduced system of residues for the given primorial. We study the relationship between asymptotic lower bounds for the values of the Jacobstal function and the maximum distance between successive primes. One algorithm for computing the Jacobstal function is given. The substantiation of the conjectures about the upper estimate of the maximum distance between successive prime numbers is given.Comment: 11 page

Celtic defensive weaponry in Britain: and its continental background

This thesis is concerned with the defensive armour of the Celtic tribes on the continent of Europe (exclusive of Iberia and much of Italy) from the 5th to approximately the 1st century B.C., or to the date of their subjugation to Roman authority. British helmets, shields and examples of body armour are, wherever possible, discussed in this context and their continental analogies noted. The evidence provided by the arms themselves has been complemented by an examination of the representations of barbarian weaponry in classical sculpture and of the descriptions by Greek and Roman writers of battles with the Celts.The helmets are divided into three main groups; the first consists of the conical helmets of La Tene I date which may owe their origin to local Bronze Age types. The second, which includes the types here labelled Castelrotto and Batina, dates mainly to the 3rd and 2nd centuries B.C., and the armourers appear to have adapted Italian helmet types current during the Celtic invasions of Northern Italy. The third group is of 1st century B.C. to 1st century A.D. date and consists of helmets of the Men and Coolus types, the former found in Southern Prance, the Alps and in Slovenia, and the latter in Eastern Prance and Rhineland Germany. Some of the helmets of Agen type may show a number of features which may be compared to Roman pieces of Haguenau-Weisenau types; other features, such as the use of enamel are more distinctly Celtic. The lack of associated finds makes it difficult to date the Coolus type firmly, but it is important because it is the class with which the British helmets have most affinity, although each British example is unique in some respect. Finally horned helmets are shown to be much rarer than was previously thought, and it is suggested that some of them were not intended for the battle-field.The adoption of the shield also appears to result from the experience of the Celts in Italy and shields are found North of the Alps only from about 300 B.C. The diversity of the surviving remains of shields does not allow such a useful division into types as was possible in the case of helmets, and the chronological groupings are also less clear-cut. The British shields, because of their frequent use of elaborate bronze decoration, are distinct from the continental examples, but an examination of the structural features of these shields and of the art styles of their decoration has made possible their inclusion into the general European framework. The representations of barbarian shields on Roman auxiliary grave stelai of the 1st century A.D. show that the long oval shape seems to have been the most common form of practical shield at this time. From the 2nd century A. D., however, in North Britain a new group of shields, more rectangular in shape with a round central boss, is isolated here for the first time; the adoption of this type may be due to the influence of both Roman and Belgic ideas.Three types of body armour are found in Celtic contexts, namely mail shirts, leather cuirasses and bronze corslets. Both mail shirts and cuirasses have the same shape and seem to have been secured by shoulder straps, decorated by some form of knob. The frequency with which this type of decoration appears on sculpture and figurines might make it possible to interpret the small decorated roundels found at the shoulders.Warriors in this light. Because of the use of perishable materials for the inner linings and crests of helmets, for shields and for many types of body armour, a number of features, such as the painted decoration on shields, can only tentatively be discussed. But the use of leather was probably much more widespread than the surviving evidence suggests. It is clear however, from the small number of helmets and of traces of body armour that the wearing of such arms was the prerogative of the chieftains and the leaders of the bands of warriors

Southern British decorated bronzes of the late pre-Roman Iron Age

This thesis is based on a study of more than 500 bronzes, described in a Catalogue, and mostly illustrated, of the late pre-Roman Iron Age from England and Wales south of Lancashire and Yorkshire. The classes of objects studied comprise: presumed and probable vehicle-fittings and horse-harness, weaponry, mirrors, vessels, 'spoons', weighing-devices, and miscellaneous other pieces including sheet mounts and ornamental studs. New classifications are proposed, and the presumed functions of the bronzes are discussed. An outline assessment is made of the techniques of manufacture, excluding data on chemical composition and physical structure. The most important technical innovations are indicated. The principles underlying the dating of the bronzes are examined, and it is concluded that previous chronologies have been over-precise, and that two phases may be discerned. Distribution-patterns are discussed; two major style-zones, a western and an eastern, are distinguished, and shown to have originated before the birth of Christ. Workshops are shown to have been located in most parts of southern Britain, C. Fox's model of workshop-distribution being rejected. Aspects of smith-organisation are considered, and directions for further research are suggested

A symmetry analysis off early mediaeval ormameimtation

The aim of this thesis is to test the appropriateness of symmetry analysis as a method for the systematic classification of Early Mediaeval ornamentation. This method is different from the traditional Montehan concept of formal classification in archaeology, in that the stylistic entities are not chosen according to the formal similarity of individual motifs and motif-elements, but according to the formal similarity of their symmetrical organisation within an ornamental pattern. It is suggested that symmetry analysis is a more objective method of classifying and analysing ornamentation, as it avoids the subjective selection of typological elements, and therefore also avoids one of the pitfalls of typological classification caused by the ambiguity of the concept of style. Washburn, the originator of this method has suggested that in this way hypotheses can be tested regarding the identity as well as the interaction or information exchange of individuals belonging to a certain cultural, ethnic or social group. In order to test the usefulness of symmetry analysis in relation to these proposals for archaeological research, garnet jewellery from the Merovingian period as well as two manuscript paintings from the Gospel-books of Lindisfarne and Kells have been analysed and compared. It was concluded that the structure of the design-fields of the different types of artefact is one of the main factors for the appearance of certain symmetries within the ornamental context of the artefacts. However, the analysis could also indicate that the method has the potential to classify decorated artefacts according to different regions and workshops of production, and even according to their different social milieu of production. It is concluded that a larger quantity of material has to be analysed in order to get conclusive results from the symmetrical analysis in relation to the latter aspects of investigation, and with regard to the other original proposals made by Washburn