This thesis discusses the classical problem of how to calculate rn(m), the number of ways to represent an integer m by a sum of n squares. To this day, there are very few formulas that allow for...Show moreThis thesis discusses the classical problem of how to calculate rn(m), the number of ways to represent an integer m by a sum of n squares. To this day, there are very few formulas that allow for easy calculation of rn(m). Here, we focus on the case when n is even, hence we can use the theory of integral weight modular forms on Γ1(4) to write down formulas for the theta function θn(q) associated to sums of n squares. In particular, we show that for only a small finite list of n can θn be written as a linear combination consisting entirely of Eisenstein series and cusp forms with complex multiplication. These give rise to “elementary” formulas for rn(m), in which knowing the prime factorization of m allows for their efficient computation. This work is related to Couveignes and Edixhoven’s forthcoming book and Peter Bruin’s forthcoming Ph.D. thesis concerning polynomial-time algorithms for calculating the prime Fourier coefficients of modular forms.Show less
Museums are the places where many people come into direct contact with archaeology for the first time. By visiting museum presentations and exhibitions the public can see and enjoy the...Show moreMuseums are the places where many people come into direct contact with archaeology for the first time. By visiting museum presentations and exhibitions the public can see and enjoy the archaeological remains of ancient civilizations and peoples. Museums are a significant and powerful instrument for public knowledge and construction of the past. The tasks of archaeological museums are not easy to define. The tasks archaeological museums had in the past have now been diminished or taken over by other institutes and organizations. The tasks that most (inter)national organisations assign to museums can be divided into collection management tasks including acquiring, conserving and researching and public tasks such as communication, exhibitioning and education. In this thesis article 9 of The ‘Convention on the Protection of the Archaeological Heritage of Europe’, usually referred to as the Malta Convention, is taken as a basis. I argue that in addition to transferring archaeological knowledge, archaeological museums should also aim to generate public involvement in archaeological heritage management. Such involvement is needed in order to make sure that the archaeological heritage, including archaeological collections, is preserved for the present as well as future generations and protected against both legal and illegal activities. In order to achieve this museums should not only offer knowledge to their public on archaeological objects, materials and cultures, but also on their value to society, the threats archaeological heritage faces, ethical issues, (inter)national legislation and the archaeological profession as a whole. Furthermore, I argue that museums should aim and have the task to serve an as broad as possible audience. Because by aiming to serve every single member of the public, museums can increase the overall appreciation of the past and archaeological knowledge. By doing this they create an enduring and far reaching basis of public awareness and support for archaeological heritage management. So, the task I state archaeological museums have is to offer knowledge and education through the presentation of archaeological collections to all members of the public on all of the different aspects of the archaeological heritage and profession in order to create public awareness and support for archaeological heritage management. The aim of my thesis is to investigate how Dutch museums fulfil their task of presenting Dutch archaeological collections to the public and, by doing so, creating public awareness and support for archaeological heritage management. I investigate this by looking at three aspects of archaeological museum presentations: what is presented, what visitor groups and target audiences do the museums aim to attract and what are the aims of the archaeological presentations. The research is an unique inventory research covering the Dutch archaeological collection of the following museums: the National Museum of Antiquities, Drents Museum, Fries Museum, Groninger Museum, Limburgs Museum, Museum Het Valkhof and Noordbrabants Museum. In order to give a complete overview of the museums’ presentations, the past, present and future permanent presentations as well as the temporary exhibitions of the seven selected museums are reviewed and compared. The data was gathered by visiting the museum presentations, by interviewing the curators of the selected museums, reading the year reports, policy plans and collections plans of the museums, and by reading general literature on museum presentation and archaeological museums as well as governmental policies, legislation on archaeological heritage and policies of (inter)national organisations. Based on the data obtained during the research it can be concluded that the archaeological presentations of the museums are very much focused on archaeological objects which are mainly used to illustrate themes relating to the daily lives of past societies, such as death, religion, utensils, arts and crafts, household, war and jewellery. The presentations are only placed in the context of the museums’ own provinces and focus on one main time period, instead of covering the wider cultural historical framework and the entire archaeological record. Temporary exhibitions on Dutch archaeology are in general not held very often. Non-archaeological themes or themes relating to the archaeology of foreign countries are preferred over them since they attract more visitors. The themes, time periods and outline of these exhibitions are very similar to those of the permanent presentations, and can thus be seen as an extension instead of an addition. It can also be concluded that teenagers, young adults, middle aged adults and the allochtonous population are the visitor groups that are currently underrepresented in the museums. The main target groups of the current and future Dutch archaeological presentations and exhibitions are school children, families and elder people. The museums therefore do not aim to provide for an as broad as possible audience. The presentations and exhibitions succeed in offering archaeological knowledge on objects, materials, themes and specific cultures to the public. However, I argue that public awareness and support for archaeological heritage management in The Netherlands are not established in this way. In order to do this themes like the archaeological profession as a whole, the value of archaeological heritage to society, the threats the archaeological record faces, national legislation and ethical issues should be included in the presentations as well. Only then is the public provided with a complete overview of all the aspects of Dutch archaeological heritage management; only then can the public place the objects in their wider context because they know where the objects come from, how they have been retrieved, what threats they face and how they are protected against both illegal and legal activities. This complete overview, or framework or context, can lead to increased public awareness and support. The aims of the museums do not include creating public awareness and support for Dutch archaeological heritage management. The archaeological profession is only paid attention to in the presentation of the Drents Museum. Furthermore, none of the presentations offer any information on the value of the archaeological heritage to society, the threats the archaeological heritage faces and ethical issues surrounding the archaeological profession. As I argued in the introduction, these are the topics that can generate public awareness and support for Dutch archaeological heritage management. The museums however focus only on the transfer of knowledge on the archaeological objects, past societies, materials and specific cultures. Furthermore, the museums do not aim to attract an as broad as possible audience but focus on school children, families and elder people. Therefore, I can conclude that the presentations in museums of Dutch archaeological collections do not fulfil their task of creating awareness and support for Dutch archaeological heritage management amongst an as broad as possible public.Show less
In 1985 is het ABC-vermoeden bedacht door de wiskundigen David Masser en Joseph Oesterl´e. Het vermoeden is een vrij eenvoudig probleem, waar geen diepe wiskunde voor nodig is om het te begrijpen....Show moreIn 1985 is het ABC-vermoeden bedacht door de wiskundigen David Masser en Joseph Oesterl´e. Het vermoeden is een vrij eenvoudig probleem, waar geen diepe wiskunde voor nodig is om het te begrijpen. Het vermoeden bewijzen echter is een heel ander verhaal. Daar gaan wij ons in deze scriptie dan ook niet aan wagen. Voordat we verder kunnen gaan, moeten we eerst enkele begrippen introduceren.Show less
In my thesis I will generalize a previous result of M. Maehara. In “Distances in a rigid unit-distance graph in the plane”1 he proved that the distances that occur between vertices in planar rigid...Show moreIn my thesis I will generalize a previous result of M. Maehara. In “Distances in a rigid unit-distance graph in the plane”1 he proved that the distances that occur between vertices in planar rigid unit-distance graphs are precisely the positive real algebraic numbers. A unit-distance graph is a framework of equal length bars which are connected in a flexible way at their endpoints. Such a framework is called rigid if it cannot be deformed without changing the length of the bars. There is also a stronger notion of rigidity which is called infinitesimal rigidity. This stronger notion asks that the framework cannot even be deformed infinitesimally without an infinitesimal change of the lengths of the bars. The picture on this page shows an example of a framework which is rigid but not infinitesimally rigid. The point with the arrows, cannot really move relative to the rest of the construction, although you can do this infinitesimally in the direction indicated by the arrows. A yet unanswered question was whether Maehara’s result also holds for infinitesimally rigid frameworks. It turned out to be true even with this stronger notion of rigidity. I will prove this in my thesis by showing that Maehara’s construction is infinitesimally rigid in most cases and give a different construction for the cases where Maehara’s construction isn’t infinitesimally rigid.Show less
Dynamics and number theory long were quite dinstinct fields of mathematics. Recently, however, progress has been made in the application of number theory to dynamics. This text seeks to elucidate a...Show moreDynamics and number theory long were quite dinstinct fields of mathematics. Recently, however, progress has been made in the application of number theory to dynamics. This text seeks to elucidate a small bit of this progress. The focus will be on the special case of discrete dynamical systems, which consist of a set X associated with a map φ : X → X. As in this text we will mainly consider X = P n (Q), the first section serves as an introduction to projective geometry. To provide the reader with some intuition, it starts out with P 1 (C) and eventually switches attention to P n (Q). The second section introduces the basic notions of discrete dynamics. In the third section, ‘height functions’ are defined. These are functions of the form h : P n (Q) → R, and they serve as the main tool in applying number theory to dynamics. As we restrict attention to projective spaces over Q, their definitions can remain quite simple; when working over an arbitrary number field K, one runs into the problem that the ring of integers of K may not be a principal ideal domain, making the definition of h substantially more complicated. For this, refer to [1]. After having defined them, some important properties of the height functions are derived. Section 4 then utilizes these properties to quickly derive some interesting theorems relating arithmetic to discrete dynamical systems.Show less
The aims of this thesis were threefold. The first and main aim was to examine urnfield – settlement relations for the Late Bronze Age and Early Iron Age within the research area of the eastern...Show moreThe aims of this thesis were threefold. The first and main aim was to examine urnfield – settlement relations for the Late Bronze Age and Early Iron Age within the research area of the eastern Netherlands. Secondly the overall picture of these relations within the eastern Netherlands was to be compared to the situation in the northern- and southern Netherlands. The third underlying aim was to illustrate the potential of the eastern Netherlands as an archaeological research area. In addition to the four already researched regions by Van Beek265 (these being south-western Salland; the northwestern and north-eastern Achterhoek and south-western Twente) north-eastern Twente was zoomed into as a fifth detail region. An examining of the present pre- and post-urnfield period settlements and burial location in combination with an inventory of the urnfield period settlements and urnfields and their locations within the physical landscape pointed out that both features underwent rather autonomous developments on the long-term scale and that both features had their own location within the physical landscape. The statement made in the previous sentence must be regarded as a trend and not as a law of Medes and Persians. Most of the settlements wandered around on the more central parts of sandy islands while many urnfields were situated more at the edges of these islands, mostly bordering on valleys that could have functioned as routes. Another substantial part of the urnfields was incorporated in an older burial site that was not necessarily situated at the edges of habitable units but also occurred more central within a territory. The urnfield of course clearly formed the constant factor within a presumed settlement territory compared to the wandering farmsteads that made use of the urnfield. Though the urnfield must have been very important to the Late Bronze Age/Early Iron communities in the bonding with the land and the ancestral world the “set-aside” locations of the urnfields suggest that they formed only a facet of life and not the centre of it. However by its monumental appearance and location on prominent spots within the landscape the urnfield must have been ever present on the background of daily life. A rough examining of the urnfields and settlements in the northern- and southern Netherlands showed that also in these regions attractive power is wielded on the urnfields by valleys present in the region. Habitation developments differ a little throughout the different regions. Where in Drenthe new terrains were colonized at the beginning of the urnfield period as a reaction on impoverishing soils and a growing population the settlements of the Meuse-Demer-Scheldt region in the south seem to have remained within the borders of the already existing territories or were founded at the cost of these territories, this despite comparable conditions as in the northern Netherlands. The eastern Netherlands shows examples of both events. Despite this difference in habitation developments urnfield – settlement relations in the northern- and southern Netherlands do not seem to have differed much from the situation in the eastern Netherlands. Though the urnfield is imputed a very central role in the southern Netherlands it seems at first hand that also here the urnfields were situated separate from the accompanying settlements instead of in the centre of them.Show less
In 2003 Arjen Doelman and Tasso J. Kaper published an article named Semistrong Pulse Interactions in a Class of Coupled Reaction-Diffusion Equations. In this article blow-up behavior was found in...Show moreIn 2003 Arjen Doelman and Tasso J. Kaper published an article named Semistrong Pulse Interactions in a Class of Coupled Reaction-Diffusion Equations. In this article blow-up behavior was found in simulations of two modified Gierer-Meinhardt systems. In this thesis this behavior in both of these systems is analyzed. First we will use rescalings to construct explicit first order expressions for the blow-up solutions. These theoretical results are illustrated by simulations.Show less
The plan area of Lanakerveld is located in South-Limburg, Maastricht. Three LBK sites, belonging to the Hezerwater cluster of settlements, have been excavated and use-wear analysis was performed on...Show moreThe plan area of Lanakerveld is located in South-Limburg, Maastricht. Three LBK sites, belonging to the Hezerwater cluster of settlements, have been excavated and use-wear analysis was performed on a selected sample of the flint assemblage. Based on the latest advances of research concerning the LBK as well as the relationship between the first farmers and the neighboring native communities in The Netherlands, the results of the functional analysis are used in combination with two previous studies from LBK sites. The fact that during these early stages of the “Neolithization” of the area the native groups remained separated for a long period is supported by making an effort to understand the circumstances and reasons of this long lasting “availability stage”. Also, it is argued that eventually the indigenous hunter gatherers adopted the “Neolithic package”, but piecemeal through a long process of adaptation. The first core argument is based on the novelty and originality of the LBK expressed in the use of flint tools in order to support the theory of a migration wave in this area. Second, an effort is made to examine and view the lifestyle, economy and culture of the incoming communities from the perspective of the indigenous groups in order to decompose the idea of the “superior farmers” versus the “favored hunter-gatherers.Show less
This thesis describes the results of an analysis of the Linearbandkeramik querns from Elsloo. A sample from the village (excavated material from the St. Jozefschool and the Riviusstraat) was...Show moreThis thesis describes the results of an analysis of the Linearbandkeramik querns from Elsloo. A sample from the village (excavated material from the St. Jozefschool and the Riviusstraat) was compared to the total assemblage from the cemetery. All stages in their biography were taken into account, from the selection of the raw materials to the way they were treated after their use-life had come to an end. The research has shown that the querns passed through an interesting life-cycle. Sandstone cobbles were collected from the gravel deposits of the river Meuse. A medium grain size was preferred. The ideal shape and size were obtained through hard percussion and the milling surface was pecked with a hammer stone to create a rough surface. The implements were subsequently employed to mill cereals, during which they were probably placed on a piece of hide or linen to catch the wheat grains and flour. Through use the milling surface became smooth, therefore it was recurrently rejuvenated. At some point their use came to an end. The grinding slabs from both assemblages differed substantially in their thickness, suggesting that exhaustion was not the sole reason why their use-life ended. The implements displayed a lot of fractures, of which some might were caused by rejuvenation, but for the majority this was not the case. From the number of fractures and their thicknesses was inferred that most of the artefacts were intentionally broken. Red ochre appeared on both assemblages and since use-wear caused by grinding this pigment was recognised on none of the tools, it must have been applied in an already powdered form. Traces were also found on some fractures, indicating that ochre was applied after they were broken. It did not occur on all implements and if it was present it seemed to be distributed over a random number of surfaces. However, in both assemblages it was mostly present on the milling surfaces. The comparison between the settlement and the cemetery assemblages revealed some interesting dissimilarities. First, there was a difference in the duration of use; the querns from the graveyard were used longer than those from the village. The second striking dissimilarity was visible in their treatment after the use-life had come to an end; the implements from the settlement were much more fragmented than those from the cemetery, while red ochre occurred more often and extensively on the graveyard assemblage. In this thesis I proposed four hypotheses to explain the peculiar practice to which the querns were subjected. The first two were applicable on both assemblages, while the others were more focused on the implements from the village. The first explanatory model linked the peculiar practice to a phase in the biography of the house. This biography was intertwined with the social life of the inhabitants and transitions to new stages could be accompanied by depositional activities. Possibly the quern fragments represent such a deposit. The second explanation linked the treatment to a specific attitude towards agriculture in LBK society. Possibly it refers to remnants of a Mesolithic way of thinking. Since milling was indirectly related to destruction of the forest it maybe was considered a dangerous activity and to appease the spirits the implements went through a special ritual at the end of their use-life. It can however also be interpreted in the opposite way and encompass an essentially Neolithic conception of the world. It seems likely that the principle of regeneration formed an essential part of Neolithic way of thinking. The activity to which the artefacts were subjected perhaps represents a deliberate offering concerned with giving sandstone back to the earth and thereby ensuring a continued supply. The third hypothesis interpreted the practice as an expression of a particular view on objects; possibly stones were considered as living subjects in LBK society. Problems like accidental breakage during rejuvenation perhaps were seen to reflect on the social relation between people and the stone, rather than on any analysis of the properties of the material itself. Maybe an accidentally broken quern was considered unusable or dangerous and therefore further destroyed and rubbed with red ochre. The fourth hypothesis connected the breakage of the implements from the settlement to a particular view on personhood. It may indicate that personhood was fractal and that the querns were imbued with a part of a person’s identity. Possibly they were deliberately fragmented in order to establish social relationships and the parts were kept as tokens of the bonds, forming inalienable links between all owners or users. It could be concluded that the treatment of the querns was informed by ways of understanding that differ from our own logic. The motivations were however rational for Linearbandkeramians, deriving from their particular perspective on the world. In this manner the peculiar ritual to which querns were subjected does not merely inform us about their roles in society, but also about LBK rationality. Obviously a deeper significance was assigned to the artefacts; their function for milling cereals was intertwined with other meanings, together encompassing their role in the lives of LBK people.Show less
