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Waring's problem is one of the two classical problems in additive number theory, the other being Goldbach's conjecture. The aims of this thesis are to provide an elementary, purely arithmetic solution of the Waring problem, to survey its vast history and to outline a few variations to it. Additive number theory studies the patterns and properties, which arise when integers or sets of integers are added. The theory saw a new surge after 1770, just before Lagrange's celebrated proof of the four-square theorem, when a British mathematician, Lucasian professor Edward Waring made the profound statement nowadays dubbed as Waring's problem: for all integers n greater than one, there exists a finite integer s such that every positive integer is the sum of s nth powers of non-negative integers. Ever since, the problem has been taken up by many mathematicians and state of the art techniques have been developed - to the point that Waring's problem, in a general sense, can be considered almost completely solved. The first section of the thesis works as an introduction to the problem. We give a profile of Edward Waring, state the problem both in its original form and using present-day language, and take a broad look over the history of the problem. The main emphasis is on the classical version of the problem, whereas the modern version is described in Section 5 with numerous other variations. In addition, generalizations to integer-valued polynomials and to general algebraic fields are described. Goldbach's conjecture is also briefly illustrated. The elementary solution of Waring's problem is presented in Sections 2 to 4. Historical perspective is carried through the thesis with the profiles of the key mathematicians to the solution. The proof presented is an improved and simplified version of Yuri Linnik's solution of Waring's problem. The second section provides the groundwork, an ingenious density argument by Lev Shnirelman, which is applied to the problem in the so called Fundamental lemma presented in Section 3. The proofs of the intermediate results needed to prove the lemma are presented in the following sections. The third section reduces the proof to an estimation of the number of solutions of a certain system of Diophantine equations. The final argument, longish induction is given at the end of the fourth section. Even though Waring's problem is solved, the progress made in the field is far from being idle. The plethora of variations and generalizations, especially Ideal Waring's problem, Modern Waring's problem and Waring–Goldbach problem are actively studied today. It is surprising how deep a problem with such a seemingly simple assertion can be. Conclusively, the challenge in this branch of mathematics is to develop new mathematical methods to prove and explain what seems so obvious.

In Cambodia, water has a special purpose as a source of life and livelihoods. Along with agriculture, fishing and forest use, industry, hydropower, navigation and tourism compete for the water resources. When rights and responsibilities related to essential and movable water are unclear, conflicts emerge easily. Therefore, water management is needed in order to plan and control the use of water resources. The international context is characterized by the Mekong River that flows through six countries. All of the countries by the river have very different roles and interests already depending on their geographical location. At the same time, water is also a tool for cooperation and peace. Locally, the water resources and related livelihoods create base for well-being, for economical and human resources in particular. They in turn are essential for the local people to participate and defend their rights to water use. They also help to construct the resource base of the state administration. Cambodia is highly dependent on the Mekong River. However, Cambodia has a volatile history whose effects can be seen for example in population structure, once suspended public institutions and weakened trust in the society. Relatively stable conditions came to the country as late as in the 1990s, therefore Cambodia for example has a weak status within the Mekong countries. This Master s thesis forms international, national and local interest groups of water use and analyzes their power relations and resources to affect water management. The state is seen as the salient actor as it has the formal responsibility of the water resources and of the coordination between the actions of different levels. In terms of water use this study focuses on production, in management on planning and in power relations on the resources. Water resources of Cambodia are seen consisting of the Mekong River and Tonle Sap Lake and the time span of the study is between the years 1991 and 2006. The material consists of semi-structured interviews collected during summer 2006 in Finland and in Cambodia as well as of literature and earlier studies. The results of the study show that the central state has difficulties to coordinate the actions of different actors because of its resource deficit and internal conflicts. The lessons of history and the vested interests of the actors of the state make it difficult to plan and to strengthen legislation. It seems that the most needed resources at the central state level are intangible as at the village level instead, the tangible resources (fulfilling the basic needs) are primarily important. The local decision-making bodies, NGOs and private sector mainly require legislation and legitimacy to support their role. However, the civil society and the international supporters are active and there are possibilities for new cooperation networks.

Hannukainen is an old mining area, at commune of Kolari in western Finnish Lapland. Rautaruukki Oy and Outokumpu Oy mined iron, copper and cold from the area in 1978 - 1990. Area is located in the Pajala-Kolari shear zone. Kaunisvaara mining project at Sweden is related to the same shear zone. Northland Mines Finland Oy is planning to restart mining of IOCG-type ore at Hannukainen area within the next couple of years. Quaternary deposits or water resources in Hannukainen area has not been examined in detail before. The mine development site is surrounded be three rivers, Valkeajoki in east, Kuerjoki in west and Äkäsjoki in south. The stratigraphical and hydrostratigraphical field studies in Hannukainen area were made in summer 2011. Stratigraphy of the Quaternary deposits was observed from the sections of Laurinoja open pit and from test pits dug to the area. Permeability of the sediment units were measured by using ring infiltration tests and calculated from grain size evaluations made from sediment samples. Groundwater was observed by evaluating the quality and representativeness of the groundwater monitoring wells, groundwater level measurements and also laboratory analyzes of main ions, trace-elements and isotope ratios of oxygen and hydrogen. In addition to chemistry samples, electric conductivity, pH, temperature and ratio of oxygen was measured with YSI multielectrode probe from rivers and open pit ponds. Groundwater - surface water interactions were observed by taking chemistry samples, measuring river bed temperatures and by installing minipiezometers. Quaternary deposits are mostly glaciofluvial and fluvial sands and gravels instead of tills. Sorted sandy sediments related to the braided river bars formed by Muonio Ice lake spillways are found from the present valleys of River Äkäsjoki, Valkeajoki and Kuerjoki. Sand and gravel sediments have good permeability and they host considerable amounts of water. Fine sediment units are thin and as discontinuous layers they host perched groundwater and confined groundwater. From the observed 23 groundwater monitoring wells only three represented the whole saturated aquifer and altogether six were other vice usable. The most remarkable problems were the lack of installation information and shortness of monitoring wells compared to the thick quaternary deposits in the area. Acid Mine Drainage -effect (AMD) of the old open pits, waste rock area and ore deposit itself could be detected from the groundwater monitoring wells in the old mining area and from the groundwater discharging to the river Äkäsjoki. Isotope ratios, river bed temperatures and hydraulic head difference in mini- piezometers reveal that groundwater is discharging from the mine development site to the rivers surrounding it. Hannukainen area is a challenging environment for mine siting. Hydrostratigraphy in the area is complicated, permeability of the Quaternary deposits is generally good, aquifer is multileveled and rivers surrounding the area get a significant amount of their water from groundwater forming in the mine development site. Most of the present groundwater monitoring wells are unusable, and in order to monitor the groundwater reliably some new monitoring wells should be installed. Major flow direction of surface and groundwater are from the mine development site to the rivers surrounding it. Classified groundwater areas should be outlined again, because present linings do not relate to true aquifers. Location of tailings area planned to the mine development site has been reconsidered.

The topic of thesis is the wave equation. The first chapter is introduction, the overview of the thesis is presented. The second chapter treats the transport equation, which is needed to solve the wave equation. In the third chapter we discuss the d’Alembert formula, and we prove the existence and uniqueness of solution. We treat the domain of dependence and region of influence. The last chapter concentrates on solving wave equations in high dimensions by Kirchhoff’s formula, method of descent and methods of spherical means.

Google Wave is a real-time communication and collaboration system introduced in 2009. The Wave system allows multiple users to view and modify hosted conversations called waves simultaneously. A wave conversation consists of a tree-like structure of messages. The messages can contain rich text, images and other attachments. Concurrency control is handled using a technique called operational transformation. It enables users to modify a wave simultaneously without acquiring any locks. Google has released Wave protocol descriptions and other technical documentation about the implementation of Wave. Google has also released a large portion of Wave source code under an open source license. Google ended the development of Google Wave in the late 2010 but the development of Wave is continued by the Apache Software Foundation. The open source version of Wave is called Wave in a Box. Wave in a Box contains the implementations of a web-based client and a Wave server. Wave is based on a federated client-server architecture. Although the Wave client is connected only to the user's own Wave service provider, the user can still communicate with users of other service providers. Wave servers are connected to each other and communicate with each other transparently to the user. In this sense the Wave architecture resembles email architecture. In this thesis we present the design for a new peer-to-peer-based Wave system called P2P-Wave. P2P-Wave was designed to be much like the original Wave. Operational transformation is used in both systems and the same functionality is available to the user in both systems. However, the systems are very different with respect to network communication. P2P-Wave has no centralized servers; all communication happens between users' own computers through the Pastry routing system. P2P-Wave also uses the Scribe multicast system and the distributed file system Past.

Maxwell’s equations are a set of equations which describe how electromagnetic fields behave in a medium or in a vacuum. This means that they can be studied from the perspective of partial differential equations as different kinds of initial value problems and boundary value problems. Because often in physically relevant situations the media are not regular or there can be irregular sources such as point sources, it’s not always meaningful to study Maxwell’s equations with the intention of finding a direct solution to the problem. Instead in these cases it’s useful to study them from the perspective of weak solutions, making the problem easier to study. This thesis studies Maxwell’s equations from the perspective of weak solutions. To help understand later chapters, the thesis first introduces theory related to Hilbert spaces, weak derivates and Sobolev spaces. Understanding curl, divergence, gradient and their properties is important for understanding the topic because the thesis utilises several different Sobolev spaces which satisfy different kinds of geometrical conditions. After going through the background theory, the thesis introduces Maxwell’s equations in section 2.3. Maxwell’s equations are described in both differential form and timeharmonic differential forms as both are used in the thesis. Static problems related to Maxwell’s equations are studied in Chapter 3. In static problems the charge and current densities are stationary in time. If the electric field and magnetic field are assumed to have finite energy, it follows that the studied problem has a unique solution. The thesis demonstrates conditions on what kind of form the electric and magnetic fields must have to satisfy the conditions of the problem. In particular it’s noted that the electromagnetic field decomposes into two parts, out of which only one arises from the electric and magnetic potential. Maxwell’s equations are also studied with the methods from spectral theory in Chapter 4. First the thesis introduces and defines a few concepts from spectral theory such as spectrums, resolvent sets and eigenvalues. After this, the thesis studies non-static problems related to Maxwell’s equations by utilising their time-harmonic forms. In time-harmonic forms the Maxwell’s equations do not depend on time but instead on frequencies, effectively simplifying the problem by eliminating the time dependency. It turns out that the natural frequencies which solve the spectral problem we study belong to the spectrum of Maxwell’s operator iA . Because the spectrum is proved to be discrete, the set of eigensolutions is also discrete. This gives the solution to the problem as the natural frequency solving the problem has a corresponding eigenvector with finite energy. However, this method does not give an efficient way of finding the explicit form of the solution.

The World Wide Web has emerged as a powerful tool and an entity that has formed a major part of everyday human life. Since web technology has become vital to humans, it is emerging as a platform where web applications mean more than they have ever meant. This demands for the user experience to be pleasurable and fast. But the disruptions in flow in traditional web applications stop the user from getting the desired experience. Initially web browsers were developed for displaying static pages linked together and user navigated to retrieve information from a server. This phenomenon is also known as multipage paradigm. It took more than two decades to evolve from a mere information-displaying entity to an application execution platform. The Ubiquitous nature of the web changed the trends, and complex web applications started emerging. These applications on user actions reload the entire web page from the server. Client and server communicate using synchronous communication mode. User action on the client triggers a call to the server and blocks the input/output until the server responds. Even after the advancement of web technologies, the web applications multipage paradigm remains the same and so do the problems of disruption in the user experience and application workflow. This thesis analyzes the design and development of modern web applications aimed at providing new single page paradigm and asynchronous communication between the client and the server to solve problems found in the multipage web application paradigm. This thesis also discusses the tools and technologies used in building single page web applications. The new single page paradigm introduces some challenges that are also considered in this thesis. Furthermore, the thesis covers that how the new paradigm can be used to develop applications on fragmented devices. Finally, the thesis describes the implementation of an example application aimed at producing a single page paradigm.

Welcome, in this thesis, some of the higher ranked, popular web content management software (CMS), namely Drupal, WordPress, Joomla and Plone, are compared by usability, from a developer’s perspective, and by performance of the resultant site build with these CMSs, to find out, among other topics, about their potential to build websites to different needs. This thesis tries to discover if a CMS exists, in this selected group, that is a clear choice above the others in both usability and performance. A substantial portion of source material for this research comes from measurements, and small demo systems built and used, in addition to any literature sources used and experience garnered from career as a web developer. In this thesis, we provide an overview of these four selected CMS; their characteristics, statistics and how they measure up to each other. And so doing, expand upon the still narrow research done in this field.

Tässä tutkielmassa vertaillaan vertaillaan keskenään web-sovellusten käyttöliittymien automaattitestauksen menetelmiä. Tutkielmassa selvitetään mitä erityistä webohjelmistojen testaamisessa on, ja miten niitä on mahdollista testata. Tutkielmassa vertaillaan myös muutamia testiautomaation tekniikoita ja niitä hyödyntäviä ohjelmointikirjastoja ja työkaluja. Cypress on Javascript-pohjainen testiautomaatio kehys web-ohjelmistojen testaamiseen. Selenium Webdriver on Wedriver protokollaa hyödyntävä selainautomaatiokirjasto, jota käytetään mm. selainohjelmistojen testaamiseen. Sikuli on automaatiokirjasto, joka perustuu kuvantunnistukseen. Näitä kirjastoja ja niiden soveltamia tekniikoita vertaillaan keskenään niiden webohjelmistojen testaamiseen soveltuvuuden, testien kehityksen ja suorituksen tehokkuuden sekä alusta- ja selaintuen näkökulmista.

Web-palvelut ovat tekniikkana suosittu tapa toteuttaa yritysten sisäiset ja ulkoiset liiketoimintaprosessit sähköisenä palveluna. Koostamisen avulla olemassa olevista web-palveluista voidaan rakentaa uusia monipuolisempia palveluja. Tässä työssä tutkitaan, minkälainen transaktionaalisuuden taso koosteisessa web-palvelussa voidaan saavuttaa käytettävissä olevilla teknologioilla. Tämä tehdään vertailemalla web-palveluiden transaktionaalisia ominaisuuksia tietokantateoriasta tuttuihin transaktion ACID-ominaisuuksiin, sarjallistuvuusteoriaan, sekä transaktiomalleihin. Työssä tutkitaan myös sitä, tarvitaanko web-palveluiden transaktionaalisessa koostamisessa ylipäätään tietokantojen teoriasta tuttuja käsitteitä ja malleja. Tutkimuksessa syvennytään niihin web-palveluiden koostamisessa käytettäviin menetelmiin ja protokolliin, jotka tarjoavat transaktionaalisia piirteitä liiketoimintaprosessien toteuttamiseksi. Nämä ovat web-arkkitehtuuripinon WS-AtomicTransaction- ja WS-BusinessActivity -protokollat. Tulosten analysoinnissa on käytetty apuna erään matkatoimiston web-palvelua. Web-palveluiden välisten transaktioiden ACID-ominaisuudet käytiin läpi kahden web-palvelun käyttötapauksen kautta. Tuloksista selvisi, että web-palveluiden välisissä transaktioissa toteutuu ainoastaan transaktion ACID-atomisuus. Samalla selvitettiin, miten tämä saavutettu transaktionaalisuuden taso käytännössä vaikuttaa web-palveluiden väliseen toimintaan. Tutkielman tuloksien vaikutuksia web-palveluiden väliseen kanssakäyntiin arvioitiin sekä palvelun toteuttajan, että loppukäyttäjän näkökulmasta. Tuloksien perusteella kävi ilmi, että vakavien ongelmien välttämiseksi toteuttajan on tunnettava sovelluksen ongelmakenttä erityisen hyvin ja ymmärrettävä se, miten koosteisen web-palvelun transaktionaalisuuden taso vaikuttaa kyseisessä ympäristössä palvelun toimintaan. Tutkielman lopussa esitellään tämän tutkielman tuloksena syntynyt eksklusiivinen malli, jonka avulla koosteeseen osallistuvien web-palveluiden välisissä transaktioissa voidaan saavuttaa globaali ACID-eristyvyys. Mallia soveltamalla web-palvelut voivat etsiä eksklusiivista kauppaa käyviä web-palveluja SOA-arkkitehtuurimallin mukaisesti UDDI-rekisteristä, ja suorittaa näiden kanssa häiriöttämästi sarjallistuvia transaktioita. ACM Computing Classification System (CCS): - Information systems~Web services - Information systems~Database transaction processing