Η παρούσα διπλωματική αναφέρεται σε ελλειπτικά προβλήματα συνοριακών συνθηκών σε κανονικά πολύγωνα, εστιάζοντας στην αρκετά γενική εξίσωση Helmholtz. Θα εφαρμοσθούν οι γνωστές υπολογιστικές μέθοδοι επίλυσης ελλειπτικών προβλημάτων (όπως η παρεμβολή με τμηματικά κυβικά πολυώνυμα) και θα αναπτυχθούν και μέθοδοι που προκύπτουν από νέες μαθηματικές αναλύσεις του προβλήματος. / In this work we deal with elliptic boundary value problems which are defined in regular polygons. The numerical results presented in the defence are derived using well established methods, such as the finite differemces and the 2d collocation, as well as a new method introduced recently which appears to yield nice results.

In this thesis we propose general and simple methods for automatic verification of parameterized systems. These are systems consisting of an arbitrary number of identical processes or components. The number of processes defines the size of the system. A parameterized system may be regarded as an infinite family of instances, namely one for each size. The aim is to perform a parameterized verification, i.e. to verify that behaviors produced by all instances, regardless of their size, comply with some safety or liveness property. In this work, we describe three approaches to parameterized verification. First, we extend the Regular Model Checking framework to systems where components are organized in tree-like structures. For such systems, we give a methodology for computing the set of reachable configurations (used to verify safety properties) and the transitive closure (used to verify liveness properties). Next, we introduce a methodology allowing the verification of safety properties for a large class of parameterized systems. We focus on systems where components are organized in linear arrays and manipulate variables or arrays of variables ranging over bounded or numerical domains. We perform backwards reachability analysis on a uniform over-approximation of the parameterized system at hand. Finally, we suggest a new approach that enables us to reduce the verification of termination under weak fairness conditions to a reachability analysis for systems with simple commutativity properties. The idea is that reachability calculations (associated with safety) are usually less expensive then termination (associated with liveness). This idea can also be used for other transition systems and not only those induced by parameterized systems.

The complex behavior of computer systems offers many challenges for formal verification. The analysis quickly becomes difficult as the number of participating processes increases. A parameterized system is a family of systems parameterized on a number n, typically representing the number of participating processes. The uniform verification problem — to check whether a property holds for each instance — is an infinite-state problem. The automated analysis of parameterized and infinite-state systems has been the subject of research over the last 15–20 years. Much of the work has focused on safety properties. Progress in verification of liveness properties has been slow, as it is more difficult in general. In this thesis, we consider verification of parameterized and infinite-state systems, with an emphasis on liveness, in the verification framework called regular model checking (RMC). In RMC, states are represented as words, sets of states as regular expressions, and the transition relation as a regular relation. We extend the automata-theoretic approach to RMC. We define a specification logic sufficiently strong to specify systems representable using RMC, and linear temporal logic properties of such systems, and provide an automatic translation from a specification into an analyzable model. We develop acceleration techniques for RMC which allow more uniform and automatic verification than before, with greater power. Using these techniques, we succeed to verify safety and liveness properties of parameterized protocols from the literature. We present a novel reachability based verification method for verification of liveness, in a general setting. We implement the method for RMC, with promising results. Finally, we develop a framework for the verification of dynamic networks based on graph transformation, which generalizes the systems representable in RMC. In this framework we verify the latest version of the DYMO routing protocol, currently being considered for standardization by the IETF.

In this master thesis it is analyzed the system of four partial fluxions of the primary row of differential equations the row of which dwindles at the point of p+1, when p = 0. The system of partial fluxions of differential equations has been solved through the modified technique of summarized degree rows. Two cases were explored: general and when the system’s ratios initial matrix is a special structure matrix. The solutions of the system dependence on the ratios, which are near partial fluxions of the searched function, were explored. There were designed four families of the detached solutions and proved that each of them depends on one optional function of y and z variables. The structure of system of solutions at the malformation points depends not only on x degree, but also on y and z variables. There is no such type of malformation in the theory of dwindle simple differential equations, therefore this analyzed dwindle of the system of partial fluxion of differential equations is called quasi-regular malformation. Keywords: differential equation, partial derivation, quasi-regular malformation.

An integer-valued polynomial is a polynomial with rational coefficients that takes an integer value when evaluated at an integer. The binomial polynomials form a regular basis for the Z-module of all integer-valued polynomials. Using the idea of a p-ordering and a p-sequence, Bhargava describes a similar characterization for polynomials that are integer-valued on some subset of the integers. This thesis focuses on characterizing the polynomials that are integer-valued on the Fibonacci numbers. For a certain class of primes p, we give a formula for the p-sequence of the Fibonacci numbers and an algorithm for finding a p-ordering using Coelho and Parry’s results on the distribution of the Fibonacci numbers modulo powers of primes. Knowing the p-sequence, we can then find a p-local regular basis for the polynomials that are integer-valued on the Fibonacci numbers using Bhargava’s methods. A regular basis can be constructed from p-local bases for all primes p.

Bakgrund: Distriktssköterskor har i Sverige haft förskrivningsrätt sedan mitten av 1990-talet. Anledningen från början var att underlätta för läkarna, framför allt på landsbygden där läkartätheten var låg. Socialstyrelsen genomförde år 2000 en uppföljning av hur förskrivningsrätten användes. Författarna ville se om och/eller hur förskrivningen har förändrats samt hur förskrivningen ser ut i Sverige jämfört med andra länder. Syfte: Syftet med studien var att beskriva distriktssköterskans erfarenheter av att förskriva läkemedel i hemsjukvården samt deras uppfattning om förskrivningsrättens framtida utveckling inom distriktssköterskeproffessionen. Metod: Författarna valde att använda en kvalitativ metod i studien och som datainsamlingsmetod användes intervjuer med öppna frågor. Vid datanalysen användes en induktiv ansats där författarna bearbetade materialet med stöd av kvalitativ innehållsanalys. Resultat: Distriktssköterskan upplever att förskrivningsrätten är en bra kompetens att ha och att den underlättar i det dagliga arbetet. Olika faktorer påverkar hur väl förskrivningen fungerar. Möjlighet till regelbunden fortbildning och kontinuitet i arbetslaget främjar förskrivningen. Däremot saknas till viss del funktionella IT-system som underlättar förskrivningen. Dessutom finns många av de läkemedel som får förskrivas redan att tillgå utan recept. Därför upplever distriktssköterskan förskrivningsrätten ibland som en otillräcklig resurs inom distriktssköterskeprofessionen som skulle kunna utnyttjas på ett effektivare sätt. / Background: Since the middle of the 1990’s district nurses in Sweden have had the ability to prescribe drugs. In the beginning the reason was to help the physicians. The Swedish Social Board monitored in the year 2000 how this prescribing was used. The authors wanted to investigate if and/or how the prescribing has changed and also how the prescribing is used in Sweden compared to other countries. Aim: The aim with the study was to describe the district nurses’ experiences of prescribing drugs in homecare nursing and their opinion of the future development of prescribing in their profession. Method: The authors chose to use a qualitative method in the study and the method of data gathering was interviews based on open questions. At the analysis of data an inductive approach was used and the authors processed the material using qualitative content analysis. Results: District nurses experience that the prescribing facilitates their daily work. Access to regular education and continuity in the working team promotes the prescribing. Some essential IT systems are missing which would facilitate prescribing. Furthermore a large amount of the drugs that can be prescribed are already available without prescription. Therefore prescribing drugs is sometimes perceived as an insufficient resource in district nurses profession that could be utilized in a more effective way.

Recently, interest in the Graphics Processing Unit (GPU) for general purpose parallel applications development and research has grown. Much of the current research on the GPU focuses on the acceleration of regular problems, as irregular problems typically do not provide the same level of performance on the hardware. We explore the potential of the GPU by investigating four problems on the GPU with regular and/or irregular properties: lookback option pricing (regular), single-source shortest path (irregular), maximum flow (irregular), and the task matching problem using multi-swarm particle swarm optimization (regular with elements of irregularity). We investigate the design, implementation, optimization, and performance of these algorithms on the GPU, and compare the results. Our results show that the regular problem achieves greater performance and requires less development effort than the irregular problems. However, we find the GPU to still be capable of providing high levels of acceleration for irregular problems.

We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.

This thesis focuses on constraints for membership in formal languages under both the systematic search and stochastic local search approaches to constraint programming (CP). Such constraints are very useful in CP for the following three reasons: They provide a powerful tool for user-level extensibility of CP languages. They are very useful for modelling complex work shift regulation constraints, which exist in many shift scheduling problems. In the analysis, testing, and verification of string-manipulating programs, string constraints often arise. We show in this thesis that CP solvers with constraints for membership in formal languages are much more suitable than existing solvers used in tools that have to solve string constraints. In the stochastic local search approach to CP, we make the following two contributions: We introduce a stochastic method of maintaining violations for the regular constraint and extend our method to the automaton constraint with counters. To improve the usage of constraints for which there exists no known constant-time algorithm for neighbour evaluation, we introduce a framework of using solution neighbourhoods, and give an efficient algorithm of constructing a solution neighbourhood for the regular constraint. In the systematic search approach to CP, we make the following two contributions: We show that there may be unwanted consequences when using a propagator that may underestimate a cost of a soft constraint, as the propagator may guide the search to incorrect (non-optimum) solutions to an over-constrained problem. We introduce and compare several propagators that compute correctly the cost of the edit-distance based soft-regular constraint. We show that the context-free grammar constraint is useful and introduce an improved propagator for it.

The work presented in this thesis discusses various formal language formalisms that extend classical formalisms like regular expressions and context-free grammars with additional abilities, most relating to order. This is done while focusing on the impact these extensions have on the efficiency of parsing the languages generated. That is, rather than taking a step up on the Chomsky hierarchy to the context-sensitive languages, which makes parsing very difficult, a smaller step is taken, adding some mechanisms which permit interesting spatial (in)dependencies to be modeled. The most immediate example is shuffle formalisms, where existing language formalisms are extended by introducing operators which generate arbitrary interleavings of argument languages. For example, introducing a shuffle operator to the regular expressions does not make it possible to recognize context-free languages like anbn, but it does capture some non-context-free languages like the language of all strings containing the same number of as, bs and cs. The impact these additions have on parsing has many facets. Other than shuffle operators we also consider formalisms enforcing repeating substrings, formalisms moving substrings around, and formalisms that restrict which substrings may be concatenated. The formalisms studied here all have a number of properties in common. They are closely related to existing regular and context-free formalisms. They operate in a step-wise fashion, deriving strings by sequences of rule applications of individually limited power. Each step generates a constant number of symbols and does not modify parts that have already been generated. That is, strings are built in an additive fashion that does not explode in size (in contrast to e.g. Lindenmayer systems). All languages here will have a semi-linear Parikh image. They feature some interesting characteristic involving order or other spatial constraints. In the example of the shuffle multiple derivations are in a sense interspersed in a way that each is unaware of. All of the formalisms are intended to be limited enough to make an efficient parsing algorithm at least for some cases a reasonable goal. This thesis will give intuitive explanations of a number of formalisms fulfilling these requirements, and will sketch some results relating to the parsing problem for them. This should all be viewed as preparation for the more complete results and explanations featured in the papers given in the appendices. / Denna avhandling diskuterar utökningar av klassiska formalismer inom formella språk, till exempel reguljära uttryck och kontextfria grammatiker. Utökningarna handlar på ett eller annat sätt omordning, och ett särskilt fokus ligger på att göra utökningarna på ett sätt som dels har intressanta spatiala/ordningsrelaterade effekter och som dels bevarar den effektiva parsningen som är möjlig för de ursprungliga klassiska formalismerna. Detta står i kontrast till att ta det större steget upp i Chomsky-hierarkin till de kontextkänsliga språken, vilket medför ett svårt parsningsproblem. Ett omedelbart exempel på en sådan utökning är s.k. shuffle-formalismer. Dessa utökar existerande formalismer genom att introducera operatorer som godtyckligt sammanflätar strängar från argumentspråk. Om shuffle-operator introduceras till de reguljära uttrycken ger det inte förmågan att känna igen t.ex. det kontextfria språket anbn, men det fångar istället vissa språk som inte är kontextfria, till exempel språket som består av alla strängar som innehåller lika många a:n, b:n och c:n. Sättet på vilket dessa utökningar påverkar parsningsproblemet är mångfacetterat. Utöver dessa shuffle-operatorer tas också formalismer där delsträngar kan upprepas, formalismer där delsträngar flyttas runt, och formalismer som begränsar hur delsträngar får konkateneras upp. Formalismerna som tas upp här har dock vissa egenskaper gemensamma. De är nära besläktade med de klassiska reguljära och kontextfria formalismerna. De arbetar stegvis, och konstruerar strängar genom successiva applikationer av individuellt enkla regler. Varje steg genererar ett konstant antal symboler och modifierar inte det som redan genererats. Det vill säga, strängar byggs additivt och längden på dem kan inte explodera (i kontrast till t.ex. Lindenmayer-system). Alla språk som tas upp kommer att ha en semi-linjär Parikh-avbildning. De har någon instressant spatial/ordningsrelaterad egenskap. Exempelvis sättet på vilket shuffle-operatorer sammanflätar annars oberoende deriveringar. Alla formalismerna är tänkta att vara begränsade nog att det är resonabelt att ha effektiv parsning som mål. Denna avhandling kommer att ge intuitiva förklaring av ett antal formalismer som uppfyller ovanstående krav, och kommer att skissa en blandning av resultat relaterade till parsningsproblemet för dem. Detta bör ses som förberedande inför läsning av de mer djupgående och komplexa resultaten och förklaringarna i de artiklar som finns inkluderade som appendix.