Direct and Inverse Scattering for Beltrami Fields

We consider an obstacle scattering problem for linear Beltrami fields. A vector field is a linear Beltrami field if the curl of the field is a constant times itself. We study the obstacles that are of Neumann type, that is, the normal component of the total field vanishes on the boundary of the obstacle. We prove the unique solvability for the corresponding exterior boundary value problem, in other words, the direct obstacle scattering model. For the inverse obstacle scattering problem, we deduce the formulas that are needed to apply the singular sources method. The numerical examples are computed for the direct scattering problem and for the inverse scattering problem.

Mathematical Aspects of Financial Markets with Frictions

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property.

On Games on Non-Wellfounded Sets and Stationary Sets

In this thesis we study a few games related to non-wellfounded and stationary sets. Games have turned out to be an important tool in mathematical logic ranging from semantic games defining the truth of a sentence in a given logic to for example games on real numbers whose determinacies have important effects on the consistency of certain large cardinal assumptions. The equality of non-wellfounded sets can be determined by a so called bisimulation game already used to identify processes in theoretical computer science and possible world models for modal logic. Here we present a game to classify non-wellfounded sets according to their branching structure. We also study games on stationary sets moving back to classical wellfounded set theory. We also describe a way to approximate non-wellfounded sets with hereditarily finite wellfounded sets. The framework used to do this is domain theory. In the Banach-Mazur game, also called the ideal game, the players play a descending sequence of stationary sets and the second player tries to keep their intersection stationary. The game is connected to precipitousness of the corresponding ideal. In the pressing down game first player plays regressive functions defined on stationary sets and the second player responds with a stationary set where the function is constant trying to keep the intersection stationary. This game has applications in model theory to the determinacy of the Ehrenfeucht-Fraisse game. We show that it is consistent that these games are not equivalent.

Numerical computation of the module of a quadrilateral

The module of a quadrilateral is a positive real number which divides quadrilaterals into conformal equivalence classes. This is an introductory text to the module of a quadrilateral with some historical background and some numerical aspects. This work discusses the following topics: 1. Preliminaries 2. The module of a quadrilateral 3. The Schwarz-Christoffel Mapping 4. Symmetry properties of the module 5. Computational results 6. Other numerical methods Appendices include: Numerical evaluation of the elliptic integrals of the first kind. Matlab programs and scripts and possible topics for future research. Numerical results section covers additive quadrilaterals and the module of a quadrilateral under the movement of one of its vertex.

Yhteistoiminnallinen opetuspaketti lukion pitkän matematiikan Polynomifunktiot-kurssilla käytettäväksi

Tutkielmassa tarkastellaan yhteistoiminnallisen oppimisen soveltamismahdollisuuksia lukion pitkän matematiikan opettamiseen. Tutkielmassa on suunniteltu yhteistoiminnallinen opetuspaketti lukion pitkän matematiikan Polynomifunktiot-kurssille. Tutkielman ensimmäisessä teoriaosassa tarkastellaan yhteistoiminnallisen oppimisen periaatteita ja esitellään neljä yhteistoiminnallisen oppimisen menetelmää: rakenteellinen lähestymistapa, tiimioppiminen ryhmässä, ryhmäavusteinen yksilöllistäminen ja palapeli. Lisäksi teoriaosassa tuodaan esille niitä tekijöitä, jotka opettajan täytyy huomioida soveltaessaan esiteltyjä menetelmiä omaan opetukseensa sekä suunnitellessaan tehtäväkokonaisuuksia. Lopuksi tarkastellaan aiemmin tehtyjä tutkimuksia yhteistoiminnallisen oppimisen soveltamisesta matematiikan opetukseen. Tutkielman toisessa teoriaosassa esitellään klassisen algebran historiaa ja polynomifunktioihin liittyviä määritelmiä, lauseita ja niiden todistuksia sekä esimerkkejä. Opetuspaketti koostuu neljästä tehtäväkokonaisuudesta: epäyhtälön ratkaiseminen, polynomilaskennan kertaus, toisen asteen polynomifunktio ja -yhtälö sekä korkeamman asteen polynomifunktioiden tutkiminen. Lisäksi opetuspaketissa on yleinen esimerkki yhteistoiminnallisen oppitunnin rakenteesta. Opetuspaketin lopussa on raportti eräässä lukiossa suoritetusta testauksesta sekä sen tuloksista. Aiempien tutkimusten sekä tämän tutkielman yhteydessä tehdyn testauksen perusteella voidaan sanoa, että yhteistoiminnallinen oppiminen soveltuu lukion matematiikan opetukseen.

Tietokoneavusteinen arviointi kurssilla Diskreetin matematiikan perusteet

Matematiikan opetuksen kehittämiseen korkeakoulutasolla on monia tapoja. Tavoitteena on parantaa opiskelijoiden opiskelukokemuksia, jotta he oppisivat paremmin. Oppimisen arvioinnin on todettu vaikuttavan oppimiseen merkittävästi. Arviointi tapahtuu yleensä sen perusteella, kuinka hyvin opiskelija menestyy kokeissa. Näihin kokeisiin liittyy kuitenkin useita ongelmia; ne koostuvat usein muutamasta tehtävästä, eivätkä siten kata koko koealuetta. Lisäksi perinteinen koetilanne on kaukana siitä ympäristöstä, jossa opittuja taitoja on tarkoitus käyttää. Tässä työssä tutkittiin Aalto-yliopiston Teknillisen korkeakoulun kurssin Diskreetin matematiikan perusteet (DMP) arviointikäytännön uudistamista. Kurssi toteutettiin sulautuvan oppimisen mallin mukaisesti osin verkossa. Arvioinnissa painotettiin jatkuvaa harjoitustehtävien tekemistä ja suurin osa näistä tehtävistä toteutetiin tietokoneavusteisina verkkotehtävinä. Käytössä oli automaattisen tarkistamisen mahdollistava STACK-järjestelmä. Työ jakaantui kahteen osaan: arvioinnissa käytettävien STACK-tehtävien laatimiseen ja empiiriseen osuuteen, jossa tutkittiin kurssin onnistumista. Tutkimuksessa keskityttiin toisaalta siihen, miten käytetty arviointimenetelmä toimi ja toisaalta siihen, millaiseksi opiskelijat menetelmän kokivat. Kurssia varten toteutettiin yhteensä 67 STACK-tehtävää, joista 46 oli käytössä kurssilla. Lisäksi kurssilla oli 26 perinteistä kirjallista tehtävää. Käytetyn arviointimenetelmän toimivuutta tutkittiin vertaamalla kurssin tuloksia vuosien 2008 ja 2009 DMP-kurssien tuloksiin. Vertailun perusteella huomattiin, että opiskelijat olivat vuonna 2010 ratkaisseet selvästi enemmän harjoitustehtäviä kuin edellisinä vuosina. Myös arvosanan 0 prosentuaalinen osuus suhteessa kaikkiin annettuihin arvosanoihin pieneni. Opiskelijoiden kokemuksien tutkimista varten laadittiin kurssikokemuskysely. Kyselyssä esitettiin väittämiä liittyen STACK-tehtävien laatuun, tavoitteiden ja vaatimusten selkeyteen, arvioinnin asianmukaisuuteen, työmäärän asianmukaisuuteen, opiskelijoiden sitoutuneisuuteen, käytännön järjestelyihin ja sulautuvaan oppimiseen liittyen. Tulokset olivat erittäin positiivisia. Kaikenkaikkiaan kokeilukurssi sujui hyvin; arvointimenetelmä toimi ja opiskelijat olivat tyytyväisiä. Vertailun ja kyselyn perusteella tärkeimmiksi kehityksen kohteiksi nousivat STACK-tehtävien automaattinen palaute, perinteisten tehtävien pisteyttäminen ja jako perinteisten tehtävien ja STACK-tehtävien välillä.

Luokan P loogisesta karakterisoinnista

Deskriptiivisessä vaativuusteoriassa tutkitaan laskennan vaativuuteen liittyviä kysymyksiä logiikan työkalujen avulla. Tällöin käsitellään tilannetta, jossa laskennan syötteenä toimivat äärelliset mallit. Tässä kehyksessä erinäisiä vaativuusluokkia voidaan karakterisoida etsimällä logiikoita, joilla on kyseistä vaativuusluokkaa vastaava ilmaisuvoima. Klassiset esimerkit tällaisista tuloksista ovat Faginin esittämä epädeterministisen polynomiaalisen ajan karakterisaatio logiikan Σ_1^1 avulla ja Immermanin, Livchakin ja Vardin esittämä deterministisen polynomiaalisen ajan karakterisaatio ensimmäisen kertaluvun inflatorisen kiintopistelogiikan avulla. Tässä opinnäytetyössä tarkastellaan Gurevichin esittämää kysymystä polynomiaalisessa ajassa ratkeavien kielten luokan P vahvasta loogisesta karakterisaatiosta. Kyseinen kysymys on yksi äärellisen malliteorian haastavimpia ongelmia. Kysymyksen esittelyyn tarvittavan peruskoneiston läpikäynnin lisäksi tässä käsi- tellään myös sen yhteyksiä laskennan vaativuusteoriassa keskeiseen P-NP-ongelmaan. Gurevichin kysymyksestä voidaan esittää myös rajoitetumpia versioita, mikäli käsitellään tilannetta, jossa laskennan syötteenä voi olla vain kiinnitetyn malliluokan K malleja. Tällöin luokan P karakterisointi helpottuu, ainakin jos luokka K on riittävän suppea. Tässä opinnäytetyössä käydään läpi Grohen esittämä tulos siitä, että mikäli luokaksi K valitaan 3-yhtenäisten tasoverkkojen luokka, niin ensimmäisen kertaluvun inflatorinen kiintopistelogiikka karakterisoi polynomiaalisessa ajassa laskettavat kielet.

Tannakian Formalism Applied to the Category of Mixed Hodge Structures

Many problems in analysis have been solved using the theory of Hodge structures. P. Deligne started to treat these structures in a categorical way. Following him, we introduce the categories of mixed real and complex Hodge structures. Category of mixed Hodge structures over the field of real or complex numbers is a rigid abelian tensor category, and in fact, a neutral Tannakian category. Therefore it is equivalent to the category of representations of an affine group scheme. The direct sums of pure Hodge structures of different weights over real or complex numbers can be realized as a representation of the torus group, whose complex points is the Cartesian product of two punctured complex planes. Mixed Hodge structures turn out to consist of information of a direct sum of pure Hodge structures of different weights and a nilpotent automorphism. Therefore mixed Hodge structures correspond to the representations of certain semidirect product of a nilpotent group and the torus group acting on it.

Computational Methods for Reconstruction and Analysis of Genome-Scale Metabolic Networks

Metabolism is the cellular subsystem responsible for generation of energy from nutrients and production of building blocks for larger macromolecules. Computational and statistical modeling of metabolism is vital to many disciplines including bioengineering, the study of diseases, drug target identification, and understanding the evolution of metabolism. In this thesis, we propose efficient computational methods for metabolic modeling. The techniques presented are targeted particularly at the analysis of large metabolic models encompassing the whole metabolism of one or several organisms. We concentrate on three major themes of metabolic modeling: metabolic pathway analysis, metabolic reconstruction and the study of evolution of metabolism. In the first part of this thesis, we study metabolic pathway analysis. We propose a novel modeling framework called gapless modeling to study biochemically viable metabolic networks and pathways. In addition, we investigate the utilization of atom-level information on metabolism to improve the quality of pathway analyses. We describe efficient algorithms for discovering both gapless and atom-level metabolic pathways, and conduct experiments with large-scale metabolic networks. The presented gapless approach offers a compromise in terms of complexity and feasibility between the previous graph-theoretic and stoichiometric approaches to metabolic modeling. Gapless pathway analysis shows that microbial metabolic networks are not as robust to random damage as suggested by previous studies. Furthermore the amino acid biosynthesis pathways of the fungal species Trichoderma reesei discovered from atom-level data are shown to closely correspond to those of Saccharomyces cerevisiae. In the second part, we propose computational methods for metabolic reconstruction in the gapless modeling framework. We study the task of reconstructing a metabolic network that does not suffer from connectivity problems. Such problems often limit the usability of reconstructed models, and typically require a significant amount of manual postprocessing. We formulate gapless metabolic reconstruction as an optimization problem and propose an efficient divide-and-conquer strategy to solve it with real-world instances. We also describe computational techniques for solving problems stemming from ambiguities in metabolite naming. These techniques have been implemented in a web-based sofware ReMatch intended for reconstruction of models for 13C metabolic flux analysis. In the third part, we extend our scope from single to multiple metabolic networks and propose an algorithm for inferring gapless metabolic networks of ancestral species from phylogenetic data. Experimenting with 16 fungal species, we show that the method is able to generate results that are easily interpretable and that provide hypotheses about the evolution of metabolism.

Data mining for telecommunications network log analysis

Telecommunications network management is based on huge amounts of data that are continuously collected from elements and devices from all around the network. The data is monitored and analysed to provide information for decision making in all operation functions. Knowledge discovery and data mining methods can support fast-pace decision making in network operations. In this thesis, I analyse decision making on different levels of network operations. I identify the requirements decision-making sets for knowledge discovery and data mining tools and methods, and I study resources that are available to them. I then propose two methods for augmenting and applying frequent sets to support everyday decision making. The proposed methods are Comprehensive Log Compression for log data summarisation and Queryable Log Compression for semantic compression of log data. Finally I suggest a model for a continuous knowledge discovery process and outline how it can be implemented and integrated to the existing network operations infrastructure.

Cross-layer Assisted TCP Algorithms for Vertical Handoff

The ever expanding growth of the wireless access to the Internet in recent years has led to the proliferation of wireless and mobile devices to connect to the Internet. This has created the possibility of mobile devices equipped with multiple radio interfaces to connect to the Internet using any of several wireless access network technologies such as GPRS, WLAN and WiMAX in order to get the connectivity best suited for the application. These access networks are highly heterogeneous and they vary widely in their characteristics such as bandwidth, propagation delay and geographical coverage. The mechanism by which a mobile device switches between these access networks during an ongoing connection is referred to as vertical handoff and it often results in an abrupt and significant change in the access link characteristics. The most common Internet applications such as Web browsing and e-mail make use of the Transmission Control Protocol (TCP) as their transport protocol and the behaviour of TCP depends on the end-to-end path characteristics such as bandwidth and round-trip time (RTT). As the wireless access link is most likely the bottleneck of a TCP end-to-end path, the abrupt changes in the link characteristics due to a vertical handoff may affect TCP behaviour adversely degrading the performance of the application. The focus of this thesis is to study the effect of a vertical handoff on TCP behaviour and to propose algorithms that improve the handoff behaviour of TCP using cross-layer information about the changes in the access link characteristics. We begin this study by identifying the various problems of TCP due to a vertical handoff based on extensive simulation experiments. We use this study as a basis to develop cross-layer assisted TCP algorithms in handoff scenarios involving GPRS and WLAN access networks. We then extend the scope of the study by developing cross-layer assisted TCP algorithms in a broader context applicable to a wide range of bandwidth and delay changes during a handoff. And finally, the algorithms developed here are shown to be easily extendable to the multiple-TCP flow scenario. We evaluate the proposed algorithms by comparison with standard TCP (TCP SACK) and show that the proposed algorithms are effective in improving TCP behavior in vertical handoff involving a wide range of bandwidth and delay of the access networks. Our algorithms are easy to implement in real systems and they involve modifications to the TCP sender algorithm only. The proposed algorithms are conservative in nature and they do not adversely affect the performance of TCP in the absence of cross-layer information.

Mining Sequential Data – in Search of Segmental Structures

Segmentation is a data mining technique yielding simplified representations of sequences of ordered points. A sequence is divided into some number of homogeneous blocks, and all points within a segment are described by a single value. The focus in this thesis is on piecewise-constant segments, where the most likely description for each segment and the most likely segmentation into some number of blocks can be computed efficiently. Representing sequences as segmentations is useful in, e.g., storage and indexing tasks in sequence databases, and segmentation can be used as a tool in learning about the structure of a given sequence. The discussion in this thesis begins with basic questions related to segmentation analysis, such as choosing the number of segments, and evaluating the obtained segmentations. Standard model selection techniques are shown to perform well for the sequence segmentation task. Segmentation evaluation is proposed with respect to a known segmentation structure. Applying segmentation on certain features of a sequence is shown to yield segmentations that are significantly close to the known underlying structure. Two extensions to the basic segmentation framework are introduced: unimodal segmentation and basis segmentation. The former is concerned with segmentations where the segment descriptions first increase and then decrease, and the latter with the interplay between different dimensions and segments in the sequence. These problems are formally defined and algorithms for solving them are provided and analyzed. Practical applications for segmentation techniques include time series and data stream analysis, text analysis, and biological sequence analysis. In this thesis segmentation applications are demonstrated in analyzing genomic sequences.

XML Messaging for Mobile Devices

In recent years, XML has been widely adopted as a universal format for structured data. A variety of XML-based systems have emerged, most prominently SOAP for Web services, XMPP for instant messaging, and RSS and Atom for content syndication. This popularity is helped by the excellent support for XML processing in many programming languages and by the variety of XML-based technologies for more complex needs of applications. Concurrently with this rise of XML, there has also been a qualitative expansion of the Internet's scope. Namely, mobile devices are becoming capable enough to be full-fledged members of various distributed systems. Such devices are battery-powered, their network connections are based on wireless technologies, and their processing capabilities are typically much lower than those of stationary computers. This dissertation presents work performed to try to reconcile these two developments. XML as a highly redundant text-based format is not obviously suitable for mobile devices that need to avoid extraneous processing and communication. Furthermore, the protocols and systems commonly used in XML messaging are often designed for fixed networks and may make assumptions that do not hold in wireless environments. This work identifies four areas of improvement in XML messaging systems: the programming interfaces to the system itself and to XML processing, the serialization format used for the messages, and the protocol used to transmit the messages. We show a complete system that improves the overall performance of XML messaging through consideration of these areas. The work is centered on actually implementing the proposals in a form usable on real mobile devices. The experimentation is performed on actual devices and real networks using the messaging system implemented as a part of this work. The experimentation is extensive and, due to using several different devices, also provides a glimpse of what the performance of these systems may look like in the future.

Indexing Heterogeneous XML for Full-Text Search

XML documents are becoming more and more common in various environments. In particular, enterprise-scale document management is commonly centred around XML, and desktop applications as well as online document collections are soon to follow. The growing number of XML documents increases the importance of appropriate indexing methods and search tools in keeping the information accessible. Therefore, we focus on content that is stored in XML format as we develop such indexing methods. Because XML is used for different kinds of content ranging all the way from records of data fields to narrative full-texts, the methods for Information Retrieval are facing a new challenge in identifying which content is subject to data queries and which should be indexed for full-text search. In response to this challenge, we analyse the relation of character content and XML tags in XML documents in order to separate the full-text from data. As a result, we are able to both reduce the size of the index by 5-6\% and improve the retrieval precision as we select the XML fragments to be indexed. Besides being challenging, XML comes with many unexplored opportunities which are not paid much attention in the literature. For example, authors often tag the content they want to emphasise by using a typeface that stands out. The tagged content constitutes phrases that are descriptive of the content and useful for full-text search. They are simple to detect in XML documents, but also possible to confuse with other inline-level text. Nonetheless, the search results seem to improve when the detected phrases are given additional weight in the index. Similar improvements are reported when related content is associated with the indexed full-text including titles, captions, and references. Experimental results show that for certain types of document collections, at least, the proposed methods help us find the relevant answers. Even when we know nothing about the document structure but the XML syntax, we are able to take advantage of the XML structure when the content is indexed for full-text search.

IP Mobility in Wireless Operator Networks

Wireless network access is gaining increased heterogeneity in terms of the types of IP capable access technologies. The access network heterogeneity is an outcome of incremental and evolutionary approach of building new infrastructure. The recent success of multi-radio terminals drives both building a new infrastructure and implicit deployment of heterogeneous access networks. Typically there is no economical reason to replace the existing infrastructure when building a new one. The gradual migration phase usually takes several years. IP-based mobility across different access networks may involve both horizontal and vertical handovers. Depending on the networking environment, the mobile terminal may be attached to the network through multiple access technologies. Consequently, the terminal may send and receive packets through multiple networks simultaneously. This dissertation addresses the introduction of IP Mobility paradigm into the existing mobile operator network infrastructure that have not originally been designed for multi-access and IP Mobility. We propose a model for the future wireless networking and roaming architecture that does not require revolutionary technology changes and can be deployed without unnecessary complexity. The model proposes a clear separation of operator roles: (i) access operator, (ii) service operator, and (iii) inter-connection and roaming provider. The separation allows each type of an operator to have their own development path and business models without artificial bindings with each other. We also propose minimum requirements for the new model. We present the state of the art of IP Mobility. We also present results of standardization efforts in IP-based wireless architectures. Finally, we present experimentation results of IP-level mobility in various wireless operator deployments.