Browsing by master's degree program "Master's Programme in Mathematics and Statistics"

Sobolev functions generalize the concept of differentiability for functions beyond classical settings. The spaces of Sobolev functions are fundamental in mathematics and physics, particularly in the study of partial differential equations and functional analysis. This thesis provides an overview of construction of an extension operator on the space of Sobolev functions on a locally uniform domain. The primary reference is Luke Rogers' work "A Degree-Independent Sobolev Extension Operator". Locally uniform domains satisfy certain geometric properties, for example there are not too thin cusps. However locally uniform domains can possess highly non-rectifiable boundaries. For instance, the interior of the Koch snowflake represents a locally uniform domain with a non-rectifiable boundary. First we will divide the interior points of the complement of our locally uniform domain into dyadic cubes and use a collection of the cubes having certain geometric properties. The collection is called Whitney decomposition of the locally uniform domain. To extend a Sobolev function to a small cube in the Whitney decomposition one approach is to use polynomial approximations to the function on an nearby piece of the domain. We will use a polynomial reproducing kernel in order to obtain a degree independent extension operator. This involves defining the polynomial reproducing kernel in sets of the domain that we call here twisting cones. These sets are not exactly cones, but have some similarity to cones. Although a significant part of Rogers' work deals extensively with proving the existence of the kernel with the desired properties, our focus will remain in the construction of the extension operator so we will discuss the polynomial reproducing kernel only briefly. The extension operator for small Whitney cubes will be defined as convolution of the function with the kernel. For large Whitney cubes it is enough to set the extension to be 0. Finally the extension operator will be the smooth sum of the operators defined for each cube. Ultimately, since the domain is locally uniform the boundary is of measure zero and no special definition for the extension is required there. However it is necessary to verify that the extension "matches" the function correctly at the boundary, essentially that their k-1-th derivatives are Lipschitz there. This concludes the construction of a degree independent extension operator for Sobolev functions on a locally uniform domain.

Solving partial differential equations using methods of probability theory has a long history. In this thesis we show that the solutions of the conductivity equation in Lipschitz domain D with Neumann boundary conditions and uniformly elliptic, measurable conductivity parameter $\kappa$, can be represented using a Feynman-Kac formula for reflecting diffusion processes X on the domain D. We begin with history and connection to statistical experiments in Chapter 1. Chapter 2 starts by introducing Banach and Hilbert spaces with spectral theory of bounded operators, together with Hölder and Sobolev spaces. Sobolev spaces provide the right properties for the boundary data and solutions. In Chapter 3, we introduce the basics of stochastic processes, martingale and continuous semimartingales. We also need the theory of Markov processes, which Hunt and Feller processes are based on. Hunt processes are because of their correspondence with Dirichlet forms to define the reflecting diffusion process X. We also introduce the concept of local time of a process. In Chapter 4 we introduce Dirichlet forms, their correspondence with self-adjoint operators and Revuz measure. In Chapter 5, we introduce the conductivity equation and the Dirichlet-to-Neumann map $\Lambda_\kappa$. The goal of the Calderón's problem is to reconstruct the conductivity parameter $\kappa$ from the map $\Lambda_\kappa$, which is a difficult, non-linear and ill-posed inverse problem. The Chapter 6 constitutes the main body of the thesis, and here we prove the Feynman-Kac formula for solutions of the conductivity equation. We use the correspondence between the Dirichlet forms and selfadjoint operators, to define a semigroup (T_t) of solutions to a abstract Cauchy equation and from the semigroup (T_t) which we can associate by the Dunford-Pettis theorem a transition density function p for the reflecting diffusion process X. We show that using De-Giorgi-Nash-Moser estimates that p is Hölder continuous and defined everywhere. We also prove p converges exponentially to stationary distribution. We generalise the concept of boundary local times using the Revuz measure, and prove the occupation formula. These results together with the Skorohod decomposition for Lipschitz conductivities is used in the four part proof of Feynman-Kac formula. In Chapter 7, we introduce the boundary trace process, which is a pure jump process corresponding to the hitting times on the boundary of the reflecting diffusion process. We state that the trace process is the infinitesimal generator of the Dirichlet-to-Neumann map -\Lambda_\kappa and thus provides a probabilistic interpetation for Calderón's problem. We end with discussion on applications of the theory and potential directions for new research. The main references of the thesis are the articles of Piiroinen and Simon ''From Feynman–Kac Formulae to Numerical Stochastic Homogenization in Electrical Impedance Tomography'' and ''Probabilistic interpretation of the Calderón problem''.

The history of set theory is a long and winding road. From its inception, set theory has grown to become its own flourishing branch of mathematics with a pivotal role in the attempt to establish a foundation for all of mathematics and as such its influence is felt in every corner of the mathematical world as it exists today. This foundational effort, in the form of establishing new set theoretic axioms, is still ongoing and a big driving force behind this movement is the many unanswered questions that remain out of reach of the set theory of today. One of the most well known of these open questions is that of the Continuum Hypothesis. In this thesis we will first dive into the history of set theory, starting by looking at the role that infinity has played in the history of mathematics. From the ancients Greeks to Cantor who finally brings infinity into mathematics in a major way through set theory. We look at the development of a foundation for mathematics through the axiomatization of set theory and then focus on the role the Continuum Hypothesis played in this effort, leading up to Gödel’s and Cohen’s proofs that showed its independence and beyond that to the research being done today. We then turn our attention to potential candidates for new axioms that would solve the Continuum Hypothesis. First we take a closer look at Gödel’s constructible universe, in which the Continuum Hypothesis is true. We look at how it is built and consider the potential results of accepting the corresponding Axiom of Constructibility as a new axiom of set theory. In the final section we examine Chris Freiling’s proposed Axioms of Symmetry, which imply the negation of the Continuum Hypothesis. After looking at Freiling’s constructions in detail we consider the arguments for and against accepting them as new axioms.

Abstract Objective The objective of this thesis is to create methods to transform the most accessible digitalized version of an apartment, the floor plan, into a format that can be analyzed by statistical modeling and use the created data to find if there are any spatial or temporal effects in the geometry of apartments floor plans. Methods The first part of the thesis was created using a mix of computer vision image manipulation methods combined with text recognition. The second portion was performed using a oneway ANOVA model. Results With the computer vision portion, we were able to successfully classify a portion of the data, however, there is a lot of room for improvement due to the recognition had a lot of room for improvement. From the created data, we were able to identify some key differences concerning our parameters, location, and year of construction. The analysis however sufferers from a quite limited dataset, where few housing corporations play a large role in the final results, so it would be wise to repeat this experiment with a more comprehensive dataset for more accurate results

In the field of insurance mathematics, it is critical to control the solvency of an insurance company. In particular, calculating the probability of ruin, which is the probability that the company’s surplus falls below zero. In this thesis a review of the fundamentals of ruin theory, the modelling process and some results and methods for the estimation of ruin probabilities is made. Most of the theorems are taken from different bibliographical sources, but a good amount of the proofs presented are original, in order to provide a more rigorous and detailed explanation. A central focus of this thesis is the Pollaczek-Khinchine formula. This formula provides a solution for the probability distribution of the maximum potential loss of an insurance company in terms of convolutions of a particular function related to the claim sizes. Apart from the theoretical results that may be derived from it and its elegance, its usefulness lies in the ideas underlying it. Specially, the idea to understand the maximum potential loss of the company as the biggest of the historical records in the loss process. Using these ideas, a recursive approach to estimating ruin probabilities is ex- plained. This approach results in an easy to program and efficient bounds method which allows for any type of claim sizes (that is, the random variables that model how big are the claims of the insureds). The only restrictions imposed come from the fact that this discussion takes place within the Poisson model. This framework allows for various claim size distributions and models the number of claims as a Poisson process. Finally, two examples of light and heavy-tailed claim size distributions are simu- lated using this recursive approach. This shows the applicability of the method and the differences between light and heavy-tailed distributions with regards to the ruin probabilities that emerge from them.

Several extensions of first-order logic are studied in descriptive complexity theory. These extensions include transitive closure logic and deterministic transitive closure logic, which extend first-order logic with transitive closure operators. It is known that deterministic transitive closure logic captures the complexity class of the languages that are decidable by some deterministic Turing machine using a logarithmic amount of memory space. An analogous result holds for transitive closure logic and nondeterministic Turing machines. This thesis concerns the k-ary fragments of these two logics. In each k-ary fragment, the arities of transitive closure operators appearing in formulas are restricted to a nonzero natural number k. The expressivity of these fragments can be studied in terms of multihead finite automata. The type of automaton that we consider in this thesis is a two-way multihead automaton with nested pebbles. We look at the expressive power of multihead automata and the k-ary fragments of transitive closure logics in the class of finite structures called word models. We show that deterministic twoway k-head automata with nested pebbles have the same expressive power as first-order logic with k-ary deterministic transitive closure. For a corresponding result in the case of nondeterministic automata, we restrict to the positive fragment of k-ary transitive closure logic. The two theorems and their proofs are based on the article ’Automata with nested pebbles capture first-order logic with transitive closure’ by Joost Engelfriet and Hendrik Jan Hoogeboom. In the article, the results are proved in the case of trees. Since word models can be viewed as a special type of trees, the theorems considered in this thesis are a special case of a more general result.

Tässä tutkielmassa tarkastellaan, miten VAR-mallien avulla tehdyt menneistykset soveltuvat Tilastokeskuksen uusimman vuonna 2021 voimaan tulleen työvoimatutkimuksen aikasarjojen taaksepäin korjaukseen vuosille 2020 - 2000. Tarkasteltavat aikasarjat ovat työikäisten miesten ja naisten kuukausittaiset työllisyys- ja työttömyysluvut. Tilastokeskus on luotettavasti taaksepäin korjannut yllä mainitut aikasarjat vuosille 2020 - 2009. Tässä tutkielmassa verrataan estimoitujen VAR-mallien menneistyksiä Tilastokeskuksen virallisiin lukuihin, joita ovat ennen vuotta 2021 voimassa olleen Tilastokeskuksen työvoimatutkimuksen mukaiset luvut sekä Tilastokeskuksen taaksepäin korjatut aikasarjat vuosille 2020 - 2009. Taaksepäin korjauksella yhdenmukaistetaan ennen 2021 olevat aikasarjat uusimman vuoden 2021 voimaan tulleen työvoimatutkimuksen mukaiseksi. Tässä tutkielmassa ei pyritä etsimään parasta mahdollista tapaa taaksepäin korjata Tilastokeskuksen uusimman vuonna 2021 voimaan tulleen työvoimatutkimuksen aikasarjoja, eikä ottamaan kantaa sihen tulisiko VAR-mallien menneistyksillä taaksepäin korjata Tilastokeskuksen uusimman vuoden 2021 voimaan tulleen työvoimatutkimuksen aikasarjoja. Käytettävissäni oli systemaattisella satunnaisotannalla poimittu Tilastokeskuksen työvoimatutkimuksen aineisto, joka sisältää kuukausittaiset tiedot vastaajien työmarkkina asemasta sekä muista keskeisistä muuttujista ajalla 2000 tammikuu - 2023 helmikuu. Lisäksi käytettävissäni oli Työ- ja elinkeinoministeriön aineisto rekisterityöttömien lukumääristä ajalla 2000 tammikuu 2023 helmikuu. Näiden aineistojen pohjalta muodostin VAR-mallit, joiden avulla menneistin aikasarjoja käyttäen ehdollista odotusarvoa, ehdolla käytettävissä oleva aineisto. Mallien estimoinnissa sekä menneistämisessä käytin eksogeenisten muuttujien havaittuja arvoja, joita ovat Työ- ja elinkeinoministeriön rekisterityöttömien lukumäärät sekä osa keskeisistä muuttujista käytettävissä olleesta Tilastokeskuksen työvoimatutkimuksen aineistosta. Mallit estimoitiin pienimmän neliösumman menetelmällä. Varmistin mallien hyvyyden tarkistaen stationaarisuuden, testaamalla homoskedastisuutta, normaalisuutta ja tutkimalla standardoituja residuaaleja ja niiden auto- ja ristikorrelaatioita. Mitä enemmän VAR-mallin estimointiin käytettiin havaintoja, sitä lähempänä aikasarjojen menneistetyt arvot ovat Tilastokeskuksen työvoimatutkimuksen virallisia lukuja, ja niissä on vähemmän radikaaleja muutoksia. VAR-mallien avulla menneistetyissä aikasarjoissa on paljon samankaltaisuutta Tilastokeskuksen työvoimatutkimuksen virallisten lukujen kanssa. Menneistykset noudattavat samankaltaista kausivaihtelua kuin viralliset luvut. Lisäksi monen aikasarjan menneistykset noudattavat samankaltaista trendiä kuin viralliset luvut, joskin tasoerot ovat aika ajoin suuria. Työ- ja elinkeinoministeriön rekisterityöttömien lukumäärät on merkitsevänä selittävänä muuttujana estimoiduissa VAR-malleissa, ja ne selittävät merkittävästi menneistyksien arvoja.

The goal of the thesis is to prove the Dold-Kan Correspondence, which is a theorem stating that the category of simplicial abelian groups sAb and the category of positively graded chain complexes Ch+ are equivalent. The thesis also goes through these concepts mentioned in the theorem, starting with categories and functors in the first section. In this section, the aim is to give enough information about category theory, so that the equivalence of categories can be understood. The second section uses these category theoretical concepts to define the simplex category, where the objects are ordered sets n = { 0 -> 1 -> ... -> n }, where n is a natural number, and the morphisms are order preserving maps between these sets. The idea is to define simplicial objects, which are contravariant functors from the simplex category to some other category. Here is also given the definition of coface and codegeneracy maps, which are special kind of morphisms in the simplex category. With these, the cosimplicial (and later simplicial) identities are defined. These identities are central in the calculations done later in the thesis. In fact, one can think of them as the basic tools for working with simplicial objects. In the third section, the thesis introduces chain complexes and chain maps, which together form the category of chain complexes. This lays the foundation for the fourth section, where the goal is to form three different chain complexes out of any given simplicial abelian group A. These chain complexes are the Moore complex A*, the chain complex generated by degeneracies DA* and the normalized chain complex NA*. The latter two of these are both subcomplexes of the Moore complex. In fact, it is later on shown that there exists an isomorphism An = NAn +DAn between the abelian groups forming these chain complexes. This connection between these chain complexes is an important one, and it is proved and used later on in the seventh section. At this point in the thesis, all the knowledge for understanding the Dold-Kan Correspondence has been presented. Thus begins the forming of the functors needed for the equivalence, which the theorem claims to exist. The functor from sAb to Ch+ maps a simplicial abelian group A to its normalized chain complex NA*, the definition of which was given earlier. This direction does not require that much additional work, since most of it was done in the sections dealing with chain complexes. However, defining the functor in the opposite direction does require some more thought. The idea is to map a chain complex K* to a simplicial abelian group, which is formed using direct sums and factorization. Forming it also requires the definition of another functor from a subcategory of the simplex category, where the objects are those of the simplex category but the morphisms are only the injections, to the category of abelian groups Ab. After these functors have been defined, the rest of the thesis is about showing that they truly do form an equivalence between the categories sAb and Ch+.

In the fields of insurance and financial mathematics, robust modeling tools are essential for accura tely assessing extreme events. While standard statistical tools are effective for data with light-tailed distributions, they face significant challenges when applied to data with heavy-tailed characteristics. Identifying whether data follow a light- or heavy-tailed distribution is particularly challenging, often necessitating initial visualization techniques to provide insights into the nature of the distribution and guide further statistical analysis. This thesis focuses on visualization techniques, employing basic visual techniques to examine the tail behaviors of probability distributions, which are crucial for understanding the implications of extreme values in financial and insurance risk assessments. The study systematically applies a series of visualizations, including histograms, Q-Q plots, P-P plots, and Hill plots. Through the interpretation of these techniques on known distributions, the thesis aims to establish a simple framework for analyzing unknown data. Using Danish fire insurance data as our empirical data, this research simulates various probability distributions, emphasizing the visual distinction between light-tailed and heavy-tailed distributions. The thesis examines a range of distributions, including Normal, Exponential, Weibull, and Power Law, each selected for its relevance in modeling different aspects of tail behavior. The mathema tical exploration of these distributions provides a standard basis for assessing their effectiveness in capturing the nature of possibility of extreme events in data. The visual analysis of empirical data reveals the presence of heavy-tailed characteristics in the Danish fire insurance data and is not very well modeled by common light-tailed models such as the Normal and Exponential distributions. These findings underscore the need for more refined approaches that better accommodate the complexities of heavy-tailed phenomena. The thesis advocates for the further use of more advanced statistical tools and extreme value theory to asses heavy-tailed behaviour more accurately. Such tools are important for developing financial and insurance models that can effectively handle the extremities present in real-world data. This thesis contributes to the understanding and application of visualization techniques in the analysis of heavy-tailed data, laying a foundation to build on with more advanced tools to have more accurate risk management practices in the financial and insurance sectors.

Tutkielman aiheena oleva Whitneyn upotuslause osoittaa todeksi sen intuitiivisen ajatuksen, että jokainen sileä monisto voidaan upottaa johonkin euklidiseen avaruuteen. Lause on nimetty Hassler Whitneyn mukaan, joka ensimmäisenä todisti sen vuonna 1936. Tutkielma jakautuu kolmeen lukuun, joista ensimmäisessä käymme lävitse tarpeellisia yleisen topologian käsitteitä ja niistä johdettuja lauseita aloittaen tärkeimmistä eli topologisen avaruuden ja kuvauksen jatkuvuuden määritelmistä. Esiteltäviin käsitteisiin kuuluvat lisäksi esimerkiksi homeomorfismi, topologinen upotus, topologian kanta, aliavaruudet ja yhtenäisyys. Luvun lopuksi tutustutaan kompaktiuteen, joka on tutkielman kannalta suuressa osassa, koska sitä käytetään myöhemmin Whitnyeyn upotuslauseen todistuksessa. Toisessa luvussa keskitytään sileiden monistojen teoriaan siinä laajuudessa kuin se tutkielman kannalta on tarpeen. Luvun aluksi määritellään monisto, joka on numeroituvan kannan omaava, lokaalisti euklidinen Hausdorffin avaruus. Sen jälkeen määritellään monistolle sileä rakenne, eli maksimaalinen sileä kartasto, jonka avulla monistosta saadaan sileä monisto. Molemmista edellä mainituista annetaan useita esimerkkejä. Lisäksi todistetaan monistojen ja sileiden monistojen ominaisuuksia. Erityisesti todistetaan esikompaktien kantojen olemassaolo kaikille monistoille. Tämän jälkeen määritellään slieä ykkösen ositus ja todistetaan sen olemassaolo sileille monistoille. Käyttäen edellistä voidaan todistaa sileiden töyssy- ja tyhjennysfunktioiden olemassaolo sileille monistoille. Näitä funktioita käytetään myöhemmin Whitneyn upotuslauseen todistuksessa. Seuraavaksi määritellään monistojen välinen sileä kuvaus ja sen jälkeen derivaatio, tangenttivektori ja tangenttiavaruus, joiden avulla voidaan määritellä sileän kuvauksen differentiaali. Tämän jälkeen siirrytään sileän immersion määritelmään, joka tehdään dfferentiaalin avulla. Sileän immersion avulla saadaan sitten määriteltyä sileä upotus. Luvun viimeisessä osiossa määritellään alimonistot ja niihin liittyvät tasojoukot. Erityisesti määritellään vielä kriittiset pisteet ja arvot. Kolmannessa ja viimeisessä luvussa todistetaan ensin joitain aputuloksia ja sen jälkeen Sardin lause, jota käytetään myöhemmin Whitneyn upotuslauseen todistamiseen. Sardin lauseen mukaan sileiden monistojen välisen sileän kuvauksen kriittisten arvojen joukko on nollamittainen. Tämän jälkeen todistetaan vielä kaksi aputulosta ennen Whitneyn upotuslauseen todistusta. Näistä jälkimmäisessä osoitetaan, että jos on olemassa sileä upotus sileältä n-monistolta jollekin euklidiselle avaruudelle, niin on olemssa sileä upotus avaruudelle R2n+1. Näiden jälkeen päästään todistamaan tutkielman päätulos Whitneyn upotuslause, jonka mukaan jokaisella reunallisella tai reunattomalla sileällä n-monistolla on olemassa vahva sileä upotus euklidiseen avaruuteen R2n+1. Todistus jakautuu kahteen osaan, joissa ensimmäisessä todistetaan tapaus, jossa sileä monisto on kompakti. Tämä tehdään rakentamalla sileä upotus sileän töyssyfunktion avulla. Todistuksen toisessa osassa jaetaan ei-kompakti sileä monisto kompakteihin alimonistoihin tyhjennysfunktion avulla. Tämän jälkeen näiden sileät upotukset yhdistetään töyssyfunktion avulla uudeksi koko moniston peittäväksi sileäksi upotukseksi.