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Browsing by study line "Matematiikka"

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  • Tiusanen, Mikko (2021)
    This thesis provides a program to compute minimal values of polynomials of degree two to get a transcendence measure for e, Napier’s (Neper’s) number. This is an indication of how close to zero some non-zero integer coefficient polynomial can come at e: Since e is transcendental, no such polynomial can actually attain zero. The thesis concentrates on the case of second degree polynomials. The program is written in the R5RS dialect of the programming language Scheme, a reasonably modern LISP version that offers integer and rational arithmetic only limited by the memory. The program is validated by comparison to results computed by hand, using another programming language, J. The program was also rewritten in the programming language C, providing the relevant rational number arithmetic by the library gmp. Performance characteristics of the programs are briefly compared. To appoximate the needed value of e, the programs use continued fractions. Relevant mathematical background for the subject is presented. Symmetries of the grid that represents the polynomial coefficients are used to speed up the computation: The symmetry with respect to the origin effectively halves the number of polynomial evaluations needed. The others employed are interesting but less significant. The idea came from a description of a chess endgame program of Ken Thompson by Jon Bentley, although it is likely to be older. These lead to employing the concept of a layer of polynomials with integer coefficients, the border of a punctured grid, in a sense. Layers turned out to rather nicely fit the handling of the accuracy requirements for e. The complexity of the program is still bounded from below by the number of points in the grid of polynomials, partitioned by the layers, making it bounded from below by a second degree polynomial in H, Omega(H²), where H is the natural number bounding the absolute values of the coefficients of the second and first power of e in the polynomial. The program is reasonably easily changed to handle any transcendental number other than e, in particular, if there is a convenient continued fraction to compute approximations to the number. Strictly speaking, the program does not compute the full transcendence measure, that is, a safe upper bound for each integer coefficient polynomial considered, but if this larger output is actually needed, only a minor change in the program is required.
  • Duevski, Teodor (2019)
    In this thesis we model the term structure of zero-coupon bonds. Firstly, in the static setting by norm optimization Hilbert space techniques and starting from a set of benchmark fixed income instruments, we obtain a closed from expression for a smooth discount curve. Moving on to the dynamic setting, we describe the stochastic modeling of the fixed income market. Finally, we introduce the Heath-Jarrow-Morton (HJM) methodology. We derive the evolution of zero-coupon bond prices implied by the HJM methodology and prove the HJM drift condition for non arbitrage pricing in the fixed income market under a dynamic setting. Knowing the current discount curve is crucial for pricing and hedging fixed income securities as it is a basic input to the HJM valuation methodology. Starting from the non arbitrage prices of a set of benchmark fixed income instruments, we find a smooth discount curve which perfectly reproduces the current market quotes by minimizing a suitably defined norm related to the flatness of the forward curve. The regularity of the discount curve estimated makes it suitable for use as an input in the HJM methodlogy. This thesis includes a self-contained introduction to the mathematical modeling of the most commonly traded fixed income securities. In addition, we present the mathematical background necessary for modeling the fixed income market in a dynamic setting. Some familiarity with analysis, basic probability theory and functional analysis is assumed.
  • Koivurova, Antti (2021)
    This thesis surveys the vast landscape of uncertainty principles of the Fourier transform. The research of these uncertainty principles began in the mid 1920’s following a seminal lecture by Wiener, where he first gave the remark that condenses the idea of uncertainty principles: "A function and its Fourier transform cannot be simultaneously arbitrarily small". In this thesis we examine some of the most remarkable classical results where different interpretations of smallness is applied. Also more modern results and links to active fields of research are presented.We make great effort to give an extensive list of references to build a good broad understanding of the subject matter.Chapter 2 gives the reader a sufficient basic theory to understand the contents of this thesis. First we talk about Hilbert spaces and the Fourier transform. Since they are very central concepts in this thesis, we try to make sure that the reader can get a proper understanding of these subjects from our description of them. Next, we study Sobolev spaces and especially the regularity properties of Sobolev functions. After briefly looking at tempered distributions we conclude the chapter by presenting the most famous of all uncertainty principles, Heisenberg’s uncertainty principle.In chapter 3 we examine how the rate of decay of a function affects the rate of decay of its Fourier transform. This is the most historically significant form of the uncertainty principle and therefore many classical results are presented, most importantly the ones by Hardy and Beurling. In 2012 Hedenmalm gave a beautiful new proof to the result of Beurling. We present the proof after which we briefly talk about the Gaussian function and how it acts as the extremal case of many of the mentioned results.In chapter 4 we study how the support of a function affects the support and regularity of its Fourier transform. The magnificent result by Benedicks and the results following it work as the focal point of this chapter but we also briefly talk about the Gap problem, a classical problem with recent developments.Chapter 5 links density based uncertainty principle to Fourier quasicrystals, a very active field of re-search. We follow the unpublished work of Kulikov-Nazarov-Sodin where first an uncertainty principle is given, after which a formula for generating Fourier quasicrystals, where a density condition from the uncertainty principle is used, is proved. We end by comparing this formula to other recent formulas generating quasicrystals.
  • Stowe, William (2023)
    Is it possible to color R^2 with 2 colors in such a way that the vertices of any unit equilateral triangle are not all the same color. This thesis seeks to answer questions of this kind in the field of Euclidean Ramsey Theory. We begin by defining that a finite configuration A is k-Ramsey in R^n if any k-coloring of R^n has a monochromatic set that is congruent to A. We both prove and disprove this property for various configurations, dimensions, and numbers of colors. This includes a discussion of the problem of finding the chromatic number of the plane, and the connection of k-Ramsey problems to immersion of unit distance graphs. We then attempt to generalize this property to different equivalence relations other than congruence and study how this affects which configurations are guaranteed monochromatic. Following from the Hales-Jewett Theorem, this line of inquiry peaks with a discussion of Gallai’s Theorem, which says that translation and scaling form a sufficient set of group actions to guarantee all configurations k-Ramsey for any k, in any dimension. We then turn our attention to the property of Ramsey-ness. A configuration A is said to be Ramsey if for any number of colors k, there exists a dimension n such that A is k-Ramsey in R^n . We show that if a configuration is Ramsey, then it must be embeddable in the surface of a sphere of some dimension. Further, we show that any brick, the Cartesian product of intervals, is Ramsey, and thus any subset of a brick is Ramsey. Finally, we prove that any triangle configuration is Ramsey.
  • Nyman, Valtteri (2022)
    Tässä tutkielmassa esitellään lyhyesti PCP-teoreema, minkä jälkeen tutkielmassa pala palalta käydään läpi teoreeman todistamiseen tarvittavia työkaluja. Tutkielman lopussa todistetaan PCP-teoreema. Vaativuusluokka PCP[O(log n), O(1)] sisältää ne ongelmat, joilla on olemassa todistus, josta vakio määrän bittejä lukien probabilistinen Turingin kone kykenee ratkaisemaan ongelman käyttäen samalla vain logaritmisen määrän satunnaisuutta suhteessa syötteen kokoon. PCP-teoreema väittää vaativuusluokan NP kuuluvan vaativuusluokkaan PCP[O(log n), O(1)]. Väritys on funktio, joka yhdistää kuhunkin joukon muuttujaan jonkin symbolin. Rajoite joillekin muuttujille on lista symboleista, joita rajoite sallii asetettavan muuttujille. Jos väritys asettaa muuttujille vain rajoitteen sallimia symboleja, rajoite on tyytyväinen väritykseen. Optimointi-ongelmat koskevat sellaisten väritysten etsimistä, että mahdollisimman moni rajoite joukosta rajoitteita on tyytyväinen väritykseen. PCP-teoreemalla on yhteys optimointi-ongelmiin, ja tätä yhteyttä hyödyntäen tutkielmassa todistetaan PCP-teoreema. Tutkielma seuraa I. Dinurin vastaavaa todistusta vuoden 2007 artikkelista The PCP Theorem by Gap Amplification. Rajoiteverkko on verkko, jonka kuhunkin kaareen liittyy jokin rajoite. Rajoiteverkkoon liittyy lisäksi aakkosto, joka sisältää ne symbolit, joita voi esiintyä verkon rajoitteissa ja värityksissä. Tutkielman päälauseen avulla kyetään kasvattamaan rajoiteverkossa olevien värityksiin tyytymättömien rajoitteiden suhteellista osuutta. Päälause takaa, että verkon koko säilyy samassa kokoluokassa, ja että verkon aakkoston koko ei muutu. Lisäksi jos verkon kaikki rajoitteet ovat tyytyväisiä johonkin väritykseen, päälauseen tuottaman verkon kaikki rajoitteet ovat edelleen tyytyväisiä johonkin väritykseen. Päälause koostetaan kolmessa vaiheessa, joita kutakin vastaa tutkielmassa yksi osio. Näistä ensimmäisessä, tutkielman osiossa 4, verkon rakenteesta muovataan sovelias seuraavia osioita varten. Toisessa vaiheessa, jota vastaa osio 6, verkon kävelyitä hyödyntäen kasvatetaan tyytymättömien rajoitteiden lukumäärää, mutta samalla verkon aakkosto kasvaa. Kolmannessa vaiheessa, osiossa 5, aakkoston koko saadaan pudotettua kolmeen sopivan algoritmin avulla. Osiossa 7 kootaan päälause ja todistetaan lausetta toistaen PCP-teoreema.
  • Karvonen, Elli (2021)
    The topological data analysis studies the shape of a space at multiple scales. Its main tool is persistent homology, which is based on other homology theory, usually simplicial homology. Simplicial homology applies to finite data in real space, and thus it is mainly used in applications. This thesis aims to introduce the theories behind persistent homology and its application, image completion algorithm. Persistent homology is motivated by the question of which scale is the most essential to study data shape. A filtration contains all scales we want to explore, and thus it is an essential tool of persistent homology. The thesis focuses on forming a filtaration from a Delaunay triangulation and its subcomplexes, alpha-complexes. We will found that these provide sufficient tools to consider homology classes birth and deaths, but they are not particularly easy to use in practice. This observation motivates to define a regional complement of the dual alpha graph. We found that its components' and essential homology classes' birth and death times correspond. The algorithm utilize this observation to complete images. The results are good and mainly as could be expected. We discuss that algorithm has potential since it does need any training or other input parameters than data. However, future studies are needed to imply it, for example, in three-dimensional data.
  • Moilanen, Eero (2022)
    In the thesis ”P-Fredholmness of Band-dominated Operators, and its Equivalence to Invertibility of Limit Operators and the Uniform Boundedness of Their Inverses”, we present the generalization of the classical Fredholm-Riesz theory with respect to a sequence of approximating projections on direct sums of spaces. The thesis is a progessive introduction to understanding and proving the core result in the generalized Fredholm-Riesz theory, which is stated in the title. The stated equivalence has been further improved and it can be generalized further by omitting either the initial condition of richness of the operator or the uniform boundedness criterion. Our focal point is on the elementary form of this result. We lay the groundwork for the classical Fredholm-Riesz theory by introducing compact operators and defining Fredholmness as invertibility on modulo compact operators. Thereafter we introduce the concept of approximating projections in infinite direct sums of Banach spaces, that is we operate continuous operators with a sequence of projections which approach the identity operator in the limit and examine whether we have convergence in the norm sense. This method yields us a way to define P-compactness, P-strong converngence and finally PFredholmness. We introduce the notion of limit operators operators by first shifting, then operating and then shifting back an operator with respect to an element in a sequence and afterwards investigating what happens in the P-strong limit of this sequence. Furthermore we define band-dominated operators as uniform limits of linear combinations of simple multiplication and shift operators. In this subspace of operators we prove that indeed for rich operators the core result holds true.
  • Jylhä, Lotta (2022)
    Pólyan lauseen mukaan verkon Z^d symmetrinen satunnaiskävely on palautuva, jos d < 3 ja poistuva, jos d ≥ 3. Alunperin Georg Pólyan todistamalle lauseelle on ajan kuluessa muodostunut erilaisia todistusmenetelmiä. Tässä tutkielmassa syvennytään näistä kahteen toisiaan täydentävään menetelmään ja todistetaan Pólyan lause niiden avulla. Luvussa 5.1 Pólyan lauseelle esitetään laskennallinen todistus, joka tarjoaa yksinkertaisen ja konkreettisen tavan tutkia säännöllisen verkon satunnaiskävelyn käyttäytymistä. Luvussa 5.2 esitettävän virtauksen teorian avulla voidaan Pólyan lauseen lisäksi tutkia satunnaiskävelyn käyttäytymistä laajemmin eri verkoissa. Tarvittavat taustatiedot verkosta, Markovin ketjusta ja satunnaiskävelystä esitetään luvuissa 2 ja 3. Pólyan lauseen todistus on jaettu kahteen eri lukuun. Lauseen todistus alkaa luvusta 5.1, jossa verkon syklien ja polkujen lukumääriä tutkimalla Pólyan lause osoitetaan verkolle Z^d, missä d ≤ 3. Kombinatorinen todistus on idealtaan yksinkertainen, mutta siinä tehtävä arvio vaatii syvällisempää perustelua. Tutkielmassa tämä arvio toteutetaan Robbinsin kaavalla, joka on tarkempi arvio kirjallisuudessa useammin käytetylle Stirlingin kaavalle. Robbinsin kaava osoitetaan luvussa 4. Luvussa 5.2 esitetään verkon virtauksen teoria, jonka avulla Pólyan lause todistetaan verkolle Z^d, missä d > 3. Verkon virtauksen ja satunnaiskävelyn yhteys löytyy virtaukseen liittyvästä energian käsitteestä. Osoittautuu, että verkon virtauksista energialtaan pienimmän virtauksen energia riippuu verkon satunnaiskävelyn käyttäytymisestä. Tulos osoitetaan ensin äärelliselle verkolle, josta se johdetaan koskemaan ääretöntä verkkoa verkkoon liittyvän kontraktion käsitteen avulla. Luvussa 6 Pólyan lauseen merkitys korostuu, kun virtauslauseen avulla osoitetaan, että satunnaiskävelyn poistuvuus säilyy verkkojen kvasi-isometriassa. Tätä varten esitetään virtauslauseen seurauksia ja tarvittavat taustatiedot kvasi-isometriasta
  • Järviniemi, Olli (2021)
    This thesis is motivated by the following questions: What can we say about the set of primes p for which the equation f(x) = 0 (mod p) is solvable when f is (i) a polynomial or (ii) of the form a^x - b? Part I focuses on polynomial equations modulo primes. Chapter 2 focuses on the simultaneous solvability of such equations. Chapter 3 discusses classical topics in algebraic number theory, including Galois groups, finite fields and the Artin symbol, from this point of view. Part II focuses on exponential equations modulo primes. Artin's famous primitive root conjecture and Hooley's conditional solution is discussed in Chapter 4. Tools on Kummer-type extensions are given in Chapter 5 and a multivariable generalization of a method of Lenstra is presented in Chapter 6. These are put to use in Chapter 7, where solutions to several applications, including the Schinzel-Wójcik problem on the equality of orders of integers modulo primes, are given.
  • Hämäläinen, Jussi (2024)
    In this thesis, we aim to introduce the reader to profinite groups. Profinite groups are defined by two characteristics: firstly, they have a topology defined on them (notably, they are compact). Secondly, they are constructed from some collection of finite groups, each equipped with a discrete topology and forming what is known as an inverse system. The profinite group emerges as an inverse limit of its constituent groups. This definition is, at this point, necessarily quite abstract. Thus, before we can really understand profinite groups we must examine two areas: first, we will study topological groups. This will give us the means to deal with groups as topological spaces. Topological groups have some characteristics that differentiate them from general topological spaces: in particular, a topological group is always a homogeneous space. Secondly, we will explore inverse systems and inverse limits, which will take us into category theory. While we could explain these concepts without categories, this thesis takes the view that category theory gives us a useful “50000-feet view” by giving these ideas a wider mathematical context. In the second chapter, we will go through preliminary information concerning group theory, general topology and category theory that will be needed later. We will begin with some basic concepts from group theory and point-set topology. These sections will mostly contain information that is familiar from the introductory university courses. The chapter will then continue by introducing some basic concepts of category theory, including inverse systems and inverse limits. For these, we will give an application by showing how the Cantor set is homeomorphic to an inverse limit of a collection of finite sets. In the third chapter, we will examine topological groups and prove some of their properties. In the fourth chapter, we will introduce an example of profinite groups: Zp, the additive group of p-adic integers. This will be expanded into a ring and then into the field Qp. We will discuss the uses of Zp and Qp and show how to derive them as an inverse limit of finite, compact groups.
  • Haltia, Alvar (2023)
    The Ising model is a classic model in statistical physics. Originally intended to model ferromagnetism, it has proven to be of great interest to mathematicians and physicists. In two dimensions it is sufficiently complex to describe interesting phenomena while still remaining analytically solvable. The model is defined upon a graph, with a random variable, called a spin, on each vertex. Other random variables may be defined as functions of spins. A classic problem of interest is the correlation of these random variables. A continuum analogue of the Ising model is possible through considering the scaling limit of the model, as the graph taken to approximate some domain e.g. in the complex plane or a torus. The core of this work is an exposition upon one method of calculating correlations of a random variable called a fermion defined in terms of spins and disorder random variables. The method is called Bosonization and associates correlations of some random variables to correlations of the Gaussian Free Field (GFF). The GFF is a random distribution, which approximately functions as a gaussian random variable whose covariance structure is given by Green's function. A result known as the Pfaffian-Hafnian identity is covered, to provide an example of an identity which may be derived using Bosonization on a continuum planar Ising model. A similar result is also presented on the Torus, using elliptic functions. These results are not original, but we present the only -- to us -- known explicit proofs based on hints from others. In the latter half of the work, Bosonization is approached using Random Current representation. Random currents give weights to each edge of the graph of the Ising model. Two other models are introduced: Alternating flows and the Dimer model. There are equivalence relations between the configuration of the Ising model, the Nesting Field of a random current and the height functions of an alternating flow and a dimer cover. Using these, correlations of random variables of the Ising model are given in terms of the height function of the Dimer model. The height function of the Dimer model is a discrete analogue of the GFF.
  • Asikainen, Timo (2023)
    Tässä työssä esitetään ja todistetaan Seifertin--van Kampenin lause. Lauseen avulla voidaan muodostaa perusryhmä topologiselle avaruudelle, joka koostuu sopivalla tavalla kahdesta tai useammasta osa-avaruudesta, joiden perusryhmät oletetaan tunnetuksi. Yleisesti perusryhmän käsite kuuluu algebrallisen topologian alaan, jossa sovelletaan abstraktin algebran käsitteitä ja menetelmiä topologisiin avaruuksiin. Perusryhmä kuvaa tärkeällä tavalla topologisen avaruuden rakennetta: topologisen avaruuden rakenteen esitys algebrallisena rakenteena, ryhmänä, on tärkeä abstraktiokeino, joka avaa merkittäviä mahdollisuuksia topologisia avaruuksia koskevalle päättelylle. Seifertin--van Kampenin lause mahdollistaa tämän päättelyn soveltamisen myös sellaisiin avaruuksiin, joiden perusryhmän muodostaminen suoraan määritelmistä lähtien ei onnistu kohtuudella. \vspace{6pt} Johdantona Seifertin--van Kampenin lauseelle työssä esitetään lauseen kannalta keskeisten algebran ja topologian käsitteiden määritelmät. Lisäksi annetaan ja osin esitetään todistukset keskeisille lauseille, joita tarvitaan Seifertin--van Kampenin lauseen todistuksessa. \vspace{6pt} Esimerkkeinä Seifertin--van Kampenin lauseen sovelluksista johdetaan kiilasumman ja erään graafin perusryhmä. Laajempana sovelluksena määritellään monistojen ja niiden erikoistapauksena kompaktien pintojen käsite ja niiden monikulmioesitys. Lopuksi esitetään ja osin todistetaan Seifertin--van Kampenin lauseen avulla kompaktien pintojen luokittelulause, jonka mukainen jokainen kompakti pinta on homeomorfinen pallon, torusten yhtenäisen summan tai projektiivisten tasojen yhtenäisen summan ja vain yhden näistä kanssa.
  • Satukangas, Eetu Aatos Elmeri (2023)
    In this thesis we prove a short time asymptotic formula for a path integral solution to the Fokker Planck heat equation on a Riemannian manifold. The result is inspired by multiple developments regarding the theory of stochastic differential equations on a Riemannian manifold. Most notably the papers by Itô (1962) which describes the stochastic differential equation, Graham (1985) which describes the probabilistic time development of the stochastic differential equation via a path integral and Anderson and Driver (1999) which proves that Graham's path integral converges to the correct notion of probability. The starting point of the thesis is a paper by R. Graham (1985) where a path integral formula for the solution of the heat equation on a Riemannian manifold is given in terms of a stochastic differential equation in Itô sense. The path integral formula contains an integrand of the exponential of an action function. The action function is defined by the given stochastic differential equation and additional integration variables denoted as the momenta of the paths appearing in the integral. The path integral is defined as the time continuum limit of a product of integrals on a discrete time lattice. The result obtained in this thesis is proven by considering the saddle point approximation of the action appearing in the finite version of Graham's path integral formula. The saddle point approximation gives a power series approximation of the action up to the second order by taking the first and second variations of the action and setting the first variation as zero. We say that the saddle point approximation is evaluated along the critical path of the action which is defined by taking the first variation as zero. The second variation of the action is called the Hessian matrix. With the saddle point approximation of the action, we obtain an asymptotic formula of the path integral which contains the exponential of the action evaluated along the critical path and the determinant of the Hessian. The main part of the proof is the evaluation of the determinant of the Hessian in the continuum limit. To this end we prove a finite dimensional version of a theorem due to R. Forman (1987), called Forman's theorem, which allows us to calculate the ratio of determinants of the Hessian parametrized by two different boundary conditions as a ratio of finite dimensional determinants. We then show that in the continuum limit the ratio of determinants of the Hessian can be written in terms of the Jacobi ow. With the Forman's theorem we then get the short time asymptotic formula by evaluating the determinants on a short time interval.
  • Härkönen, Robert Mattias (2021)
    Plane algebraic curves are defined as zeroes of polynomials in two variables over some given field. If a point on a plane algebraic curve has a unique tangent line passing through it, the point is called simple. Otherwise, it is a singular point or a singularity. Singular points exhibit very different algebraic and topological properties, and the objective of this thesis is to study these properties using methods of commutative algebra, complex analysis and topology. In chapter 2, some preliminaries from classical algebraic geometry are given, and plane algebraic curves and their singularities are formally defined. Curves and their points are linked to corresponding coordinate rings and local rings. It is shown that a point is simple if and only if its corresponding local ring is a discrete valuation ring. In chapter 3, the Newton-Puiseux algorithm is introduced. The algorithm outputs fractional power series known as Puiseux expansions, which are shown to produce parametrizations of the local branches of a curve around a singular point. In chapter 4, Puiseux expansions are used to study the topology of complex plane algebraic curves. Around singularities, curves are shown to have an iterated torus knot structure which is, up to homotopy, determined by invariants known as Puiseux pairs.
  • Oksa, Ella (2024)
    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.
  • Uotila, Valter Johan Edvard (2022)
    In this work, I prove the theorem of Bröcker and Scheiderer for basic open semi-algebraic sets. The theorem provides an upper bound for a stability index of a real variety. The theory is based on real closed fields which generalize real numbers. A real variety is a subset of a real closed field that is defined by polynomial equalities. Every semi-algebraic set is defined by a boolean combination of polynomial equations and inequalities of the sign conditions involving a finite number of polynomials. The basic semi-algebraic sets are those semi-algebraic sets that are defined solely by the sign conditions. In other words, we can construct semi-algebraic sets from the basic semi-algebraic sets by taking the finite unions, intersections, and complements of the basic semi-algebraic sets. Then the stability index of a real variety indicates the upper bound of numbers of polynomials that are required to express an arbitrary semi-algebraic subset of the variety. The theorem of Bröcker and Scheiderer shows that such upper bound exists and is finite for basic open semi-algebraic subsets of a real variety. This work aims to be detailed in the proofs and represent sufficient prerequisites and references. The first chapter introduces the topic generally and motivates to study the theorem. The second chapter provides advanced prerequisites in algebra. One of such results is the factorial theorem of a total ring of fractions. Other advanced topics include radicals, prime ideals, associative algebras, a dimension of a ring, and various quotient structures. The third chapter defines real closed fields and semi-algebraic sets that are the fundamental building blocks of the theory. The third chapter also develops the theory of quadratic forms. The main result of this chapter is Witt’s cancellation theorem. We also shortly describe the Tsen-Lang theorem. The fourth chapter is about Pfister forms. Pfister forms are special kinds of quadratic forms that we extensively use in the proof of the main theorem. First, we define general Pfister forms over fields. Then we develop their theory over the fields of rational functions. Generally, Pfister forms share multiple similar properties as quadratic forms. The fifth chapter represents one- and two-dimensional examples of the main theorem. These examples are based on research that is done on constructive approaches to the theorem of Bröcker and Scheiderer. The examples clarify and motivate the result from an algorithmic perspective. Finally, we prove the main theorem of the work. The proof is heavily based on Pfister forms.
  • Tamminen, Eeli (2022)
    A convex function, which Hessian determinant equals to one, defined in a convex and bounded polygon is called a surface tension. Moreover at the boundary of the given polygon it is demanded that the function is piece-wise affine. This setting originates from the theory of dimer models but in this thesis the system is studied as such. The boundary condition gives us points of interest i.e. the corners of the polygon and so called quasi-frozen points, where the boundary function is not differentiable. With a suitable homeomorphism one can map the unit disc to the polygon in question. In this setting an explicit formula for the gradient of the surface tension is derived. Furthermore the values of the gradient in corner and quasi-frozen points are derived as limits from, which as a corollary the directed derivatives of the surface tension are studied.
  • Hednäs, Mats (2023)
    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.
  • Penttinen, Jussi (2021)
    HMC is a computational method build to efficiently sample from a high dimensional distribution. Sampling from a distribution is typically a statistical problem and hence a lot of works concerning Hamiltonian Monte Carlo are written in the mathematical language of probability theory, which perhaps is not ideally suited for HMC, since HMC is at its core differential geometry. The purpose of this text is to present the differential geometric tool's needed in HMC and then methodically build the algorithm itself. Since there is a great introductory book to smooth manifolds by Lee and not wanting to completely copy Lee's work from his book, some basic knowledge of differential geometry is left for the reader. Similarly, the author being more comfortable with notions of differential geometry, and to cut down the length of this text, most theorems connected to measure and probability theory are omitted from this work. The first chapter is an introductory chapter that goes through the bare minimum of measure theory needed to motivate Hamiltonian Monte Carlo. Bulk of this text is in the second and third chapter. The second chapter presents the concepts of differential geometry needed to understand the abstract build of Hamiltonian Monte Carlo. Those familiar with differential geometry can possibly skip the second chapter, even though it might be worth while to at least flip through it to fill in on the notations used in this text. The third chapter is the core of this text. There the algorithm is methodically built using the groundwork laid in previous chapters. The most important part and the theoretical heart of the algorithm is presented here in the sections discussing the lift of the target measure. The fourth chapter provides brief practical insight to implementing HMC and also discusses quickly how HMC is currently being improved.
  • Yli-Seppälä, Oona (2023)
    Reshetnyakin lauseen mukaan kvasisäännöllinen kuvaus, joka ei ole vakio, on avoin, diskreetti ja suunnansäilyttävä. Suunnansäilyttävät kuvaukset määritellään topologisen asteen avulla ja siksi tässä tutkielmassa Reshetnyakin lauseen todistuksessa keskeistä on kvasisäännöllisen kuvauksen osoittaminen suunnansäilyttäväksi. Topologiselle asteelle on tässä tutkielmassa valittu analyyttinen lähestymistapa. Luvussa 3 topologinen aste määritellään jatkuvasti differentioituville funktioille ja luvussa 4 siirretään topologisen asteen määritelmä ja keskeiset tulokset Sobolev-funktioille. Reshetnyakin lauseen todistuksen runkona käytetään Titus–Youngin lausetta, jonka mukaan jatkuvat, kevyet ja suunnansäilyttävät kuvaukset ovat avoimia ja diskreettejä. Titus–Youngin lause esitellään ja todistetaan luvussa 5. Luvussa 6 esitellään kvasisäännölliset kuvaukset. Alaluvussa 6.1 tarkastellaan kvasisäännöllisten kuvausten ja elliptisten osittaisdifferentiaaliyhtälöiden yhteyttä. Osittaisdifferentiaaliyhtälöiden avulla voidaan osoittaa kvasisäännöllinen kuvaus kevyeksi. Viimeisessä luvussa esitellään Reshetnyakin lause ja osoitetaan, että kvasisäännöllinen kuvaus, joka ei ole vakio, on kevyt ja suunnansäilyttävä. Tällöin kvasisäännöllisen kuvauksen avoimuus ja diskreettisyys seuraa Titus–Youngin lauseesta.