Browsing by study line "Matematik"

Tutkielma käsittelee invariantin aliavaruuden ongelmaa. Päälähteenä toimii Isabelle Chalendarin ja Jonathan Partingtonin kirja Modern Approaches to the Invariant-Subspace Problem. Invariantin aliavaruuden ongelmassa kysytään, onko kompleksisessa Banachin avaruudessa X jokaisella jatkuvalla lineaarisella operaattorilla T olemassa suljettu aliavaruus A, joka on invariantti (T(A) ⊂ A) ja ei-triviaali (A 6= {0} ja A 6= X). Invariantin aliavaruuden ongelma on vielä avoin kompleksiselle ääretönulotteiselle separoituvalle Hilbertin avaruudelle. Tutkielma koostuu neljästä luvusta. Ensimmäisessä luvussa käydään läpi tarvittavia määritelmiä ja teorioita sekä pohjustetaan tulevia kappaleita. Toisessa luvussa määritellään Banachin algebra ja kompaktit operaattorit sekä esitetään Schauderin kiintopistelause ja päätuloksena Lomonosovin lause, jonka korollaarina saadaan, että kompaktilla operaattorilla, joka ei ole nollaoperaattori, on ei-triviaali invariantti aliavaruus. Lomonosovin lause on esitetty Chalendarin ja Partingtonin kirjan luvussa 6. Kolmannessa luvussa siirrytään Hilbertin avaruuksiin ja tutkitaan normaaleja operaattoreita. Päätuloksena todistetaan, että normaalilla operaattorilla, joka ei ole nollaoperaattori, on ei-triviaali hyperinvariantti aliavaruus. Tätä varten määritellään spektraalisäteen ja spektraalimitan käsitteet sekä näihin liittyviä tuloksia. Normaalit operaattorit löytyvät Chalendarin ja Partingtonin kirjan luvusta 3. Neljäs luku käsittelee minimaalisia vektoreita. Luvussa esitetään Hahn-Banachin, Eberlein-Smulyan ja Banach-Alaoglun lauseet sekä sovelletaan minimaalisia vektoreita invariantin aliavaruuden ongelmaan. Minimaalisten vektoreiden avulla saadaan esimerkiksi uusi ja erilainen todistus sille, että kompaktilla operaattorilla, joka ei ole nollaoperaattori, on ei-triviaali invariantti aliavaruus. Chalendarin ja Partingtonin kirja käsittelee minimaalisia vektoreita luvussa 7.

The question of how much one logic can express compared to another can be measured with formula size, and important results have been reached with formula size games. These games can separate two classes of structures from each other within a given number of moves. Since formula size can also be expressed through extended syntax trees, we are interested in seeing what attributes or benefits games or trees have in different situations. First-order logic and its fragments are particularly interesting. This thesis discusses formula size games and analyses their use in known succinctness results between fragments of first-order logic and also between first-order logic and modal logic. While extended syntax trees may be preferred for results between fragments of first-order logic, the formula size game can be easily constructed for different languages. We find that both methods have advantages depending on the two logics that are compared to each other.

Tutkielman päämääränä on esitellä ja todistaa Milnorin lause (John Milnor, 1968) geometrisen ryhmäteorian alalta. Milnorin lause on olennainen osa äärellisesti viritettyjen ratkeavien ryhmien kasvun luokittelua. Se kertoo, että äärellisesti viritetyt ratkeavat ryhmät joko kasvavat eksponentiaalisesti tai ovat polysyklisiä. Polysyklisten ryhmien kasvun tiedetään olevan joko polynomista tai eksponentiaalista. Näin ollen äärellisesti viritetyt ratkeavat ryhmät kasvavat joko polynomisesti tai eksponentiaalisesti. Tutkielman ensimmäinen luku on johdantoa ja toinen luku on esitietoja. Tutkielman kolmannessa luvussa esitellään ryhmät ja aakkostot. Erityisesti esitellään, mitä tarkoittaa ajatella ryhmän alkioita jonkin aakkoston sanoina. Lisäksi määritellään vapaat ryhmät ja ryhmien esitykset. Tämän jälkeen neljännessä luvussa ryhmiin määritellään ryhmän Cayley graafin avulla sanametriikaksi kutsuttu metriikka. Todistetaan, että eri virittäjäjoukkojen suhteen muodostetut sanametriikat ovat keskenään bilipschitzekvivalentit. Lopulta määritellään ryhmien kasvu ja todistetaan, että ryhmän kasvu ei riipu valitusta virittäjäjoukosta. Viidennessä luvussa esitellään ratkeavat ryhmät, nilpotentit ryhmät ja polysykliset ryhmät ja muutamia konkreettisia esimerkkejä näistä ryhmistä. Lisäksi esitellään näiden ryhmien keskeisiä ominaisuuksia ja niiden välisiä suhteita. Todistetaan esimerkiksi, että jokainen nilpotentti ja polysyklinen ryhmä on myös ratkeava ryhmä. Kuudennessa luvussa todistetaan tutkielman päätulos, Milnorin lause. Se tapahtuu induktiolla ratkeavalle ryhmälle ominaisen subnormaalin laskevan jonon pituuden suhteen. Lisäksi esitellään ja todistetaan tarvittavia aputuloksia. Luvun lopussa esitellään Wolfin lause (Joseph Wolf, 1968) ja yhdistetään Milnorin ja Wolfin lauseet yhdeksi tulokseksi, Milnor-Wolfin lauseeksi. Milnor-Wolfin lauseen nojalla äärellisesti viritettyjen ratkeavien ryhmien kasvu saadaan luokiteltua.

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.

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.

This thesis considers crossing probabilities in 2D critical percolation, and modular forms. In particular, I give an exposition on the theory on modular forms, percolation theory and complex analysis that is needed to characterise the crossing probabilities by means of modular forms. These results are not mine, but I review them and present full proofs which are omitted in the literature. In the special case of 2 dimensions, the percolation theory admits a lot of symmetries due to its conformal invariance at the criticality. This makes its study especially fruitful. There are various types of percolation, but let us consider for example a critical bond percolation on a square lattice. Mark each edge in the lattice black (open) or white (closed) with equal probability, and each edge independently. The probability that there is cluster of connected black edges which is attached to both left and right side of the rectangle, is the horizontal crossing probability. Note that there is always either such a black cluster connecting the left and right sides or a white cluster connecting the upper and lower sides of the rectangle in the dual lattice. This gives us a further symmetry. The crossing probability at the scaling limit, where the mesh size of the square lattice goes to zero, is given by Cardy-Smirnov’s formula. This formula was first derived unrigorously by Cardy, but in 2001 it was proved by Smirnov in the case of a triangular site percolation. I present an alternative expression for the Cardy-Smirnov’s formula in terms of modular forms. In particular, I show that Cardy-Smirnov’s formula can be written as an integral of Dedekind’s eta function restricted to the positive imaginary axis. For this, one needs first that the conformal cross ratio for a rectangle corresponds to the values of the modular lambda function at the positive imaginary axis. This follows by using Schwartz reflection to the conformal map from the rectangle to the upper half plane given by Riemann mapping theorem, and finding an explicit expression for the construction using Weierstrass elliptic function. Using the change of basis for the period module and uniqueness of analytic extension, it follows that the analytic extension for the conformal cross ratio is invariant with respect to the congruent subgroup of the modular group of level 2, and is indeed the modular lambda function. Now, one may reformulate the hypergeometric differential equation satisfied by Cardy-Smirnov’s formula as a function on lambda. Using the symmetries of lambda function, one can deduce the relation to Dedekind’s eta. Lastly, I show how Cardy-Smirnov’s formula is uniquely characterised by two assumptions that are related to modular transformation. The first assumption arises from the symmetry of the problem, but there is not yet physical argument for the second.

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.

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.

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.

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.