Browsing by study line "Matematik"
Now showing items 21-40 of 56
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(2021)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.
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(2022)In this thesis we study the article by J. Bourgain and C. Demeter called A study guide for the l^2 decoupling theorem. In particular, we hope to give an in detail exposition to certain results from the aforementioned research article so that this text combined with the master’s thesis On the l^2 decoupling theorem by Jaakko Sinko covers the l^2 decoupling theorem comprehensively in the range 2 ≤ p ≤ 2n/(n−1). The results in this text also self-sufficiently motivate the use of the extension operator and explain why it is possible to prove linear decouplings with multilinear estimates. We begin the thesis by giving the basic notation and highlighting some useful results from analysis and linear algebra that are later used in the thesis. In the second chapter we introduce and prove a certain multilinear Kakeya inequality, which asserts an upper bound for the overlap of neighbourhoods of nearly axis parallel lines in R^n that point in different directions. In the next chapter this is applied to prove a multilinear cube inflation inequality, which is one of the main mechanisms in the proof of the l^2 decoupling theorem. In the fourth chapter we study two forms of linear decoupling. One that is defined by an extension operator and one that defined via Fourier restriction. The main result of this chapter is that the former is strong enough to produce decoupling inequalities that are of the latter form. The fifth chapter is reserved for comparing linear and multilinear decouplings. Here we use the main result of the previous chapter to prove that multilinear estimates can produce linear decouplings, if the lower dimensional decoupling constant is somehow contained. This paves the way for the induction proof of the l^2 decoupling theorem.
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(2024)This thesis is motivated by the necessity to develop a numerical model for the computation of airflow within simulated ship systems, with particular emphasis on the starting air system. The starting air system serves as a conduit for compressed air, fulfilling various functions through a network of pipelines, most notably supplying air for engine initiation. The thesis comprises a theoretical section and a computational case study. In the theoretical segment, the study delves into compressible flow phenomena within multi-dimensional domains by formulating the conservation laws for mass, momentum, and energy. These laws encapsulate the entirety of compressible airflow phenomena. Subsequently, these conservation laws are reduced to a one-dimensional domain, wherein they are expressed as hyperbolic partial differential equations suitable for one-dimensional pipeline analysis. Following the exploration of the mathematical underpinnings of compressible airflow, the focus shifts to numerical methods capable of generating approximate solutions for the partial differential equations governing one-dimensional compressible flow in straight, cylindrical pipelines, as expounded in the thesis's theoretical section. Given the absence of analytical solutions for these equations, numerical methods and computational solutions become imperative. In this thesis, the method of lines is employed to transform the partial differential equations into ordinary differential equations through the discretization of spatial coordinates using a centered midpoint approximation. The classical explicit fourth-order Runge-Kutta method is then applied to solve the resulting ordinary differential equations. The selected Runge-Kutta method undergoes theoretical scrutiny, and a symbolic derivation for this method is provided. In the computational case study, the thesis directs its attention to a scenario wherein a highly pressurized air vessel discharges into ambient conditions through a straight, cylindrical pipe. This case study is designed with a typical marine starting air system in mind and seeks to answer the question of the achievable volumetric flow under normal conditions, given varying air vessel pressures that define the boundary conditions for this problem. A dedicated software was developed for the case study, employing the chosen Runge-Kutta method. Airflow parameters—pipe pressure, temperature, and volumetric flow under normal conditions—are collected during the simulation run-time, chosen to extend until the pressure vessel reaches ambient pressure. The results of the case study are presented and discussed in the concluding section of the thesis.
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(2024)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.
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(2021)In topology, one often wishes to find ways to extract new spaces out of existing spaces. For example, the suspension of a space is a fundamental technique in homotopy theory. However, in recent years there has been a growing interest in extracting topological information out of discrete structures. In the field of topological data-analysis one often considers point clouds, which are finite sets of points embedded in some R^m. The topology of these sets is trivial, however, often these sets have more structure. For example, one might consider a uniformly randomly sampled set of points from a circle S1. Clearly, the resulting set of points has some geometry associated to it, namely the geometry of S1. The use of certain types of topological spaces called Vietoris-Rips and Cech complexes allows one to study the "underlying topology" of point clouds by standard topological means. This in turn enables the application of tools from algebraic topology, such as homology and cohomology, to be applied to point clouds. Vietoris-Rips and Cech complexes are often not metrizable, even though they are defined on metric spaces. The purpose of this thesis is to introduce a homotopy result of Adams and Mirth concerning Vietoris-Rips metric thickenings. In the first chapter, we introduce the necessary measure theory for the main result of the thesis. We construct the 1-Wasserstein distance, and prove that it defines a metric on Polish spaces. We also note, that the 1-Wasserstein distance is a metric on general metric spaces. In the sequel, we introduce various complexes on spaces. We study simplicial complexes on R^n and introduce the concept of a realization. We then prove a theorem on the metrizability of a realization of a simplicial complex. We generalize simplicial complexes to abstract simplicial complexes and study the geometric realization of some complexes. We prove a theorem on the existence of geometric realizations for abstract simplicial complexes. Finally, we define Vietoris-Rips and Cech complexes, which are complexes that are formed on metric spaces. We introduce the nerve lemma for Cech complexes, and prove a version of it for finite CW-complexes. The third chapter introduces the concept of reach, which in a way measures the curvature of the boundary of a subset of R^n. We prove a theorem that characterizes convex, closed sets of R^n by their reach. We also introduce the nearest point projection map π, and prove its continuity. In the final chapter, we present some more measure theory, which leads to the definitions of Vietoris-Rips and Cech metric thickenings. The chapter culminates in constructing an explicit homotopy equivalence between a metric space X of positive reach and its Vietoris-Rips metric thickening.
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(2021)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.
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(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.
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(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.
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(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.
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(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.
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(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.
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(2024)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.
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(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.
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(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.
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(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
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(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.
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(2024)Tässä tutkielmassa tutustutaan proäärellisiin ryhmiin, siis topologisiin ryhmiin, jotka ovat isomorfisia äärellisten topologisten ryhmien muodostaman inverssisysteemin rajan kanssa. Tutkielman alussa esitetään topologisten ryhmien yleistä teoriaa, sekä tutustutaan inverssisysteemeihin yleisesti esittämällä kokoelma näiden ominaisuuksia. Tämän jälkeen tutkielmassa siirrytään käsittelemään proäärellisiä ryhmiä ja ja esitetään tärkeä karakterisaatio, jonka mukaan topologinen ryhmä on proäärellinen jos ja vain jos se on kompakti ja täysin epäyhtenäinen. Tästä seuraa Blairen lauseen nojalla, että jokainen proäärellinen ryhmä on Blairen avaruus. Tutkielman lopuksi käsitellään proäärellisiä täydellistymiä, erityisesti kokonaislukujen pro-p täydellistymämme, siis p-adisille luvuille, annetaan oma kappaleensa, jossa esitetään näiden konstruktio inverssisysteemin rajana sekä äärettömän pitkinä luonnollisina lukuina, sekä esitetään näiden ominaisuuksia. Näistä tärkeimpänä esitetään Henselin lemma, jonka avulla p-adisten lukujen polynomille löydetään juuri, kunhan sille on annettu riittähän hyvä arvio. Tällä tuloksella on käyttöä myös modulaariaritmeriikassa. Viimeisenä tutkielmassa esitetään kokonaislukujen proäärellinen täydellistymä
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(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.
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(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.
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(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.
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