Browsing by master's degree program "Magisterprogrammet i matematik och statistik"

In my mathematics master's thesis we dive into the wave equation and its inverse problem and try to solve it with neural networks we create in Python. There are different types of artificial neural networks. The basic structure is that there are several layers and each layer contains neurons. The input goes to all the neurons in the first layer, the neurons do calculations and send the output to all the neurons in the next layer. In this way, the input data goes through all the neurons and changes and the last layer outputs this changed data. In our code we use operator recurrent neural network. The biggest difference between the standard neural network and the operator recurrent neural network is, that instead of matrix-vector multiplications we use matrix-matrix multiplications in the neurons. We teach the neural networks for a certain number of times with training data and then we check how well they learned with test data. It is up to us how long and how far we teach the networks. Easy criterion would be when a neural network has learned the inversion completely, but it takes a lot of time and might never happen. So we settle for a situation when the error, the difference between the actual inverse and the inverse calculated by the neural network, is as small as we wanted. We start the coding by studying the matrix inversion. The idea is to teach the neural networks to do the inversion of a given 2-by-2 real valued matrix. First we deal with networks that don't have the activation function ReLU in their layers. We seek a learning rate, a small constant, that speeds up the learning of a neural network the most. After this we start comparing networks that don't have ReLU layers to networks that do have ReLU layers. The hypothesis is that ReLU assists neural networks to learn quicker. After this we study the one-dimensional wave equation and we calculate its general form of solution. The inverse problem of the wave equation is to recover wave speed c(x) when we have boundary terms. Inverse problems in general do not often have a unique solution, but in real life if we have measured data and some additional a priori information, it is possible to find a unique solution. In our case we do know that the inverse problem of the wave equation has a unique solution. When coding the inverse problem of the wave equation we use the same approach as with the matrix inversion. First we seek the best learning rate and then start to compare neural networks with and without ReLU layers. The hypothesis once again is that ReLU supports the learning of the neural networks. This turns out to be true and happens more clearly with wave equation than with matrix inversion. All the teaching was run on one computer. There is a chance to get even better results if a more powerful computer is used.

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.

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.

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.

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.

Let L_N denote the maximal number of points in a rate 1 Poisson process on the plane which a piecewise linear increasing path can pass through while traversing from (0, 0) to (N, N). It is well-known that the expectation of L_N / N converges to 2 as N increases without bound. A perturbed version of this system can be introduced by superimposing an independent one-dimensional rate c > 0 Poisson process on the main diagonal line {x = y} of the plane. Given this modification, one asks whether and if so, how, the above limit might be affected. A particular question of interest has been whether this modified system exhibits a nontrivial phase transition in c. That is, whether there exists a threshold value c_0 > 0 such that the limit is preserved for all c < c_0 but lost outside this interval. Let L^c_N denote the maximal number of points in the system perturbed by c > 0 which an increasing piecewise linear path can pass through while traversing from (0, 0) to (N, N). In 2014, Basu, Sidoravicius, and Sly showed that there is no such phase transition and that, for all c > 0, the expectation of L^c_N / N converges to a number strictly greater than 2 as N increases without bound. This thesis gives an exposition of the arguments used to deduce this result.

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.

The quantification of carbon dioxide emissions pose a significant and multi-faceted problem for the atmospheric sciences as a part of the research regarding global warming and greenhouse gases. Emissions originating from point sources, referred to as plumes, can be simulated using mathematical and physical models, such as a convection-diffusion plume model and a Gaussian plume model. The convection-diffusion model is based on the convection-diffusion partial differential equation describing mass transfer in diffusion and convection fields. The Gaussian model is a special case or a solution for the general convection-diffusion equation when assumptions of homogeneous wind field, relatively small diffusion and time independence are made. Both of these models are used for simulating the plumes in order to find out the emission rate for the plume source. An equation for solving the emission rate can be formulated as an inverse problem written as y=F(x)+ε where y is the observed data, F is the plume model, ε is the noise term and x is an unknown vector of parameters, including the emission rate, which needs to be solved. For an ill-posed inverse problem, where F is not well behaved, the solution does not exist, but a minimum norm solution can be found. That is, the solution is a vector x which minimizes a chosen norm function, referred to as a loss function. This thesis focuses on the convection-diffusion and Gaussian plume models, and studies both the difference and the sensibility of these models. Additionally, this thesis investigates three different approaches for optimizing loss functions: the optimal estimation for linear model, Levenberg–Marquardt algorithm for non-linear model and adaptive Metropolis algorithm. A goodness of different fits can be quantified by comparing values of the root mean square errors; the better fit the smaller value the root mean square error has. A plume inversion program has been implemented in Python programming language using the version 3.9.11 to test the implemented models and different algorithms. Assessing the parameters' effect on the estimated emission rate is done by performing sensitivity tests for simulated data. The plume inversion program is also applied for the satellite data and the validity of the results is considered. Finally, other more advanced plume models and improvements for the implementation will be discussed.

It is important that the financial system retains its functionality throughout the macroeconomic cycle. When people lose their trust in banks the whole economy can face dire consequences. Therefore accurate and stable predictions of the expected losses of borrowers or loan facilities are vital for the preservation of a functioning economy. The research question of the thesis is: What effect does the choice of calibration type have on the accuracy of the probability of default values predictions. The research question is attempted to be answered through an elaborate simulation of the whole probability of default model estimation exercise, with a focus on the rank order model calibration to the macroeconomic cycle. Various calibration functions are included in the study to offer more diversity in the results. Furthermore, the thesis provides insight into the regulatory environment of financial institutions, presenting relevant articles from accords, regulations and guidelines by international and European supervisory agents. In addition, the thesis introduces statistical methods for model calibration to the long-run average default rate. Finally, the thesis studies the effect of calibration type on the probability of default parameter estimation. The investigation itself is done by first simulating the data and then by applying multiple different calibration functions, including two logit functions and two Bayesian models to the simulated data. The simulation exercise is repeated 1 000 times for statistically robust results. The predictive power was measured using mean squared error and mean absolute error. The main finding of the investigation was that the simple grades perform unexpectedly well in contrast to raw predictions. However, the quasi moment matching approach for the logit function generally resulted in higher predictive power for the raw predictions in terms of the error measures, besides against the captured probability of default. Overall, simple grades and raw predictions yielded similar levels of predictive power, while the master scale approach lead to lower numbers. It is reasonable to conclude that the best selection of approaches according to the investigation would be the quasi moment matching approach for the logit function either with simple grades or raw predictions calibration type, as the difference in the predictive power between these types was minuscule. The calibration approaches investigated were significantly simplified from actual methods used in the industry, for example, calibration exercises mainly focus on the derivation of the correct long-run average default rate over time and this study used only the central tendency of the portfolio as the value.

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.