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Browsing by study line "Matematik"

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  • Hirvonen, Minna (2020)
    Several extensions of first-order logic are studied in descriptive complexity theory. These extensions include transitive closure logic and deterministic transitive closure logic, which extend first-order logic with transitive closure operators. It is known that deterministic transitive closure logic captures the complexity class of the languages that are decidable by some deterministic Turing machine using a logarithmic amount of memory space. An analogous result holds for transitive closure logic and nondeterministic Turing machines. This thesis concerns the k-ary fragments of these two logics. In each k-ary fragment, the arities of transitive closure operators appearing in formulas are restricted to a nonzero natural number k. The expressivity of these fragments can be studied in terms of multihead finite automata. The type of automaton that we consider in this thesis is a two-way multihead automaton with nested pebbles. We look at the expressive power of multihead automata and the k-ary fragments of transitive closure logics in the class of finite structures called word models. We show that deterministic twoway k-head automata with nested pebbles have the same expressive power as first-order logic with k-ary deterministic transitive closure. For a corresponding result in the case of nondeterministic automata, we restrict to the positive fragment of k-ary transitive closure logic. The two theorems and their proofs are based on the article ’Automata with nested pebbles capture first-order logic with transitive closure’ by Joost Engelfriet and Hendrik Jan Hoogeboom. In the article, the results are proved in the case of trees. Since word models can be viewed as a special type of trees, the theorems considered in this thesis are a special case of a more general result.
  • Nyberg, Jonne (2020)
    Spectral theory is a powerful tool when applied to differential equations. The fundamental result being the spectral theorem of Jon Von Neumann, which allows us to define the exponential of an unbounded operator, provided that the operator in question is self-adjoint. The problem we are considering in this thesis, is the self-adjointness of the Schr\"odinger operator $T = -\Delta + V$, a linear second-order partial differential operator that is fundamental to non-relativistic quantum mechanics. Here, $\Delta$ is the Laplacian and $V$ is some function that acts as a multiplication operator. We will study $T$ as a map from the Hilbert space $H = L^2(\mathbb{R}^d)$ to itself. In the case of unbounded operators, we are forced to restrict them to some suitable subspace. This is a common limitation when dealing with differential operators such as $T$ and the choice of the domain will usually play an important role. Our aim is to prove two theorems on the essential self-adjointness of $T$, both originally proven by Tosio Kato. We will start with some necessary notation fixing and other preliminaries in chapter 2. In chapter 3 basic concepts and theorems on operators in Hilbert spaces are presented, most importantly we will introduce some characterisations of self-adjointness. In chapter 4 we construct the test function space $D(\Omega)$ and introduce distributions, which are continuous linear functionals on $D(\Omega).$ These are needed as the domain for the adjoint of a differential operator can often be expressed as a subspace of the space of distributions. In chapter 5 we will show that $T$ is essentially self-adjoint on compactly supported smooth functions when $d=3$ and $V$ is a sum consisting of an $L^2$ term and a bounded term. This result is an application of the Kato-Rellich theorem which pertains to operators of the form $A+B$, where $B$ is bounded by $A$ in a suitable way. Here we will also need some results from Fourier analysis that will be revised briefly. In chapter 6 we introduce some mollification methods and prove Kato's distributional inequality, which is important in the proof of the main theorem in the final chapter and other results of similar nature. The main result of this thesis, presented in chapter 7, is a theorem originally conjectured by Barry Simon which says that $T$ is essentially self-adjoint on $C^\infty_c(\mathbb{R}^d)$, when $V$ is a non-negative locally square integrable function and $d$ is an arbitrary positive integer. The proof is based around mollification methods and the distributional inequality proven in the previous chapter. This last result, although fairly unphysical, is somewhat striking in the sense that usually for $T$ to be (essentially) self-adjoint, the dimension $d$ restricts the integrability properties of $V$ significantly.
  • Kurki, Joonas (2021)
    The goal of the thesis is to prove the Dold-Kan Correspondence, which is a theorem stating that the category of simplicial abelian groups sAb and the category of positively graded chain complexes Ch+ are equivalent. The thesis also goes through these concepts mentioned in the theorem, starting with categories and functors in the first section. In this section, the aim is to give enough information about category theory, so that the equivalence of categories can be understood. The second section uses these category theoretical concepts to define the simplex category, where the objects are ordered sets n = { 0 -> 1 -> ... -> n }, where n is a natural number, and the morphisms are order preserving maps between these sets. The idea is to define simplicial objects, which are contravariant functors from the simplex category to some other category. Here is also given the definition of coface and codegeneracy maps, which are special kind of morphisms in the simplex category. With these, the cosimplicial (and later simplicial) identities are defined. These identities are central in the calculations done later in the thesis. In fact, one can think of them as the basic tools for working with simplicial objects. In the third section, the thesis introduces chain complexes and chain maps, which together form the category of chain complexes. This lays the foundation for the fourth section, where the goal is to form three different chain complexes out of any given simplicial abelian group A. These chain complexes are the Moore complex A*, the chain complex generated by degeneracies DA* and the normalized chain complex NA*. The latter two of these are both subcomplexes of the Moore complex. In fact, it is later on shown that there exists an isomorphism An = NAn +DAn between the abelian groups forming these chain complexes. This connection between these chain complexes is an important one, and it is proved and used later on in the seventh section. At this point in the thesis, all the knowledge for understanding the Dold-Kan Correspondence has been presented. Thus begins the forming of the functors needed for the equivalence, which the theorem claims to exist. The functor from sAb to Ch+ maps a simplicial abelian group A to its normalized chain complex NA*, the definition of which was given earlier. This direction does not require that much additional work, since most of it was done in the sections dealing with chain complexes. However, defining the functor in the opposite direction does require some more thought. The idea is to map a chain complex K* to a simplicial abelian group, which is formed using direct sums and factorization. Forming it also requires the definition of another functor from a subcategory of the simplex category, where the objects are those of the simplex category but the morphisms are only the injections, to the category of abelian groups Ab. After these functors have been defined, the rest of the thesis is about showing that they truly do form an equivalence between the categories sAb and Ch+.
  • Boughdiri, Larbi (2022)
    The topic of thesis is the wave equation. The first chapter is introduction, the overview of the thesis is presented. The second chapter treats the transport equation, which is needed to solve the wave equation. In the third chapter we discuss the d’Alembert formula, and we prove the existence and uniqueness of solution. We treat the domain of dependence and region of influence. The last chapter concentrates on solving wave equations in high dimensions by Kirchhoff’s formula, method of descent and methods of spherical means.
  • Milén, Hannu (2023)
    The thesis consists of presenting and analysing the original proof for the Embedding Theorem that Hassler Whitney gave in his 1936 article Differentiable Manifolds. The embedding theorem states that given an m-dimensional Cr-differentiable (r ≥ 1) manifold M, it is possible to embed it in Euclidean space Rn, if n ≥ 2m + 1. Embedding is defined as a mapping f : M → Rn which is Cr-smooth, bijective immersion that is homeomorphism to its image f[M]. Whitney’s proof rests on few important novel concepts and a series of lemmas in relation to them. These concepts include the concept of the k-extent of a set, a sort of a k-dimensional measure in an n-dimensional space; the concept of Cr-function g : M → N approximating (f, M, r, η), where f is a Cr-function f : M → N, η an error function; and the concept of (f, r, η)-properties defined for such g. Outstanding lemmas of general nature are Lemma 7: If f : M → N is a Cr-map and A ⊂ M is of finite (zero) k-extent, then f[A] is of finite (zero) k-extent. Lemma 8: For open sets R and R′ of Rm and Rh, if {Tα} is a h-parameter family of C1-maps of R ⊂ Rm into Rn, and A ⊂ R and B ⊂ Rn closed subsets, such that A is of finite k-extent and B of zero (h − k)-extent, then for some α ∈ R′, Tα[A] does not intersect B. Lemma 9: If f : M → N is a Cr-map, η positive continuous function in M, Ω1, Ω2, . . . are (f, r, η)- properties, then there is a Cr-map F : M → N which approximates (f, M, r, η) and has properties Ω1, Ω2, . . . . Lemmas 11 and 12 then show that bijectivity and immersion property are the logical sum of countable number of (f, r, η)-properties. These facts are used in finding an embedding F : M → Rn by perturbing a given smooth function f : M → Rn. Detailed treatment of all proofs is provided. Adjustments to the proofs are made where deemed necessary; auxiliary assumptions are made where they seem to be required. Clarifications and proofs are given to facts noted but not proven in the article
  • Pim, Jonathan (2022)
    Wolfe’s Theorem states that there is an isometric isomorphism between the space of flat k-cochains and the flat differential k-forms in R^n . The flat forms are the space of essentially bounded differential forms with an essentially bounded weak exterior derivative. The flat cochains are the dual space of the flat chains which are geometric objects based on finite linear combinations of k-simplices. In this sense, Wolfe’s Theorem connects geometry and analysis. After proving Wolfe’s Theorem, we give two corollaries: that the isomorphism from Wolfe’s Theorem can be concretely approximated by convolution with smooth mollifiers, and a version of Stokes’ Theorem for flat chains. Our method for proving Wolfe’s Theorem involves isometrically embedding the flat chains, as well as a predual of the flat forms, into the space of flat currents. By way of some approximation theorems in the space of flat currents, the images of these two embeddings coincide. Thus, the flat chains are isomorphic to that predual. This isomorphism lifts to their dual spaces giving Wolfe’s Theorem.