Browsing by study line "Mathematics"

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.

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.

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.

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.

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+.

Tutkielman aiheena oleva Whitneyn upotuslause osoittaa todeksi sen intuitiivisen ajatuksen, että jokainen sileä monisto voidaan upottaa johonkin euklidiseen avaruuteen. Lause on nimetty Hassler Whitneyn mukaan, joka ensimmäisenä todisti sen vuonna 1936. Tutkielma jakautuu kolmeen lukuun, joista ensimmäisessä käymme lävitse tarpeellisia yleisen topologian käsitteitä ja niistä johdettuja lauseita aloittaen tärkeimmistä eli topologisen avaruuden ja kuvauksen jatkuvuuden määritelmistä. Esiteltäviin käsitteisiin kuuluvat lisäksi esimerkiksi homeomorfismi, topologinen upotus, topologian kanta, aliavaruudet ja yhtenäisyys. Luvun lopuksi tutustutaan kompaktiuteen, joka on tutkielman kannalta suuressa osassa, koska sitä käytetään myöhemmin Whitnyeyn upotuslauseen todistuksessa. Toisessa luvussa keskitytään sileiden monistojen teoriaan siinä laajuudessa kuin se tutkielman kannalta on tarpeen. Luvun aluksi määritellään monisto, joka on numeroituvan kannan omaava, lokaalisti euklidinen Hausdorffin avaruus. Sen jälkeen määritellään monistolle sileä rakenne, eli maksimaalinen sileä kartasto, jonka avulla monistosta saadaan sileä monisto. Molemmista edellä mainituista annetaan useita esimerkkejä. Lisäksi todistetaan monistojen ja sileiden monistojen ominaisuuksia. Erityisesti todistetaan esikompaktien kantojen olemassaolo kaikille monistoille. Tämän jälkeen määritellään slieä ykkösen ositus ja todistetaan sen olemassaolo sileille monistoille. Käyttäen edellistä voidaan todistaa sileiden töyssy- ja tyhjennysfunktioiden olemassaolo sileille monistoille. Näitä funktioita käytetään myöhemmin Whitneyn upotuslauseen todistuksessa. Seuraavaksi määritellään monistojen välinen sileä kuvaus ja sen jälkeen derivaatio, tangenttivektori ja tangenttiavaruus, joiden avulla voidaan määritellä sileän kuvauksen differentiaali. Tämän jälkeen siirrytään sileän immersion määritelmään, joka tehdään dfferentiaalin avulla. Sileän immersion avulla saadaan sitten määriteltyä sileä upotus. Luvun viimeisessä osiossa määritellään alimonistot ja niihin liittyvät tasojoukot. Erityisesti määritellään vielä kriittiset pisteet ja arvot. Kolmannessa ja viimeisessä luvussa todistetaan ensin joitain aputuloksia ja sen jälkeen Sardin lause, jota käytetään myöhemmin Whitneyn upotuslauseen todistamiseen. Sardin lauseen mukaan sileiden monistojen välisen sileän kuvauksen kriittisten arvojen joukko on nollamittainen. Tämän jälkeen todistetaan vielä kaksi aputulosta ennen Whitneyn upotuslauseen todistusta. Näistä jälkimmäisessä osoitetaan, että jos on olemassa sileä upotus sileältä n-monistolta jollekin euklidiselle avaruudelle, niin on olemssa sileä upotus avaruudelle R2n+1. Näiden jälkeen päästään todistamaan tutkielman päätulos Whitneyn upotuslause, jonka mukaan jokaisella reunallisella tai reunattomalla sileällä n-monistolla on olemassa vahva sileä upotus euklidiseen avaruuteen R2n+1. Todistus jakautuu kahteen osaan, joissa ensimmäisessä todistetaan tapaus, jossa sileä monisto on kompakti. Tämä tehdään rakentamalla sileä upotus sileän töyssyfunktion avulla. Todistuksen toisessa osassa jaetaan ei-kompakti sileä monisto kompakteihin alimonistoihin tyhjennysfunktion avulla. Tämän jälkeen näiden sileät upotukset yhdistetään töyssyfunktion avulla uudeksi koko moniston peittäväksi sileäksi upotukseksi.

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.