Thèse--Univ. de Paris. / "Index bibliographique": p. [657]-659; "Appendice bibliographique": p. [660]

The field of infinite series is so large that any investigation into that field must necessarily be limited to a particular phase. An attempt has been made to develop a number of tests having a wide range of applications. Particular emphasis has been placed on tests for series of positive terms.

The purpose of this paper is to examine certain questions concerning infinite series. The first chapter introduces several basic definitions and theorems from calculus. In particular, this chapter contains the proofs for various convergence tests for series of real numbers. The second chapter deals primarily with the equivalence of absolute convergence, unconditional convergence, bounded multiplier convergence, and c0 multiplier convergence for series of real numbers. Also included in this chapter is a proof that an unconditionally convergent series may be rearranged so that it converges to any real number desired. The third chapter contains a proof of the Silverman-Toeplitz Theorem together with several applications.

This thesis attempts to establish properties of Hölder and Cesàro summable series analogous to those of ordinary convergent series and also to establish properties that are possibly different from those of convergent series.

Webben växer och webbapplikationer blir mer och mer komplexa och innehållsrika. Med detta så ska dessa växande applikationer även kunna presenteras på ett tilltalande och effektivt sätt. Jättarna Facebook, Twitter, Instagram och små bloggar har alla börjat använda sig av en teknik för att kunna dynamiskt ladda in ytterligare innehåll på en sida utan att behöva ladda om den. Denna teknik kallas för Infinite Scroll eller oändlig scrollning. Infinite Scroll kan resultera i enormt stort innehåll vilket tidigare studier visar på att det bidrar till bristande prestanda för rendering och manipulering av innehållet. Arbetet kommer utföra ett experiment där målet är att undersöka hur två populära SPA-ramverk AngularJS och ReactJS hanterar detta problem.

In this thesis we use a diffeo-geometric framework based on manifolds hat are locally modeled on ``convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold M, we construct a weak symplectic structure on each leaf I_w of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangians is equivalent to a heuristic weak symplectic structure of Weinstein. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf I_w consisting of positive weighted isotropics onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space I_w can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.

This thesis investigates the small time asymptotics of solutions of stochastic equations in infinite dimensions. In this abstract H denotes a separable Hilbert space, A denotes a linear operator on H generating a strongly continuous semigroup and (W(t))t???0 denotes a separable Hilbert space-valued Wiener process. In chapter 2 we consider the mild solution (Xx(t))t???[0,1] of a stochastic initial value problem dX = AX dt + dW t ??? (0, 1] X(0) = x ??? H , where the equation has an invariant measure ??. Under some conditions L(Xx(t)) has a density k(t, x, ??) with respect to ?? and we can find the limit limt???0 t ln k(t, x, y). For infinite dimensional H this limit only provides the lower bound of a large deviation principle (LDP) for the family of continuous trajectory-valued random variables { t ??? [0, 1] ??? Xx(??t) : ?? ??? (0, 1]}. In each of chapters 3, 4 and 5 we find an LDP which describes the small time asymptotics of the continuous trajectories of the solution of a stochastic initial value problem. A crucial role is played by the LDP associated with the Gaussian trajectory-valued random variable of the noise. Chapter 3 considers the initial value problem dX(t) = (AX(t) + F(t,X(t))) dt + G(X(t)) dW(t) t ??? (0, 1] X(0) = x ??? H, where drift function F(t, ??) is Lipschitz continuous on H uniformly in t ??? [0, 1] and diffusion function G is Lipschitz continuous, taking values that are Hilbert-Schmidt operators. Chapter 4 considers an equation with dissipative drift function F defined on a separable Banach space continuously embedded in H; the solution has continuous trajectories in the Banach space. Chapter 5 considers a linear initial value problem with fractional Brownian motion noise. In chapter 6 we return to equations with Wiener process noise and find a lower bound for liminft???0 t ln P{X(0) ??? B,X(t) ??? C} for arbitrary L(X(0)) and Borel subsets B and C of H. We also obtain an upper bound for limsupt???0 t ln P{X(0) ??? B,X(t) ??? C} when the equation has an invariant measure ??, L(X(0)) is absolutely continuous with respect to ?? and the transition semigroup is holomorphic.

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This thesis deals with various problems about the normal and subnormal structure of infinite groups. We first consider the relationship between the number of normal subgroups of a group G and of a subgroup H of finite index in G. We prove Theorem 1.5 There exists a finitely generated group G which has a subgroup H of index 2 such that H has continuously many normal subgroups and G has only countably many normal subgroups. Proposition 1.7 Let k be an infinite cardinal. Then there exists a group G of cardinality k that has only 12 normal subgroups but which contains a subgroup H of index 2 having k normal subgroups. We then consider partially ordered sets and investigate the subnormal structure of generalized wreath products. We deal with the question whether the number of subnormal subgroups of an infinite group is determined by the number of its n-step subnormal subgroups for an integer n. We prove Theorem 5.3 Let G be a group. Then G has finitely many subnormal subgroups if and only if it has finitely many 2-step subnormal subgroups. Theorem 5.5 Let m and n be infinite cardinals such that m ≤ n. Then there exists a group G with the following properties: (1) The cardinality of G is n. (2) The number of normal subgroups of G is <mathematical symbol>. (3) The number of 2-step subnormal subgroups of G is m. (4) The number of 3-step subnormal subgroups of G is 2ⁿ. Finally we consider characteristically simple groups with countably many normal subgroups. We construct a new type of characteristically simple groups: Corollary 6.15 Let ∧ be a partially ordered set such that for λ,<mathematical symbol>∊∧ there exists an automorphism a of ∧ such that <mathematical symbol> ≤ λa. Let <mathematical symbol>(∧) be the distributive lattice of semi-ideals of ∧. Then there exists a group G with the following properties: (1) |G| ≤ max(<mathematical symbol> of |<mathematical symbol>(∧)|). (2) All subnormal subgroups of G are normal in G. (3) The lattice of normal subgroups of G is isomorphic to <mathematical symbol> (∧). (4) The group G is characteristically simple.

In this thesis a set, Ω, of cardinality N_K and a group acting on Ω, with N_K+1 orbits on the power set of Ω, is found for every infinite cardinal N_K. Let W_K denote the initial ordinal of cardinality N_K. Define N := {α₁α₂ . . . α_n∣ 0 < n < w, α_j ∈ w_K for j = 1, . . .,n, α_n a successor ordinal} R := {ϰ ∈ N ∣ length(ϰ) = 1 mod 2} and let these sets be ordered lexicographically. The order types of N and R are Κ-types (countable unions of scattered types) which have cardinality N_K and do not embed w*₁. Each interval in N or R embeds every ordinal of cardinality N_K and every countable converse ordinal. N and R then embed every K-type of cardinality N_K with no uncountable descending chains. Hence any such order type can be written as a countable union of wellordered types, each of order type smaller than w^w_k. In particular, if α is an ordinal between w^w_k and w_K+1, and A is a set of order type α then A= ⋃_n<wA_n where each A_n has order type wⁿ_k. If X is a subset of N with X and N - X dense in N, then X is orderisomorphic to R, whence any dense subset of R has the same order type as R. If Y is any subset of R then R is (finitely) piece- wise order-preserving isomorphic (PWOP) to R ⋃^. Y. Thus there is only one PWOP equivalence class of N_K-dense K-types which have cardinality N_K, and which do not embed w*₁. There are N_K+1 PWOP equivalence classes of ordinals of cardinality N</sub>K</sub>. Hence the PWOP automorphisms of R have N_K+1 orbits on θ(R). The countably piece- wise orderpreserving automorphisms of R have N₀ orbits on R if ∣k∣ is smaller than w₁ and ∣k∣ if it is not smaller.