This thesis is concerned with the solution of linear operator equations by projection methods known as minimal residual (MR) and orthogonal residual (OR) methods. We begin with a rather abstract framework of approximation by orthogonal and oblique projection in Hilbert space. When these approximation schemes are applied to sequences of nested spaces, with a simple requirement relating trial and test spaces in case of the OR method, one can derive at this rather general level the basic relations which have been proved for many specific Krylov subspace methods for solving linear systems of equations in the literature. The crucial quantities with which we describe the behavior of these methods are angles between subspaces. By replacing the given inner product with one that is basis-dependent, one can also incorporate methods based on non-orthogonal bases such as those based on the non-Hermitian Lanczos process for solving linear systems. In fact, one can show that any reasonable approximation method based on a nested sequence of approximation spaces can be interpreted as an MR or OR method in this way. When these abstract approximation techniques are applied to the solution of linear operator equations, there are three generic algorithmic formulations, which we identify with some algorithms in the literature. Specializing further to Krylov trial and test spaces, we recover the well known Krylov subspace methods. Moreover, we show that our general framework also covers in a natural way many recent generalizations of Krylov subspace methods, which employ techniques such as augmentation, deflation, restarts and truncation. We conclude with a chapter on error and residual bounds, deriving some old and new results based on the angles framework. This work provides a natural and consistent framework for the sometimes confusing plethora of methods of Krylov subspace type introduced in the last 50 years.

This thesis is a study of four cities and their city-plans. The purpose of the paper is to understand similarities and differences between cities in northern Greece during the Classical period. This is done with a comparative method and Kevin Lynch’s theory of the image of the city. The method and the theory are the foundation for the study. The research question used to be able to fulfil the purpose of the paper is Which similarities and differences exist in the construction of Classical cities in northern Greece and why does these similarities and differences exist? The general plans of the chosen cities are studied, not individual buildings and remains. Aspects such as roads, the placement of city walls, agora, public buildings, and residential areas are compared in the paper. The four cities that were studied in the paper were Amphipolis, Olynthus, Pella, and Thasos. All located in Macedonia or on Chalcidice and Thasos. They were selected based on a few criteria. All were known cities from the Classical period, had been excavated to quite a large extent and were not only religious places or burial grounds. The cities are not exact copies of each other. Olynthus and Pella are built after the Hippodamian plan, but Amphipolis and Thasos are built over time with an organic city-plan. Buildings, structures, and central places are the same in the different cities but how the cities are structured and how buildings are placed vary. The terrain, the landscape, traditions, philosophical ideas could all be part of the cause why the four compared cities have been structured differently. The land the cities have been built on vary which affect how a city can be planned and built. The result of the thesis is that the cities in northern Greece have similarities in what types of buildings and structures can be found in them. How the city is structured and organized are the differences in the city-plans and to which degree depends on multiple factors.

A k × n array with entries from a q-letter alphabet is called a t-covering array if each t × n submatrix contains amongst its columns each one of the gt different words of length t that can be produced by the q letters. In the present article we use a probabilistic approach based on an appropriate Markov chain embedding technique, to study a t-covering problem where, instead of looking at all possible t ×n submatrices, we consider only submatrices of dimension t ×n with its rows being consecutive rows of the original k × n array. Moreover, an exact formula is established for the probability distribution function of the random variable, which enumerates the number of deficient submatrices (i.e., submatrices with at least one missing word, amongst their columns), in the case of a k × n binary matrix (q = 2) obtained by realizing kn Bernoulli variables.

Pólya and Schur completely characterized all real-rootedness preserving linear operators acting on the standard monomial basis in their famous work from 1914. The corresponding eigenvalues are from then on known as multiplier sequences. In 2009 Borcea and Br\"and\'en gave a complete characterization for general linear operators preserving real-rootedness (and stability) via the symbol. Relying heavily on these results, in this thesis, we are able to completely characterize multiplier sequences for generalized Laguerre bases. We also apply our methods to reprove the characterization of Hermite multiplier sequences achieved by Piotrowski in 2007.

In this thesis, we provide convergent numerical solutions to non-linear forward-BSDEs (Backward Stochastic Differential Equations). Applications in mathematical finance, financial economics and financial econometrics are discussed. Numerical examples show the effectiveness of our methods.

In 1991, A. Máté precisely calculated the first-order asymptotic behavior of the sequence of Christoffel functions associated with Szego measures on the unit circle. Our principal goal is the abstraction of his result in two directions: We compute the translated asymptotics, limn λn(µ, x + a/n), and obtain, as a corollary, a universality limit for the fairly broad class of Szego weights. Finally, we prove Máté’s result for measures supported on smooth curves in the plane. Our proof of the latter derives, in part, from a precise estimate of certain weighted means of the Faber polynomials associated with the support of the measure. Finally, we investigate a variety of applications, including two novel applications to ill-posed problems in Hilbert space and the mean ergodic theorem.

We will introduce three new classes of graphs; namely bipartite dot product graphs, probe dot product graphs, and combinatorial orthogonal graphs. All of these representations were inspired by a vector representation known as a dot product representation. Given a bipartite graph G = (X, Y, E), the bipartite dot product representation of G is a function ƒ : X ∪ Y → Rk and a positive threshold t such that for any κ ∈ Χ and γ ∈ Υ , κγ ∈ ε if and only if f(κ) · f(γ) ≥ t. The minimum k such that a bipartite dot product representation exists for G is the bipartite dot product dimension of G, denoted bdp(G). We will show that such representations exist for all bipartite graphs as well as give an upper bound for the bipartite dot product dimension of any graph. We will also characterize the bipartite graphs of bipartite dot product dimension 1 by their forbidden subgraphs. An undirected graph G = (V, E) is a probe C graph if its vertex set can be parti-tioned into two sets, N (nonprobes) and P (probes) where N is independent and there exists E' ⊆ N × N such that G' = (V, E ∪ E) is a C graph. In this dissertation we introduce probe k-dot product graphs and characterize (at least partially) probe 1-dot product graphs in terms of forbidden subgraphs and certain 2-SAT formulas. These characterizations are given for the very different circumstances: when the partition into probes and nonprobes is given, and when the partition is not given. Vectors κ = (κ1, κ2, . . . , κn)T and γ = (γ1, γ2, . . . , γn)T are combinatorially orthogonal if |{i : κiγi = 0}| ≠ 1. An undirected graph G = (V, E) is a combinatorial orthogonal graph if there exists ƒ : V → Rn for some n ∈ Ν such that for any u, υ &Isin; V , uv ∉ E iff ƒ(u) and ƒ(v) are combinatorially orthogonal. These representations can also be limited to a mapping g : V → {0, 1}n such that for any u,v ∈ V , uv ∉ E iff g(u) · g(v) = 1. We will show that every graph has a combinatorial orthogonal representation. We will also state the minimum dimension necessary to generate such a representation for specific classes of graphs.

Indiana University-Purdue University Indianapolis (IUPUI) / We consider the behavior of zeros of random polynomials of the from \begin{equation*} P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z) \end{equation*} as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第18400号 / 情博第515号 / 新制||情||91(附属図書館) / 31258 / 京都大学大学院情報学研究科数理工学専攻 / (主査)准教授 辻本 諭, 教授 中村 佳正, 教授 梅野 健 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

京都大学 / 0048 / 新制・論文博士 / 博士(理学) / 乙第12910号 / 論理博第1546号 / 新制||理||1590(附属図書館) / 32120 / ソウル大学大学院数学科 / (主査)教授 中島 啓, 教授 小野 薫, 教授 向井 茂 / 学位規則第4条第2項該当 / Doctor of Science / Kyoto University / DFAM