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This thesis is comprised of three logically independent parts. As the title suggests, each part is related to vector bundles on abelian varieties. We first use Brill-Noether theory to study the geometry of a general curve in its canonical embedding. We prove that there is no $g$ for which the canonical embedding of a general curve of genus $g$ lies on the Segre embedding of any product of three or more projective spaces. We then consider non-degenerate line bundles on abelian varieties. Central to our work is Mumford's index theorem. We give an interpretation of this theorem, and then prove that non-degenerate line bundles, with nonzero index, exhibit positivity analogous to ample line bundles. As an application, we determine the asymptotic behaviour of families of cup-product maps. Using this result, we prove that vector bundles, which are associated to these families, are asymptotically globally generated. To illustrate our results, we consider explicit examples. We also prove that simple abelian varieties, for which our results apply in all possible instances, exist. This is achieved by considering a particular class of abelian varieties with real multiplication. The final part of this thesis concerns the theory of theta and adelic theta groups. We extend and refine work of Mumford, Umemura, and Mukai. For example, we determine the structure and representation theory of theta groups associated to a class of vector bundles which we call simple semi-homogeneous vector bundles of separable type. We also construct, and clarify functorial properties enjoyed by, adelic theta groups associated to line bundles. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-06-24 17:14:21.687

Being among the most popular and efficient classification and regression methods currently available, implementations of support vector machines exist in almost every popular programming language. Currently four R packages contain SVM related software. The purpose of this paper is to present and compare these implementations. (authors' abstract)

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 1995

by Peter Au. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 87-88). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminaries --- p.7 / Chapter 2.1 --- Recession and Conjugate Functions --- p.7 / Chapter 2.2 --- Directional derivative and Subgradient --- p.10 / Chapter 2.3 --- Well-Posedness and E-subgradient --- p.15 / Chapter 2.4 --- Exact Penalization --- p.17 / Chapter 3 --- Some Recent Results on Error Bounds --- p.20 / Chapter 3.1 --- Hoffman's Error Bound --- p.20 / Chapter 3.2 --- Extension of Hoffman's Error Bound to Polynomial Systems --- p.26 / Chapter 3.2.1 --- An Error Bound to Polynomial Systems --- p.28 / Chapter 3.2.2 --- Error Bound for Convex Quadratic Inequali- ties Systems --- p.30 / Chapter 3.3 --- Error Bounds for a Convex Inequality --- p.41 / Chapter 3.3.1 --- Unconstrained Case --- p.42 / Chapter 3.3.2 --- Constrained Case --- p.47 / Chapter 3.4 --- Error Bounds for System of Convex Inequalities --- p.55 / Chapter 3.4.1 --- Unconstrained Case --- p.56 / Chapter 3.4.2 --- Constrained Case --- p.60 / Chapter 4 --- Some Recent Results on Certain Proper Efficient Points --- p.63 / Chapter 4.1 --- Scalarization of Henig Proper Efficient Points --- p.63 / Chapter 4.1.1 --- Preliminaries --- p.64 / Chapter 4.1.2 --- Scalarization by Monotone Minkowski Func- tionals --- p.68 / Chapter 4.1.3 --- Scalarization by Continuous Norms --- p.73 / Chapter 4.2 --- Pareto Optimizing and Scalarly Stationary Sequence --- p.75 / Bibliography

Lee, Hon Leung. / "August 2011." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 98-101) and index. / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Preliminaries --- p.11 / Chapter 2.1 --- Functional analysis --- p.11 / Chapter 2.2 --- Convex analysis --- p.14 / Chapter 2.3 --- Relative interiors --- p.19 / Chapter 2.4 --- Multifunctions --- p.21 / Chapter 2.5 --- Variational analysis --- p.22 / Chapter 3 --- A unified notion of optimality --- p.29 / Chapter 3.1 --- Basic notions of minimality --- p.29 / Chapter 3.2 --- A unified notion --- p.32 / Chapter 4 --- Separation theorems --- p.38 / Chapter 4.1 --- Zheng and Ng fuzzy separation theorem --- p.38 / Chapter 4.2 --- Extremal principles and other consequences --- p.43 / Chapter 5 --- Necessary conditions for the unified notion of optimality --- p.49 / Chapter 5.1 --- Local asymptotic closedness --- p.49 / Chapter 5.2 --- First order necessary conditions --- p.56 / Chapter 5.2.1 --- Introductory remark --- p.56 / Chapter 5.2.2 --- Without operator constraints --- p.59 / Chapter 5.2.3 --- With operator constraints --- p.66 / Chapter 5.3 --- Comparisons with known necessary conditions --- p.74 / Chapter 5.3.1 --- Finite-dimensional setting --- p.74 / Chapter 5.3.2 --- Zheng and Ng's work --- p.76 / Chapter 5.3.3 --- Dutta and Tammer's work --- p.80 / Chapter 5.3.4 --- Bao and Mordukhovich's previous work --- p.81 / Chapter 6 --- A weak notion: approximate efficiency --- p.84 / Chapter 6.1 --- Approximate minimality --- p.85 / Chapter 6.2 --- A scalarization result --- p.86 / Chapter 6.3 --- Variational approach --- p.94 / Bibliography --- p.98 / Index --- p.102

In this thesis we study two important issues in vector optimization problem (VOP). The first is on the scalarization; here we provide some merit functions for VOP and analyze their error bound property. The second is on generalization of Ekeland's variational principle; here this famous result in variational analysis is now extended from the original setting for scalar-valued functions to that of vector-valued functions. This generalization enable us to study the error bound property for systems of functions instead of that for a single function. / Liu Chun-guang. / "June 2006." / Adviser: Kung-fu Ng. / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6440. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (p. 92-94). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

In this thesis we study the moduli of holomorphic vector bundles over a non-compact complex space X, which will mainly be of dimension 2 or 3 and which contains a distinguished rational curve ℓ ⊂ X. We will consider the situation in which X is the total space of a holomorphic vector bundle on CP1 and ℓ is the zero section. While the treatment of the problem in this full generality requires the study of complex analytic spaces, it soon turns out that a large part of it reduces to algebraic geometry. In particular, we prove that in certain cases holomorphic vector bundles on X are algebraic. A key ingredient in the description of themoduli are numerical invariants that we associate to each holomorphic vector bundle. Moreover, these invariants provide a local version of the second Chern class. We obtain sharp bounds and existence results for these numbers. Furthermore, we find a new stability condition which is expressed in terms of these numbers and show that the space of stable bundles forms a smooth, quasi-projective variety.