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This study covers means, mean value theorems of the differential calculus, and mean value theorems of integral calculus.

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by Nga-Wai Liu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Bibliography: leaves 161-166. / Chapter Chapter 0 --- Introduction / Chapter 0.1 --- Historical background I ´ؤ The Atiyah-Singer index theorem --- p.1 / Chapter 0.2 --- Historical background II ´ؤGauge theory --- p.3 / Chapter 0.3 --- Arrangement of the thesis --- p.5 / Chapter Chapter 1 --- Fredholm operators / Chapter 1.1 --- Basic propetries --- p.7 / Chapter 1.2 --- Compact operators --- p.8 / Chapter 1.3 --- Homotopy- invariance of the index --- p.9 / Chapter 1.4 --- Family of Fredholm operators ´ؤ Index bundle --- p.13 / Chapter 1.5 --- Wiener-Hopf operators --- p.19 / Chapter Chapter 2 --- K-theory / Chapter 2.1 --- K-theory of compact spaces --- p.24 / Chapter 2.2 --- K-theory with compact support --- p.28 / Chapter 2.3 --- Bott periodicity theorem --- p.32 / Chapter 2.4 --- Difference construction --- p.44 / Chapter 2.5 --- Thom isomorphism theorem on K-theory --- p.51 / Chapter Chapter 3 --- Operators on manifolds / Chapter 3.1 --- Differential operators on Euclidean spaces --- p.54 / Chapter 3.2 --- Differential operators on manifolds --- p.55 / Chapter 3.3 --- Pseudodifferential operators on Euclidean spaces --- p.58 / Chapter 3.4 --- Pseudodifferential operators on manifolds --- p.62 / Chapter 3.5 --- Elliptic operators --- p.70 / Chapter 3.6 --- Tensor products --- p.76 / Chapter Chapter 4 --- Atiyah-Singer index theorem / Chapter 4.1 --- The topological index --- p.84 / Chapter 4.2 --- The analytical index --- p.87 / Chapter 4.3 --- The Atiyah-Singer index theorem --- p.89 / Chapter 4.4 --- Characteristic classes --- p.95 / Chapter 4.5 --- Thorn isomorphisms --- p.98 / Chapter 4.6 --- Cohomological formulation of the topological index --- p.101 / Chapter Chapter 5 --- Geometric preliminaries / Chapter 5.1 --- "Connections on principal bundles, and associated bundles" --- p.104 / Chapter 5.2 --- Gauge transformations --- p.109 / Chapter 5.3 --- Riemannian geometry --- p.112 / Chapter 5.4 --- Bochner-Weitzenboch formula --- p.116 / Chapter 5.5 --- Characteristic classes via curvature forms --- p.121 / Chapter 5.6 --- Holonomy --- p.126 / Chapter Chapter 6 --- Gauge theory / Chapter 6.1 --- The Yang-Mills functionals --- p.128 / Chapter 6.2 --- Instantons on S4 --- p.131 / Chapter 6.3 --- Moduli of self-dual connections --- p.142 / Chapter 6.4 --- Manifold structure for Moduli of self-dual connections --- p.153 / References --- p.161

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<p> In this thesis we study the representation theorems for evenly positive definite functions on Euclidean spaces. A generalization of the concept of evenness on R^n to a concept of symmetry on any locally compact abelian group is given. In addition, a result analogous to the Weil-Povzner-Raikov Theorem is obtained for the representation of symmetrically positive definite functions on locally compact abelian groups.</p> / Thesis / Master of Science (MSc)

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