We study the existence of flows in infinite networks and extend basic theorems due to Gale and Hoffman and to Ford and Fulkerson. The classical approach to finite networks uses a constructive combinatorical algorithm that has become known as the labelling algorithm. Our approach to infinite networks involves Hahn--Banach type theorems on the existence of certain linear functionals. Thus the main tools are from the theory of functional and convex analysis. In Chapter II, we discuss sublinear and linear functionals on real vector spaces in the spirit of the work of K"{o}nig. In particular, a generalization of K"{o}nig's minimum theorem is established. Our theory leads to some useful interpolation results. We also establish a variant of the main interpolation theorem in the context of convex cones. We reformulate the results of Ford--Fulkerson and Gale--Hoffman in terms of certain additive and biadditive set functions. In Chapter III, we show that the space of all additive set functions may be canonically identified with the dual space of a space of certain step functions and that the space of all biadditive set functions may be identified with the dual space of a space of certain step functions in two variables. Our work an additive set functions is in the spirit of classical measure theory, while the case of biadditive set functions resembles the theory of product measures. In Chapter IV, we develop an extended version of the Gale--Hoffman theorem on the existence of flows in infinite networks in a setting of measure-theoretic flavor. This general flow theorem is one of our central results. We discuss, as an application of our flow theorem, a Ford--Fulkerson type result on maximal flows and minimal cuts in infinite networks containing sources and sinks. In addition, we present applications to flows in locally finite networks and to the existence of antisymmetric flows under certain natural conditions. We conclude with a discussion of the case of triadditive set functions. In the appendix, we review briefly the classical theory of maximal flows and minimal cuts in networks with finitely many nodes.

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本论文着重研究了带有Lp模球约束以及二次约束的多项式优化问题的计算复杂度以及关于此类问题的近似算法。在本论文中，利用张量对称化的技巧，我们首次证明了当P∈ [2 ，∞] ，任意高阶的带有Lp模球约束的多项式优化问题均为NP 困难。借助模的对偶性质，我们将这类优化问题转化为求解凸体半径的问题，从而使得我们获得了之前研究所无法使用的算法工具。具体来说，利用计算凸几何的算法工具，对于Lp模球约束的多项式优化问题，我们得到了近似比为[附圖]的确定性多项式时间近似算法，其中d为目标多项式的阶次， n 为问题的维度。使用随机算法，我们将近似比进一步提高为此类问题的己知最优值。[附圖]。此外，我们发展了计算凸几何当中对于凸体半径的计算方法，从而设计出了一种对二次约束多项式优化问题近似比为[附圖]的近似算法，其中m为问题的约束个数。我们的结果涵盖并提高了之前关于此类问题的研究结果。我们相信在本论文中使用的新的算法工具，将在今后的多项式优化问题研究中得到更广泛的应用。 / In this thesis, we present polynomial time approximation algorithms for solving various homogeneous polynomial optimization problems and their multilinear relaxations. Specifically, for the problems with Lp ball constraint, where P∈ [2 ，∞], by reducing them to that of determining the Lq-diameter of certain convex body, we show that they can be approximated to within a factor of [with formula] in deterministic polynomial time, where q = p=(p - 1) is the conjugate of p, n is the number of variables, and d is the degree of the polynomial. We further show that with the help of randomization, the approximation guarantee can be improved to [with formula], which is independent of p and is currently the best for the aforementioned problems. Moreover, we extend the argument of deterministic algorithm mentioned above to solve the quadratically constrained polynomial optimization problems. In particular, for any intersection of ellipsoids K, we can, in polynomial time, construct a random polytope P, which satisfies [with formula]. Then, by reducing the problem to that of evaluating the maximum polytopal norm [with formula] induced by P, over certain convex body, we can approximate the quadratically constrained problem within a factor of [with formula] in polynomial time. Our results unify and generalize those in the literature, which focus either on the quadratic case or the case where [with formula]. We believe that the wide array of tools used in this thesis will have further applications in the study of polynomial optimization problems. / Detailed summary in vernacular field only. / Hou, Ke. / On title page "p" is subscript. / Thesis (Ph.D.) Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 106-111). / Abstracts also in Chinese.

<p>This thesis is about complex convexity. We compare it with other notions of convexity such as ordinary convexity, linear convexity, hyperconvexity and pseudoconvexity. We also do detailed study about ℂ-convex Hartogs domains, which leads to a definition of ℂ-convex functions of class <i>C</i>¹. The study of Hartogs domains also leads to characterization theorem of bounded ℂ-convex domains with <i>C</i>¹ boundary that satisfies the interior ball condition. Both the method and the theorem is quite analogous with the known characterization of bounded ℂ-convex domains with <i>C</i>² boundary. We also show an exhaustion theorem for bounded ℂ-convex domains with <i>C</i>² boundary. This theorem is later applied, giving a generalization of a theorem of L. Lempert concerning the relation between the Carathéodory and Kobayashi metrics.</p>

This thesis is about complex convexity. We compare it with other notions of convexity such as ordinary convexity, linear convexity, hyperconvexity and pseudoconvexity. We also do detailed study about ℂ-convex Hartogs domains, which leads to a definition of ℂ-convex functions of class C1. The study of Hartogs domains also leads to characterization theorem of bounded ℂ-convex domains with C1 boundary that satisfies the interior ball condition. Both the method and the theorem is quite analogous with the known characterization of bounded ℂ-convex domains with C2 boundary. We also show an exhaustion theorem for bounded ℂ-convex domains with C2 boundary. This theorem is later applied, giving a generalization of a theorem of L. Lempert concerning the relation between the Carathéodory and Kobayashi metrics.

A series of experiments were carried out to measure the effects of convex surface curvature on film cooling. In the first series of experiments cooling holes were positioned along the vane such that their non-dimensional curvature parameter, 2r/d, was matched. Single row of holes with the same diameter were placed at high and moderate curvature position along a turbine vane resulting in 2r/d = 28 and 40, accordingly. A third row of holes was installed on the vane at the same location as the moderate curvature row with a larger hole diameter, resulting in 2r/d = 28, matching the high curvature row. Adiabatic temperature measurements were then carried out for blowing ratios of M = 0.30 to 1.60 tested at a density ratio of DR = 1.20. The results indicated that there was some scaling of performance present with matching 2r/d, but there was not an exact matching of performance. The second series of experiments focused on the effects of a changing surface curvature downstream of injection. Two row of holes were positioned along the vane surface such that the local radius of curvature and hole diameters were equivalent, with one row positioned upstream of the maximum curvature point and the other downstream of the maximum curvature point. Adiabatic temperature measurements were carried out for blowing ratios of M = 0.30 to 1.60 and tested at a density ratio of DR = 1.20. The results show that the change in curvature downstream plays a significant role in the performance of film cooling and that the local surface curvature is insufficient in capturing its effects. Additional experiments were carried out to measure the effects of the approaching boundary layer influence on film cooling as well as the effect of injection angle at a weakly convex surface. / text

The problem of computing the minimum distance between two convex hulls has applications to many areas including robotics, computer graphics and path planning. Moreover, determining the minimum distance between two convex hulls plays a significant role in support vector machines (SVM). In this study, a new algorithm for finding the minimum distance between two convex hulls is proposed and investigated. A convergence of the algorithm is proved and applicability of the algorithm to support vector machines is demostrated. The performance of the new algorithm is compared with the performance of one of the most popular algorithms, the sequential minimal optimization (SMO) method. The new algorithm is simple to understand, easy to implement, and can be more efficient than the SMO method for many SVM problems.