We study the space of complete Riemannian metrics of nonnegative curvature on the sphere equipped with C^{k+\alpha} topology. We show the space is homogenous for k>=2. If k is infinite, we show that the space is homeomorphic to the separable Hilbert space. We also prove for finite k, the space minius any compact subset is weakly contractible.

Positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory, and they have recently been extended to the setting of free noncommutative function theory. In this paper, we develop the subject further in a number of directions. We give a characterization of completely positive noncommutative kernels in the setting of Hilbert C*-modules and Hilbert W*-modules. We prove an Arveson-type extension theorem for completely positive noncommutative kernels, and we show that a uniformly bounded noncommutative kernel can be decomposed into a linear combination of completely positive noncommutative kernels. / Ph. D.

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In this thesis, we study weighted composition operatorsT(xn)=(£fnX£m(n)) between sequence spaces(c0,c,l1,lp), and more precisely, the sufficient and necessary condition that they are compact. First,we obtain some results of weighted composition operators beingcompact, weakly compact and completely continuous on c0 spaces. Then, we extend then to c,l1,and lp(1<p<¡Û) spaces. Finally, we obtain the condition that an operator from c0, c or lp into c0, c, or lq is compact, weakly compact or completely continuous.

Rectilinear crossing number of a graph is the number of crossing edges in a drawing with all straight line edges. The problem of drawing an n-vertex complete graph such that its rectilinear crossing number is minimum is known to be an NP-Hard problem. In this thesis, we present a heuristic that attempts to achieve the theoretical lower bound value of the rectilinear crossing number of a n+1 vertex complete graph from that of n vertices. Our algorithm accepts an optimal or near-optimal rectilinear drawing of Kn graph as input and tries to place a new node such that the crossing number is minimized. Based on prior optimal drawings of Kn, we make an empirical observation that the optimal drawings are triangular in shape. The proposed heuristic has three steps: (1) Given the optimal or near-optimal drawing of Kn, the outer triangle is determined; (2) A set of candidate positions for the (n+1)th node is determined by ensuring none of them are collinear with two or more nodes in the graph; and (3) the best drawing with least rectilinear crossing number is chosen based on the drawings corresponding to the candidate position. A loose bound on the worst-case time complexity of the proposed algorithm is O(n7). The heuristic is not guaranteed to yield optimal solution as the search space is constrained by the input graph. In our experimental results, we obtained optimal results for complete graphs of up to n=27.

A large portion of research in science and engineering, as well as in business, concerns one similar problem: how to make things "better”? Once properly modeled (although usually a highly nontrivial task), this kind of questions can be approached via a mathematical optimization problem. Optimal solution to a mathematical optimization problem, when interpreted properly, might corresponds to new knowledge, effective methodology or good decisions in corresponding application area. As already proved in many success stories, research in mathematical optimization has a significant impact on numerous aspects of human life. Recently, it was discovered that a large amount of difficult optimization problems can be formulated as copositive programming problems. Famous examples include a large class of quadratic optimization problems as well as many classical combinatorial optimization problems. For some more general optimization problems, copositive programming provides a way to construct tight convex relaxations. Because of this generality, new knowledge of copositive programs has the potential of being uniformly applied to these cases. While it is provably difficult to design efficient algorithms for general copositive programs, we study copositive programming from two standard aspects, its relaxations and its separation problem. With regard to constructing computational tractable convex relaxations for copositive programs, we develop direct constructions of two tensor relaxation hierarchies for the completely positive cone, which is a fundamental geometric object in copositive programming. We show connection of our relaxation hierarchies with known hierarchies. Then we consider the application of these tensor relaxations to the maximum stable set problem. With regard to the separation problem for copositive programming. We first prove some new results in low dimension of 5 x 5 matrices. Then we show how a separation procedure for this low dimensional case can be extended to any symmetric matrices with a certain block structure. Last but not least, we provide another approach to the separation and relaxations for the (generalized) completely positive cone. We prove some generic results, and discuss applications to the completely positive case and another case related to box-constrained quadratic programming. Finally, we conclude the thesis with remarks on some interesting open questions in the field of copositive programming.

Semigroups of completely positive maps arise naturally both in noncommutative stochastic processes and in the dynamics of open quantum systems. Since its inception in the 1970's, the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup. In quantum dynamics, this amounts to embedding a given open system inside some closed system, while in noncommutative probability, it corresponds to the construction of a Markov process from its transition probabilities. In addition to the existence of dilations, one is interested in what properties of the original semigroup (unitality, various kinds of continuity) are preserved. Several authors have proved the existence of dilations, but in general, the dilation achieved has been non-unital; that is, the unit of the original algebra is embedded as a proper projection in the dilation algebra. A unique approach due to Jean-Luc Sauvageot overcomes this problem, but leaves unclear the continuity of the dilation semigroup. The major purpose of this thesis, therefore, is to further develop Sauvageot's theory in order to prove the existence of continuous unital dilations. This existence is proved in Theorem 6.4.9, the central result of the thesis. The dilation depends on a modification of free probability theory, and in particular on a combinatorial property akin to free independence. This property is implicit in some Sauvageot's original calculations, but a secondary goal of this thesis is to present it as its own object of study, which we do in chapter 3.

Through von Neumann's reduction theory, the classification of injective von Neumann algebras acting on separable Hilbert spaces translates into the classification of injective factors. In his proof of the uniqueness of the injective type II₁ factor, Connes showed an alternate characterization of the approximately inner automorphisms of type II₁ factors. Moreover, he conjectured that this characterization could be extended to all types of factors acting on separable Hilbert spaces. In this thesis, we present a general toolbox containing the basic notions needed to study von Neumann algebras, before describing our work concerning Connes' conjecture in the case of type IIIλ factors. We have obtained partial results towards the proof of a modified version of this conjecture.

We study the structure of the set of extreme points of the compact convex set of matrix-valued holomorphic functions with positive real part on a finitely-connected planar domain 𝐑 normalized to have value equal to the identity matrix at some prescribed point t₀ ∈ 𝐑. This leads to an integral representation for such functions more general than what would be expected from the result for the scalar-valued case. After Cayley transformation, this leads to a integral Agler decomposition for the matrix Schur class over 𝐑 (holomorphic contractive matrix-valued functions over 𝐑). Application of a general theory of abstract Schur-class generated by a collection of test functions leads to a transfer-function realization for the matrix Schur-class over 𝐑, extending results known up to now only for the scalar case. We also explain how these results provide a new perspective for the dilation theory for Hilbert space operators having 𝐑 as a spectral set. / Ph. D.

abstract: This thesis aims to investigate the capacity and bit error rate (BER) performance of multi-user diversity systems with random number of users and considers its application to cognitive radio systems. Ergodic capacity, normalized capacity, outage capacity, and average bit error rate metrics are studied. It has been found that the randomization of the number of users will reduce the ergodic capacity. A stochastic ordering framework is adopted to order user distributions, for example, Laplace transform ordering. The ergodic capacity under different user distributions will follow their corresponding Laplace transform order. The scaling law of ergodic capacity with mean number of users under Poisson and negative binomial user distributions are studied for large mean number of users and these two random distributions are ordered in Laplace transform ordering sense. The ergodic capacity per user is defined and is shown to increase when the total number of users is randomized, which is the opposite to the case of unnormalized ergodic capacity metric. Outage probability under slow fading is also considered and shown to decrease when the total number of users is randomized. The bit error rate (BER) in a general multi-user diversity system has a completely monotonic derivative, which implies that, according to the Jensen's inequality, the randomization of the total number of users will decrease the average BER performance. The special case of Poisson number of users and Rayleigh fading is studied. Combining with the knowledge of regular variation, the average BER is shown to achieve tightness in the Jensen's inequality. This is followed by the extension to the negative binomial number of users, for which the BER is derived and shown to be decreasing in the number of users. A single primary user cognitive radio system with multi-user diversity at the secondary users is proposed. Comparing to the general multi-user diversity system, there exists an interference constraint between secondary and primary users, which is independent of the secondary users' transmission. The secondary user with high- est transmitted SNR which also satisfies the interference constraint is selected to communicate. The active number of secondary users is a binomial random variable. This is then followed by a derivation of the scaling law of the ergodic capacity with mean number of users and the closed form expression of average BER under this situation. The ergodic capacity under binomial user distribution is shown to outperform the Poisson case. Monte-Carlo simulations are used to supplement our analytical results and compare the performance of different user distributions. / Dissertation/Thesis / M.S. Electrical Engineering 2012