Mathematical optimization

A cross-layer design architecture featuring a new network
stack component called a controller is presented. The
controller takes system status information from the protocol
components and uses it to tune the behavior of the network
stack to a given performance objective. A controller design
strategy using a machine learning algorithm and a simulator
is proposed, implemented, and tested. Results show the
architecture and design strategy are capable of producing a
network stack that outperforms the existing protocol stack for
arbitrary performance objectives. The techniques presented
give network designers the flexibility to easily tune the
performance of their networks to suit their application. This
cognitive networking architecture has great potential for high
performance in future wireless networks.

In this dissertation we present a computational approach to Conley's Decomposition
Theorem, which gives a global decomposition of dynamical systems, and
introduce an explicit numerical algorithm with computational complexity bounds
for computing global dynamical structures of a continous map including attractorrepeller
pairs and Conley's Lyapunov function. The approach is based on finite spatial
discretizations and combinatorial multivalued maps. The method is successful
in exhibiting approximations of attractor-repeller pairs, invariant sets, and Conley's
Lyapunov function. We used the C++ language to code the algorithm.

This dissertation studies two independent problems, one is about graph labeling
and the other problem is related to connectivity condition in a simple graph.
Graph labeling is a rapidly developing area of research in graph theory, having connections with a variety of application-oriented areas such as VLSI optimization, data
structures and data representation. Furthermore, the connectivity conditions in a simple graphs may help us to study the new aspects of ad hoc networks, social networks and web graphs. In chapter 2, we study path systems, reduced path systems and how to construct a super edge-graceful tree with any number of edges using path systems. First, we give an algorithm to reduce a labeled path system to a smaller labeled path system of a different type. First, we investigate the cases (m, k) = (3; 5) and
(m, k) = (4; 7), where m is the number of paths and 2k is the length of each path, and then we give a generalization for any k, m = 3 and m = 4. We also describe a procedure to construct a super-edge-graceful tree with any number of edges.

In wireless orthogonal frequency division multiple-access (OFDMA) standards,
subcarriers are grouped into chunks and a chunk of subcarriers is made as the minimum allocation unit for subcarrier allocation. We investigate the chunk-based resource allocation for OFDMA downlink, where data streams contain packets with diverse bit-errorrate (BER) requirements. Supposing that adaptive transmissions are based on a number of discrete modulation and coding modes, we derive the optimal resource allocation scheme that maximizes the weighted sum of average user rates under the multiple BER and total power constraints. With proper formulation, the relevant optimization problem is cast as an integer linear program (ILP). We can rigorously prove that the zero duality gap holds for the formulated ILP and its dual problem. Furthermore, it is shown that the optimal strategy for this problem can be obtained through Lagrange dual-based gradient iterations with fast convergence and low computational complexity per iteration. Relying on the stochastic optimization tools, we further develop a novel on-line algorithm capable of dynamically learning the underlying channel distribution and asymptotically approaching the optimal strategy without knowledge of intended wireless channels a priori. In addition, we extend the proposed approach to maximizing the a-fair utility functions of average user rates, and show that such a utility maximization can nicely balance the trade-off between the total throughput and fairness among users.

A second-order Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lower-order derivatives play a key role in forcing certain types of dynamics. However, the application of these techniques requires an analytic restriction on the Lagrangian that it satisfy a twist property. In this dissertation we approach this problem from the point of view of curve shortening in an effort to remove the twist condition. In classical curve shortening a family of curves evolves with a velocity which is normal to the curve and proportional to its curvature. The evolution of curves with decreasing action is more general, and in the first part of this dissertation we develop some results for curve shortening flows which shorten lengths with respect to a Finsler metric rather than a Riemannian metric. The second part of this dissertation focuses on analytic methods to accommodate the fact that the Finsler metric for second-order Lagrangian system has singularities. We prove the existence of simple periodic solutions for a general class of systems without requiring the twist condition. Further; our results provide a frame work in which to try to further extend the topological forcing theorems to systems without the twist condition.

A zero knowledge identification protocol is an interactive proof system that allows a person to prove that he knows a secret key associated with his identity without revealing the secret key. This type of protocol is the topic of a fairy tale, by Gustavus Simmons called the King's Dilemma, about a king and the problem he has with thieves impersonating his tax collectors. It describes a zero-knowledge identification protocol that will rid the king of his problem. I present this system, the motivation for this thesis, and the transformations from this protocol, that uses lead weights and containers, to protocols that use mathematical elements. The security of these protocols is determined by the complexity of the underlying mathematical problem, such as the knapsack and discrete logarithm problem, and three properties: completeness, soundness, and zero knowledge.

Model reduction of large-scale systems over a specified frequency range of operation is studied in this research and reported in this dissertation. Frequency-domain balanced structures with integration of singular perturbation are proposed for model reduction of large-scale continuous-time as well as discrete-time systems. This method is applied to both open-loop as well as closed-loop systems. It is shown that the response of reduced systems closely resemble that of full order systems within a specified frequency range of operation. Simulation experiments for the model reduction of several large-scale, continuous and discrete-time systems demonstrate the superiority of the proposed technique over the previously available methods.

In this dissertation, we study some open problems concerning the existence or non-existence of some combinatorial designs. We give the construction or proof of non-existence of some Steiner systems, large sets of designs, and graph designs, with prescribed automorphism groups.

The main objective of this thesis is to simulate, evaluate and discuss three standard methodologies of calculating Value-at-Risk (VaR) : Historical simulation, the Variance-covariance method and Monte Carlo simulations. Historical simulation is the most common nonparametric method. The Variance-covariance and Monte Carlo simulations are widely used parametric methods. This thesis defines the three aforementioned VaR methodologies, and uses each to calculate 1-day VaR for a hypothetical portfolio through MATLAB simulations. The evaluation of the results shows that historical simulation yields the most reliable 1-day VaR for the hypothetical portfolio under extreme market conditions. Finally, this paper concludes with a suggestion for further studies : a heavy-tail distribution should be used in order to imporve the accuracy of the results for the two parametric methods used in this study.

In this work, we investigate input-to-state stability (ISS) and other related stability properties for control systems with time-delays. To overcome the complexity caused by the presence of the delays, we adopt a Razumikhin approach. The underlying idea of this approach is to treat the delayed variables as system uncertainties. The advantage of this approach is that one works in the more familiar territory of stability analysis for delay-free systems in the context of ISS instead of carrying out stability analysis on systems of functional differential equations. Our first step is to provide criteria on ISS and input-to-input stability properties based on the Razumikhin approach. We then turn our attention to large-scale interconnected systems. It has been well recognized that the small-gain theory is a powerful tool for stability analysis of interconnected systems. Using the Razumikhin approach, we develop small-gain theorems for interconnected systems consisting of two or more subs ystems with time-delays present either in the interconnection channels or within the subsystems themselves. As an interesting application, we apply our results to an existing model for hematopoesis, a blood cell production process,and improve the previous results derived by linear methods. Another important stability notion in the framework of ISS is the integral ISS (iISS) property. This is a weaker property than ISS, so it supplies to a larger class of systems. As in the case of ISS, we provide Razumikhin criteria on iISS for systems with delays. An example is presented to illustrate that though very useful in practice, the Razumikhin approach only provides sufficient conditions, not equivalent conditions. Finally, we address stability of time-varying systems with delays in the framework of ISS.