Nonlinear optics represents a significant area of research and technology concerned with the modification of material optical properties using light. The interaction between light and such materials gives rise to a multitude of nonlinear optical effects, including second har-monic generation, third harmonic generation, high harmonic generation, and sum frequency generation. This thesis focuses on a specific and relevant nonlinear phenomenon within this field, namely the nonlinear Kerr effect, which involves the modification of a material’s re-fractive index through the exposure to an intense beam of light. The nonlinear Kerr effect holds promise for various applications, such as self-phase modulation in laser technology and the utilization of optical solitons in telecommunications. However, the limited availability of materials with sufficiently strong Kerr effects often restricts the practical application of this effect across different industries. Concurrently, optical time-varying systems play crucial roles in modern technologies, in-cluding optical modulators, LiDAR systems, and adaptive cameras. These systems involve the dynamic modification of optical properties. To achieve ultra-fast modulation of light properties, it is beneficial to explore materials with ultra-fast modulation speeds of the op-tical refractive index for integration into time-varying systems. While electro-optical effects represent the most common methods for achieving high-speed modulation of the effective refractive index, the utilization of all-optical methods, such as the nonlinear Kerr effect, presents an alternative approach. Nevertheless, the absence of simultaneous high speed and large nonlinear Kerr response in the majority of well-established materials restricts the utilization of the Kerr effect in time-varying systems.This thesis focuses on the study of a group of materials known as epsilon-near-zero (ENZ) materials, where the real part of the permittivity vanishes at a specific wavelength referred to as the ENZ wavelength. Specifically, indium-tin-oxide (ITO), a transparent conducting oxide, is investigated, with its ENZ wavelength falling within the infrared region of the elec-tromagnetic spectrum. ITO has been shown to possess a record-breaking large nonlinear Kerr effect with sub-picosecond response times, making it an excellent candidate for all-optical time-varying systems. The primary objective of this research is to investigate the applications of this large, fast nonlinear response and, where possible, enhance its effective-ness. One notable application of rapid and substantial modifications in the refractive index of a material is adiabatic wavelength conversion of light. In one project, a thin layer of ITO is subjected to a pump-probe setup, where an intense pump beam of light triggers the nonlinear response of ITO, causing the refractive index to rapidly change while a probe beam passes through the modulated system. Consequently, the wavelength of the probe beam undergoes conversion. Furthermore, it has been demonstrated that the nonlinear response of ITO can be sig-nificantly enhanced in the presence of a plasmonic metasurface. Metasurfaces consist of two-dimensional arrays of sub-wavelength scattering objects capable of manipulating the vectorial properties of light. In another project, we design a gradient metasurface composed of gold placed over ITO, enabling the diffraction of incident light into various diffraction orders depending on the ratio between the wavelength of light and the periodicity of the metasurface. This unique property is utilized to dynamically steer the diffraction orders of the probe beam, achieving wavelength conversion by exciting the nonlinear response of the ITO substrate with a second pump beam. Additionally, we investigate the interaction of resonance modes in an amorphous silicon metasurface, known as Mie modes, with an inherently dark mode in a thin layer of ITO known as the ENZ mode. Through experimental and analytical approaches, we demonstrate that two fundamental Mie modes, electric dipole resonance and magnetic dipole resonance, can strongly couple with the ENZ mode. This strong coupling creates a highly complex system with a large and rapid nonlinear response, enabling the manipulation of light on sub-picosecond timescales. In our final main project, we delve into investigating the nonlinear response of ITO nanoparticles. To accomplish this, we put forth a numerical recursive approach that allows us to incorporate the significant nonlinear Kerr effect of ITO into inherently linear simulation environments. Subsequently, we employ this proposed method to extract the scattering pattern of sub-wavelength antennas fabricated from ITO in both linear and nonlinear optical regimes. Our objective is to explore the potential applications of ITO nanoantennas in various fields. Moreover, this thesis encompasses other projects related to ENZ materials. We investi-gate the nonlinear response of an artificially created ENZ medium by stacking subsequent layers of materials with negative and positive permittivities within the visible range of the electromagnetic spectrum. Additionally, we explore the nonlinear response of nanoparticles made of ITO. Lastly, we present our investigations into the strong coupling of the ENZ mode in a thin layer of ITO with surface plasmon polaritons in a layer of gold in contact with ITO.

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The objective of this research is to develop time-varying signal processing methods for rapidly varying non-stationary signals based on time-varying autoregressive (TVAR) modeling, and to apply such methods to frequency-modulated (FM) interference rejection in direct-sequence spread spectrum (DSSS) communications. For fast varying non-stationary signal processing, such as the task to reject an FM interference that could chirp over the entire DSSS bandwidth in a symbol interval, an explicit description of the variation is necessary to form a time-varying filter. This is realized using the TVAR model, which is an autoregressive model whose coefficients are time-varying with the variation modeled as a linear combination of a set of known functions of time. In DSSS communications, when the strength of an interference - which could be a hostile jammer or overlaid communication signal - possibly exceeds the inherent spread spectrum processing gain, interference rejection is necessary to secure a usable bit-error-rate. The contributions of this research include: a) revealed the advantageous performance of TVAR model based instantaneous frequency estimation (TVAR-IF), which is expected to change the prevailing opinion that regards TVAR-IF as a poor estimator; b) proposed a time-varying Prony method to improve TVAR-IF at low SNR; c) proposed to use TVAR-IF for time-varying FIR notch filter based FM jammer suppression in DSSS communications; d) developed TVAR model based time-varying optimum filters, including the TVAR based Kalman filter (TVAR-KF) and the TVAR based Wiener filter (TVAR-WF); e) developed a TVAR-WF based formulation of FM interference soft-cancellation in DSSS communications; and f) proposed a TVAR based linear prediction error (TVAR-LPE) filter for soft-cancellation of FM interference in DSSS communications. For the interference rejection problem, our TVAR-IF controlled notch filter yields high processing gain close to that using the known IF and much higher than that using the WVD based IF estimate. Furthermore, unlike the IF based notch filter approaches, the proposed soft-cancellation methods utilize the full spectral information captured by the TVAR model. Our soft-cancellation approaches, including TVAR-WF and TVAR-LPE, maintain at least the DSSS system performance expected when no filtering is used, even under estimated conditions. The latter is in contrast to the notch filter based approaches, which may cause deterioration of overall system performance at low jammer-to-signal ratios. / Ph. D.

This dissertation addresses fundamental theoretical problems relevant to flight control for aerial vehicles and weapons in highly uncertain dynamical environment. The approach taken in this dissertation is the L1 adaptive control, which is elaborated from its design perspective for output feedback solution and is extended to time-varying reference systems to support augmentation of gain-scheduled baseline controllers. Compared to conventional adaptive controllers, L1 control has the following advantages: i) it has guaranteed uniformly bounded transient response for system's both signals, input and output; ii) it enables fast adaptation while maintains a bounded away from zero time-delay margin. The proposed adaptive control approach can recover the nominal performance of the flight control systems in the presence of rapid variation of uncertainties. Furthermore, the benefit of L1 adaptive control is its promise for development of theoretically justified tools for Verification and Validation (V&V) of adaptive systems. Adaptive control for uncertain systems usually needs to handle two types of uncertainties: matched and unmatched uncertainties. Both of these two uncertainties will appear in practical flight control problems. In this dissertation, adaptive approaches which can compensate for these two types of uncertainties will be discussed respectively. Two architectures of L1 adaptive control, namely L1 state feedback adaptive control and L1 output feedback adaptive control, are studied. The state feedback adaptive control is applied for compensation of matched uncertainties. Although the state feedback scheme is capable of handling certain type of unmatched uncertainties, such approach is not explored in this dissertation. On the other hand, the output feedback approach is mainly aimed to solve problems in the presence of unmatched uncertainties. The dissertation first discusses the state feedback L1 adaptive control for time-invariant reference systems. The adaptive controller is designed to augment an existing baseline controller. The closed loop system of the plant and the baseline controller is time-invariant. This closed loop system, which is a Linear Time Invariant (LTI) system, determines the dynamics of the reference system. The adaptive feedback can compensate for nonlinear state- and time-dependent uncertainty with uniformly bounded transient response. In this dissertation we discuss the Multi-Input Multi-Output (MIMO) extension of the method. Two flight control examples,Unmanned Combat Aerial Vehicle (UCAV) and Aerial Refueling Autopilot, are considered in the presence of nonlinear uncertainties and control surface failures. The L1 adaptive controller without any redesign leads to scaled response for system's both signals, input and output, dependent upon changes in the initial conditions, system parameters and uncertainties. The time-delay margin analysis for these two examples verifies the theoretical claims. Next, the output feedback approach is studied. The adaptive output feedback controller can be applied to reference systems that do not verify the Strict Positive Real (SPR) condition for their input-output transfer function. In this dissertation, specific design guidelines are presented that render the approach suitable for practical applications. A missile autopilot design example is given to demonstrate the benefits of the design approach. Finally, the L1 state feedback adaptive controller is extended to time-varying reference systems. The adaptive controller intends to augment a gain-scheduled baseline controller. The reference system, which is determined by the closed loop system of the plant and the baseline gain-scheduled controller, is time-varying. The adaptive controller with time-varying reference system is proved to have guaranteed performance bounds similar to those obtained for the case of linear time-invariant reference systems. With this result, the aerial refueling application can be extended to a complete scenario, which includes a racetrack maneuver for an aircraft. The concluding chapter discusses the challenging issues for future research. / Ph. D.

A deterministic change to a time-varying queueing model is described as either changing the number of entities, the queue capacity, or the number of servers in the system at selected times. We use a surrogate distribution for N(t), the number of entities in the system at time t, to approximate deterministic changes to the Ph(t)/Ph(t)/1/c and the Ph(t)/M(t)/s/c queueing models. We develop a solution technique to minimize the number of state probabilities to be approximated. / Master of Science

In this thesis, ties and time-varying covariates in survival analysis are investigated. There are two types of ties: ties between event times (Type 1 ties) and ties between event times and the time that discrete time-varying covariates change or "jump"(Type 2 ties). The Cox proportional hazards model is one of the most important regression models for survival analysis. Methods for including Type 1 ties and time-varying covariates in the Cox proportional hazards model are well established in previous studies, but Type 2 ties have been ignored in the literature. This thesis discusses the effect of Type 2 ties on Cox's partial likelihood, the current default method to treat Type 2 ties in statistical packages SAS and R (called Fail before Jump in this thesis), and proposes alternative methods (Random and Equally Weighted) for Type 2 ties. A simulation study as well as an analysis of data sets from real research both suggest that both Random and Equally Weighted methods perform better than the other two methods. Also the effect of the percentages of Type 1 and Type 2 ties on these methods for handling both types of ties is discussed. / NSERC

This thesis consists of four self-contained papers related to the change of market structure and the quality of equity market. In Paper [I] we found, by using of a Flexible Dynamic Component Correlations (FDCC) model, that the creation of a common cross-border stock trading platform has increased the long-run trends in conditional correlations between foreign and domestic stock market returns. In Paper [II] we study whether the creation of a uniform Nordic and Baltic stock trading platform has affected weak-form information efficiency. The results indicate that the stock market consolidations have had a positive effect on the information efficiency and turnover for an average firm. The merger effects are, however, asymmetrically distributed in the sense that relatively large (small) firms located on relatively large (small) markets experience an improved (reduced) information efficiency and turnover. Although the results indicate that changes in the level of investor attention (measured by turnover) may explain part of the changes in information efficiency, they also lend support to the hypothesis that merger effects may partially be driven by changes in the composition of informed versus uninformed investors following a stock. Paper [III] analyzes whether the measured level of trust in different countries can explain bilateral stock market correlations. One finding is that generalized trust among nations is a robust predictor for stock market correlations. Another is that the trust effect is larger for countries which are close to each other. This indicates that distance mitigates the trust effect. Finally, we confirm the effect of trust upon stock market correlations, by using particular trust data (bilateral trust between country A and country B) as an alternative measurement of trust. In Paper [IV] we present the impact of the stock market mergers that took place in the Nordic countries during 2000 – 2007 on the probabilities for stock price jumps, i.e. for relatively extreme price movements. The main finding is that stock market mergers, on average, reduce the likelihood of observing stock price jumps. The effects are asymmetric in the sense that the probability of sudden price jumps is reduced for large and medium size firms whereas the effect is ambiguous for small size firms. The results also indicate that the market risk has been reduced after the stock market consolidations took place.

"This dissertation proposes a vibration-based approach to detect and monitor structural damage by tracking the instantaneous modal parameters. A change in the instantaneous modal parameters indicates change in the structural health condition. In contrast to many existing structural health monitoring schemes, the proposed approach is less model dependent and works well for both sudden and evolving damage, general loading conditions and complex structures. The instantaneous modal parameters, including modal frequency, mode shape vector and modal damping ratio, are introduced as a bridge between the system properties and time varying vibration modes. The theoretical background of the time-varying vibration modes is developed. It has been shown that for slowly time-varying systems such modes exist and the instantaneous modal parameters have a clear physical interpretation and can be identified from free and forced vibration responses. A set of known techniques are used in an innovative way to identify the instantaneous modal parameters. Applicability of the identification techniques depends on the nature and availability of measurement data. Wavelet ridge method is used to identify the instantaneous modal frequencies and normalized instantaneous mode shape vectors from free vibration data. Wavelet packet sifting technique in conjunction with Hilbert transform and confidence index is proposed to identify the normalized instantaneous mode shape vector from both free and forced vibration data. Time-varying Kalman filter is integrated with the wavelet packet sifting technique to identify the instantaneous modal frequencies and the instantaneous modal damping ratios from free and forced vibration data. The proposed approach has been validated using both simulation and experimental data. The simulation data is obtained from a multi-degree-of-freedom system with time varying stiffness under different loading conditions. Experimental data include both impact testing data from the ASCE benchmark study and shaking-table test data of a full-size two-story wooden building structure, conducted at DPRI, Kyoto University, Japan. It has been shown that the proposed approach can successfully detect and monitor damage and, therefore, has great potential for real applications."

One critical step to successfully integrate wireless data networks to the high-speed wired backbone is the design of network control policies that efficiently utilize resources to provide Quality of Service (QoS) to the users in the integrated networks. Such a design has remained a challenge since wireless networks are time-varying in nature, not only in terms of user/packet arrivals but also in terms of physical channel conditions and access opportunities. In this thesis, we study the stochastic control of time-varying networks to design efficient scheduling and resource allocation policies. In particular, in Chapter 3, we focus on a broad class of control policies that work based on a pick-and-compare principle for networks with time-varying channels. By trading the throughput for complexity and memory requirement, these policies require less complexity compared to the well-investigated throughput-optimal Generalized Maximum Weight Matching (GMWM) policy and also require only linear-memory storage with the number of data-flows. Through Lyapunov analysis tools, we characterize the stability region and delay performance of the studied policies and show how they vary in response to the channel variations. In Chapter 4, we go into further detail and consider the problem of network control from a new perspective through which we carefully incorporate the time-efficiency of underlying scheduling algorithms. Specifically, we develop a policy that dynamically adjusts the time given to the available scheduling algorithms according to queue-backlog and channel correlations. We study the resulting stability region of developed policy and show that the region is at least as large as the one for any static policy. Finally, motivated by the current under-utilization of wireless spectrum, in Chapter 5, we investigate the control of cognitive radio networks as a special example of networks that provide time-varying access opportunities. We assume that users dynamically join and leave the network and may have different utility functions, or could collaborate for a common purpose. We develop a policy that performs joint admission and resource control and works for any user load, either inside or outside the capacity region. Through Lyapunov Optimization techniques, we show that the developed policy can achieve a utility performance arbitrarily close to the optimality with a tradeoff in the average service delay of admitted users.

One critical step to successfully integrate wireless data networks to the high-speed wired backbone is the design of network control policies that efficiently utilize resources to provide Quality of Service (QoS) to the users in the integrated networks. Such a design has remained a challenge since wireless networks are time-varying in nature, not only in terms of user/packet arrivals but also in terms of physical channel conditions and access opportunities. In this thesis, we study the stochastic control of time-varying networks to design efficient scheduling and resource allocation policies. In particular, in Chapter 3, we focus on a broad class of control policies that work based on a pick-and-compare principle for networks with time-varying channels. By trading the throughput for complexity and memory requirement, these policies require less complexity compared to the well-investigated throughput-optimal Generalized Maximum Weight Matching (GMWM) policy and also require only linear-memory storage with the number of data-flows. Through Lyapunov analysis tools, we characterize the stability region and delay performance of the studied policies and show how they vary in response to the channel variations. In Chapter 4, we go into further detail and consider the problem of network control from a new perspective through which we carefully incorporate the time-efficiency of underlying scheduling algorithms. Specifically, we develop a policy that dynamically adjusts the time given to the available scheduling algorithms according to queue-backlog and channel correlations. We study the resulting stability region of developed policy and show that the region is at least as large as the one for any static policy. Finally, motivated by the current under-utilization of wireless spectrum, in Chapter 5, we investigate the control of cognitive radio networks as a special example of networks that provide time-varying access opportunities. We assume that users dynamically join and leave the network and may have different utility functions, or could collaborate for a common purpose. We develop a policy that performs joint admission and resource control and works for any user load, either inside or outside the capacity region. Through Lyapunov Optimization techniques, we show that the developed policy can achieve a utility performance arbitrarily close to the optimality with a tradeoff in the average service delay of admitted users.