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In this thesis, we focus on the numerical approximation of SDEs with a drift which is not globally Lipschitz, and corresponding sensitivity calculations. First, we propose an adaptive timestep construction for an Euler- Maruyama approximation of these SDEs over a finite time interval. It is proved that if the timestep is bounded appropriately, then over a finite time interval the numerical approximation is stable, and the expected number of timesteps is finite. Moreover, we extend this scheme to ergodic SDEs with contractivity over an infinite time interval and prove that the bound for moments and the strong error of the numerical solution are uniform in T, which allow us to introduce the adaptive multilevel Monte Carlo (MLMC) method to further improve the efficiency. Next, we apply a new MLMC method for the ergodic SDEs without contractivity. By introducing a change of measure technique, we simulate the paths with contractivity and add the Radon-Nikodym derivative to the estimator. It is shown that the variance of the new level estimators increases linearly in T, which is a great reduction compared with the exponential increase in the standard MLMC. Lastly, we derive a new pathwise sensitivity estimator for chaotic SDEs by introducing a spring term between the original and perturbed SDEs. The variance of the new estimator increases only linearly in time T; compared with the exponential increase of the standard pathwise estimator. We also consider using this estimator for the SDE with Richardson-Romberg extrapolation on the volatility parameter to approximate the sensitivities of the invariant measure of chaotic ODEs. All of the analysis is supported by numerical results.

The quantum nature of nuclei plays an important role in materials and molecules containing light mass elements, for example in enzymes and water, which are crucial to life and technology. Yet in materials modelling, nuclear quantum effects (NQEs) are often ignored. Using the path integral theory, NQEs in hydrogen bonds and reaction rates for various systems described with density functional theory (DFT) have been examined. A review of the theoretical background, including DFT, path integral molecular dynamics, and the instanton rate theory, is given in Chapter 2. In Chapter 3, the NQEs on the binding energy of DNA base pairs are investigated, revealing that NQEs strengthen the binding of base pairs at room temperature. A counter-intuitive temperature dependence where NQEs are less significant at low temperature was also discovered and explained through a picture of competing quantum effects. In Chapters 4 and 5, hydrogen diffusion on metal surfaces is studied. Calculations show that ``broad-top" diffusion barriers exist on many metal surfaces, and that these have a distinct tunnelling behaviour compared to conventional ``parabolic-top" barriers. The theoretical and experimental implications of this unique behaviour are discussed. In Chapter 5, an improved multidimensional instanton rate theory is tested and applied to hydrogen diffusion on Pd(110), a complex potential energy surface that includes an interesting diffusion pathway over a second order saddle point. Finally, tunnelling assisted water diffusion on Pd(111) is studied using the instanton rate theory combined with DFT. It is found that tunnelling, including the tunnelling of oxygen atoms, plays a role in changing the water dimer diffusion mechanism on this surface at below 40 K. Water monomer and dimer diffusion on a variety of transition metal (111) surfaces have also been studied in this work.

In relay-assisted direct-sequence code-division multiple-access (DS-CDMA) systems, the distance between the relay and the destination receiver may be significantly shorter than that between the source transmitter and the destination receiver. Therefore, the transmission power of the relay may be significantly reduced in comparison to that of the source transmitter. In this thesis, we investigate the dependence of the achievable bit error ratio (BER) performance of DS-CDMA systems on the specific locations of the relays as well as on the power-sharing among the source transmitters and relays, when considering different propagation pathloss exponents. This thesis is focused on the class of repetition-based cooperation aided schemes, including both amplify-and-forward (AF) as well as decode-and-forward (DF) schemes, with an emphasis on lowcomplexity AF schemes. In our study, the signals received at the destination receiver from the source transmitters as well as from the relays are detected based on a range of diversity combining schemes having a relatively low-complexity. Specifically, the maximal ratio combining (MRC), the maximum signal-to-interference-plus-noise ratio (MSINR) and the minimum mean-square error (MMSE) principles are considered. We propose a novel cooperation aided DS-CDMA uplink scheme, where all the source mobile terminals (MTs) share a common set of relays for the sake of achieving relay diversity. As shown in our study, this low-complexity AF-based cooperation strategy is readily applicable to the challenging scenario where each source MT requires the assistance of several separate relays in order to achieve relay diversity. Another novel cooperation scheme is proposed for the downlink of DS-CDMAsystems, where the downlink multiuser interference (MUI) is suppressed with the aid of transmitter preprocessing, while maintaining the relay diversity order facilitated by the specific number of relays employed, despite using simple matched-filter (MF) based receivers. The transmitter preprocessing schemes considered include both the zero-forcing (ZF) and the MMSE-assisted arrangements, which belong to the class of linear transmitter preprocessing schemes. Furthermore, these transmitter preprocessing schemes are operated under the assumption that the base station’s transmitter employs explicit knowledge about the spreading sequences assigned to the destination MTs, but requires no knowledge about the downlink channels. Our study demonstrates that the proposed relay-assisted DS-CDMA systems using transmitter preprocessing are capable of substantially mitigating the downlink MUI, despite using low-complexity MF receivers.

Mobile micro-robots that can navigate in liquid environment have drawn a great interest due to their possibility in serving as a maneuverable in vivo cargo for medical uses, for instance, biosensing, bioimaging, microsurgery, or drug delivery. This dissertation describes the development and investigation of an untethered, bubble-powered micro drone that can navigate in3-D spaces equipped with a controllable drug releasing function in three parts: (1) design, fabrication, and testing of the 3-D micro drone, (2) development of the drug releasing mechanism and its integration with the micro drone, and (3) examination of the ability of drone in the tissue-like environments including analyses of the acoustic signal transmission through soft materials. First, propulsion of the micro drone was achieved by microstreaming generated by acoustically oscillating bubbles trapped inside one-end-open micro tubes. A two-photon polymerization 3-D printer was utilized to create a complex 3-D structure of the drone that has at least three different groups of micro tubes aligned in three different directions for three independent thrusts. By carefully selecting the lengths of the micro tubes and arranging them into different directions, the microstreaming flow can navigate the micro drone in 3-D spaces. In addition, a deliberate mass-distribution across the body of the micro drone was made to generate restoration in orientation under disturbances or during actuations such that the micro drone always returns to the initial posture. Second, a dosage-controllable, remotely activated drug release system is achieved by a micro tube that has ratchet structure on its interior wall which traps a liquid droplet in the middle of the tube as a representative of liquid drug. As the tube with the droplet is immersed into liquid, air bubbles automatically form at both sides of the droplet. When the bubbles are excited by an external acoustic field, mass exchange between the liquid droplet and the surrounding liquid occurs. The amount of the dosage can be precisely tuned by the duty ratio and the number of cycles. The releasing system was integrated with the drone demonstrating an on-demand controllable releasing function while swimming. Last, the acoustic signal transmission was modeled by ANSYS package to explore the possibility of remote actuations of micro drones inside human body. It was found that the bending waves in the rigid layer excited by piezo actuators significantly dissipate when the rigid layer is in direct contact with a soft (phantom) tissue. However, by coupling a proper medium between the rigid layer and the tissue, the acoustic signal can be effectively transmitted to a target across the soft tissue. All the above results imply that the proposed drone and actuation mechanism have become one step closer to their practical application.

This thesis presents a systematic investigation of advanced dual-fuel low temperature combustion (LTC) strategy, reactivity-controlled compression ignition (RCCI), on a modified diesel engine to obtain high thermal efficiency and low engine-out emissions using hydrous ethanol as the primary fuel. Direct use of hydrous ethanol in internal combustion engines has been shown to provide significant energy savings in the bio-ethanol production process, substantially improving life-cycle energy use and economics compared to commercially available anhydrous ethanol. While many studies have reported high thermal efficiency and extremely low soot and NOX emissions without exhaust aftertreatment systems in RCCI engines fueled with petroleum-based fuels, limited work has been done regarding the characteristics of performance and emissions of RCCI operation using hydrous ethanol. In this work, a systematic experimental investigation has been conducted to explore the operability region, characterize the performance and emissions, and optimize the key operating parameters of RCCI operation using hydrous ethanol as the primary fuel. It has been shown that hydrous ethanol with water content of up to 25% by volume can be effectively used in RCCI operation with ethanol energy fraction up to 75% over a wide engine load range. However, intake charge heating is needed at low loads and exhaust gas recirculation (EGR) is required under high load conditions. Response surface methodology (RSM) has been shown to be an efficient approach of identifying and optimizing the key engine operating parameters of RCCI operation with reduced experimentation. The performance and emissions characteristics of RCCI combustion have been shown to be different and subject to the influence of different engine operating parameters under different engine operating conditions. NOX and soot emissions have been considerably reduced with the optimal settings of operating parameters generated from the RSM models. The fumigated low reactivity fuel has been shown to primarily contribute to the relatively high hydrocarbon emissions from the RCCI combustion. The particulate matter (PM) emissions from RCCI combustion have been shown to be primarily composed of semi-volatile organic compounds and a smaller fraction of solid carbonaceous particles. The characteristics of RCCI PM emissions vary when using different low reactivity fuels and they are highly sensitive to dilution conditions such as dilution ratio and sampling system temperature.

The square-planar tetracyanonickelate(II) anion was intercalated into 2:1 and 3:1 Mg-Al layered double hydroxide systems (LDHs). In the 2:1 material, the anion holds itself at an angle of about 30° to the layers, whereas in the 3:1 material it lies more or less parallel to the layers. This is confirmed by orientation effects in the infrared spectra of the intercalated materials and by X-ray diffraction (XRD) data. The measured basal spacings for the intercalated LDH hosts are approximately 11 Å for the 2:1 and approximately 8 Å for the 3:1. The IR of the 2:1 material shows a slight splitting in the ν(CN) peak, which is suppressed in that compound's oriented IR spectrum, indicating that at least some of the intercalated anion's polarization is along the z-axis. This effect is not seen in the 3:1 material. A comparison between chloride LDHs and nitrate LDHs was made with respect to intercalation of tetracyanonickelate(II) anions. Both XRD data and atomic absorption spectroscopy (AAS) data of the LDH tetracyanonickelates confirms that there are no significant differences between the products from the two types of starting materials. The presence of a weak ν(NO) peak in the IR spectra of those samples made from nitrate parents indicates the presence of small amounts of residual [NO3]- in those systems. Small amounts of Cl- present in the chloride-derived samples, while perhaps detectable using AAS, would not be detectable in this manner. An attempted synthesis of Mg-Al LDH carbonates starting from reduced Mg and Al was unsuccessful due to pH constraints on hydroxide solubility in the solvent system used (water). The pH required to precipitate Al(OH)3 in the system was too high to allow precipitation of Mg(OH)2. Consequently, we found it impossible to have both of the required metal hydroxides present simultaneously in the system. An additional synthesis using a halogen as an oxidizing agent also failed to produce material of any characterizable quality.