Photonic crystals have allowed unprecedented control of light and have allowed bringing new functionalities on chip. Photonic crystal waveguides (PCWs), which are linear defects in a photonic crystal, have unique features that distinguish these waveguides from other waveguides. The unique features include very large dispersion, existence of slow light, and the possibility of tailoring the dispersion properties for guiding light. In my research, I have overcome some of the challenges in using slow light in PCWs. In this work, I have demonstrated (i) high efficiency coupling of light into slow group velocity modes of a PCW, (ii) large bandwidth high transmission and low dispersion bends in PCWs, (iii) accurate modeling of pulse propagation in PCWs, (iv) high efficiency absorbing boundary conditions for dispersive slow group velocity modes of PCWs. To demonstrate the utility of slow light in designing high efficiency devices, I have demonstrated refractive index sensors using slow light in PCWs. In the end, a few high efficiency devices based on slow light in PCWs are mentioned. The remaining issues in the widespread use of PCW are also discussed in the last chapter.

Structure-property relationships for two-photon absorption (2PA) in branched organic chromophores is a topic of current interest, as is the design of chromophores with advantageous properties for two-photon laser scanning microscopy (2PLSM). The main goals of this dissertation were to study and explain the one-photon absorption (1PA) and 2PA properties of cruciform chromophores based on 1,4-distyryl-2,5-bis(phenylethynyl)benzene with varying electron donor (D) and acceptor (A) groups, and to characterize the 2PLSM-relevant response of some of these chromophores and a set of dipolar chromophores to binding with zinc ions. The compounds were studied by 1PA, fluorescence and 2PA spectroscopy. A ππ* exciton model was developed to explain the spectral properties of the 1,4-distyryl-2,5-bis(phenylethynyl)benzene cruciform with no D or A groups or with four identical D groups at the termini of the linear arms of the chromophore. This model indicated that there is some coupling and mixing of the lowest excited states e of the linear arms, leading to splitting of the 1PA spectrum of the cruciform. There was little coupling or mixing of the higher excited states e′ accessed in 2PA, leading to a two-band 2PA spectrum for the chromophore, in contrast to cruciform compounds in the literature with identical conjugated arms, which have one visible 2PA band. For cruciforms with D groups on the styryl arm and A character on the terminal phenyls of the phenylethynyl arms (D/A cruciforms), the ππ* exciton model was complemented with a charge-transfer (CT) exciton model describing interactions of charge-transfer pathways between the D and A groups. This model explained the broadness of the 1PA band of D/A cruciforms as well as the two 2PA bands observed for these chromophores. The fluorescence and 2PA spectral responses to binding of Zn²⁺ ions to the D or A groups of some cruciform compounds were also assessed, to provide insight into the design of new analyte-sensing cruciforms for 2PLSM that take advantage of enhancement or reduction of D/A character upon analyte binding. It was found that canceling charge donation from the D groups in differing D/A cruciforms resulted in fluorescence and 2PA spectra nearly indistinguishable from each other, suggesting that turn-off of D groups is not an optimal modality of 2PLSM analyte sensing in cruciforms. Binding Zn²⁺ to A groups was shown to result in an increase in the D/A character of the cruciform, with fluorescence peak energies that changed depending on the location of the A group. It is suggested that the use of non-binding donors and analyte-binding A groups in differing patterns on the arms could be a valuable design motif to achieve 2PLSM sensor compounds based on this cruciform structure. The 2PA spectra of a set of dipolar Zn²⁺ sensing dyes designed for ratiometric imaging in 2PLSM were also studied. These dyes had moderate 2PA strength, with redshifts of fluorescence 2PA spectra on Zn²⁺ binding. The isosbestic point of 2PA of most chromophores was within the range of 2PLSM excitation sources commonly used, rendering these dyes good candidates for use in ratiometric sensing in 2PLSM.

This dissertation is a contribution to the the study of the longtime dynamics of evolutionary equations in unbounded domains. It is of particular interest to prove the existence of global attractors for solutions of such equations. Th this end one need in general some type of asymtotical compactness. In the case the evolutionary PDE is defined on a bounded domain, asymptotical compactness follows from the regularity estimates and the compactnes of Sobolev embeddings and therefore the existence of attractors has been established for most of the disipative equations of mathematocal physics in a bounded domain. The problem is more challenging when the domain is unbounded since the Sobolev embeddings are no longer comapct, so that the usual regularity estimates may not be sufficient.To overcome this obstacle of compactness, A.V. Babin and M.I. Vishik introduced some weighted Sobolev spaces. In their pioneering paper, Proc. Roy. Soc. Edinb.

Soft trim absorbing parts (i.e., headliners, backwalls, side panels, etc.) are normally comprised of different layers including films, adhesives, foams and fibers. Several approaches to determine the complex wavenumber and characteristic impedance for porous sound absorbing materials are surveyed and the advantages and disadvantages of each approach are discussed. It is concluded that the recently documented three-point method produces the smoothest results. It is also shown that measurement of the flow resistance and the use of empirical equations is sufficient for many common materials. Following this, the transfer impedance of covers, adhesives, and densified layers are measured using an impedance difference approach. The transfer matrix method was then used to predict the sound absorption of a multi-layered materal which included a perforated cover, fiber layers, and an adhesive. The predicted results agree well with measurement.

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This thesis consists of a summary of four papers dealing with resonances of Dirac operators on Euclidean 3-space. In Paper I we show that the Complex Absorbing Potential (CAP) method is valid in the semiclassical limit for resonances sufficiently close to the real line if the potential is smooth and compactly supported. In Paper II  we continue the investigations initiated in Paper I but here we study clouds of resonances close to the real line and show that in some sense the CAP method remains valid also for multiple resonances. In Paper III we study perturbations of Dirac operators with smooth decaying scalar potentials  and show that these possess many resonances near certain points related to the maximum and the minimum of the potential. In Paper IV we show a trace formula of Poisson type for Dirac operators having compactly supported potentials which is related to resonances. The techniques mainly stem from complex function theory and scattering theory.

The nonlinear optical (NLO) properties were investigated in various extended　π-conjugated chromophores: cyanine and alkyne carbocations; porphyrin dimers; and squaraine compounds that possess electronic, double resonance, and vibronic based NLO properties. In summary: (i) It was demonstrated that the alkyne carbocations have very similar optical properties to traditional cyanine dyes. Our theoretical results establish that the alkyne carbocations, in spite of their significant degree of bond-length alternation, behave in the same way as cyanine dyes. (ii) The nature of the -bridge in porphyrin dimers tunes the electronic coupling strength, which in turn determines the splitting of the energy levels and the (non)linear optical properties. (iii) We have shown that the origin of the lowest TPA-active states in squaraines is dependent on the nature of substituent donor moiety, changing from predominantly electronic to vibronic in character. For all squaraines containing indolinylidenemethyl donors, a vibronic origin for the TPA peak, and the energy and lineshape of the experimentally observed lowest TPA peak in these compounds were confirmed.

Organisms have movements that are usually modeled by particles’ random walks. Under some mathematical technical assumptions the movements are described by diffusion equations. However, empirical data often show that the movements are not simple random walks. Instead, they are correlated random walks and are described by telegraph equations. This thesis considers telegraph equations with and without bias corresponding to correlated random walks with and without bias. Analytical solutions to the equations with absorbing boundary conditions and their mean passage times are obtained. Numerical simulations of the corresponding correlated random walks are also performed. The simulation results show that the solutions are approximated very well by the corresponding correlated random walks and the mean first passage times are highly consistent with those from simulations on the corresponding random walks. This suggests that telegraph equations can be a good model for organisms with the movement pattern of correlated random walks. Furthermore, utilizing the consistency of mean first passage times, we can estimate the parameters of telegraph equations through the mean first passage time, which can be estimated through experimental observation. This provides biologists an easy way to obtain parameter values. Finally, this thesis analyzes the velocity distribution and correlations of movement steps of amoebas, leaving fitting the movement data to telegraph equations as future work.

The objective of the study is to apply Markov chain analysis to analyze student flow through King Abdul Aziz University (KAU) in Saudi Arabia, and to predict important metrics such as graduation and dropout rates. This objective arises from examination of the policies of KAU University. We begin with background information detailing the subject of study, then move into a general outline of stochastic processes. We then use these methods to construct a specific matrix of transition probabilities with data from the university student population. Finally, we discuss the calculation of the possibilities of transition between each level of study and the average time a student takes to complete each stage. The study uses Markov chains with these outcomes to analyze student retention data from the Department of Mathematics at KAU. From this analysis, the study will provide university policy recommendations that can be generalized to examine other universities.

The majority of research on high strain-rate effects in timber structures has been limited to the study of the load-bearing members in isolation. Limited work has been conducted on timber connections and full-scale timber assemblies under blast loading, and these have generally been constrained to qualitative observations. In North America, the increasing prevalence of mid- and high-rise timber structures makes them susceptible to blast effects. In addition, questions remain on how to design and optimize these timber assemblies, including the connections, against blast loads, due in part to the limitations on comprehensive design provisions. The effects of far-field blast explosions were simulated using the University of Ottawa shock tube. A total of fifty-eight dynamic tests were conducted on connection-level and full-scale specimens. The research program aimed to investigate the behaviour of heavy-timber connections when subjected to simulated blast loads. The experimental results showed that connections with a main failure mechanism consisting of wood crushing experienced significant increases in dynamic peak load when compared to the static peak load. In contrast, connections where steel yielding and rupturing occurred experienced no statistically significant increase in dynamic peak load. Full-scale glulam specimens with bolted connections designed to yield via wood crushing and bolt bending performed better than those with overdesigned connections. Bolted connections which failed in splitting led to premature failure of the glulam assembly. Reinforcement with self-tapping screws allowed these bolted joints to fail in a combination of bolt yielding and wood crushing, and provided more ductility when compared to unreinforced specimens. Specially designed energy-absorbing connections significantly increased the energy dissipation capabilities of the timber assemblies. The basis of these connections was to allow for connection yielding while delaying failure of the wood member. This was achieved via elastoplastic connection behaviour, which effectively limited the load imparted onto the wood member. Based on the experimental results, limitations in the current Canadian blast provisions were highlighted and discussed. A two-degree-of-freedom blast analysis software was developed and validated using full-scale and connection-level experimental results and was found to adequately capture the system response with reasonable accuracy. Sensitivity analyses regarding the applicability of using single-degree-of-freedom analysis were presented and discussed.