In this thesis, we show that every periodic virtual knot can be realized as the closure of a periodic virtual braid. If K is a q-periodic virtual knot with quotient K_*, then the knot group G_{K_*} is a quotient of G_K and we derive an explicit q-symmetric Wirtinger presentation for G_K, whose quotient is a Wirtinger presentation for G_{K_*}. When K is an almost classical knot and q=p^r, a prime power, we show that K_* is also almost classical, and we establish a Murasugi-like congruence relating their Alexander polynomials modulo p. This result is applied to the problem of determining the possible periods of a virtual knot $K$. For example, if K is an almost classical knot with nontrivial Alexander polynomial, our result shows that K can be p-periodic for only finitely many primes p. Using parity and Manturov projection, we are able to apply the result and derive conditions that a general q-periodic virtual knot must satisfy. The thesis includes a table of almost classical knots up to 6 crossings, their Alexander polynomials, and all known and excluded periods. / Thesis / Doctor of Philosophy (PhD)

In this thesis, two separate problems are examines. First, sinusoidal signals are quite prevalent in practical applications. For example, any machine driven by a rotary shaft will exhibit periodic behaviour. For this reason, the estimation of sinusoidal parameters is studied extensively in the literature. Often in practical applications, there are unmodeled disturbances to the system, and the incoming measurements are noisy. Thus, estimation of the parameters of a sinusoidal signal in real-time for these conditions is of interest, calling for the use of a filter-based approach such as the Extended Kalman Filter. Considering the sinusoidal signal in its complex form, a novel approach is proposed resulting in a complex-valued filter. The resulting complex Extended Kalman Filter’s performance is evaluated in various test environments and is compared to standard approaches to the estimation problem using a Discrete Fourier Transform and standard Extended Kalman Filter. Results show that the complex Extended Kalman Filter outperforms the standard approaches in some cases in both accuracy and convergence rate. Second, research on hybrid systems has seen a large growth in interest in recent years. This is largely due to the increase of natural systems where discrete mode dynamics interact with continuous state dynamics. Switched systems are a subclass of hybrid systems that restrict their definition to continuous dynamic systems that interact with dis- crete switching events. Controller synthesis for such systems is no trivial task. Given the current trend in Artificial Intelligence and Machine Learning approaches, Dynamic Programming is explored as a means to approximate optimal control policies for switched systems. Discussions of discretization of the system’s state space are presented, followed by a high-level overview of an algorithm that leverages Dynamic Programming to find the approximated optimal control policies. Finally, the algorithm is applied to several examples to demonstrate its effectiveness. / Thesis / Master of Applied Science (MASc)

Difference equations arise in many fields of mathematics, both as discrete analogs of continuous behavior (analysis, numerical approximations) and as independent models for discrete behavior (population dynamics, economics, biology, ecology, etc.). In recent years, many models - especially in mathematical biology - are based on higher order nonlinear difference equations. As a result, there has been much focus on the existence of periodic solutions of certain classes of these equations and the asymptotic behavior of these periodic solutions. In this dissertation, we study the existence and global attractivity of both periodic and quasiperiodic solutions of two different higher order nonlinear difference equations. Both equations arise in biological applications.

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Input-output systems with time periodic parameters are commonly found in nature (e.g., oceanic movements) and engineered systems (e.g., vibrations due to gyroscopic forces in vehicles). In a broader sense, periodic behaviors can arise when there is a dynamic equi- librium between inertia and various balancing forces. A classic example is a structure in a steady wind or current that undergoes large oscillations due to vortex shedding or flutter. Such phenomena can have either positive or negative outcomes, like the efficient operation of wind turbines or the collapse of the Tacoma Narrows Bridge. While the systems mentioned here are typically all modeled as systems of nonlinear partial differential equations, the pe- riodic behaviors of interest typically form part of a stable "center manifold," the analysis of which prompts linearization around periodic solutions. The linearization produces linear, time periodic partial differential equations. Discretization in the spatial dimension typically produces large scale linear time-periodic systems of ordinary differential equations. The need to simulate responses to a variety of inputs motivates the development of effective model re- duction tools. We seek to address this need by investigating partial Floquet transformations, which serve to simultaneously remove the time dependence of the system and produce effec- tive reduced order models. In this thesis we describe the time-periodic analogs of important concepts for time invariant model reduction such as the transfer function and the H2 norm. Building on these concepts we present an algorithm which converges to the dominant poles of an infinite dimensional operator. These poles may then be used to produce the partial Floquet transform. / Master of Science / Systems that exhibit time periodic behavior are commonly found both in nature and in human-made structures. Often, these system behaviors are a result of periodic forces, such as the Earth's rotation, which leads to tidal forces and daily temperature changes affecting atmospheric and oceanic movements. Similarly, gyroscopic forces in vehicles can cause no- ticeable vibrations and noise. In a broader sense, periodic behaviors can arise when there's a dynamic equilibrium between inertia and various balancing forces. A classic example is a structure in a steady wind or current that undergoes large oscillations due to vortex shedding or flutter. Such phenomena can have either positive or negative outcomes, like the efficient operation of wind turbines or the collapse of the Tacoma Narrows Bridge. Linear Time-Periodic (LTP) systems are crucial in understanding, simulating, and control- ling such phenomena, even in situations where the fundamental dynamics are non-linear. This importance stems from the fact that the periodic behaviors of interest typically form part of a stable "center manifold," especially under minor disturbances. In natural systems, the absence of this stability would mean these oscillatory patterns would not be commonly observed, and in engineered systems, they would not be desirable. Additionally, the process of deriving periodic solutions from nonlinear systems often involves solving large scale linear periodic systems, raising the question of how to effectively reduce the complexity of these models, a question we address in this thesis.

Yes / An offset quad-element, two-port, high-gain, and multiband multiple-input multiple-output (MIMO) planar antenna based on a log-periodic dipole array (LPDA) for Ku/K-band wireless communications is proposed, in this paper. A single element antenna has been designed starting from Carrel's theory and then optimized with a 50-Ω microstrip feed-line with two orthogonal branches that results mainly in a broadside radiation pattern and improves diversity parameters. For experimental confirmation, the designed structure is printed on an RT-5880 substrate with a thickness of 1.57 mm. The total substrate dimensions of the MIMO antenna are 55 × 45 mm2. According to the measured results, the designed structure is capable of working at 1.3% (12.82-12.98 GHz), 3.1% (13.54-13.96 GHz), 2.3% (14.81-15.15 GHz), 4.5% (17.7-18.52 GHz), and 4.6% (21.1-22.1 GHz) frequency bands. Additionally, the proposed MIMO antenna attains a peak gain of 4.2-10.7 dBi with maximum element isolation of 23.5 dB, without the use of any decoupling structure. Furthermore, the analysis of MIMO performance metrics such as the envelope correlation coefficient (ECC) and mean effective gain (MEG) validates good characteristics, and field correlation performance over the operating band. The proposed design is an appropriate option for multiband MIMO applications for various wireless systems in Ku/K-bands.

Sensing in harsh environments is always in great need. Although many sensors and sensing systems are reported, such as optical fiber sensors and acoustic sensors, they all have drawbacks. In this dissertation, fused quartz and sapphire acoustic waveguides and sensors are developed for high temperature and heavy gamma radiation. The periodic structure, acoustic fiber Bragg grating (AFBG), is the core sensor structure in this dissertation. To better analyze the propagation of acoustic waves, the acoustic coupled more analysis is proposed. It could solve for the reflection spectrum of the AFBG with at most 2.1% error. For the waveguide, the fused quartz "suspended core" waveguide is designed. It achieved strong acoustic energy confinement so surface perturbations no longer affected the wave propagation. Single crystal sapphire fiber features low acoustic loss, and survivability under high temperature. It is also chosen as an acoustic waveguide. AFBGs are fabricated in both waveguides. The fused quartz suspended core AFBG is shown to sense temperature up to 1000 C and to have stable reading at 700 C for 14 days. The sapphire AFBG as a temperature sensor works up to 1500 C and also provides continuous stable reading at 1100 C for 12 days. Both waveguides with AFBGs are then tested under long-term gamma radiation. Despite some fluctuations from radiation-related causes, the readings of both sensors generally remain stable. Given the experimental observations, the fused quartz AFBG waveguide and the sapphire AFBG waveguide are shown to work well in high temperature and gamma radiations. / Doctor of Philosophy / Sensing in harsh environments, like high temperature, high pressure, and corrosive environment, is always in great need. Efficient and safe operation of instruments like nuclear reactors could be better secured. Although many sensors and sensing systems are reported, such as optical fiber sensors and acoustic sensors, they all have drawbacks so new designs are constantly in need. In this dissertation, silica (a glass commonly acquired by melting sand) and sapphire (used in iphone screens due to its transparency and hardness) acoustic waveguides and sensors are developed. A periodic structure known as acoustic fiber Bragg grating (AFBG) is the core sensor structure in this dissertation. A calculation method is proposed first. Acoustic wave needs a waveguide to propagate somewhere further, and a new waveguide structure is made to keep the acoustic energy within the very center of the waveguide, so any change on the outer surface does not affect the wave inside. Also, sapphire has good acoustic property and is used. The AFBGs are fabricated in both waveguides. These sensing waveguides are shown to work at >1000 C temperature and provide stable reading for more than 10 days. Long term exposure to gamma radiation for weeks or months resulted in stable performances. Therefore, it is concluded that silica and sapphire waveguide sensors are successfully developed for high temperature and nuclear radiation applications.

Over the course of the last century, electronic structure theory (or, alternatively, computational quantum chemistry) has grown from being a fledgling field to being a "full partner with experiment" [Goddard Science 1985, 227 (4689), 917--923]. Numerous instances of theory matching experiment to very high accuracy abound, with one excellent example being the high-accuracy ab initio thermochemical data laid out in the 2004 work of Tajti and co-workers [Tajti et al. J. Chem. Phys. 2004, 121, 11599] and another being the heats of formation and molecular structures computed by Feller and co-workers in 2008 [Feller et al. J. Chem. Phys. 2008, 129, 204105]. But as the authors of both studies point out, this very high accuracy comes at a very high cost. In fact, at this point in time, electronic structure theory does not suffer from an accuracy problem (as it did in its early days) but a cost problem; or, perhaps more precisely, it suffers from an accuracy-to-cost ratio problem. We can compute electronic energies to nearly any precision we like, as long as we are willing to pay the associated cost. And just what are these high computational costs? For the purposes of this work, we are primarily concerned with the way in which the computational cost of a given method scales with the system size; for notational purposes, we will often introduce a parameter, N, that is proportional to the system size. In the case of Hartree-Fock, a one-body wavefunction-based method, the scaling is formally N⁴, and post-Hartree-Fock methods fare even worse. The coupled cluster singles, doubles, and perturbative triples method [CCSD(T)], which is frequently referred to as the "gold standard" of quantum chemistry, has an N⁷ scaling, making it inapplicable to many systems of real-world import. If highly accurate correlated wavefunction methods are to be applied to larger systems of interest, it is crucial that we reduce their computational scaling. One very successful means of doing this relies on the fact that electron correlation is fundamentally a local phenomenon, and the recognition of this fact has led to the development of numerous local implementations of conventional many-body methods. One such method, the DLPNO-CCSD(T) method, was successfully used to calculate the energy of the protein crambin [Riplinger, et al. J. Chem. Phys 2013, 139, 134101]. In the following work, we discuss how the local nature of electron correlation can be exploited, both in terms of the occupied orbitals and the unoccupied (or virtual) orbitals. In the case of the former, we highlight some of the historical developments in orbital localization before applying orbital localization robustly to infinite periodic crystalline systems [Clement, et al. 2021, Submitted to J. Chem. Theory Comput.]. In the case of the latter, we discuss a number of different ways in which the virtual space can be compressed before presenting our pioneering work in the area of iteratively-optimized pair natural orbitals ("iPNOs") [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589]. Concerning the iPNOs, we were able to recover significant accuracy with respect to traditional PNOs (which are unchanged throughout the course of a correlated calculation) at a comparable truncation level, indicating that our improved PNOs are, in fact, an improved representation of the coupled cluster doubles amplitudes. For example, when studying the percent errors in the absolute correlation energies of a representative sample of weakly bound dimers chosen from the S66 test suite [Řezác, et al. J. Chem. Theory Comput. 2011, 7 (8), 2427--2438], we found that our iPNO-CCSD scheme outperformed the standard PNO-CCSD scheme at every truncation threshold (τ_PNO) studied. Both PNO-based methods were compared to the canonical CCSD method, with the iPNO-CCSD method being, on average, 1.9 times better than the PNO-CCSD method at τ_PNO = 10⁻⁷ and more than an order of magnitude better for τ_PNO < 10⁻¹⁰ [Clement, et al. J. Chem. Theory Comput 2018, 14 (9), 4581--4589]. When our improved PNOs are combined with the PNO-incompleteness correction proposed by Neese and co-workers [Neese, et al. J. Chem. Phys. 2009, 130, 114108; Neese, et al. J. Chem. Phys. 2009, 131, 064103], the results are truly astounding. For a truncation threshold of τ_PNO = 10⁻⁶, the mean average absolute error in binding energy for all 66 dimers from the S66 test set was 3 times smaller when the incompleteness-corrected iPNO-CCSD method was used relative to the incompleteness-corrected PNO-CCSD method [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589]. In the latter half of this work, we present our implementation of a limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) based Pipek-Mezey Wannier function (PMWF) solver [Clement, et al. 2021 }, Submitted to J. Chem. Theory Comput.]. Although orbital localization in the context of the linear combination of atomic orbitals (LCAO) representation of periodic crystalline solids is not new [Marzari, et al. Rev. Mod. Phys. 2012, 84 (4), 1419--1475; Jònsson, et al. J. Chem. Theory Comput. 2017, 13} (2), 460--474], to our knowledge, this is the first implementation to be based on a BFGS solver. In addition, we are pleased to report that our novel BFGS-based solver is extremely robust in terms of the initial guess and the size of the history employed, with the final results and the time to solution, as measured in number of iterations required, being essentially independent of these initial choices. Furthermore, our BFGS-based solver converges much more quickly and consistently than either a steepest ascent (SA) or a non-linear conjugate gradient (CG) based solver, with this fact demonstrated for a number of 1-, 2-, and 3-dimensional systems. Armed with our real, localized Wannier functions, we are now in a position to pursue the application of local implementations of correlated many-body methods to the arena of periodic crystalline solids; a first step toward this goal will, most likely, be the study of PNOs, both conventional and iteratively-optimized, in this context. / Doctor of Philosophy / Increasingly, the study of chemistry is moving from the traditional wet lab to the realm of computers. The physical laws that govern the behavior of chemical systems, along with the corresponding mathematical expressions, have long been known. Rapid growth in computational technology has made solving these equations, at least in an approximate manner, relatively easy for a large number of molecular and solid systems. That the equations must be solved approximately is an unfortunate fact of life, stemming from the mathematical structure of the equations themselves, and much effort has been poured into developing better and better approximations, each trying to balance an acceptable level of accuracy loss with a realistic level of computational cost and complexity. But though there has been much progress in developing approximate computational chemistry methods, there is still great work to be done. Many chemical systems of real-world import (particularly biomolecules and potential pharmaceuticals) are simply too large to be treated with any methods that consistently deliver acceptable accuracy. As an example of the difficulties that come with trying to apply accurate computational methods to systems of interest, consider the seminal 2013 work of Riplinger and co-workers [Riplinger, et al. J. Chem. Phys. 2013, 139, 134101]. In this paper, they present the results of a calculation performed on the protein crambin. The method used was DLPNO-CCSD(T), an approximation to the "gold standard" computational method CCSD(T). The acronym DLPNO-CCSD(T) stands for "`domain-based local pair natural orbital coupled cluster with singles, doubles, and perturbative triples." In essence, this method exploits the fact that electron-electron interactions ("electron correlation") are a short-range phenomenon in order to represent the system in a mathematically more compact way. This focus on the locality of electron correlation is a crucial piece in the effort to bring down computational cost. When talking about computational cost, we will often talk about how the cost scales with the approximate system size N. In the case of CCSD(T), the cost scales as N⁷. To see what this means, consider two chemical systems A and B. If system B is twice as large as system A, then the same calculation run on both systems will take 2⁷ = 128 times longer on system B than on system A. The DLPNO-CCSD(T) method, on the other hand, scales linearly with the system size, provided the system is sufficiently large (we say that it is "asymptotically linearly scaling"), and so, for our example systems A and B, the calculation run on system B should only take twice as long as the calculation run on system A. But despite the favorable scaling afforded by the DLPNO-CCSD(T) method, the time to solution is still prohibitive. In the case of crambin, a relatively small protein with 644 atoms, the calculation took a little over 30 days. Clearly, such timescales are unworkable for the field of biochemical research, where the focus is often on the interactions between multiple proteins or other large biomolecules and where many more data points are required. In the work that follows, we discuss in more detail the genesis of the high costs that are associated with highly accurate computational methods, as well as some of the approximation techniques that have already been employed, with an emphasis on local correlation techniques. We then build off this foundation to discuss our own work and how we have extended such approximation techniques in an attempt to further increase the possible accuracy to cost ratio. In particular, we discuss how iteratively-optimized pair natural orbitals (the PNOs of the DLPNO-CCSD(T) method) can provide a more accurate but also more compact mathematical representation of the system relative to static PNOs [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589]. Additionally, we turn our attention to the problem of periodic infinite crystalline systems, a class of materials less commonly studied in the field of computational chemistry, and discuss how the local correlation techniques that have already been applied with great success to molecular systems can potentially be applied in this domain as well [Clement, et al. 2021, Submitted to J. Chem. Theory Comput.].

This dissertation discusses a new approach to spline approximation. A periodic spline approximation 𝑓_M,m,N(x) = Σ_k=1^Nα_kΦ_M,k(x) to a periodic function 𝑓(x) is determined by requiring < Φ_m,j, 𝑓 - 𝑓_M,m,N > = 0 for j = 1,...,N, where the Φ_L,k's are the unique periodic spline basis functions of order 𝐿. Error estimates, examples and some relationships to wavelets are given for the case M - m = 2μ. The case M - m = 2µ + 1 is briefly discussed but not completely explored. / Ph. D.