The purpose of this paper is to develop an invariant integral for a compact topological group and, then to use that integral to prove the fundamental Peter-Weyl Theorem.

The fumonisins are a class of polyketide mycotoxins produced by Fusarium verticilliodes (formerly Fusarium monoliforme) which commonly affects maize. Ingestion of these toxins has been associated with leukoencephalomalacia in equine species, pulmonary oedema in swine, hepatocarcinogenesis in rats and have been linked to oesophageal cancer in humans. The structurally related AAL toxins are host specific mycotoxins produced by Alternaria alternata f. sp. lycopersici, producing stem canker disease in susceptible tomato cultivars. Examination of the C-11-C-20 fragment of the fumonisin B1 backbone [(2S,3S,5R,10R,12S,14S, 15R,16R)-2-amino-3,5,10,14,15-hydroxy-12,16-dimethyleicosane] and the C-10-C-17 fragment of the AAL toxin TA backbone[(2S,4S,5R,11S,13S,14R,15R)-1-amino-2,4,5,13,14-hydroxy-11,15- dimethylheptadecane], reveals four common stereogenic centres, with the only difference between the two fragments being the length of the alkyl chain. It is thought that the position and configuration of these four stereogenic centres is conserved among all members of the fumonisin and AAL classes of toxins. Retrosynthetic analysis of the backbones reveals a common intermediate aldehyde, which can be synthesised from methyl (S)-3-hydroxy-2-methylpropionate. A simple synthetic route to access the C-11-C-20 fragment for the fumonisins and the C-10-C-17 fragment of the AAL toxins was devised utilising Sharpless asymmetric epoxidation and an Evans aldol reaction as key transformations. In practice, it was found that although the Sharpless asymmetric epoxidation produced the desired epoxide in low enantiomeric excess, the two diastereomers produced could be separated by two consecutive flash chromatography silica gel columns. In pursuit of a more efficient method for introduction of the stereogenic centre in the target, other synthetic routes and key transformations were considered. Jacobsen’s kinetic resolution of terminal racemic epoxides was explored, requiring a terminal alkene from which the racemic epoxide was synthesised. An attempt to synthesise the terminal alkene from the appropriate tosylate and vinyl-MgBr, mediated by copper (I) iodide, failed. The synthetic route was redesigned, and the terminal alkene was synthesised by two one-carbon additions: the first a nucleophilic substitution with cyanide, and the second a Wittig olefination. The resolution of the terminal epoxide was also unsuccessful with no significant kinetic resolution occurring. Sharpless asymmetric dihydroxylation was also investigated; however, this reaction too failed to produce products of high diastereomeric excess. As a consequence, it was decided to pursue the asymmetric epoxidation route as the diastereomeric products could at least be separated. The second key transformation, the Evans aldol reaction, also provided an interesting result. When the aldol reaction was attempted with benzaldehyde and enolates derived from (4R,5S)-3-butanoyl-4-methyl-5-phenyl-oxazolidin-2-one and (4R,5S)-3-hexanoyl-4-methyl-5-phenyl-oxazolidin-2-one, the butanoyl derivative was found to give the expected Evans syn product, while the hexanoyl derivative was found to give the non-Evans syn product, with proof provided by single crystal X-ray diffraction analysis. It is proposed that the aldol reaction with the hexanoyl derivative does not proceed through the expected Zimmerman-Traxler-type transition state, but rather through an open chain transition state similar to that seen for asymmetric alkylation reactions. Synthesis of the pentanoyl derivative, and subjecting it to the same aldol reaction gave the expected syn Evans product, as deduced from spectroscopic properties. When the aldol reaction was attempted with the appropriate aldehyde intermediate, it was found that the dibutylboron triflate in the reaction medium caused the cleavage of the O-TBS ether protection, resulting in the formation of (3S,5R)-3-(4-methoxybenzyloxy)-5-methyl-tetrahydropyran-2-ol, before the aldehyde could undergo the aldol reaction. In order to avoid this problem, it is suggested that an alternative protecting group strategy using a more robust protecting group, such as a benzyl group which is stable to Lewis acids, could be substituted for the O-TBS group. Copyright / Dissertation (MSc)--University of Pretoria, 2012. / Chemistry / unrestricted

Baseilhac and Benedetti have created a quantum hyperbolic knot invariant similar to the colored Jones polynomial. Their invariant is based on the polyhedral decomposition of the knot complement into ideal tetrahedra. The edges of the tetrahedra are assigned cross ratios based on their interior angles. Additionally, these edges are decorated with charges and flattenings which can be determined by assigning weights to the longitude and meridian of the boundary torus of a neighborhood of the knot. Baseilhac and Benedetti then use a summation of matrix dilogarithms to get their invariants. This thesis investigates these invariants for the figure eight knot. In fact, it will be shown that the volume of the complete hyperbolic structure of the knot serves as an upper bound for the growth of the invariants.

The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection, or homoclinic web. In this thesis, the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way, we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddle-focus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycle-to-cycle homoclinics in the spatial problem are robust with respect to small changes in energy. The method uses high order Fourier-Taylor and Chebyshev series approximations in conjunction with the parameterization method, a general functional analytic framework for invariant manifolds. Tools that admit a natural notion of a-posteriori error analysis. Finally, we develop and implement a validation algorithm which we later use to obtain Theorems confirming the existence of homoclinic dynamics. This approach, known as the Radii polynomial, is a contraction mapping argument which can be applied to both the parameterized manifold and the Chebyshev arcs. When the Theorem applies, it guarantees the existence of a true solution near the approximation and it provides an upper bound on the C0 norm of the truncation error. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2021. / FAU Electronic Theses and Dissertations Collection

Let G be a Lie group acting on a homogeneous space G/K. The center of the universal enveloping algebra of the Lie algebra of G maps homomorphically into the center of the algebra of differential operators on G/K invariant under the action of G. In the case that G is a Jacobi Lie group of rank 2, we prove that this homomorphism is surjective and hence that the center of the invariant differential operator algebra is the image of the center of the universal enveloping algebra. This is an extension of work of Bringmann, Conley, and Richter in the rank 1case.

The larger area of my thesis is Algebra; more specifically, my work belongs to the following two major branches of Algebra: \emph{representation theory} and \emph{invariant theory}. In brief, the objective of representation theory is to investigate algebraic objects through their actions on vector spaces; this allows the well-developed toolkit of linear algebra to be brought to bear on complex algebraic problems. The theory has played a crucial role in nearly every subdiscipline of pure mathematics. Outside of pure mathematics, representation theory has been successfully used, for instance, in the study of symmetries of physical systems and in describing molecular structures in physical chemistry. Invariant theory is another classical algebraic theme permeating virtually all areas of pure mathematics and some areas of applied mathematics as well, notably coding theory. The theory studies actions of algebraic objects, traditionally groups and Lie algebras, on algebras, that is, vector spaces that are equipped with a multiplication. \bigskip The representation theory of (associative) algebras provides a useful setting in which to studymany aspects of the two most classical flavors of representation theory under a common umbrella: representations of groups and of Lie algebras. However, it turns out that general algebras fail to capture certain features of group representations and the same can be said for representations of Lie algebras as well. The additional structure that is needed in order to access these features is naturally provided by the important class of \emph{Hopf algebras}. Besides unifying the representation theories of groups and of Lie algebras, Hopf algebras serve a similar purpose in invariant theory, allowing for a simultaneous treatment of group actions (by automorphisms) and Lie algebras (by derivations) on algebras. More importantly, actions of Hopf algebras have the potential of capturing additional aspects of the structure of algebras they act on, uncovering features that cannot be accessed by ordinary groups or Lie algebras. \bigskip Presently, the theory of Hopf algebras is still nowhere near thelevel that has been achieved for groups and for Lie algebras over the course of the past century and earlier. This thesis aims to make a contribution to the representation and invariant theories of Hopf algebras, focusing for the most part on Hopf algebras that are not necessarily finite dimensional. Specifically, the contributions presented here can be grouped under two headings: \smallskip \noindent\qquad(i) \textbf{ Invariant Theory:} Hopf algebra actions and prime spectra, and\smallskip \noindent\qquad(ii)\textbf{ Representation Theory:} the adjoint representation of a Hopf algebra. \smallskip In the work done under the heading (i), we were able to use the action of cocommutative Hopf algebras on other algebras to "stratify" the prime spectrum of the algebra being acted upon, and then express each stratum in terms of the spectrum of a commutative domain. Additionally, we studied the transfer of properties between an ideal in the algebra being acted upon, and the largest sub-ideal of that ideal, stable under the action. We were able to achieve results for various families of acting Hopf algebras, namely \emph{cocommutative} and \emph{connected} Hopf algebras.\\The main results concerning heading (ii) concerned the subalgebra of locally finite elements of a Hopf algebra, often called the finite part of the Hopf algebra. This is a subalgebra containing the center that was used successfully to study the ring theoretic properties of group algebras, Lie algebras, and other classical structures. We prove that the finite is not only a subalgebra, but a coideal subalgebra in general, and in the case of (almost) cocommuative Hopf algebra, it is indeed a Hopf subalgebra. The results in this thesis generalize earlier theorems that were proved for the prototypical special classes of Hopf algebras: group algebras and enveloping algebras of Lie algebras. / Mathematics

<p> Several test statistics, which are known, can be used for testing for exponentiality. A new test statistic TE is proposed. TE is based on a censored sample and is similar to Tiku's T statistic for testing for normality. The distribution of TE tends to normality with increasing sample size. Besides, TE is easy to compute and is both origin and scale invariant. The power of TE for non-exponential distributions is comparable with Shapiro & Wilk statistic W-exponential. </p> / Thesis / Master of Science (MSc)

Batch reactors are often used to produce high quality products because any batch that does not meet quality speci cations can be easily discarded. However, for high-value products, even a few wasted batches constitute substantial economic loss. Fortunately, databases of historical data that can be exploited to improve operation are often readily available. Motivated by these considerations, this thesis addresses the problem of direct, data-based quality control for batch processes. Speci cally, two novel datadriven modeling and control strategies are proposed. The rst approach addresses the quality modeling problem in two steps. To begin, a partial least squares (PLS) model is developed to relate complete batch trajectories to resulting batch qualities. Next, the so called missing-data problem, encountered when using PLS models partway through a batch, is addressed using a data-driven, multiple-model dynamic modeling approach relating candidate input trajectories to future output behavior. The resulting overall model provides a causal link between inputs and quality and is used in a model predictive control scheme for direct quality control. Simulation results for two di erent polymerization reactors are presented that demonstrate the e cacy of the approach. The second strategy presented in this thesis is a state-space motivated, timeinvariant quality modeling and control approach. In this work, subspace identi cation methods are adapted for use with transient batch data allowing state-space dynamic models to be identifi ed from historical data. Next, the identifi ed states are related through an additional model to batch quality. The result is a causal, time-independent model that relates inputs to product quality. This model is applied in a shrinking horizon model predictive control scheme. Signi cantly, inclusion of batch duration as a control decision variable is permitted because of the time-invariant model. Simulation results for a polymerization reactor demonstrate the superior capability and performance of the proposed approach. / Thesis / Doctor of Philosophy (PhD) / High-end chemical products, ranging from pharmaceuticals to specialty plastics, are key to improving quality of life. For these products, production quality is more important than quantity. To produce high quality products, industries use a piece of equipment called a batch reactor. These reactors are favorable over alternatives because if any single batch fails to meet a quality specifi cation, it can be easily discarded. However, given the high-value nature of these products, even a small number of discarded batches is costly. This motivates the current work which addresses the complex topic of batch quality control. This task is achieved in two steps: first methods are developed to model prior reactor behavior. These models can be applied to predict how the reactor will behave under future operating policies. Next, these models are used to make informed decisions that drive the reaction to the desired end product, eliminating o -spec batches.

The invariant theory of finite reflection groups has rich connections to geometry, topology, representation theory, and combinatorics. We consider finite reflection groups acting on vector spaces over fields of arbitrary characteristic, where many arguments of classical invariant theory break down. When the characteristic of the underlying field is positive, reflections may be nondiagonalizable. A group containing these so-called transvections has order which is divisible by the characteristic of the underlying field, so is in the modular setting. In this thesis, we examine the action on differential derivations, which include products of differential forms and derivations, and identify the structure of the set of invariants under the action of groups fixing a single hyperplane, groups with maximal transvection root spaces acting on vector spaces over prime fields, as well as special linear groups and general linear groups over finite fields.

Atmospheric sensing tasks, including measuring the thermodynamic state (pressure, temperature, and humidity) and kinematic state (wind velocity) of the atmospheric boundary layer (ABL) can aid in numerical weather prediction, help scientists assess climatological and topological features over a region, and can be incorporated into flight path planning and control of small aircraft. Small uncrewed aerial vehicles (UAVs) are becoming an attractive platform for atmospheric sensing tasks as they offer increased maneuverability and are low-cost instruments when compared to traditional atmospheric sensing methods such as ground-based weather stations and weather balloons. In situ measurements using a UAV can be obtained for the thermodynamic state of the ABL using dedicated sensors that directly measure pressure, temperature, and humidity whereas the kinematic state (wind velocity) can be measured directly, using, for example, a five-hole Pitot probe or a sonic anemometer mounted on an aircraft, or indirectly. Indirect measurement methods consider the dynamics of the aircraft and use measurements from its operational sensor suite to infer wind velocity. This work is concerned with the design of the invariant extended Kalman filter (invariant EKF) for indirect wind estimation using a small, fixed-wing uncrewed aerial vehicle. Indirect wind estimation methods are classified as model-based or model-free, where the model refers to the aerodynamic force and moment model of the considered aircraft. The invariant EKF is designed for aerodynamic model-free wind estimation using a fixed-wing UAV in horizontal-plane flight and the full six degree of freedom UAV. The design of the invariant EKF relies on leveraging the symmetries of the dynamic system in the estimation scheme to obtain more accurate estimates where convergence of the filter is guaranteed on a larger set of trajectories when compared to conventional estimation techniques, such as the conventional extended Kalman filter (EKF). The invariant EKF is applied on both simulated and experimental flight data to obtain wind velocity estimates where it is successful in providing accurate wind velocity estimates and outperforms the conventional EKF. Overall, this work demonstrates the feasibility and effectiveness of implementing an invariant EKF for aerodynamic model-free indirect wind estimation using only the available measurements from the operational sensor suite of a UAV. / Doctor of Philosophy / Atmospheric sensing tasks, such as obtaining measurements of the pressure, temperature, humidity, and wind velocity of the atmospheric boundary layer (ABL), the lowest part of the atmosphere, have historically been dominated by the use of ground-based weather stations and deployment of weather balloons. Uncrewed aerial vehicles (UAVs) are emerging as an attractive, cost-effective platform for measuring desired quantities in the ABL. A UAV provides increased maneuverability when compared to fixed ground-based sensors and weather balloons as it can fly in different patterns and over any specified region within physical limits. Measurements of the ABL can help atmospheric scientists improve numerical weather prediction by providing more temporally and spatially dense data, in addition to helping assess climatological or topological features such as how the flow of wind varies over different types of terrain. A UAV can measure wind velocity directly or indirectly. Direct wind velocity measurements require mounting a dedicated wind sensor on a UAV and indirect measurement methods require only knowledge of the UAV's motion model with measurements from sensors already onboard to support automated flight. This work is concerned with designing an estimator for indirect wind velocity estimation using a small, fixed-wing UAV and only measurements from its operational sensor suite. The estimator, the invariant extended Kalman filter, leverages the symmetries of the system to provide estimates of the state or extended state of the system which can include position, velocity, and wind velocity. A system with symmetry is one that is unchanged by actions or transformations such as translation and rotation. The knowledge that the system remains unchanged under some transformations is used in the design of the invariant EKF. This estimator is then implemented for indirect wind estimation on both simulated and experimental flight data where it, in general, outperforms a conventional estimation method–the extended Kalman filter. The work presented in this dissertation demonstrates the effectiveness of implementing an invariant EKF for indirect wind estimation using a small, fixed-wing UAV and measurements from its operational sensor suite.