An algebra $A$ over a field $\K$ is said to be \textit{invertible} if it has a basis $\B$ consisting only of of units; if $\B^{-1}$ is again a basis, $A$ is \textit{invertible-2}, or \textit{I2}. The question of when an invertible algebra is necessarily I2 arises naturally. The study of these algebras was recently initiated by Lòpez-Permouth, Moore, Szabo, Pilewski \cite{lopezIJM}, \cite{lopez1}. In this work, we prove several positive results on this problem, answering also some questions and generalizing a few results from these papers. We show that every field is an I2 algebra over any subfield, and that any subalgebra of the rational functions field $K(x)$ that strictly contains $K[x]$, with $K$ an algebraically closed field, has a symmetric basis $\B=\B^{-1}$. Using this, we expand the class of examples of algebras known to be invertible or I2 with several classes, such as semiprimary rings over fields $K\neq \F_2$ satisfying some additional mild conditions. We also show that every commutative, finitely generated, invertible algebra without zero divisors is almost I2 in the sense that it becomes I2 after localization at a single element.

This research aims to refresh and reinterpret the radical theory of associative rings using the base radical and base semisimple class constructions. It also endeavours to generalise some results about ideals of rings in terms of accessible subrings. A characterisation of accessible subrings is included. By applying the base radical and base semisimple class constructions to many of the known results in established radical theory a number of gaps are uncovered and closed, with the goal of making the theory more accessible to advanced undergraduate and graduate students and mathematicians in related fields, and to open up new areas of investigation. After a literature review and brief reminder of algebra rudiments, the useful properties of accessible subrings and the U and S operators independent from radical class connections are described. The section on accessible subrings illustrates that replacing ideals with accessible subrings is indeed possible for a number of results and demonstrates its usefulness. The traditional radical and semisimple class definitions are included and it is shown that the base radical and base semisimple class constructions are equivalent. Diagrams illustrating the constructions support the definitions. From then on, all radical and semisimple classes mentioned are understood to have the base radical and base semisimple class form. Subject to the constraints of this work, many known results of traditional radical theory are reinterpreted with new proofs, illustrating the potential to simplify the understanding of radical theory using the base radical and base semisimple class constructions. Along with reinterpreting known results, new results emerge giving further insight to radical theory and its intricacies. Accessible subrings and the U and S operators are integrated into the development. The duality between the base radical and base semisimple class constructions is demonstrated in earnest. With a measure of the theory presented, the new constructions are applied to examples and concrete radicals. Context is supported by establishing the relationship between some well-known rings and the radical and related classes of interest. The title of the thesis, Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions, reflects the understanding that reinterpreting the entirety of radical theory is beyond the scope of this work. The conclusion includes an outlook listing further research that time did not allow.

This paper examined a few different types of interacting galaxies and the implications the data has on cosmological theories. An analysis of the Holmberg Effect, Holmberg (1969)’s observation that companion galaxies tend to be near the poles of edge-on galaxies rather than in the planes, was done by reexamining the spirals viewed by Holmberg. Only using radial velocity confirmed companions to test for this effect showed that if anything, the opposite of the Holmberg effect is true. However, due to small number statistics, more edge-on spirals and their companions would have to be viewed in order to determine if this is an actual physical effect of anisotropic companion distribution around primary galaxies. It is important to discover if there is a preferred distribution, because it could point to the distribution of dark matter around the primary galaxies, and have interesting impacts on large-scale universal structure and cosmology theory.

Scientists and engineers measure dielectric constant because it gives them better understanding of materials and helps them to know how to integrate these materials into their design processes;it also helps them to shorten design life cycle,and aside these two functions,it has numerous uses all of which cannot be enumerated in this section.Owing to its usefulness,various measurement methods of dielectric constant of materials have been developed over the years.Each method has its limitations which affect the accuracy of the measurement;these limitations range from frequency,temperature,and  mearsurement environment to material under test.   In this thesis,four most common methods of measuring dielectric constant were discussed and the most accurate one,the resonant method,was chosen and worked on .The project was executed by making a mathematical analysis of the ring resonator which was later simulated in HFSS to get results which would be comparable to ones obtained in  laboratory measurements.   The ring was fabricated and taken to the laboratory for measurement.Two monopole antennas were connected to the two ports of a VNA with one antenna serving as the transmitter and the other serving as the receiver.   The resonant frequencies obtained were combined with the geometric parameters of the ring resonator and that of the MUT in equations written into MATLAB scripts;this equations were used to extract the dielectric constant of the MUT.

A helicopter rotor in descent flight encounters its own wake, resulting in a doughnut-shaped ring around the rotor disk, known as the Vortex Ring State (VRS). Flight in VRS condition can be dangerous as it may cause uncommanded drop in descent rate, loss of control effectiveness, power settling, excessive thrust and torque fluctuations, and vibration. As simple momentum theory is no longer valid for a rotor in VRS, modeling of rotor inflow in VRS continues to challenge researchers, especially for flight simulation applications. In this dissertation, a simplified inflow model, called the ring vortex model, is developed for a helicopter rotor operating in descent condition. By creating a series of vortex rings near the rotor disk, the ring vortex model addresses the strong flow interaction between the rotor wake and the surrounding airflow in descent flight. In addition, the total mass flow parameter in the existing inflow models is augmented to create a steady state transition between the helicopter and the windmill branches. With the ring vortex model, rotor inflow can now be adequately predicted over a wide range of descent rates. Validations of the ring vortex model for helicopter rotors are conducted extensively in axial and inclined descent. Effects from blade taper, blade twist, and rotor thrust are also investigated with further application of the finite-state inflow model. The ring vortex model is applied to a single main-rotor helicopter. The main effort is to establish VRS boundary based on heave stability criterion. In addition, two important phenomena observed in the descent flight tests are addressed in the dynamic simulation, including uncommanded drop in descent rate and loss of collective control effectiveness. The ring vortex model is further applied to a side-by-side rotor configuration. Lateral thrust asymmetry on the side-by-side rotor configuration can be reproduced through uneven distribution of vortex rings at the two rotors. Two important issues are investigated, including the impact of vortex rings on lateral thrust deficit and on lateral AFCS limit.

Compared with Nd:YAG, the traditional high power solid state laser gain medium, Yb:YAG has less quantum defect, no excited state absorption, and longer fluorescence lifetime, which makes it suitable for Q-switched laser. In addition, concentration quenching is absent in Yb:YAG, higher concentration of active ion makes the thickness of gain medium thinner. For ring cavities, the necessity of symmetrical beam path is important, a thinner Yb:YAG crystal can reduce the shift of optical beam path and avoids cavity unstability. Thus, Yb:YAG is suitable for the two-mirror ring cavity. In this study, a compact and efficient Yb:YAG ring laser with 50.3% slope efficiency was demonstrated. And the Yb:YAG ring laser performances influenced by thermal effect was analyzed and compared to that of Nd:YAG ring laser. The polarization of ring lasers with different configurations were also discussed. In manufacturing process, the coating design on gain medium and laser mirrors were introduced. The advantages of passively Q-switched laser are efficient, compact, simple setup and no complicated driving circuits. They make passively Q-switched laser suitable for various applications, such as nonlinear optics, medical treatment, micromachining, material processing, and range finder. Due to spontaneous noise from the gain medium, conventional passively Q-switched laser has large timing jitter. This study is to build up a passively Q-switched Yb:YAG/Cr4+:YAG ring laser with lower timing jitter. At present, a Q-switched ring laser with a peak power of 208 W and a pulse width of 33 ns, was developed. Its slope efficiency is 18.1% with a timing jitter of 11.9%. To our knowledge, this is the first passively Q-switched Yb:YAG/Cr4+:YAG ring laser.

Synthesis of a polymer with benzoxazine units in the main chain backbone by a trifunctional phenol, a difunctional amine, and paraformaldehyde was achieved. Thermal, mechanical and spectroscopic characterization and the viscosity properties of the synthesized polymer were studied. In the first step of this study, a fast and feasible method for the synthesis of the benzoxazine precursors was developed since some methods mentioned in the literature about the synthesis of the benzoxazine derivatives last long time. The second step was to polymerize the benzoxazine precursors thermally. The curing of benzoxazine precursors was done via ring opening polymerization at 150 oC and a final polymerization was observed at about 250 oC. 1H NMR, 13C NMR and FT-IR spectroscopies revealed the characteristic peaks for the formation of benzoxazine ring. Among them, 13C NMR gave important clue on the formation of the benzoxazine. The thermal characterization of the benzoxazine precursors and the polymers indicated that the ring opening polymerization of these precursors started at around 110 oC and a final polymerization was about at 230 oC. Differential Scanning Calorimetry thermograms of the polybenzoxazine indicated a secondary transition at around 270 oC. An onset decomposition of the benzoxazine oligomers started around 100 oC in Thermal Gravimetric Analysis thermograms performed under N2 atmosphere and two major maximum weight losses were observed at 273 oC and 439 oC. However, polybenzoxazine showed a starting degradation at about 260 oC and the maximum weight loss temperatures were seen at 296 oC and 465 oC. Viscosity variation of the reaction mixture was studied by Ubbelohde Viscometer at 30 oC. Viscosity results indicated that the increase in the intrinsic viscosity of the reaction mixture till 50th minute and followed by a decrease due to possible branching and the intra-crosslinking of the benzoxazine oligomers. Mechanical properties of the polymer films, prepared by compression molding at 180 0C, were investigated. Test results showed that low tensile strength whereas comparatively high elongation.

This dissertation is devoted to our synthetic studies towards the total synthesis of the natural product Nakadomarin A, and Manzamine A using the Pauson-Khand reaction as the key step. Chapter 1 reviews past work using Pauson-Khand technology. Chapter 2 reviews the N-alkyl piperidine family of natural products. Chapter 3 reviews published total syntheses of Manzamine A and Nakadomarin A. Chapter 4 explores our work using the Pauson-Khand reaction to form the ABC rings of Nakadomarin A and subsequent B ring expansion to form the ABC ring core of Manzamine A. Chapter 5 explores our approaches to the furan portion, as well as, our approaches to the macrocyclic F ring. Finally Chapter 6 contains the description of the experiments performed along with relevant analytical data. / text

Thesis (Ph. D.)--Ohio University, November, 2002. / Title from PDF t.p. Includes bibliographical references (leaves 164-171).

x, 68 p. : ill. (some col.) / Motivated by the Adams spectral sequence for computing stable homotopy groups, Priddy defined a class of algebras called Koszul algebras with nice homological properties. Many important algebras arising naturally in mathematics are Koszul, and the Koszul property is often tied to important structure in the settings which produced the algebras. However, the strong defining conditions for a Koszul algebra imply that such algebras must be quadratic. A very natural generalization of Koszul algebras called K 2 algebras was recently introduced by Cassidy and Shelton. Unlike other generalizations of the Koszul property, the class of K 2 algebras is closed under many standard operations in ring theory. The class of K 2 algebras includes Artin-Schelter regular algebras of global dimension 4 on three linear generators as well as graded complete intersections. Our work comprises two distinct projects. Each project was motivated by an aspect of the theory of Koszul algebras which we regard as sufficiently powerful or fundamental to warrant an interpretation for K 2 algebras. A very useful theorem due to Backelin and Fröberg states that if A is a Koszul algebra and I is a quadratic ideal of A which is Koszul as a left A -module, then the factor algebra A/I is a Koszul algebra. We prove that if A is Koszul algebra and A I is a K 2 module, then A/I is a K 2 algebra provided A/I acts trivially on Ext A ( A/I,k ). As an application of our theorem, we show that the class of sequentially Cohen-Macaulay Stanley-Reisner rings are K 2 algebras and we give examples that suggest the class of K 2 Stanley-Reisner rings is actually much larger. Another important recent development in ring theory is the use of A ∞ -algebras. One can characterize Koszul algebras as those graded algebras whose Yoneda algebra admits only trivial A ∞ -structure. We show that, in contrast to the situation for Koszul algebras, vanishing of higher A ∞ -structure on the Yoneda algebra of a K 2 algebra need not be determined in any obvious way by the degrees of defining relations. We also demonstrate that obvious patterns of vanishing among higher multiplications cannot detect the K 2 property. This dissertation includes previously unpublished co-authored material. / Committee in charge: Dr. Brad Shelton, Chair; Dr. Victor Ostrik, Member; Dr. Nicholas Proudfoot, Member; Dr. Arkady Vaintrob, Member; Dr. David Boush, Outside Member