Lobachevski Illuminated: Content, Methods, and Context of the Theory of Parallels

In the 1820\u27s, Nikolai Ivanovich Lobachevski discovered and began to explore the world\u27s first non-Euclidean geometry. This crucial development in the history of mathematics was not recognized as such in his own lifetime. When his work finally found a sympathetic audience in the late 19th century, it was reinterpreted in the light of various intermediate developments (particularly Riemann\u27s conception of geometry), which were foreign to Lobachevski\u27s own way of thinking about the subject. Because our modern understanding of his work derives from these reinterpretations, many of Lobachevski\u27s most striking ideas have been forgotten. To recover them, I have produced an illuminated version of Lobachevski\u27s most accessible work, Geometrische Untersuchungen zur Theorie der Parallellinien (Geometric Investigations on the Theory of Parallels), a book that he published in 1840. I have produced a new English version of this work, together with extensive mathematical, historical, and philosophical commentary. The commentary expands and explains Lobachevski\u27s often cryptic statements and proofs, while linking the individual propositions of his treatise to the related work of his predecessors (including Gerolamo Saccheri, J.H. Lambert, and A.M. Legendre), his contemporaries (including J·nos Bolyai and Karl Friedrich Gauss), and his followers (including Eugenio Beltrami, Henri PoincarÈ, and David Hilbert). This dissertation supplies the contemporary reader with all of the tools necessary to unlock Lobachevski\u27s rich, beautiful, but generally inaccessible world

A Comparative Study of the Introduction of Advanced Placement and International Baccalaureate Mathematics Courses in Ningbo, China

Recently, in an attempt to promote globalization and internationalization, the Chinese government has introduced AP (Advanced Placement) and IB (International Baccalaureate) programs in Chinese high school. Although it is one of the biggest countries to introduce international programs to its secondary education in an effort to prepare more students to go overseas to pursue their higher education, China does not have much research focusing on introduced foreign academic programs. The purpose of this study is to fill in some gaps in the research while providing a better understanding of the depth behind the phenomenon of introducing the AP program and the IB diploma program in China and that introduction’s impact on the existing general Chinese high school program in mathematics. Multiple sources of data were collected and used to make various kinds of analysis such as contextual analysis, cross-curricula comparisons and statistical analysis. The findings illustrated the differences and similarities between the AP program and the IB program in their respective schools in Ningbo, China. It further examined the differences and similarities between the AP program/ IB program and the intended Chinese high school program in mathematics education. It also explored mathematics instructors’ perceptions of the imported AP and IB programs in China. Limitations of this study include the absence of some test scores, the relatively small sample size and the circumscribed selection of interviewees. This study provides a guide to help Chinese students and their parents decide on a learning program based on individual preferences. Also, the results of this study indicate that a considerable gap exists between secondary education and higher education in Chinese mathematics, and also points to possible limitations for individualized learning. The findings imply the need to consider curricular reform and suggest that local teachers and non-local teachers who teach in the imported programs consider reinforcing their teaching by learning from each other. Policymakers need to make adjustments to consider local conditions when introducing international programs so as to offer the most suitable program possible to native students

Enriched and Isogeometric Boundary Element Methods for Acoustic Wave Scattering

This thesis concerns numerical acoustic wave scattering analysis. Such problems have been solved with computational procedures for decades, with the boundary element method being established as a popular choice of approach. However, such problems become more computationally expensive as the wavelength of an incident wave decreases; this is because capturing the oscillatory nature of the incident wave and its scattered field requires increasing numbers of nodal variables. Authors from mathematical and engineering backgrounds have attempted to overcome this problem using a wide variety of procedures. One such approach, and the approach which is further developed in this thesis, is to include the fundamental character of wave propagation in the element formulation. This concept, known as the Partition of Unity Boundary Element Method (PU-BEM), has been shown to significantly reduce the computational burden of wave scattering problems. This thesis furthers this work by considering the different interpolation functions that are used in boundary elements. Initially, shape functions based on trigonomet- ric functions are developed to increase continuity between elements. Following that, non-uniform rational B-splines, ubiquitous in Computer Aided Design (CAD) soft- ware, are used in developing an isogeometric approach to wave scattering analysis of medium-wave problems. The enriched isogeometric approach is named the eXtended Isogeometric Boundary Element Method (XIBEM). In addition to the work above, a novel algorithm for finding a uniform placement of points on a unit sphere is presented. The algorithm allows an arbitrary number of points to be chosen; it also allows a fixed point or a bias towards a fixed point to be used. This algorithm is used for the three-dimensional acoustic analyses in this thesis. The new techniques developed within this thesis significantly reduce the number of degrees of freedom required to solve a problem to a certain accuracy—this reduc- tion is more than 70% in some cases. This reduces the number of equations that have to be solved and reduces the amount of integration required to evaluate these equations

Calculus Volume 2

Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 2 covers integration, differential equations, sequences and series, and parametric equations and polar coordinates.https://commons.erau.edu/oer-textbook/1001/thumbnail.jp

Flight Mechanics/Estimation Theory Symposium

The conference emphasizes orbital position estimation and trajectory calculation methods that consider flight mechanical parameters