Schwartz functions on Nash manifolds

In this paper we extend the notions of Schwartz functions, tempered functions and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case classically known properties of Schwartz functions on $R^n$ and build some additional tools which are important in representation theory.Comment: 35 pages, LaTex. v3:minor changes + reference to results of duClou

Non-orthogonal geometric realizations of Coxeter groups

We define in an axiomatic fashion a \emph{Coxeter datum} for an arbitrary Coxeter group $W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear paring without any integrality and non-degeneracy requirements. These representations are not required to be embeddings of $W$ in the orthogonal group of any vector space, and they give rise to a pair of inter-related root systems generalizing the classical root systems of Coxeter groups. We obtain comparison results between these non-orthogonal root systems and the classical root systems. Further, we study the equivalent of the Tits cone in these non-orthogonal representations, and we show that strong results on the geometry in the equivalent of the Tits cone can be obtained

Roses Are Red, Violets Are Bluehow Poetry In Science Can Help Students Learn Something New

This study was an attempt to examine how poetry integrated with science could assist eighth graders in the memorization of key science vocabulary words. Furthermore, it would investigate if student attitude, interest, and motivation would improve with the use of the poetry. Instruction was adjusted to implement poetry into astronomy lessons. Memorization activities such as poems, chanting, and repetition were used to help students remember the vocabulary and the definitions. Pre/post tests were used to interpret if the poetry did assist in the memorization of the astronomy vocabulary. Science interest surveys and science attitude surveys were used to interpret if the use of the poetry helped to increase student interests in and attitudes toward science. This study was intended to be a first step toward proving how poetry could benefit students in the areas of memorization, attitude, and interest of science; and if successful, perhaps could be used to assist in other subjects as well

Branching integrals and Casselman phenomenon

Let $G$ be a real semisimple Lie group, $K$ its maximal complex subgroup, and $G_C$ its complexification. It is known that all the $K$-finite matrix elements on $G$ admit holomorphic continuation to branching functions on $G_C$ having singularities at the a prescribed divisor. We propose a geometric explanation of this phenomenon. The note also contsins a general survey of holomorphic continuations of infinite-dimensional representations.Comment: 13pp, an addendum is adde

Morse matchings on polytopes

We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of the half cube; this procedure may be of independent interest for other highly symmetric polytopes

Doctor of Philosophy

dissertationThis research project improved general chemistry performance through four major efforts: 1) identifying factors predictive of students failing general chemistry, 2) adjusting course structure, 3) changing course curriculum to align with chemical education research, and 4) implementing metacognitive training. Low math ability and poor attendance at discussion sections most significantly contributed to students failing the course. Course prerequisites were then implemented and attendance in discussion became required, resulting in a 3% increase in final exam median score and a semester decrease in student enrollment. Yearly enrollment did not change in response to the addition of course prerequisites. Greater structure was added to the established flipped classroom through online homework questions, which linked students to videos and offered tutorials on the topic. Practice tests were added to this system, resulting in a 4% increase in median final exam score and a 6% improvement in pass rate. Metacognitive training within practice tests was added to the homework, with students predicting their ability, receiving feedback of their chemistry ability, and developing study plans. This training overcame low-achieving students' tendency to overestimate their chemistry ability but did not improve assessment scores. Regular metacognitive training was implemented within weekly quizzes, in addition to practice tests, including the system of prediction, feedback, and study plans. Student tendency to overestimate assessment scores was overcome and scores on each course exam improved. After factoring out other effects, overall final exam average improved by 4%, with the bottom quartile of students improving by 10%. This training was implemented over the year of general chemistry, improving the bottom quartile's final exam score by 2.7% for each semester of training received. Analysis of this training indicated that regular completion of study plans and accurate prediction of end-of-semester assessment scores correlated to positive semester trajectory. By the end of this study, student enrollment rates, the number of students passing the course, and the percent pass rates for General Chemistry I and II reached the highest levels seen since the outset. Final exam scores generally increased over the study, with a minor decline witnessed in the most recent semester of General Chemistry I