Eigenbundles, Quaternions, and Berry's Phase

Given a parameterized space of square matrices, the associated set of eigenvectors forms some kind of a structure over the parameter space. When is that structure a vector bundle? When is there a vector field of eigenvectors? We answer those questions in terms of three obstructions, using a Homotopy Theory approach. We illustrate our obstructions with five examples. One of those examples gives rise to a 4 by 4 matrix representation of the Complex Quaternions. This representation shows the relationship of the Biquaternions with low dimensional Lie groups and algebras, Electro-magnetism, and Relativity Theory. The eigenstructure of this representation is very interesting, and our choice of notation produces important mathematical expressions found in those fields and in Quantum Mechanics. In particular, we show that the Doppler shift factor is analogous to Berry's Phase.Comment: 22 pages, also found on http://math.purdue.edu/~gottlie

Conjugacy classes of solutions to equations and inequations over hyperbolic groups

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results to enumerate the immutable subgroups of a torsion-free hyperbolic group.Comment: 28 pages; referee's comments incorporated; to appear in the Journal of Topolog

Evaluating the Georgia HOPE Scholarship Program: Impact on Students Attending Public Colleges and Universities

Two years after starting college, recipients of Georgia's HOPE scholarship program are more likely to still be enrolled in college, have higher grade point averages (GPA), and have earned more credit hours than their counterparts. The Council for School Performance, housed in the Applied Research Center in the School of Policy Studies at Georgia State University, conducted the first assessment of the impact of the HOPE scholarship on college performance. After following the 1994-95 HOPE recipients into their third year of college, the results show a positive impact of the program on all three outcomes included in the study.HOPE provides Georgia high school graduates who earn an overall high school GPA of 3.0 or higher with free tuition, fees, and a book allowance at public colleges and universities. Only HOPE scholars with a high school GPA between 3.0 and 3.16 were selected for this evaluation. This allowed researchers to isolate the effect of the HOPE scholarship on the recipients by selecting a comparison group with similar characteristics. The comparison group was matched by their core high school GPA (includes academic courses only) and institution type. The students in the comparison group did not receive the HOPE scholarship because they did not apply or did not meet all of the HOPE eligibility requirements.Two questions were analyzed in this evaluation: (1) Does HOPE motivate higher levels of performance and higher rates of persistence among students in college? (2) Does HOPE allow students greater choice in selecting institutions of higher education? Other factors such as institution type, sex, race, and high school preparation were included in this analysis because they also affect college performance. This study compares students with similar backgrounds to isolate the impact of HOPE on college performance. In future studies, we will examine another potential impact of HOPE, its effect on high school performance

Does size matter? Experiences and perspectives of BIM implementation from large and SME construction contractors

This paper presents the findings of a qualitative study into the experiences and perspectives of large and SME construction contractors towards the implementation of Building Information Modelling (BIM) within their organisations. Results of the survey were statistically analysed to test for similarity and significant variations between the two groups. The results confirmed that both groups were equally aware of the perceived benefits of BIM, but found that the largest barriers to implementation were the costs associated with the technology and training requirements. Significant differences between the groups included plans to implement BIM and concerns with legal and commercial barriers

Chen ranks and resonance

The Chen groups of a group $G$ are the lower central series quotients of the maximal metabelian quotient of $G$. Under certain conditions, we relate the ranks of the Chen groups to the first resonance variety of $G$, a jump locus for the cohomology of $G$. In the case where $G$ is the fundamental group of the complement of a complex hyperplane arrangement, our results positively resolve Suciu's Chen ranks conjecture. We obtain explicit formulas for the Chen ranks of a number of groups of broad interest, including pure Artin groups associated to Coxeter groups, and the group of basis-conjugating automorphisms of a finitely generated free group.Comment: final version, to appear in Advances in Mathematic

Fixed Point Indices and Manifolds with Collars

This paper concerns a formula which relates the Lefschetz number L(f) for a map f:M --> M' to the fixed point index I(f) summed with the fixed point index of a derived map on part of the boundary of M. Here M is a compact manifold and M' is M with a collar attached.Comment: Accepted for publication in Fixed Point Theory and Applications as part of the proceedings of the Newfoundland conference on fixed points, 200