Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

This paper gives $n$-dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space \sM_{\dd}^n of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, ACC-coordinates, as those $(n+2) \times (n+2)$ real matrices \bW with \bW^T \bQ_{D,n} \bW = \bQ_{W,n} where $Q_{D,n} = x_1^2 +... + x_{n+2}^2 - \frac{1}{n}(x_1 +... + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + ... + 2x_{n+2}^2$, and \bQ_{D,n} and \bQ_{W,n} are their corresponding symmetric matrices. There are natural actions on the parameter space \sM_{\dd}^n. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions one can find rational Apollonian cluster ensembles (all curvatures rational) and strongly rational Apollonian sphere ensembles (all ACC-coordinates rational).Comment: 37 pages. The third in a series on Apollonian circle packings beginning with math.MG/0010298. Revised and extended. Added: Apollonian groups and Apollonian Cluster Ensembles (Section 4),and Presentation for n-dimensional Apollonian Group (Section 5). Slight revision on March 10, 200

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)$\times$(center) is an integer vector. This series of papers explain such properties. A {\em Descartes configuration} is a set of four mutually tangent circles with disjoint interiors. We describe the space of all Descartes configurations using a coordinate system \sM_\DD consisting of those $4 \times 4$ real matrices \bW with \bW^T \bQ_{D} \bW = \bQ_{W} where \bQ_D is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and \bQ_W of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the parameter space \sM_\DD. We observe that the Descartes configurations in each Apollonian packing form an orbit under a certain finitely generated discrete group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups, the dual Apollonian group and the super-Apollonian group, which have nice geometrically interpretations. We show these groups are hyperbolic Coxeter groups.Comment: 42 pages, 11 figures. Extensively revised version on June 14, 2004. Revised Appendix B and a few changes on July, 2004. Slight revision on March 10, 200

Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. It observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a super-packing. The circles in a super-packing never cross each other but are nested to an arbitrary depth. Certain Apollonian packings and super-packings are strongly integral in the sense that the curvatures of all circles are integral and the curvature$\times$centers of all circles are integral. We show that (up to scale) there are exactly 8 different (geometric) strongly integral super-packings, and that each contains a copy of every integral Apollonian circle packing (also up to scale). We show that the super-Apollonian group has finite volume in the group of all automorphisms of the parameter space of Descartes configurations, which is isomorphic to the Lorentz group $O(3, 1)$.Comment: 37 Pages, 11 figures. The second in a series on Apollonian circle packings beginning with math.MG/0010298. Extensively revised in June, 2004. More integral properties are discussed. More revision in July, 2004: interchange sections 7 and 8, revised sections 1 and 2 to match, and added matrix formulations for super-Apollonian group and its Lorentz version. Slight revision in March 10, 200

Testing the Efficacy of INSIGHTS on Student Disruptive Behavior, Classroom Management, and Student Competence in Inner City Primary Grades

A prevention trial tested the efficacy of INSIGHTS into Children’s Temperament as compared to a Read Aloud attention control condition in reducing student disruptive behavior and enhancing student competence and teacher classroom management. Participants included 116 first and second grade students, their parents, and their 42 teachers in six inner city schools. Teachers completed the Sutter-Eyberg Student Behavior Inventory (SESBI) and the Teacher’s Rating Scale of Child’s Actual Competence and Social Acceptance (TRS) at baseline and again upon completion of the intervention. Boys participating in INSIGHTS, compared with those in the Read Aloud program, showed a significant decline in attentional difficulties and overt aggression toward others. Teachers in INSIGHTS, compared to those in the attention control condition, reported significantly fewer problems managing the emotional-oppositional behavior, attentional difficulties, and covert disruptive behavior of their male students. They also perceived the boys as significantly more cognitively and physically competent

Charge Carriers for Next-Generation Redox Flow Batteries

Increasing the volumetric energy density of redox flow batteries beyond that of the archetypal all-vanadium system requires the development of highly soluble charge carriers that can store multiple electrons per charge cycle. In this review article we will describe the design and performance of a range of new charge carriers for flow batteries, with an emphasis on those with multi-electron redox properties. These include fullerene derivatives, multifunctional organic systems, metal coordination complexes, and polyoxometalates. Our discussion will include an evaluation of the fundamental physical and electrochemical properties of the charge carriers and their impact on battery performance and energy density

Linking Life Skills and Norms with adolescent substance use and delinquency in South Africa

We examined factors targeted in two popular prevention approaches with adolescent drug use and delinquency in South Africa. We hypothesized adolescent life skills to be inversely related, and perceived norms to be directly related to later drug use and delinquency. Multiple regression and a relative weights approach were conducted for each outcome using a sample of 714 South African adolescents ages 15 to 19 years (M = 15.8 years, 57% female). Perceived norms predicted gateway drug use. Conflict resolution skills (inversely) and perceived peer acceptability (directly) predicted harder drug use and delinquency. The “culture of violence” within some South African schools may make conflict resolution skills more salient for preventing harder drug use and delinquency.Department of HE and Training approved lis

An Essential Staphylococcus Aureus Cell Division Protein Directly Regulates FtsZ Dynamics

Binary fission has been well studied in rod-shaped bacteria, but the mechanisms underlying cell division in spherical bacteria are poorly understood. Rod-shaped bacteria harbor regulatory proteins that place and remodel the division machinery during cytokinesis. In the spherical human pathogen Staphylococcus aureus, we found that the essential protein GpsB localizes to mid-cell during cell division and co-constricts with the division machinery. Depletion of GpsB arrested cell division and led to cell lysis, whereas overproduction of GpsB inhibited cell division and led to the formation of enlarged cells. We report that S. aureus GpsB, unlike other Firmicutes GpsB orthologs, directly interacts with the core divisome component FtsZ. GpsB bundles and organizes FtsZ filaments and also stimulates the GTPase activity of FtsZ. We propose that GpsB orchestrates the initial stabilization of the Z-ring at the onset of cell division and participates in the subsequent remodeling of the divisome during cytokinesis

Substance abuse and HIV risk behaviours amongst primary health care service users in Cape Town

Objective: To document prevalence of, and association between, substance use and HIV risk behaviours among primary care patients.Method: Cross-sectional survey. Four primary care clinics in Cape Town. We selected clinics using stratified sampling, and systematically selected 131 patients from attendance logs. We assessed substance use with the Alcohol, Smoking and Substance Involvement Screening Test, and HIV risk with items addressing injection drug use, blood-sharing rituals, and sexual risk behaviours. Results: Substances most used at hazardous levels were tobacco (28.2%) and alcohol (14.8%). Among possible HIV risk factors, highest prevalence was participation in blood-sharing rituals (25%), and having had an STI (19.8%). An association between substance use and sexual risk behaviours was only found among those aged 18-24. Conclusion: In younger patients, presence of substance use or HIV risk behaviours increases the probability that the other is present. Keywords: substance abuse, HIV risk behaviours, primary care South African Psychiatry Review Vol. 8(4) 2005: 160-16

Translational research in South Africa: evaluating implementation quality using a factorial design

Background: HealthWise South Africa: Life Skills for Adolescents (HW) is an evidence- based substance use and sexual risk prevention program that emphasizes the positive use of leisure time. Since 2000, this program has evolved from pilot testing through an efficacy trial involving over 7,000 youth in the Cape Town area. Beginning in 2011, through 2015, we are undertaking a new study that expands HW to all schools in the Metro South Education District. Objective: This paper describes a research study designed in partnership with our South African collaborators that examines three factors hypothesized to affect the quality and fidelity of HW implementation: enhanced teacher training; teacher support, structure and supervision; and enhanced school environment. Methods: Teachers and students from 56 schools in the Cape Town area will participate in this study. Teacher observations are the primary means of collecting data on factors affecting implementation quality. These factors address the practical concerns of teachers and schools related to likelihood of use and cost-effectiveness, and are hypothesized to be "active ingredients" related to high-quality program implementation in real-world settings. An innovative factorial experimental design was chosen to enable estimation of the individual effect of each of the three factors. Results: Because this paper describes the conceptualization of our study, results are not yet available. Conclusions: The results of this study may have both substantive and methodological implications for advancing Type 2 translational research