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

Sexual function in 16- to 21-year-olds in Britain

Purpose: Concern about young people's sexuality is focused on the need to prevent harmful outcomes such as sexually transmitted infections and unplanned pregnancy. Although the benefit of a broader perspective is recognized, data on other aspects of sexuality, particularly sexual function, are scant. We sought to address this gap by measuring the population prevalence of sexual function problems, help seeking, and avoidance of sex in young people. Methods: A cross-sectional stratified probability sample survey (Natsal-3) of 15,162 women and men in Britain (response rate: 57.7%), using computer-assisted self-interviews. Data come from 1875 (71.9%) sexually active, and 517 sexually inactive (18.7%), participants aged 16–21 years. Measures were single items from a validated measure of sexual function (the Natsal-SF). Results: Among sexually active 16- to 21-year-old participants, 9.1% of men and 13.4% of women reported a distressing sexual problem lasting 3 months or more in the last year. Most common among men was reaching a climax too quickly (4.5%), and among women was difficulty in reaching climax (6.3%). Just over a third (35.5%) of men and 42.3% of women reporting a problem had sought help, but rarely from professional sources. Among those who had not had sex in the last year, just >10% of young men and women said they had avoided sex because of sexual difficulties. Conclusions: Distressing sexual function problems are reported by a sizeable minority of sexually active young people. Education is required, and counseling should be available, to prevent lack of knowledge, anxiety, and shame progressing into lifelong sexual difficulties

Sustainable Food Sector Curriculum Co-Create, EdTech 2021

Sustainable Food Curriculum Co-create – Empowering the Student Voice This paper will outline the current National Forum for Enhancement of Teaching and Learning in Higher Education- SATLE Funded Technological University Dublin project ‘Sustainable Food Curriculum Co-Create’. The project’s goal is to build capacity for integrating sustainability learning outcomes across Food programmes through educator professional development and co-creation with students. This is being achieved through the piloting of Food sector-focused sustainability modules with co-learning for academic staff and students (5 ECTS CPD in Educating for Food Sustainability for academic staff and 5 ECTS Elective module in Food Sector Sustainability for students). Curriculum design involved several stakeholders in Education for Sustainable Development from across the Food System including enterprise and state organisations who advised on sector-specific sustainability issues, food experts from across several Schools, and sustainability experts from across the University and beyond. It also was informed by industry publications and policy frameworks. A co-creation approach was adopted as sustainable development is complex, requiring transdisciplinary problem-solving where knowledge emerges between established fields, providing space for alternative perspectives and innovative solutions. It should be influenced by viewpoints of all stakeholders, including students, enterprise, regulatory bodies, communities, and social partners [1]. Curriculum delivery included peer-lectures from staff participating on the module, lectures from experts from across the University, as well as non-academic experts who were invited to deliver key industry related elements. The student voice is captured through co-creation workshops and reflective exercises. All participants in the modules were asked to complete a reflection-in-action, which was designed to document learning transformation in knowledge, values and behaviours. All co-creation workshops were facilitated online using MS TEAMS breakout rooms. The first workshop focused on evaluating the embeddedness of sustainability within individual module descriptor by applying the AASHE-STARS [2] criteria for sustainability-focused and sustainability-related courses. Guided by the SDSN’s SDG keywords [3], a mapping of selected module’s learning outcomes, indicative syllabi, assessments and learning resources to the SDGs was also undertaken by students and staff and opportunities for enhancement were identified. Each breakout room contained a mix of staff and students who were familiar with the respective modules for evaluation. An additional workshop was run to evaluate the circularity of module delivery using SusTEACH [4]. Finally, a third workshop focused on design thinking as a problem-solving process, supported by creative online tools, for addressing super-wicked problems posed by the sustainability challenge. The assessment strategy of the Food Sector sustainability modules involved (1) reflection of personal development throughout the module, as well as on improving performance in the UN supported online multiple-choice question sustainability literacy test (SULITEST); (2) artifacts generated during the workshops and (3) for academic staff, reviewing one of their module descriptors to embed sustainability learning outcomes. Whilst staff and students were each enrolled on different versions of the programme, the underlying ethos was on co-learning. As partners equally vested in a transformative learning process for a rapidly evolving field of study, the paper will highlight the potential for impact that the project will have on catalysing action to review the School’s food education provision. [1] QAA (2014) Education for sustainable Development: Guidelines for UK Higher Education Providers. Higher Education Academy and UK QAA. [2] Technical Manual - The Sustainability Tracking, Assessment & Rating System (aashe.org) [3] Universities & SDGs | SDSN Australia, New Zealand and Pacific (ap-unsdsn.org) [4] http://www9.open.ac.uk/SusTeach/index.ht

Creating the DIS Speaking Test for Placement: Considering Validity, Reliability, and Practicality

Placement testing presents a number of challenges for university language programs. Although there are a variety of benefits for creating one in-house, for practical purposes, commercial assessment tools are often chosen to accomplish the task of dividing students into groups of similar abilities. The goals and objectives for each program should inform the types of testing involved and appropriately delineating student abilities should typically be achieved through measuring all or a combination of several skills: reading, writing, speaking, listening, grammar and vocabulary. This paper details the steps used in the Department of International Studies (DIS) when creating the DIS Speaking Test for Placement, a speaking component integrated into the existing placement structure. This procedure is described and evaluated to show how developers tried to adhere to principles of reliability and validity in concert with practical issues. This paper does not purport that the test is in fact valid or reliable as further research will need to be conducted to better establish the degree to which those principles were achieved; however, the researchers do hope that this study can serve as an example of one department’s attempt, highlight their efforts to identify where immediate improvements can be made to benefit the test’s raters, and contribute to further dialogue about speaking placement test creation.論

Rapid climate change results in long-lasting spatial homogenization of phylogenetic diversity

Scientific understanding of biodiversity dynamics, resulting from past climate oscillations and projections of future changes in biodiversity, has advanced over the past decade. Little is known about how these responses, past or future, are spatially connected. Analyzing the spatial variability in biodiversity provides insight into how climate change affects the accumulation of diversity across space. Here, we evaluate the spatial variation of phylogenetic diversity of European seed plants among neighboring sites and assess the effects of past rapid climate changes during the Quaternary on these patterns. Our work shows a marked homogenization in phylogenetic diversity across Central and Northern Europe linked to high climate change velocity and large distances to refugia. Our results suggest that the future projected loss in evolutionary heritage may be even more dramatic, as homogenization in response to rapid climate change has occurred among sites across large landscapes, leaving a legacy that has lasted for millennia