Construction of spherical cubature formulas using lattices

We construct cubature formulas on spheres supported by homothetic images of shells in some Euclidian lattices. Our analysis of these cubature formulas uses results from the theory of modular forms. Examples are worked out on the sphere of dimension n-1 for n=4, 8, 12, 14, 16, 20, 23, and 24, and the sizes of the cubature formulas we obtain are compared with the lower bounds given by Linear Programming

A new approach to professional learning for academics teaching in next generation learning spaces

Abstract The promise of Next Generation Learning Spaces appears to remain unfulfilled. This chapter explores why and how the design of professional learning for academics teaching in such spaces can and should be transformed. It takes a fresh look at why old professional development is failing and proposes a new way to engage academics in their own professional learning. Rather than continuing with traditional professional development that is most often, ad hoc, formal and centrally driven, comprising mandated professional development workshops and a website that may only be visited once, the chapter explores the move from 'old' professional development to 'new' professional learning. It draws on the fields of organisational theory, cognitive theory and behavioural economics. New professional learning is characterised by a 'pull' rather than a 'push' philosophy. Academic staff themselves drive their own learning, choosing what, when and how they want to learn to become better teachers. Multiple and various learning opportunities embedded in day to day work are just-in-time, self-directed, performance-driven and evaluated within an organisational system. In this way the institutional setting influences behaviour by 'nudging' habits and setting defaults resulting in academics making the 'right' decisions and doing the 'right' thing. By addressing the compelling issue of how to enhance academic staff teaching capability, this chapter can help university leaders to think beyond the professional development approaches of yesterday. Aligning with this new direction will result in enhanced learning and teaching in the future

Which finitely generated Abelian groups admit isomorphic Cayley graphs?

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.Comment: 16 pages; v2: added reference, reformulated quasi-convexity, v3: small corrections; to appear in Geometriae Dedicat

Assessing professional skills development at a third year level

Employers, professional bodies, instructors and students themselves recognise the need for graduates to leave university with a good understanding of both disciplinary content as well as a range of highly developed professional skills. Many universities have responded to this need by encouraging the development of such skills in their programs of study. Curtin Business School has implemented the Professional Skills Project that aims to integrate the teaching and assessing of professional skills into the units of the Bachelor of Commerce degree program.As part of this initiative, the first author successfully won a grant to implement a project that focussed on supporting students' development of their presentation and written communication skills in his third year management unit. These skills were specifically selected based on the lecturer's previous experiences of teaching the unit which showed that while students in CBS are given opportunities to develop their of professional skills throughout the course, there seemed to be a significant variation in students' skill levels. The variation in skill development was particularly apparent in the assessment of group presentations and written assignments in semester one 2001. Thus, opportunities for students to develop these skills were integrated into the unit and data on students' perceptions of their skill development were obtained by them completing a questionnaire at the beginning and end of the unit.In this paper, we outline how the skills were taught and assessed, present the data on the changes in students' perceptions of their skill levels, and discuss the implications for teaching and assessing presentation and written communication skills in the context of the discipline

An elementary approach to toy models for D. H. Lehmer's conjecture

In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However, Bannai-Miezaki showed that none of the nonempty shells of the integer lattice \ZZ^2 in \RR^2 is a spherical 4-design, and that none of the nonempty shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none of the nonempty shells of the integer lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \QQ(\sqrt{-1}) and \QQ(\sqrt{-3}) is a spherical 2-design. In the proof, the theory of modular forms played an important role. Recently, Yudin found an elementary proof for the case of \ZZ^{2}-lattice which does not use the theory of modular forms but uses the recent results of Calcut. In this paper, we give the elementary (i.e., modular form free) proof and discuss the relation between Calcut's results and the theory of imaginary quadratic fields.Comment: 18 page

Caffeine intake, plasma caffeine level, and kidney function: a Mendelian randomization study

Caffeine is a psychoactive substance widely consumed worldwide, mainly via sources such as coffee and tea. The effects of caffeine on kidney function remain unclear. We leveraged the genetic variants in the CYP1A2 and AHR genes via the two-sample Mendelian randomization (MR) framework to estimate the association of genetically predicted plasma caffeine and caffeine intake on kidney traits. Genetic association summary statistics on plasma caffeine levels and caffeine intake were taken from genome-wide association study (GWAS) meta-analyses of 9876 and of >47,000 European ancestry individuals, respectively. Genetically predicted plasma caffeine levels were associated with a decrease in estimated glomerular filtration rate (eGFR) measured using either creatinine or cystatin C. In contrast, genetically predicted caffeine intake was associated with an increase in eGFR and a low risk of chronic kidney disease. The discrepancy is likely attributable to faster metabolizers of caffeine consuming more caffeine-containing beverages to achieve the same pharmacological effect. Further research is needed to distinguish whether the observed effects on kidney function are driven by the harmful effects of higher plasma caffeine levels or the protective effects of greater intake of caffeine-containing beverages, particularly given the widespread use of drinks containing caffeine and the increasing burden of kidney disease

Measuring the quality of a quantum reference frame: the relative entropy of frameness

In the absence of a reference frame for transformations associated with a group G, any quantum state that is non-invariant under the action of G may serve as a token of the missing reference frame. We here introduce a novel measure of the quality of such a token: the relative entropy of frameness. This is defined as the relative entropy distance between the state of interest and the nearest G-invariant state. Unlike the relative entropy of entanglement, this quantity is straightforward to calculate and we find it to be precisely equal to the G-asymmetry, a measure of frameness introduced by Vaccaro et al. It is shown to provide an upper bound on the mutual information between the group element encoded into the token and the group element that may be extracted from it by measurement. In this sense, it quantifies the extent to which the token successfully simulates a full reference frame. We also show, that despite a suggestive analogy from entanglement theory, the regularized relative entropy of frameness is zero and therefore does not quantify the rate of interconversion between the token and some standard form of quantum reference frame. Finally, we show how these investigations yield a novel approach to bounding the relative entropy of entanglement.Comment: 12 pages; many improvements in v2 including a weakening of the assumptions of the main theorem and better upper bounds for both the relative entropy of frameness for arbitrary compact Lie groups and the relative entropy of entanglement. Published versio

New Dimensions for Wound Strings: The Modular Transformation of Geometry to Topology

We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in hep-th/0510044. Milnor's theorem relates negative sectional curvature on a compact Riemannian manifold to exponential growth of its fundamental group, which translates in string theory to a higher effective central charge arising from winding strings. This exponential density of winding modes is related by modular invariance to the infrared small perturbation spectrum. Using self-consistent approximations valid at large radius, we analyze this correspondence explicitly in a broad set of time-dependent solutions, finding precise agreement between the effective central charge and the corresponding infrared small perturbation spectrum. This indicates a basic relation between geometry, topology, and dimensionality in string theory.Comment: 28 pages, harvmac big. v2: references and KITP preprint number added, minor change

On three-manifolds dominated by circle bundles

We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of the product of the two-sphere and the circle. This characterization can also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. We also determine which three-manifolds are dominated by non-trivial circle bundles, and which three-manifold groups are presentable by products.Comment: 12 pages; to appear in Math. Zeitschrift; ISSN 1103-467

Property (RD) for Hecke pairs

As the first step towards developing noncommutative geometry over Hecke C*-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the subgroup H in a Hecke pair (G,H) is finite, we show that the Hecke pair (G,H) has (RD) if and only if G has (RD). This provides us with a family of examples of Hecke pairs with property (RD). We also adapt Paul Jolissant's works in 1989 to the setting of Hecke C*-algebras and show that when a Hecke pair (G,H) has property (RD), the algebra of rapidly decreasing functions on the set of double cosets is closed under holomorphic functional calculus of the associated (reduced) Hecke C*-algebra. Hence they have the same K_0-groups.Comment: A short note added explaining other methods to prove that the subalgebra of rapidly decreasing functions is smooth. This is the final version as published. The published version is available at: springer.co