Elliptic CR-manifolds and shear invariant ODE with additional symmetries

We classify the ODEs that correspond to elliptic CR-manifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can be only 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable istropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold

Free CR distributions

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions $n$ and codimensions $n^2$ are among the very few possibilities of the so called parabolic geometries. Indeed, the homogeneous model turns out to be \PSU(n+1,n)/P with a suitable parabolic subgroup $P$. We study the geometric properties of such real $(2n+n^2)$-dimensional submanifolds in $\mathbb C^{n+n^2}$ for all $n>1$. In particular we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry

Invariants of elliptic and hyperbolic CR-structures of codimension 2

We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves ``almost'' like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of ``semi-flat'' manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection. In addition, for real-analytic manifolds we describe certain local normal forms that do not require passing to bundles, but in many ways agree with the structure of the parallelism.Comment: 42 pages, see also http://wwwmaths.anu.edu.au/research.reports/97mrr.htm

ONLINE EXAMS IN UNDERGRADUATE MATHEMATICS AT UNE DURING COVID-19 ISOLATION

The University of New England (UNE) has been a champion in online teaching even prior to the COVID-19 pandemic. Nonetheless, moving all teaching to a purely online modality presented challenges. A key goal is to have fair examinations that satisfy academic standards and avoid plagiarism. A fair exam guarantees that a student has achieved the learning outcomes and is prepared for further studies or capable of problem solving. We discuss our experiences in creating fair online examinations. Invigilated paper exams were replaced by a combination of online exams using Möbius, an e-learning platform, as well as other “take-home” alternative assessments. Möbius has the ability to create interactive exercises with support for mathematical notation. We found Möbius is suitable for standard skills amenable to multiple choice or numerical answers. However, for more advanced topics Möbius presented additional challenges not present for paper exams and this limited its usefulness. We conclude that upper level undergraduate classes are better tested in a paper or oral examination. The necessity to run all exams online in this trimester caused us to rethink what we ultimately want to achieve in a fair examination as well as to look for alternative examination formats for the future

Applying the Seven Principles for Good Practice in Undergraduate Education: Improving Research Writing Skills in a Writing-Emphasis Health Counseling Course

It is frequently noted by college professors that students are ill prepared to conduct proper research. In order to address this problem, the authors discuss the use of a set of instructions to enable students to reach their writing potential. The Seven Principles for Good Practice in Undergraduate Education by Chickering and Gamson (1987) formulate a set of guidelines to improve teaching and learning. This paper will list the seven principles followed by the approach that the authors took in adhering to the principles

Computer Security: Just How Secure Is Your Computer?

Security has become a matter of utmost importance since the aftermath of September 11th, especially in the area of computer systems. There are many options that a health educator can do as an individual to help secure the computer system at their worksite and in their home. This article is a brief overview of some of the precautions that should be taken on a daily basis to protect your computer systems, files, data, and other pertinent information

CR embeddings of CR manifolds

We improve results of Baouendi, Rothschild and Treves and of Hill and Nacinovich by finding a much weaker sufficient condition for a CR manifold of type (n, k) to admit a local CR embedding into a CR manifold of type (n+ ℓ, k- ℓ). While their results require the existence of a finite dimensional solvable transverse Lie algebra of vector fields, we require only a finite dimensional extension