Remarks on CR-manifolds of codimension 22 in C\sp 4

summary:The aim of the article is to give a conceptual understanding of Kontsevich's construction of the universal element of the cohomology of the coarse moduli space of smooth algebraic curves with given genus and punctures. \par In a first step the author presents a toy model of tree graphs coloured by an operad $\cal P$ for which the graph complex and the universal cycle will be constructed. The universal cycle has coefficients in the operad for $\Omega({\cal P}^*)$-algebras with trivial differential over the (dual) cobar construction $\Omega({\cal P}^*)$. If $\cal P$ is Koszul the explicit form of the universal cycle will be presented. In a second step the author then considers general $\cal P$-coloured graphs over cyclic operads $\cal P$. The construction of the graph complex and the universal class in the cohomology of the graph complex resembles the previous constructions for tree graphs. \par The coefficients of the universal cohomology class are elements in the

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

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