Series of nilpotent orbits

We organize the nilpotent orbits in the exceptional complex Lie algebras into series using the triality model and show that within each series the dimension of the orbit is a linear function of the natural parameter a=1,2,4,8, respectively for f_4,e_6,e_7,e_8. We also obtain explicit representatives in a uniform manner. We observe similar regularities for the centralizers of nilpotent elements in a series and graded components in the associated grading of the ambient Lie algebra. More strikingly, for a greater than one, the degrees of the unipotent characters of the corresponding Chevalley groups, associated to these series through the Springer correspondance are given by polynomials which have uniform expressions in terms of a.Comment: 20 pages, revised version with more formulas for unipotent character

Infant EEG activity as a biomarker for autism: a promising approach or a false promise?

The ability to determine an infant's likelihood of developing autism via a relatively simple neurological measure would constitute an important scientific breakthrough. In their recent publication in this journal, Bosl and colleagues claim that a measure of EEG complexity can be used to detect, with very high accuracy, infants at high risk for autism (HRA). On the surface, this appears to be that very scientific breakthrough and as such the paper has received widespread media attention. But a close look at how these high accuracy rates were derived tells a very different story. This stems from a conflation between "high risk" as a population-level property and "high risk" as a property of an individual. We describe the approach of Bosl et al. and examine their results with respect to baseline prevalence rates, the inclusion of which is necessary to distinguish infants with a biological risk of autism from typically developing infants with a sibling with autism. This is an important distinction that should not be overlooked

Representations of the exceptional and other Lie algebras with integral eigenvalues of the Casimir operator

The uniformity, for the family of exceptional Lie algebras g, of the decompositions of the powers of their adjoint representations is well-known now for powers up to the fourth. The paper describes an extension of this uniformity for the totally antisymmetrised n-th powers up to n=9, identifying (see Tables 3 and 6) families of representations with integer eigenvalues 5,...,9 for the quadratic Casimir operator, in each case providing a formula (see eq. (11) to (15)) for the dimensions of the representations in the family as a function of D=dim g. This generalises previous results for powers j and Casimir eigenvalues j, j<=4. Many intriguing, perhaps puzzling, features of the dimension formulas are discussed and the possibility that they may be valid for a wider class of not necessarily simple Lie algebras is considered.Comment: 16 pages, LaTeX, 1 figure, 9 tables; v2: presentation improved, typos correcte

Primality-testing Mersenne numbers

Report of a systematic review of Mersenne numbers 2^p-1 for p < 62982

A unified approach on Springer fibers in the hook, two-row and two-column cases

We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the Springer fiber. We solve the problem in the hook, two-row and two-column cases. We provide two main characterizations which are common to the three cases, and which involve dominance relations between Young diagrams and combinatorial algorithms. Then, for these three cases, we deduce topological properties of the components and their intersections.Comment: 42 page

A real-time articulatory visual feedback approach with target presentation for second language pronunciation learning

International audienceArticulatory information can support learning or remediating pronunciation of a second language (L2). This paper describes an electromagnetic articulometer-based visual-feedback approach using an articulatory target presented in real-time to facilitate L2 pronunciation learning. This approach trains learners to adjust articulatory positions to match targets for a L2 vowel estimated from productions of vowels that overlap in both L1 and L2. Training of Japanese learners for the American English vowel /ae/ that included visual training improved its pronunciation regardless of whether audio training was also included. Articulatory visual feedback is shown to be an effective method for facilitating L2 pronunciation learning