7 research outputs found
Overview of (pro-)Lie group structures on Hopf algebra character groups
Character groups of Hopf algebras appear in a variety of mathematical and
physical contexts. To name just a few, they arise in non-commutative geometry,
renormalisation of quantum field theory, and numerical analysis. In the present
article we review recent results on the structure of character groups of Hopf
algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild
assumptions on the Hopf algebra or the target algebra the character groups
possess strong structural properties. Moreover, these properties are of
interest in applications of these groups outside of Lie theory. We emphasise
this point in the context of two main examples: The Butcher group from
numerical analysis and character groups which arise from the Connes--Kreimer
theory of renormalisation of quantum field theories.Comment: 31 pages, precursor and companion to arXiv:1704.01099, Workshop on
"New Developments in Discrete Mechanics, Geometric Integration and
Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spai
Age-related changes in global motion coherence: conflicting haemodynamic and perceptual responses
Our aim was to use both behavioural and neuroimaging data to identify indicators of perceptual decline in motion processing. We employed a global motion coherence task and functional Near Infrared Spectroscopy (fNIRS). Healthy adults (n = 72, 18-85) were recruited into the following groups: young (n = 28, mean age = 28), middle-aged (n = 22, mean age = 50), and older adults (n = 23, mean age = 70). Participants were assessed on their motion coherence thresholds at 3 different speeds using a psychophysical design. As expected, we report age group differences in motion processing as demonstrated by higher motion coherence thresholds in older adults. Crucially, we add correlational data showing that global motion perception declines linearly as a function of age. The associated fNIRS recordings provide a clear physiological correlate of global motion perception. The crux of this study lies in the robust linear correlation between age and haemodynamic response for both measures of oxygenation. We hypothesise that there is an increase in neural recruitment, necessitating an increase in metabolic need and blood flow, which presents as a higher oxygenated haemoglobin response. We report age-related changes in motion perception with poorer behavioural performance (high motion coherence thresholds) associated with an increased haemodynamic response
Pre- and Post-Lie Algebras: The Algebro-Geometric View
We relate composition and substitution in pre- and post-Lie algebras to algebraic geometry. The Connes-Kreimer Hopf algebras and MKW Hopf algebras are then coordinate rings of the infinite-dimensional affine varieties consisting of series of trees, resp. Lie series of ordered trees. Furthermore we describe the Hopf algebras which are coordinate rings of the automorphism groups of these varieties, which govern the substitution law in pre- and post-Lie algebras.acceptedVersio