A combinatorial model for reversible rational maps over finite fields

We study time-reversal symmetry in dynamical systems with finite phase space, with applications to birational maps reduced over finite fields. For a polynomial automorphism with a single family of reversing symmetries, a universal (i.e., map-independent) distribution function R(x)=1-e^{-x}(1+x) has been conjectured to exist, for the normalized cycle lengths of the reduced map in the large field limit (J. A. G. Roberts and F. Vivaldi, Nonlinearity 18 (2005) 2171-2192). We show that these statistics correspond to those of a composition of two random involutions, having an appropriate number of fixed points. This model also explains the experimental observation that, asymptotically, almost all cycles are symmetrical, and that the probability of occurrence of repeated periods is governed by a Poisson law.Comment: LaTeX, 19 pages with 1 figure; to be published in Nonlinearit

Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.Comment: 29 pages, 3 figure

The association between stress and mood across the adult lifespan on default mode network

Aging of brain structure and function is a complex process characterized by high inter- and intra-individual variability. Such variability may arise from the interaction of multiple factors, including exposure to stressful experience and mood variation, across the lifespan. Using a multimodal neuroimaging and neurocognitive approach, we investigated the association of stress, mood and their interaction, in the structure and function of the default mode network (DMN), both during rest and task-induced deactivation, throughout the adult lifespan. Data confirmed a decreased functional connectivity (FC) and task-induced deactivation of the DMN during the aging process and in subjects with lower mood; on the contrary, an increased FC was observed in subjects with higher perceived stress. Surprisingly, the association of aging with DMN was altered by stress and mood in specific regions. An increased difficulty to deactivate the DMN was noted in older participants with lower mood, contrasting with an increased deactivation in individuals presenting high stress, independently of their mood levels, with aging. Interestingly, this constant interaction across aging was globally most significant in the combination of high stress levels with a more depressed mood state, both during resting state and task-induced deactivations. The present results contribute to characterize the spectrum of FC and deactivation patterns of the DMN, highlighting the crucial association of stress and mood levels, during the adult aging process. These combinatorial approaches may help to understand the heterogeneity of the aging process in brain structure and function and several states that may lead to neuropsychiatric disorders.The work was supported by SwitchBox-FP7-HEALTH-2010-Grant 259772-2 and by ON.2, O NOVO NORTE, North Portugal Regional Operational Programme 2007/2013, of the National strategic Reference Framework (NSRF) 2007/2013, through the European Regional Development Fund (ERDF)info:eu-repo/semantics/publishedVersio