Symmetry reduction of Brownian motion and Quantum Calogero-Moser systems

Let $Q$ be a Riemannian $G$-manifold. This paper is concerned with the symmetry reduction of Brownian motion in $Q$ and ramifications thereof in a Hamiltonian context. Specializing to the case of polar actions we discuss various versions of the stochastic Hamilton-Jacobi equation associated to the symmetry reduction of Brownian motion and observe some similarities to the Schr\"odinger equation of the quantum free particle reduction as described by Feher and Pusztai. As an application we use this reduction scheme to derive examples of quantum Calogero-Moser systems from a stochastic setting.Comment: V2 contains some improvements thanks to referees' suggestions; to appear in Stochastics and Dynamic

Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization

We develop Hamilton-Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton-Jacobi theory to the Hamiltonized systems. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton-Jacobi equation associated with the Hamiltonized system also solve the nonholonomic Hamilton-Jacobi equation associated with the original Chaplygin system. The results are illustrated through several examples.Comment: Accepted for publication in Journal of Geometry and Physic

Collisionless kinetic theory of rolling molecules

We derive a collisionless kinetic theory for an ensemble of molecules undergoing nonholonomic rolling dynamics. We demonstrate that the existence of nonholonomic constraints leads to problems in generalizing the standard methods of statistical physics. In particular, we show that even though the energy of the system is conserved, and the system is closed in the thermodynamic sense, some fundamental features of statistical physics such as invariant measure do not hold for such nonholonomic systems. Nevertheless, we are able to construct a consistent kinetic theory using Hamilton's variational principle in Lagrangian variables, by regarding the kinetic solution as being concentrated on the constraint distribution. A cold fluid closure for the kinetic system is also presented, along with a particular class of exact solutions of the kinetic equations.Comment: Revised version; 31 pages, 1 figur

AP2γ: a new player on adult hippocampal neurogenesis regulation

Since the recognition that the mammalian brain retains the ability to generate newborn neurons with functional relevance throughout life, the matrix of molecular regulators that govern adult neurogenesis has been the focus of much interest. In a recent study published in Molecular Psychiatry, we demonstrate Activating Protein 2γ (AP2γ), a transcription factor previously implicated in cell fate determination in the developing cortex, as a novel player in the regulation of glutamatergic neurogenesis in the adult hippocampus. Using distinct experimental approaches, we showed that AP2γ is specifically present in a subpopulation of transient amplifying progenitors, where it acts as a crucial promoter of proliferation and differentiation of adult-born glutamatergic granule neurons. Strikingly, deficiency of AP2γ in the adult brain compromises the generation of new glutamatergic neurons, with impact on the function of cortico-limbic circuits. Here, we share our view on how AP2γ integrates the transcriptional orchestration of glutamatergic neurogenesis in the adult hippocampus, and consequently, how it emerges as a novel molecular candidate to study the translation of environmental pressures into alterations of brain neuroplasticity in homeostatic, but also in neuropathological contexts.Bial Foundation (427/14); Northern Portugal Regional Operational Programme (NORTE 2020); European Regional Development Fund (FEDER) (projects NORTE-01-0145-FEDER-000013 e NORTE-01-0145-FEDER-000023); Competitiveness Factors Operational Programme (COMPETE)info:eu-repo/semantics/publishedVersio

Integrable Euler top and nonholonomic Chaplygin ball

We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the nonholonomic Chaplygin ball.Comment: 25 pages, LaTeX with AMS fonts, final versio

LR and L+R Systems

We consider coupled nonholonomic LR systems on the product of Lie groups. As examples, we study $n$-dimensional variants of the spherical support system and the rubber Chaplygin sphere. For a special choice of the inertia operator, it is proved that the rubber Chaplygin sphere, after reduction and a time reparametrization becomes an integrable Hamiltonian system on the $(n-1)$--dimensional sphere. Also, we showed that an arbitrary L+R system introduced by Fedorov can be seen as a reduced system of an appropriate coupled LR system.Comment: 18 pages, 1 figur

Oligodendrocyte heterogeneity in the mouse juvenile and adult central nervous system

Oligodendrocytes have been considered as a functionally homogeneous population in the central nervous system (CNS). We performed single-cell RNA sequencing on 5072 cells of the oligodendrocyte lineage from 10 regions of the mouse juvenile and adult CNS. Thirteen distinct populations were identified, 12 of which represent a continuum from Pdgfra(+) oligodendrocyte precursor cells (OPCs) to distinct mature oligodendrocytes. Initial stages of differentiation were similar across the juvenile CNS, whereas subsets of mature oligodendrocytes were enriched in specific regions in the adult brain. Newly formed oligodendrocytes were detected in the adult CNS and were responsive to complex motor learning. A second Pdgfra(+) population, distinct from OPCs, was found along vessels. Our study reveals the dynamics of oligodendrocyte differentiation and maturation, uncoupling them at a transcriptional level and highlighting oligodendrocyte heterogeneity in the CNS

Regional Innovation and Research Policy Outlook: Policy Practices in Eight European Regions

This publication is part of a project called 'Practical Regional Research and Innovation policy in Action. The Efficient Tools for Regional Catching-up in New Member States'