An Invitation to Singular Symplectic Geometry

In this paper we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to $b$-symplectic manifolds: $b$-contact manifolds.Comment: 14 pages, 1 figur

Cotangent models for group actions on bb-Poisson manifolds

In this article we give a normal form of a $b$-symplectic form in the neighborhood of a compact orbit of a Lie group action on a $b$-symplectic manifold. We establish cotangent models for Poisson actions on $b$-Poisson manifolds and a $b$-symplectic slice theorem. We examine interesting particular instances of group actions on $b$-symplectic manifolds preserving the Poisson structure. Also, we revise the notion of cotangent lift and twisted $b$-cotangent lift introduced in [KM] and provide a generalization of the twisted $b$-cotangent lift. We introduce the notion of $b$-Lie group and the associated $b$-symplectic structures in its $b$-cotangent bundle together with their reduction theory.Comment: 19 pages, 1 figur

Strain-induced structural instability in FeRh

We perform density functional calculations to investigate the structure of the inter-metallic alloy FeRh under epitaxial strain. Bulk FeRh exhibits a metamagnetic transition from a low-temperature antiferromagnetic (AFM) phase to a ferromagnetic (FM) phase at 350K, and its strain dependence is of interest for tuning the transition temperature to the room-temperature operating conditions of typical memory devices. We find an unusually strong dependence of the structural energetics on the choice of exchange-correlation functional, with the usual local density approximation (LDA) yielding the wrong ground-state structure, and generalized gradient (GGA) extensions being in better agreement with the bulk experimental structure. Using the GGA we show the existence of a metastable face-centered-cubic (fcc)-like AFM structure that is reached from the ground state body-centered-cubic (bcc) AFM structure by following the epitaxial Bain path. We predict that this metastable fcc-like structure has a significantly higher conductivity than the bcc AFM phase. We show that the behavior is well described using non-linear elasticity theory, which captures the softening and eventual sign change of the orthorhombic shear modulus under compressive strain, consistent with this structural instability. Finally, we predict the existence of an additional unit-cell-doubling lattice instability, which should be observable at low temperature.Comment: 10 pages, 7 figure

b-Structures on Lie groups and Poisson reduction

Motivated by the group of Galilean transformations and the subgroup of Galilean transformations which fix time zero, we introduce the notion of a b-Lie group as a pair where G is a Lie group and H is a codimension-one Lie subgroup. Such a notion allows us to give a theoretical framework for transformations of space-time where the initial time can be seen as a boundary. In this theoretical framework, we develop the basics of the theory and study the associated canonical b-symplectic structure on the b-cotangent bundle ¿ together with its reduction theory. Namely, we extend the minimal coupling procedure to ¿ and prove that the Poisson reduction under the cotangent lifted action of H by left translations can be described in terms of the Lie Poisson structure on ¿ (where is the Lie algebra of H) and the canonical b-symplectic structure on ¿ , where is viewed as a one-dimensional b-manifold having as critical hypersurface (in the sense of b-manifolds) the identity element.Peer ReviewedPostprint (published version

Effects of optogenetic and visual stimulation on gamma activity in the visual cortex

Studying brain functions and activity during gamma oscillations can be a challenge because it requires careful planning to create the necessary conditions for a controlled experiment. Such an experiment consists of placing the brain into a gamma state and investigating cognitive processing with a careful design. Cortical oscillations in the gamma frequency range (30-80 Hz) play an essential role in a variety of cognitive processes, including visual processing and cognition. The present study aims to investigate the effects of a visual stimulus on the primary visual cortex under gamma oscillations. Specifically, we sought to explore the behavior of gamma oscillations triggered by optogenetic stimulation in the II and IV layers of the visual cortex, both with and without concurrent visual stimulation. Our results show that optogenetic stimulation increases the power of gamma oscillation in both layers of the visual cortex. However, the combined stimuli resulted in a reduction of gamma power in layer II and an increase and reinforcement in gamma power in layer IV. Modelling the results with the Wilson-Cowan model suggests changes in the input of the excitatory population due to the combined stimuli. In addition, our analysis of the data using the Lempel-Ziv complexity method supports our interpretations from the modeling. Thus, our results suggest that optogenetic stimulation enhances low gamma power in both layers of the visual cortex, while simultaneous visual stimulation has differing effects on the two layers, reducing gamma power in layer II and increasing it in layer IV