On the stability of Hamiltonian relative equilibria with non-trivial isotropy

We consider Hamiltonian systems with symmetry, and relative equilibria with isotropy subgroup of positive dimension. The stability of such relative equilibria has been studied by Ortega and Ratiu and by Lerman and Singer. In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subspace of the Lie algebra of the symmetry group, with different splittings giving different criteria. In this note we remove this splitting construction and so provide a more general and more easily computed criterion for stability. The result is also extended to apply to systems whose momentum map is not coadjoint equivariant

Some Integrals Involving Bessel Functions

AbstractA number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber′s second exponential integral and Laplace transforms of products of three Bessel functions

Point vortices on the hyperbolic plane

We investigate some properties of the dynamical system of point vortices on the hyperboloid. This system has noncompact symmetry SL(2, R) and a coadjoint equivariant momentum map J. The relative equilibrium conditions are found and the trajectories of relative equilibria with non-zero momentum value are described. We also provide the classification of relative equilibria and the stability criteria for a number of cases, focusing on N=2, 3. Contrary to the system on the sphere, relative equilibria with non-compact momentum isotropy subgroup are found, and are used to illustrate the different stability types of relative equilibria.Comment: To appear in J. Mathematical Physic

Spatial Autocorrelation Analysis for New FUA Inner Strategic Asset: A Case Study of the Metropolitan City of Milan, Italy

Functional urban areas represent integrated urban contexts whose territories are economically interconnected. They, therefore, include a central city and all the municipalities that make up the commuting area for work reasons. The economic energies and settlement transformations that characterize these territories have been consolidated over time. The current geographic conformation, as defined today, does not provide information on each municipality's rank (role) in the overall functioning. In this perspective, the work presented examines the demographic and urban dynamics that have affected the FUA of Milan in the last 60 years and then evaluates the presence of possible homogeneous geographic clusters (hot and cold spots) through spatial correlation techniques. Statistic validation was performed through the ANOVA and subsequent posthoc analysis (Tukey-Kramer method). Results show a new configurational asset within the FUA of Milan, which could provide a new key to interpreting the territory, aimed at identifying homogeneous areas to adopt new and more effective forms of strategic planning

Critical points of higher order for the normal map of immersions in R^d

We study the critical points of the normal map v : NM -> Rk+n, where M is an immersed k-dimensional submanifold of Rk+n, NM is the normal bundle of M and v(m, u) = m + u if u is an element of NmM. Usually, the image of these critical points is called the focal set. However, in that set there is a subset where the focusing is highest, as happens in the case of curves in R-3 with the curve of the centers of spheres with contact of third order with the curve. We give a definition of r-critical points of a smooth map between manifolds, and apply it to study the 2 and 3-critical points of the normal map in general and the 2-critical points for the case k = n = 2 in detail. In the later case we analyze the relation with the strong principal directions of Montaldi (1986) [2]. (C) 2011 Elsevier B.V. All rights reserved.Work partially supported by CAPES (BEX 4533/06-2).Monera, M.; Montesinos-Amilibia, A.; Moraes, S.; Sanabria Codesal, E. (2012). Critical points of higher order for the normal map of immersions in R^d. Topology and its Applications. 159:537-544. https://doi.org/10.1016/j.topol.2011.09.029S53754415

Point vortices on the sphere: a case with opposite vorticities

We study systems formed of 2N point vortices on a sphere with N vortices of strength +1 and N vortices of strength -1. In this case, the Hamiltonian is conserved by the symmetry which exchanges the positive vortices with the negative vortices. We prove the existence of some fixed and relative equilibria, and then study their stability with the ``Energy Momentum Method''. Most of the results obtained are nonlinear stability results. To end, some bifurcations are described.Comment: 35 pages, 9 figure

Stability of relative equilibria with singular momentum values in simple mechanical systems

A method for testing $G_\mu$-stability of relative equilibria in Hamiltonian systems of the form "kinetic + potential energy" is presented. This method extends the Reduced Energy-Momentum Method of Simo et al. to the case of non-free group actions and singular momentum values. A normal form for the symplectic matrix at a relative equilibrium is also obtained.Comment: Partially rewritten. Some mistakes fixed. Exposition improve

Deformation of geometry and bifurcation of vortex rings

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.Comment: 26 page

Memory for single items, word pairs, and temporal order of different kinds in a patient with selective hippocampal lesions

One kind of between-list and two kinds of within-list temporal order memory were examined in a patient with selective bilateral hippocampal lesions. This damage disrupted memory for all three kinds of temporal order memory, but left item and word pair recognition relatively intact. These findings are inconsistent with claims that (1) hippocampal lesions, like those of the medial temporal lobe (MTL) cortex, disrupt item and word pair recognition, and that (2) hippocampal lesions disrupt temporal order memory and item recognition to the same degree. Not only was word pair recognition intact in the patient, but further evidence indicates that her recognition of other associations between items of the same kind is also spared so retrieval of such associations cannot be sufficient to support within-list temporal order recognition. Rather, as other evidence indicates that the patient is impaired at recogni-tion of associations between different kinds of information, within-list (and possibly between-list) temporal order memory may be impaired by hippocampal lesions because it critically depends on re-trieving associations between different kinds of information