James Croll was a pioneer in studies of the impact of the slowly changing orbital dynamics of the Earth on climate change. His book Climate and Time in their Geological Relations (1875) was far ahead of its time in seeking correlations between climate change, the occurrence of ice ages and perturbations to the Earth's orbit about the Sun. The astronomical cycles he discovered are now called ‘Milankovitch Cycles’ after the Serbian scientist whose research was first published in the Handbuch der Klimatologie in 1930. The celestial mechanical and astronomical background to Croll's research is the focus of this essay. The development of the understanding of the impact of perturbations of the elliptical planetary orbits by other bodies in the solar system paralleled new mathematical techniques, many of which were developed in association with celestial mechanical problems. The central contributions of many of the major mathematicians of the late 18th and 19th Centuries, including Euler, Lagrange, Laplace and Le Verrier, are highlighted. Although Croll's contributions faded from view for several generations, his pioneering insights have now been demonstrated to have been basically correct.

The purpose of this investigation was to analyze the influence of stimulus orientation on the responses of individual neurons in the monkey's lateral geniculate nucleus (LGN). Our specific goals were to assess the prevalence and the degree of orientation tuning in the monkey LGN and to determine if the preferred stimulus orientations of LGN neurons varied as a function of receptive-field position. The primary motivation for this research was to gain insight into the receptive-field configuration of LGN neurons and consequently into the neural mechanisms which determine the spatial organization of LGN receptive fields in primates.

In both the parvocellular and magnocellular layers, the responses of the majority of individual neurons to sine-wave gratings varied as a function of stimulus orientation. The influence of stimulus orientation was, however, highly dependent on the spatial characteristics of the stimulus; the greatest degree of orientation bias was observed for spatial frequencies higher than the cell's optimal spatial frequency. On a population basis, the degree of orientation bias was similar for all major classes of LGN neurons (e.g. ON vs. OFF center; parvocellular vs. magnocellular) and did not vary systematically with receptive-field eccentricity. At a given receptive-field location, LGN neurons, particularly cells in the parvocellular laminae, tended to prefer either radially oriented stimuli or stimuli oriented more horizontally than their polar axis. Our analyses of the orientation-dependent changes in spatial-frequency response functions, which was based on the Soodak et al., (1987; Soodak, 1986) two-dimensional, difference-of-Gaussian receptive-field model, suggested that the orientation bias in LGN neurons was due to an elongation of the receptive-field center mechanism which in some cases appeared to consist of multiple subunits. Direct comparisons of the orientation-tuning characteristics of LGN cells and their retinal inputs (S potentials) indicated that the orientation bias in the monkey LGN reflects primarily the functional properties of individual retinal ganglion cells. We conclude that orientation sensitivity is a significant property of subcortical neurons in the primate's geniculo-cortical pathway.

It is still an open question whether a compact embedded hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in $\mathbb{R}^3$. In a recent paper, Alías and Malacarne (Rev. Mat. Iberoamericana18 (2002), 431–442) have shown that this is true for the case of hypersurfaces in $\mathbb{R}^{n+1}$ with constant scalar curvature, and more generally, hypersurfaces with constant higher-order $r$-mean curvature, when $r\geq2$. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold $\bar{M}$, where we will consider a general geometric configuration consisting of an immersed hypersurface into $\bar{M}$ with boundary on an oriented hypersurface $P$ of $\bar{M}$. For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of $P$, as well as the geometry of $P$ as a hypersurface of $\bar{M}$. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature (the Euclidean space $\mathbb{R}^{n+1}$, the hyperbolic space $\mathbb{H}^{n+1}$, and the sphere $\mathbb{S}^{n+1}$). In particular, we are able to extend the symmetry results given in the recent paper mentioned above to the case of hypersurfaces with constant higher-order $r$-mean curvature in the hyperbolic space and in the sphere.

We propose a new formulation of the 3D Boltzmann non linear operator, without assuming Grad's angular cutoff hypothesis, and for intermolecular laws behaving as 1/rs , with s> 2. It involves natural pseudo differential operators, under a form which is analogous to the Landau operator. It may be used in the study of the associated equations, and more precisely in the non homogeneous framework.