Summary Self-motion triggers complementary visual and vestibular reflexes supporting image-stabilization and balance. Translation through space produces one global pattern of retinal image motion (optic flow), rotation another. We show that each subtype of direction-selective ganglion cell (DSGC) adjusts its direction preference topographically to align with specific translatory optic flow fields, creating a neural ensemble tuned for a specific direction of motion through space. Four cardinal translatory directions are represented, aligned with two axes of high adaptive relevance: the body and gravitational axes. One subtype maximizes its output when the mouse advances, others when it retreats, rises, or falls. ON-DSGCs and ON-OFF-DSGCs share the same spatial geometry but weight the four channels differently. Each subtype ensemble is also tuned for rotation. The relative activation of DSGC channels uniquely encodes every translation and rotation. Though retinal and vestibular systems both encode translatory and rotatory self-motion, their coordinate systems differ.

We investigate isometric immersions of disks with constant negative curvature into R 3 , and the minimizers for the bending energy, i.e. the L 2 norm of the principal curvatures over the class of W 2,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H 2 into R 3 . In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the L ∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W 2,2 , and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates.

We present and summarize the results of recent studies on non-Euclidean plates with imposed constant negative Gaussian curvature in both the Föppl -von Kármán and Kirchhoff approximations. Motivated by experimental results we focus on annuli with a periodic profile. We show that in the Föppl -von Kármán approximation there are only two types of global minimizers -flat and saddle shaped deformations with localized regions of stretching near the boundary of the annulus. We also show that there exists local minimizers with n-waves that have regions of stretching near their lines of inflection. In the Kirchhoff approximation we show that there exist exact isometric immersions with periodic profiles. The number of waves in these configurations is set by the condition that the bending energy remains finite and grows approximately exponentially with the radius of the annulus. For large radii, these shape are energetically favorable over saddle shapes and could explain why wavy shapes are selected by crocheted models of the hyperbolic plane.PACS numbers: 02.40.-k, 68.60.-p, 68.90.+g, 89.75.kd When subjected to localized lateral swelling or growth, thin elastic sheets can form periodic rippling patterns as observed in torn plastic sheets [1], along the edges of leaves [2] and in swelling hydrogels [3]. Since the bending energy of thin sheets is much weaker than stretching, the morphology of these structures is a result of the sheet bending to relieve in-plane strain. Recently, there has been interest in using swelling as a mechanism for shaping flat sheets into desired three-dimensional structures [3,4]. This type of specified morphogenesis has been experimentally realized by varying the radial shrinkage factor of thermally responsive gel disks [3] and by directly controlling the local swelling at specific points through halftone gel lithography [4].The equilibrium configuration of such sheets can be modelled as the minimum of a non-Euclidean elastic energy that measures strains from a fixed two-dimensional Riemannian metric g defined on the midsurface of the sheet D ⊂ R 2 [5]. The intuition behind this model is that g encodes how the local swelling changes the intrinsic distance between material coordinates and the body deforms to be as "close as possible" to an isometric immersion of g. In this model, deformations with vanishing stretching energy correspond to isometric immersions of g into R 3 and in the vanishing thickness limit a particular isometric immersion, provided one exists, is selected by the bending energy [6].This letter summarizes two studies of swelling thin elastic sheets that we presented in [7] and [8]. In these works we considered the Föppl -von Kármán (FvK) and Kirchhoff approximations when D is an annulus with inner radius r 0 and outer radius R and g has corresponding Gaussian curvature K 0 that is constant and negative. In geodesic polar coordinates (ρ, θ) the metric takes the form g = dr 2 +|K 0 | −1 sinh 2 ( |K 0 |)dθ 2 and isometric immersions of g correspond to (local) isometric immersions ...

Abstract. In sea-ice-covered areas, the sea ice floe size distribution (FSD) plays an important role in many processes affecting the coupled sea–ice–ocean–atmosphere system. Observations of the FSD are sparse – traditionally taken via a painstaking analysis of ice surface photography – and the seasonal and inter-annual evolution of floe size regionally and globally is largely unknown. Frequently, measured FSDs are assessed using a single number, the scaling exponent of the closest power-law fit to the observed floe size data, although in the absence of adequate datasets there have been limited tests of this “power-law hypothesis”. Here we derive and explain a mathematical technique for deriving statistics of the sea ice FSD from polar-orbiting altimeters, satellites with sub-daily return times to polar regions with high along-track resolutions. Applied to the CryoSat-2 radar altimetric record, covering the period from 2010 to 2018, and incorporating 11 million individual floe samples, we produce the first pan-Arctic climatology and seasonal cycle of sea ice floe size statistics. We then perform the first pan-Arctic test of the power-law hypothesis, finding limited support in the range of floe sizes typically analyzed in photographic observational studies. We compare the seasonal variability in observed floe size to fully coupled climate model simulations including a prognostic floe size and thickness distribution and coupled wave model, finding good agreement in regions where modeled ocean surface waves cause sea ice fracture.