Grid cells on steeply sloping terrain: evidence for planar rather than volumetric encoding

Neural encoding of navigable space involves a network of structures centred on the hippocampus, whose neurons –place cells – encode current location. Input to the place cells includes afferents from the entorhinal cortex, which contains grid cells. These are neurons expressing spatially localised activity patches, or firing fields, that are evenly spaced across the floor in a hexagonal close-packed array called a grid. It is thought that grid cell grids function to enable the calculation of distances. The question arises as to whether this odometry process operates in three dimensions, and so we queried whether grids permeate three-dimensional space – that is, form a lattice – or whether they simply follow the environment surface. If grids form a three-dimensional lattice then a tilted floor should transect several layers of this lattice, resulting in interruption of the hexagonal pattern. We model this prediction with simulated grid lattices and show that on a 40-degree slope the firing of a grid cell should cover proportionally less of the surface, with smaller field size and fewer fields and reduced hexagonal symmetry. However, recording of grid cells as animals foraged on a 40-degree-tilted surface found that firing of grid cells was almost indistinguishable, in pattern or rate, from that on the horizontal surface, with if anything increased coverage and field number, and preserved field size. It thus appears unlikely that the sloping surface transected a lattice. However, grid cells on the slope displayed slightly degraded firing patterns, with reduced coherence and slightly reduced symmetry. These findings collectively suggest that the grid cell component of the metric representation of space is not fixed in absolute three-dimensional space but is influenced both by the surface the animal is on and by the relationship of this surface to the horizontal, supporting the hypothesis that the neural map of space is multi-planar rather than fully volumetric

The Schwarzian derivative and the Wiman-Valiron property

Consider a transcendental meromorphic function in the plane with finitely many critical values, such that the multiple points have bounded multiplicities and the inverse function has finitely many transcendental singularities. Using the Wiman-Valiron method it is shown that if the Schwarzian derivative is transcendental then the function has infinitely many multiple points, the inverse function does not have a direct transcendental singularity over infinity, and infinity is not a Borel exceptional value. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method

Global Estimation of Range Resolved Thermodynamic Profiles from MicroPulse Differential Absorption Lidar

We demonstrate thermodynamic profile estimation with data obtained using the MicroPulse DIAL such that the retrieval is entirely self contained. The only external input is surface meteorological variables obtained from a weather station installed on the instrument. The estimator provides products of temperature, absolute humidity and backscatter ratio such that cross dependencies between the lidar data products and raw observations are accounted for and the final products are self consistent. The method described here is applied to a combined oxygen DIAL, potassium HSRL, water vapor DIAL system operating at two pairs of wavelengths (nominally centered at 770 and 828 nm). We perform regularized maximum likelihood estimation through the Poisson Total Variation technique to suppress noise and improve the range of the observations. A comparison to 119 radiosondes indicates that this new processing method produces improved temperature retrievals, reducing total errors to less than 2 K below 3 km altitude and extending the maximum altitude of temperature retrievals to 5 km with less than 3 K error. The results of this work definitively demonstrates the potential for measuring temperature through the oxygen DIAL technique and furthermore that this can be accomplished with low-power semiconductor-based lidar sensors

From Pre-Elite to Elite: The Pathway Travelled by Adolescent Golfers

This study employed interpretative phenomenological analysis (IPA) to explore the lived experiences of eight high performing adolescent golfers who had all successfully travelled the path from novice to elite level status. By means of semi-structured qualitative interviews, participants answered questions centred on four key areas which explored their journey from preelite to elite adolescent status: initial involvement and continued participation in golf, the meaning of golf, golf environment and social support. Two super-ordinate themes emerged from participants accounts: Early Pre-elite Sporting Experiences and Strategic Approaches to Develop Adolescent Golfing Excellence. The study provides key insights into individual, social and environmental factors that enabled pre-elite adolescent golfers to make a successful transition to the elite pathway, and highlights plausible factors that may make a difference whether an athlete becomes elite or not. The findings will help coaches, policy makers and sport psychologists more effectively support emerging talent in golf

Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Tsuji related to the Denjoy-Carleman-Ahlfors theorem.Comment: 17 page

Permutable entire functions satisfying algebraic differential equations

It is shown that if two transcendental entire functions permute, and if one of them satisfies an algebraic differential equation, then so does the other one.Comment: 5 page

2D Signal Estimation for Sparse Distributed Target Photon Counting Data

In this study, we explore the utilization of maximum likelihood estimation for the analysis of sparse photon counting data obtained from distributed target lidar systems. Specifically, we adapt the Poisson Total Variation processing technique to cater to this application. By assuming a Poisson noise model for the photon count observations, our approach yields denoised estimates of backscatter photon flux and related parameters. This facilitates the processing of raw photon counting signals with exceptionally high temporal and range resolutions (demonstrated here to 50 Hz and 75 cm resolutions), including data acquired through time-correlated single photon counting, without significant sacrifice of resolution. Through examination involving both simulated and real-world 2D atmospheric data, our method consistently demonstrates superior accuracy in signal recovery compared to the conventional histogram-based approach commonly employed in distributed target lidar applications

Baker's conjecture for functions with real zeros

Baker's conjecture states that a transcendental entire functions of order less than 1/2 has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits consisting of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1. Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1