Generalized Ehrhart polynomials

Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of P(n) to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in P(n) is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related.Comment: 18 pages, no figures; v2: Sections 4 and 5 added, proofs and exposition have been expanded and clarifie

Role Playing Learning for Socially Concomitant Mobile Robot Navigation

In this paper, we present the Role Playing Learning (RPL) scheme for a mobile robot to navigate socially with its human companion in populated environments. Neural networks (NN) are constructed to parameterize a stochastic policy that directly maps sensory data collected by the robot to its velocity outputs, while respecting a set of social norms. An efficient simulative learning environment is built with maps and pedestrians trajectories collected from a number of real-world crowd data sets. In each learning iteration, a robot equipped with the NN policy is created virtually in the learning environment to play itself as a companied pedestrian and navigate towards a goal in a socially concomitant manner. Thus, we call this process Role Playing Learning, which is formulated under a reinforcement learning (RL) framework. The NN policy is optimized end-to-end using Trust Region Policy Optimization (TRPO), with consideration of the imperfectness of robot's sensor measurements. Simulative and experimental results are provided to demonstrate the efficacy and superiority of our method

Three-dimensional structure of basal body triplet revealed by electron cryo-tomography.

Basal bodies and centrioles play central roles in microtubule (MT)-organizing centres within many eukaryotes. They share a barrel-shaped cylindrical structure composed of nine MT triplet blades. Here, we report the structure of the basal body triplet at 33 Å resolution obtained by electron cryo-tomography and 3D subtomogram averaging. By fitting the atomic structure of tubulin into the EM density, we built a pseudo-atomic model of the tubulin protofilaments at the core of the triplet. The 3D density map reveals additional densities that represent non-tubulin proteins attached to the triplet, including a large inner circular structure in the basal body lumen, which functions as a scaffold to stabilize the entire basal body barrel. We found clear longitudinal structural variations along the basal body, suggesting a sequential and coordinated assembly mechanism. We propose a model in which δ-tubulin and other components participate in the assembly of the basal body

Chaotic Dynamics of Trans-Neptunian Objects Perturbed by Planet Nine

Observations of clustering among the orbits of the most distant trans-Neptunian objects (TNOs) has inspired interest in the possibility of an undiscovered ninth planet lurking in the outskirts of the solar system. Numerical simulations by a number of authors have demonstrated that, with appropriate choices of planet mass and orbit, such a planet can maintain clustering in the orbital elements of the population of distant TNOs, similar to the observed sample. However, many aspects of the rich underlying dynamical processes induced by such a distant eccentric perturber have not been fully explored. We report the results of our investigation of the dynamics of coplanar test-particles that interact with a massive body on an circular orbit (Neptune) and a massive body on a more distant, highly eccentric orbit (the putative Planet Nine). We find that a detailed examination of our idealized simulations affords tremendous insight into the rich test-particle dynamics that are possible. In particular, we find that chaos and resonance overlap plays an important role in particles' dynamical evolution. We develop a simple mapping model that allows us to understand, in detail, the web of overlapped mean-motion resonances explored by chaotically evolving particles. We also demonstrate that gravitational interactions with Neptune can have profound effects on the orbital evolution of particles. Our results serve as a starting point for a better understanding of the dynamical behavior observed in more complicated simulations that can be used to constrain the mass and orbit of Planet 9.Comment: Revised to address referree comments; accepted to A

Information-theoretic classification of SNOMED improves the organization of context-sensitive excerpts from Cochrane Reviews

The emphasis on evidence based medicine (EBM) has placed increased focus on finding timely answers to clinical questions in presence of patients. Using a combination of natural language processing for the generation of clinical excerpts and information theoretic distance based clustering, we evaluated multiple approaches for the efficient presentation of context-sensitive EBM excerpts