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