Automatic Classification of Restricted Lattice Walks

We propose an experimental mathematics approach leading to the computer-driven discovery of various structural properties of general counting functions coming from enumeration of walks

A Case Study in Meta-AUTOMATION: AUTOMATIC Generation of Congruence AUTOMATA For Combinatorial Sequences

This article is a sequel to a recent article by Eric Rowland and Reem Yassawi, presenting yet another approach to the fast determination of congruence properties of `famous' combinatorial sequences. The present approach can be taught to a computer, and our beloved servant, Shalosh B. Ekhad, was able to generate many new theorems, for famous sequences, of course, but also for many obscure ones!Comment: 17 pages, accompanied by Maple package

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

In this paper, two methods for approximating the stabilizing solution of the Hamilton–Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton–Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.

Formulas for Continued Fractions. An Automated Guess and Prove Approach

We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial conditions. This is used to generate the first few coefficients and from there a conjectured formula. This formula is then proved automatically thanks to a linear recurrence satisfied by some remainder terms. Extensive experiments show that this simple approach and its straightforward generalization to difference and $q$-difference equations capture a large part of the formulas in the literature on continued fractions.Comment: Maple worksheet attache

Hierarchically Structured Reinforcement Learning for Topically Coherent Visual Story Generation

We propose a hierarchically structured reinforcement learning approach to address the challenges of planning for generating coherent multi-sentence stories for the visual storytelling task. Within our framework, the task of generating a story given a sequence of images is divided across a two-level hierarchical decoder. The high-level decoder constructs a plan by generating a semantic concept (i.e., topic) for each image in sequence. The low-level decoder generates a sentence for each image using a semantic compositional network, which effectively grounds the sentence generation conditioned on the topic. The two decoders are jointly trained end-to-end using reinforcement learning. We evaluate our model on the visual storytelling (VIST) dataset. Empirical results from both automatic and human evaluations demonstrate that the proposed hierarchically structured reinforced training achieves significantly better performance compared to a strong flat deep reinforcement learning baseline.Comment: Accepted to AAAI 201