We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of a...

This paper presents a novel testbed capable of rotating a target to follow an arbitrary prescribed rotational trajectory, such as the tumbling motion of a satellite undergoing torque-free dynamics. This work is part of a larger research endeavor investigating technology applicable to autonomous rendezvous and docking with a tumbling satellite. This testbed provides unoccluded visibility of the ...

We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Introduction Consider a distribution on a manifold M, i.e. subbundle of the tangent bundle Π ⊂ TM . Non-holonomic Riemannian metric is a Riemannian metric g ∈ SΠ on this bundle. We call the pair (Π, g) sub-Riemannian structure. A curve γ : [0, 1] → M is called horizontal if γ̇ is a sect...

This paper presents a path planning algorithm for autonomous navigation of non-holonomic mobile robots in complex environments. The irregular contour of obstacles is represented by segments. The goal of the robot is to move towards a known target while avoiding obstacles. The velocity constraints, robot kinematic model and non-holonomic constraint are considered in the problem. The optimal path...

Non-holonomic vehicles are of immense practical value and increasingly subject to automation. However, controlling them accurately, e.g., when parking, is known be challenging for automatic control methods, including model predictive (MPC). Combining results from MPC theory sub-Riemannian geometry in the form homogeneous nilpotent system approximations, this paper proposes a comprehensive, read...

In order to find a general framework for different combinatorial identities and identities that involve special functions, Bernshtein introduced in [1] a notion of holonomic functions. Let us recall this definition (in formulating it, we will follow [10]). A discrete holonomic function f(n) of one variable is defined as a solution of a homogeneous, linear difference equation with polynomial coe...

We apply the holonomic gradient method (HGM) introduced by [9] to the calculation of orthant probabilities of multivariate normal distribution. The holonomic gradient method applied to orthant probabilities is found to be a variant of Plackett’s recurrence relation ([14]). However an implementation of the method yields recurrence relations more suitable for numerical computation than Plackett’s...

In this article we present new results for families of orthogonal polynomials and special functions, that are determined by algorithmical approaches. In the first section, we present new results, especially for discrete families of orthogonal polynomials, obtained by an application of the celebrated Zeilberger algorithm. Next, we present algorithms for holonomic families f(n, x) of special func...