Related papers: Generalizations of the Hanoi Towers Problem

The purpose of this paper is to show that: when a single particle moving under 3-proper time (three-dimensional time), the trajectories of a classical particle are equivalent to a quantum field with spin. Three-proper time models are built…

For many fundamental problems in computational topology, such as unknot recognition and $3$-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms…

The main theme of the article is the study of discrete systems of material points subjected to constraints not only of a geometric type (holonomic constraints) but also of a kinematic type (nonholonomic constraints). The setting up of the…

"Dominos" are special entities consisting of a hard dimer-like kernel surrounded by a soft hull and governed by local interactions. "Soft hull" and "hard kernel" mean that the hulls can overlap while the kernel acts under a repulsive…

We prove the existence of a number of smooth periodic motions $u_*$ of the classical Newtonian $N$-body problem which, up to a relabeling of the $N$ particles, are invariant under the rotation group ${\cal R}$ of one of the five Platonic…

We consider discretization of the 'geometric cover problem' in the plane: Given a set $P$ of $n$ points in the plane and a compact planar object $T_0$, find a minimum cardinality collection of planar translates of $T_0$ such that the union…

In this paper we first extend the diminishing stepsize method for nonconvex constrained problems presented in [4] to deal with equality constraints and a nonsmooth objective function of composite type. We then consider the particular case…

A theoretical framework is established for explicitly calculating rigid kernels of self-similar regular branch groups. This is applied to a new infinite family of branch groups in order to provide the first examples of self-similar, branch…

Three-body and n-body problems in celestial mechanics are age-old and challenging puzzles. In recent years, several breakthroughs are made in finding periodic orbits for three-body problem. And Bohua Sun proposed a conjecture on Kepler's…

The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length $n$ with displacement one is in…

We present a novel, general, optimally fast, incremental way of searching for a universal algorithm that solves each task in a sequence of tasks. The Optimal Ordered Problem Solver (OOPS) continually organizes and exploits previously found…

Some classical graph problems such as finding minimal spanning tree, shortest path or maximal flow can be done efficiently. We describe slight variations of such problems which are shown to be NP-complete. Our proofs use straightforward…

We study the complexity of a class of problems involving satisfying constraints which remain the same under translations in one or more spatial directions. In this paper, we show hardness of a classical tiling problem on an N x N…

We present a homotopic approach to solving challenging, optimization-based motion planning problems. The approach uses Homotopy Optimization, which, unlike standard continuation methods for solving homotopy problems, solves a sequence of…

Although rare, collisions of two or more bodies in the N-body problem are apparent obstacles at which Newton's Law of Gravity ceases to make sense. Without understanding the nature of collisions, a complete understanding of the N-body…

The three body problem is a special case of the n body problem where one takes the initial positions and velocities of three point masses and attempts to predict their motion over time according to Newtonian laws of motion and universal…

I discuss the physical basis of classical mechanics, such as expressed commonly using the framework of Newton's Principia. Newton's formulation of the laws of motion is seen to have quite a few ambiguities and shortcomings. Therefore I…

The pebble motion problem (PMP) asks whether one configuration of labeled pebbles on a graph can be transformed into another by moving pebbles to adjacent unoccupied vertices. It is a fundamental model of graph reconfiguration and is…

We call a continuous path of polygons decreasing if the convex hulls of the polygons form a decreasing family of sets. For an arbitrary polygon of more than three vertices, we characterize the polygons contained in it that can be reached by…

We consider three problems of selecting optimal gun barrel direction (or those of selecting optimal semiaxis position) when firing an unguided artillery projectile on the assumption that the gun barrel semiaxis can move in a connected…