Construction Through Decomposition: A Linear Time Algorithm for the N-queens Problem

Configuring N mutually non-attacking queens on an N-by-N chessboard is a contemporary problem that was first posed over a century ago. Over the past few decades, this problem has become important to computer scientists by serving as the standard example of backtracking search methods. A related problem, placing the N queens on a toroidal board, has been discussed in detail by Polya and Chandra, Their work focused on characterizing the solvable cases and finding regular solutions, board setups that solve the problem while arranging the queens in a regular pattern. This paper describes a new linear time divide-and-conquer algorithm that solves both problems. When applied to the toroidal problem, the algorithm yields a family of non-regular solutions based on number factorization. These solutions, in turn, can be modified to solve the standard N-queens problem for board sizes which are unsolvable on a torus

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. We also prove applications of these results to dynamical billiards, mathematical physics and number theory. In the space of affine lattices $ASL_2(\mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum $\mathcal{H}(1,1)$ of translation surfaces. For these curves (and more in general curves which are well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff genericity) we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, which was recently explored by Dragovic and Radnovic, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. This generalizes a phenomenon recently discovered by Fraczek and Schmoll which could so far only be proved for random periodic configurations. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers, which extends previous work by Elkies and McMullen, is also obtained.Comment: To appear in Journal of Modern Dynamic

A Surprisingly Non-attractiveness of Commercial Poison Baits to Newly Established Population of White-Footed Ant, Technomyrmex brunneus (Hymenoptera: Formicidae), in a Remote Island of Japan

The white-footed ant, Technomyrmex brunneus, was newly introduced and established in a remote island of Japan and has caused unacceptable damage to the daily life of residents. To establish proper control measures, the present study investigated whether T. brunneus is effectively attracted to commercially available poison baits used to exterminate common household pest ants and the Argentine ant in Japan. Cafeteria experiments using three types of nontoxic baits and eight types of commercial poison baits for ants were conducted in the field, and the attractiveness was compared among the baits. The liquid poison bait “Arimetsu,” which consists of 42.6% water, 55.4% sugar, and 2.0% borate, and nontoxic 10% (w/v) sucrose water showed the highest attractiveness. On the other hand, other commercial poison baits were not as attractive. Therefore, sucrose liquid is the most effective attractive component to use in poison baits for T. brunneus

Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry

We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalisation of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries with constant flag curvature always map to conformally flat spacetimes and that the Finsler condition maps to either a causality condition or it breaks down at an ergo-surface in the spacetime picture. The gauge equivalence in this network of relations is considered as well as the connection to analogue models and the viewpoint of magnetic flows. We provide a variety of examples.Comment: 37 pages, 6 figure