Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Let $m,n\ge 2$, $m\le n$. It is well-known that the number of (two-dimensional) threshold functions on an $m\times n$ rectangular grid is {eqnarray*} t(m,n)=\frac{6}{\pi^2}(mn)^2+O(m^2n\log{n})+O(mn^2\log{\log{n}})= \frac{6}{\pi^2}(mn)^2+O(mn^2\log{m}). {eqnarray*} We improve the error term by showing that $$ t(m,n)=\frac{6}{\pi^2}(mn)^2+O(mn^2). $

Solving challenging grid puzzles with answer set programming

We study four challenging grid puzzles, Nurikabe, Heyawake, Masyu, Bag Puzzle, interesting for answer set programming (ASP) from the viewpoints of representation and computation: they show expressivity of ASP, they are good examples of a representation methodology, and they form a useful suite of benchmarks for evaluating/improving computational methods for nontight programs

Analysis of an air-spaced patch antenna near 1800 MHz

Microstrip antennas are a type of printed antenna which consists of a patch on top of a grounded substrate. A major limitation for the performance of the patch antenna is the dielectric substrate. The idea of using air as dielectric was therefore considered to overcome that limitation because air has the lowest permittivity and no loss. The goal of this work is to build an air-spaced patch antenna, with the minimum resonant frequency at 1800 MHz and with a return loss of at least 10 dB. This work is novel because the air-spaced patch antenna has not been extensively studied. Existing literature on patch antennas with dielectric were used for the design of the antenna (dimensions of the patch, ground plane and height) and to understand the principles of operation of microstrip patch antennas in general. Simulations using the NEC code and experiments in the RF laboratory were used for this air-spaced patch antenna study. The Numerical Electromagnetic Code (NEC) was used as the simulation tool in this work. The air-spaced patch antenna was simulated to find a trend for the variation of the return loss and impedance with the resonant frequency. Simulation also helped determine cases that will not be meaningful to explore in the experiment. The experiment was done in the RF laboratory of Marquette University College of Engineering. Two procedures were used to calculate the patch dimensions using two different sources ([2], [3]). They lead to two patch antennas that were tested. For each antenna, the height of the dielectric substrate and the recess feed distance were varied. Antenna 2 (procedure 2 – [3]) provided the best results with a resonant frequency of 1800 MHz and a return loss of 21 dB. It was found that the error between experimental and simulation resonant frequency is generally 5% or less. This error increases as the dielectric height increases, and as the recess distance increases. Simulation results roughly follow the experimental results trend

On FPL configurations with four sets of nested arches

The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a,b,c,d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a formula as a sum of determinants. This is made quite explicit when min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates the numbers of configurations with b=d.Comment: 22 pages, TeX, 16 figures; a new formula for a generating function adde

Asymptotics for numbers of line segments and lines in a square grid

We present an asymptotic formula for the number of line segments connecting q+1 points of an nxn square grid, and a sharper formula, assuming the Riemann hypothesis. We also present asymptotic formulas for the number of lines through at least q points and, respectively, through exactly q points of the grid. The well-known case q=2 is so generalized

Shape of Cosmic String Loops

Complicated cosmic string loops will fragment until they reach simple, non-intersecting ("stable") configurations. Through extensive numerical study we characterize these attractor loop shapes including their length, velocity, kink, and cusp distributions. We find that an initial loop containing M harmonic modes will, on average, split into 3M stable loops. These stable loops are approximately described by the degenerate kinky loop, which is planar and rectangular, independently of the number of modes on the initial loop. This is confirmed by an analytic construction of a stable family of perturbed degenerate kinky loops. The average stable loop is also found to have a 40% chance of containing a cusp. We examine the properties of stable loops of different lengths and find only slight variation. Finally we develop a new analytic scheme to explicitly solve the string constraint equations.Comment: 11 pages, 19 figures. See http://www.phys.cwru.edu/projects/strings/ for more information, movies, code, etc. Minor clarification suggested by referee. Accepted for publication in Phys. Rev.