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The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2016 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.

Using a simple three-country model of international duopoly, this study analyzes the optimal choice of rules of origin (ROO) in a free trade area/agreement (FTA) when firms from outside the FTA must undertake foreign direct investment (FDI) in FTA countries and conduct part of their production process within the FTA to comply with the ROO. FDI causes spillovers of the superior production technology from a non-FTA firm to its competitor within the FTA, depending on how much of the production process is shifted to the FTA area. In this situation, this study predicts that as the degree of multilateral trade liberalization before formation of the FTA is higher, the optimal ROO tends to be less stringent.

For positive integers $m$ and $n$, we denote by $\mathrm{BH}(m,n)$ the set of all $H\in M_{n\times n}(\mathbb{C})$ such that $HH^\ast=nI_n$ and each entry of $H$ is an $m$-th root of unity where $H^\ast$ is the adjoint matrix of $H$ and $I_n$ is the identity matrix. For $H_1,H_2\in \mathrm{BH}(m,n)$ we say that $H_1$ is \textit{equivalent} to $H_2$ if $H_1=PH_2 Q$ for some monomial matrices $P, Q$ whose nonzero entries are $m$-th roots of unity. In this paper we classify $\mathrm{BH}(17,17)$ up to equivalence by computer search., Comment: 9 pages

This paper studies two‐dimensional cellular automata ca−90(m,n) having states 0 and 1 and working on a square lattice of size (m−1)×(n−1). All their dynamics, driven by the local transition rule 90, can be simply formulated by representing their configurations with Laurent polynomials over a finite field F2={0,1}. The initial configuration takes the next configuration to a particular configuration whose cells all have the state 1. This paper answers the question of whether the initial configuration lies on a limit cycle or not, and, if that is the case, some properties on period lengths of such limit cycles are studied.

We analyze the Hunter vs. Rabbit game on a graph, which is a model of communication in adhoc mobile networks. Let G be a cycle graph with N nodes. The hunter can move from a vertex to a vertex along an edge. The rabbit can jump from any vertex to any vertex on the graph. We formalize the game using the random walk framework. The strategy of the rabbit is formalized using a one dimensional random walk over . We classify strategies using the order O(k −β−1) of their Fourier transformation. We investigate lower bounds and upper bounds of the probability that the hunter catches the rabbit. We found a constant lower bound if β∈(0,1) which does not depend on the size N of the graph. We show the order is equivalent to O(1/logN) if β=1 and a lower bound is 1/N (β−1)/β if β∈(1,2]. These results help us to choose the parameter β of a rabbit strategy according to the size N of the given graph. We introduce a formalization of strategies using a random walk, theoretical estimation of bounds of a probability that the hunter catches the rabbit, and also show computing simulation results.