Nobuaki Mutoh

Enumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry

We describe computer algorithms that can enumerate and display, for a given n > 0 (in theory, of any size), all n -ominoes, n -iamonds, and n -hexes that can tile the plane using only rotations; these sets necessarily contain all such tiles that are fundamental domains for p4, p3, and p6 isohedral tilings. We display the outputs for small values of n . This expands on earlier work [3].

The Polyhedra of Maximal Volume Inscribed in the Unit Sphere and of Minimal Volume Circumscribed about the Unit Sphere

In this paper, we consider two classes of polyhedra. One is the class of polyhedra of maximal volume with n vertices that are inscribed in the unit sphere of R 3 . The other class is polyhedra of minimal volume with n vertices that are circumscribed about the unit sphere of R 3 . We construct such polyhedra for n up to 30 by a computer aided search and discuss some of their properties.

Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry

Mathematics Department PPHAC Moravian College, 1200 Main Street,Bethlehem, 18018-6650 PA, USA; E-Mail: schattdo@moravian.edu* Author to whom correspondence should be addressed; E-Mail: fukuda@kitasato-u.ac.jp;Tel.: +81-42-778-8046; Fax: +81-42-778-8268.Received: 4 August 2011; in revised form: 29 November 2011 / Accepted: 2 December 2011 /Published: 12 December 2011Abstract: We describe computer algorithms that produce the complete set of isohedraltilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and thetilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilingsare of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with p3m1, p4m,or p6m symmetry groups that have polyominoes or polyiamonds as fundamental domains.We display the algorithms’ output and give enumeration tables for small values of n. Thisexpands earlier works [1,2] and is a companion to [3].Keywords: polyominoes; polyiamonds; isohedral tilings; two-dimensional symmetrygroups; fundamental domainsClassification: MSC 52C20, 05B50, 68U05

Maximin distance for n points in a unit square or a unit circle

Given n points inside a unit square (circle), let d n (c n ) denote the maximum value of the minimum distance between any two of the n points. The problem of determining d n (c n ) and identifying the configuration of that yields d n (c n ) has been investigated using geometric methods and computer-aided methods in a number of papers. We investigate the problem using a computer-aided search and arrive at some approximations which improve on earlier results for n=59, 73 and 108 for the unit square, and also for n=70, 73, 75 and 77, ⋯ , 80 for the unit circle. The associated configurations are identified for all the above-mentioned improved results.

Uniform coverings of 2-paths in the complete bipartite directed graph

Let G be a directed graph and H be a subgraph of G. A D(G, H, λ ) design is a multiset 𝒟 of subgraphs of G each isomorphic to H so that every directed 2-path in G lies in exactly λ subgraphs in 𝒟. In this talk, we show that there exists a D(Kn,n*,C→2n,1) design for every n≥2, where Kn,n* is the complete bipartite directed graph and C→2n is a directed Hamilton cycle in Kn,n*.

Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D 2

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have pmm, pmg, pgg or cmm symmetry [1]. These symmetry groups are members of the crystal class D2 among the 17 two-dimensional symmetry groups [2]. We display the algorithms’ output and give enumeration tables for small values of n. This work is a continuation of our earlier works for the symmetry groups p3, p31m, p3m1, p4, p4g, p4m, p6, and p6m [3–5].

Improving Search Efficiency of Incremental Variable Selection by Using Second-Order Optimal Criterion

We address the problem of improving search efficiency of incremental variable selection. As one application, we focus on generalized linear models that are linear with respect to their parameters, but their objective functions are not restricted to a standard sum of squared error. In this paper, we present a method for incrementally selecting a set of relevant variables together with a newly proposing criterion based on second-order optimality for our models. In our experiments using a synthetic dataset with tens of thousands of variables, we show that the proposed method was able to completely restore the relevant variables. Moreover, the method substantially improved the search efficiency in comparison to a conventional calculation method. Furthermore, it is shown that we obtained promissing initial results using a real dataset in health-checkup.

Visibility of Disks on the Lattice Points

It is known that the probability with which a lattice point is visible from the origin is 1/ζ(2) = π 2/6. In this paper, instead of points, we place circular disks with the constant radius d on every lattice point, and calculate the longest distance to the farthest visible disk from the origin and the probability with which a disk is visible by using a computer.