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This article builds on Thurston's height functions. His tiling algorithm is reinterpreted using lattice theory and then generalized in order to generate any tiling of a hole-free region. Combined with a natural encoding of tilings by words,…
In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups…
Consider two circles, externally tangential,and with integer radii R1, R2; and with R1>R2.The two circles have three tangent lines in common, one of them being T1T2. If M is the midpoint of T1T2, and K the point of intersection of the lines…
A novel code construction algorithm is presented to find all the possible code families for code reconfiguration in an OCDMA system. The algorithm is developed through searching all the complete subgraphs of a constructed graph. The…
Here we examine some Erdos-Falconer-type problems in vector spaces over finite fields involving right angles. Our main goals are to show that a) a subset A of F_q^d of size >> q^[(d+2)/3] contains three points which generate a right angle,…
One approach to achieving correct finite element assembly is to ensure that the local orientation of facets relative to each cell in the mesh is consistent with the global orientation of that facet. Rognes et al. have shown how to achieve…
We investigate the problem of partitioning a rectilinear polygon $P$ with $n$ vertices and no holes % with no holes into rectangles using disjoint line segments drawn inside $P$ under two optimality criteria. In the minimum ink partition,…
In this paper, we propose a fast and convergent algorithm to solve unassigned distance geometry problems (uDGP). Technically, we construct a novel quadratic measurement model by leveraging $\ell_0$-norm instead of $\ell_1$-norm in the…
A general construction yielding infinitely many families of $D(m^2)$-triples of triangular numbers is presented. Moreover, each triple obtained from this construction contains the same triangular number $T_n$.
Counting the number of triangles in a graph has many important applications in network analysis. Several frequently computed metrics like the clustering coefficient and the transitivity ratio need to count the number of triangles in the…
In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner's questions have been generalized and some of them are already solved. In the…
A dual approach to defining the triangle sequence (a type of multidimensional continued fraction algorithm, initially developed in NT/9906016) for a pair of real numbers is presented, providing a new, clean geometric interpretation of the…
We consider the problem of finding minimum-link rectilinear paths in rectilinear polygonal domains in the plane. A path or a polygon is rectilinear if all its edges are axis-parallel. Given a set $\mathcal{P}$ of $h$ pairwise-disjoint…
The well-known three distance theorem states that there are at most three distinct gaps between consecutive elements in the set of the first n multiples of any real number. We generalise this theorem to higher dimensions under a suitable…
This paper proposes an algorithm for clipping line segment against an axis-aligned rectangular window. The conventional algorithms for line segment clipping treat the clipping boundary and/or the line segment to be clipped as line. The…
A basic and an improved ear clipping based algorithm for triangulating simple polygons and polygons with holes are presented. In the basic version, the ear with smallest interior angle is always selected to be cut in order to create fewer…
A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph $G$ in a metric space $X$ is a drawing of $G$ in $X$ in which every…
The existence problem for maximal sets of equiangular lines (or SICs) in complex Hilbert space of dimension $d$ remains largely open. In a previous publication (arXiv:2112.05552) we gave a conjectural algorithm for how to construct a SIC if…
This manuscript introduces Diophantine labeling, a new way of labeling of the vertices for finite simple undirected graphs with some divisibility condition on the edges. Maximal graphs admitting Diophantine labeling are investigated and…
A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary…