We present a methodology for constructing weavings from 2-colorings of the plane. In particular, we consider tilings T of the plane by triangles and their corresponding triangle groups G. We derive 2-colorings of T using the index 2 subgroups of G.

The aim of this study was to analyze a representative sample of Philippine indigenous textiles in order to capture the range of symmetries and color symmetries present. This paper examines the existence of symmetries in finite designs, and classifies the plane-group and frieze-group symmetry types of the repeated patterns in woven textiles. The tendency of a particular symmetry to be more or less common than another can indicate relationships between the symmetries and the weaving technique or the culture that produced them. This paper will also examine designs and patterns with color symmetry found in these textiles. The sample consisted of 588 repeated patterns and finite designs in textiles (389 plane, 166 frieze and 33 finite) culled from well known museums in the Philippines, personal collections of scholars, existing literature on Philippine textiles and field visits.research papers

In this paper, we study extensions of trivial difference sets in dihedral groups. Such relative difference sets have parameters of the form (uλ, u, uλ, λ) or (uλ + 2, u, uλ + 1, λ) and are called semiregular or affine type, respectively. We show that there exists no nontrivial relative difference set of affine type in any dihedral group. We also show a connection between semiregular relative difference sets in dihedral groups and Menon-Hadamard difference sets.In the last section of the paper, we consider (m, u, k, λ) difference sets of general type in a dihedral group relative to a non-normal subgroup. In particular, we show that if a dihedral group contains such a difference set, then m is neither a prime power nor product of two distinct primes.

This work develops, in graph-theoretic terms, a methodology for systematically constructing weavings of overlapping nets derived from 2-colorings of the plane. From a 2-coloring, two disjoint simple, connected graphs called nets are constructed. The union of these nets forms an overlapping net, and a weaving map is defined on the intersection points of the overlapping net to form a weaving. Furthermore, a procedure is given for the construction of mixed overlapping nets and for deriving weavings from them.

Quick recovery of water services immediately after an earthquake is critical. This is to minimize hazards to environmental sanitation and consequent health problems caused by the lack of potable water supply. It is necessary therefore that water lifeline operators establish restoration strategies to deal with damage scenarios in their respective concession areas specifically during extreme seismic events. The recent 6.7 magnitude earthquake in Surigao City due to the movement of the Philippine Fault Zone: Surigao segment underscored this need. However due to the complexity of the network a systematic restoration sequence that minimizes restoration time and maximizes delivery of water service should be employed. In this research, the authors employed Horn's algorithm to determine the optimal restoration strategy of a pipeline network in Surigao City, Philippines. The repair sequence starts with the determination of a minimal spanning tree of the given pipeline network. The water source is designated as the root of this tree while the nodes represent the water demand at specific areas. The edges of the tree structure representing the pipelines connect the nodes. The assigned numeric value or weight of an edge (link) denotes the time to repair that specific pipeline. This value is a function of the length of the pipeline. The results show that an optimal job sequence may be carried out by considering maximal ratios of expanding family trees within the network. A least penalty function is a consequence of the optimal repair job sequence.