In this paper, we study deployment patterns to achieve full coverage and -connectivity under different ratios of the sensor communication range (denoted by ) to the sensing range (denoted by ) for homogeneous wireless sensor networks (WSNs). In particular, we propose new patterns for 3and 5-connectivity. We also discover that there exists a hexagon-based universally elemental pattern that can g...

Grid and block-based motion vector estimation techniques are proposed for motion tracking in video sequences. The grid technique is referred to the hexagon-diamond pattern. While, block-based technique is referred to 16 × 16 pixels of blocks in a single frame in video sequences. The hexagon and diamond pattern is applied onto the 16 × 16 pixels blocks in a single frame for motion tracking purpo...

A hexagon quadrangle system of order n and index [HQS (n)] is a pair (X,H), where X is a finite set of n vertices and H is a collection of edge disjoint hexagon quadrangles (called blocks) which partitions the edge set of Kn, with vertex set X. A hexagon quadrangle system is said to be a 4-nesting [N(4) − HQS] if the collection of all the 4-cycles contained in the hexagon quadrangles is a /2-fo...

A long-standing conjecture in combinatorics, made by Erdős and Simonovits, is that the maximum number of edges in an n-vertex graph without a hexagon is asymptotically 1 2n 4/3 as n→∞. This conjecture corresponds to the asymptotic optimality of constructions of generalized quadrangles as a source of dense hexagon-free graphs. In this paper, we construct a counterexample to this conjecture. For ...

PURPOSE This study aimed to evaluate stress distribution on peri-implant bone simulating the influence of platform switching in external and internal hexagon implants using three-dimensional finite element analysis. MATERIALS AND METHODS Four mathematical models of a central incisor supported by an implant were created: External Regular model (ER) with 5.0 mm × 11.5 mm external hexagon implan...

We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the dual graph of a quartered lozenge hexagon, and then use Lindström-GesselViennot methodology to find the number of tilings of a quartered lozenge hexagon.

The well-known dual pair of Napoleon equilateral triangles intrinsic to each triangle is extended to infinite sequences of them, shown to be special cases of infinite regular hexagon sequences on each triangle. A set of hexagon-to-hexagon transformations, the hex operators, is defined for this purpose, a set forming an abelian monoid under function composition. The sequences result from arbitra...

By investigating the construction of the split Cayley generalized hexagon, H(2), we get that there do not exist five distinct hexagon lines each a distance two apart from each other. From this we prove that the smallest Paige loop has a covering number of seven and that its automorphism group permutes these coverings transitively.