‎The distinguishing number (resp. index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$‎ ‎such that $G$ has an vertex labeling (resp. edge labeling) with $d$ labels that is preserved only by a trivial‎ ‎automorphism‎. ‎For any $n in mathbb{N}$‎, ‎the $n$-subdivision of $G$ is a simple graph $G^{frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$...

Both the four-point and the uniform cubic B-spline refinement (i.e. subdivision) schemes double the number of vertices of a closed-loop polygonal curve P and respectively produce sequences of vertices fk and bk. The Js refinement proposed here produces vertices vk=(1–s)fk+sbk. Iterative applications of Js yield a family of curves parameterized by s. It includes the four-point curve (J0), the un...

A set S of vertices of a graphG = (V,E) is a dominating set if every vertex of V (G)\S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Velammal ...

Both the four-point and the uniform cubic B-spline refinement (i.e. subdivision) schemes double the number of vertices of a closed-loop polygonal curve P and produce sequences of vertices fj and bj respectively. We proposed to analyze a subdivision scheme Js that blends the rules of these two refinement methods to produce vertices of the form vj=(1–s)fj+sbj. Iterative applications of Js yield a...

In a seminal paper [5], Jos Stam gave a method for evaluating Catmull-Clark subdivision surfaces [1] at parameter values near an interior extraordinary vertex (EV). The basic idea is to subdivide recursively until the (u, v) parameter to be evaluated is contained in a regular 4×4 grid of control points which define a bicubic B-spline patch. The subdivision steps can be computed very efficiently...

The practice of tablet splitting as a way to reduce prescription medication costs has become increasingly prevalent. The United States Pharmacopeial Convention (USPC) has no standards for the subdivision characteristics of scored tablets. Literature results show that many tablets on the US market exhibit unacceptable subdivision characteristics. The European Pharmacopoeia (EP) provides requirem...

It is observed that the Powell-Sabin 12-split triangle is reen-able since the same split of the 4 similar subtriangles of a triangle contains the lines of split of the original triangle. This property of the split is the key to the existence of a subdivision scheme, for the evaluation of the C 1 quadratic spline on the split which interpolates function and gradient values at the 3 vertices of t...

In recent years, several matrix-valued subdivisions have been proposed for triangular meshes. The matrix-valued subdivisions can simulate and extend the traditional scalar-valued subdivision, such as loop and 3 subdivision. In this paper, we study how to construct the matrix-valued subdivision wavelets, and propose the novel biorthogonal wavelet based on matrix-valued subdivisions on multiresol...

The objective of this study was to evaluate the dentoskeletal changes consequent to orthodontic treatment in subjects with Class II subdivision malocclusions, treated with asymmetric extractions, compared with a normal-occlusion control group. The sample consisted of 3 groups, with 30 subjects in each: normal-occlusion subjects (group 1), untreated Class II subdivision subjects (group 2), and C...