Narrow Results

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant.

Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2, 3, 4, or 4 (respectively) and conversely.

We are thus led to conjecture that for each prime p there exists a unique positive integer, m_p, with the following property. For any link L of non-null determinant and any modulus r such that p is the least prime divisor of the determinant of L and the modulus r, the minimum number of colors of L modulo r is m_p.

Due to the biocompatibility, mechanical strength and esthetics properties, titanium–porcelain prosthetic plays an important role in prosthodontics. However, weak bonding strength and considerable thickness of the porcelain restrict its application. Whether we can find a method to increase the bonding strength and reduce the thickness of the porcelain is an acute problem. In this study, ceramic coatings with similar color of nature teeth are fabricated on the surface of pure titanium by micro-arc oxidation (MAO). The colors, thickness and bonding strength of the coatings can be controlled by adding ferrous sulfate into the electrolytes. These new coatings can be used in titanium–porcelain prosthetic to substitute the opaque and dentin porcelain which can enhance the bonding strength and decrease the thickness of the porcelain compared with the conventional technique.

Diagnosing the quality of components in fault-tolerant computer systems often requires numerous tests with limited resources. It is usually the case that repeated tests on a selected, limited number of components are performed and the results are taken into account so as to infer a diagnostic property of the computer system as a whole. In this paper we abstract fault-tolerant testing as the following problem concerning the color of the majority in a set of colored balls. Given a set of balls each colored with one of two colors, the majority problem is to determine whether or not there is a majority in one of the two colors. In case there is such a majority, the aim is to output a ball of the majority color, otherwise to declare that there is no majority. We propose algorithms for solving the majority problem by repeatedly testing only k-tuple queries. Namely, successive answers of an oracle (which accepts as input only k-tuples) to a sequence of k-tuple queries are assembled so as to determine whether or not the majority problem has a solution. An issue is to design an algorithm which minimizes the number of k-tuple queries needed in order to solve the majority problem on any possible input of n balls. In this paper we consider three querying models: Output, Counting, and General, reflecting the amount and type of information provided by the oracle on each test for a k-tuple.