Melham's Conjecture on Odd Power Sums of Fibonacci Numbers

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at $1$, and will be an integer polynomial after multiplying it by a product of the first consecutive Lucas numbers of odd order. This presents an affirmative answer to a conjecture of Melham.Comment: 15page

Deposition of Ti/TiC Composite Coatings on Implant Structures Using Laser Engineered Net Shaping

A new method of depositing hard and wear resistant composite coatings on metal-onmetal bearing surfaces of titanium implant structures is proposed and demonstrated. The method consists of depositing a Ti/TiC composite coating (~ 2.5 mm thick) on titanium implant bearing surfaces using Laser Engineered Net Shaping (LENS®). Defect-free composite coatings were successfully produced at various amounts of the reinforcing TiC phase with excellent interfacial characteristics using a mixture of commercially pure Ti and TiC powders. The coatings consisted of a mixture of coarser unmelted/partially melted (UMC) TiC particles and finer, discreet resolidified (RSC) TiC particles uniformly distributed in the titanium matrix. The amounts of UMC and RSC were found to increase with increasing TiC content of the original powder mixture. The coatings exhibited a high level of hardness, which increased with increasing TiC content of the original powder mixture. Fractographic studies indicated that the coatings, even at 60 vol.% TiC, do not fail in a brittle manner. Various aspects of LENS® deposition of Ti/TiC composite coatings are addressed and a preliminary understanding of structure-property-fracture correlations is presented. The current work shows that the proposed approach to deposit composite coatings using laser-based metal deposition processes is highly-effective, which can be readily utilized on a commercial basis for manufacture of high-performance implants.Mechanical Engineerin

Division and the Giambelli Identity

Given two polynomials f(x) and g(x), we extend the formula expressing the remainder in terms of the roots of these two polynomials to the case where f(x) is a Laurent polynomial. This allows us to give new expressions of a Schur function, which generalize the Giambelli identity.Comment: 9 pages, 1 figur