152,657 research outputs found
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 , 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
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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
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