Discrete mechanics Based on Finite Element Methods

Discrete Mechanics based on finite element methods is presented in this paper. We also explore the relationship between this discrete mechanics and Veselov discrete mechanics. High order discretizations are constructed in terms of high order interpolations.Comment: 14 pages, 0 figure

Efficient Approximations for the Online Dispersion Problem

The dispersion problem has been widely studied in computational geometry and facility location, and is closely related to the packing problem. The goal is to locate n points (e.g., facilities or persons) in a k-dimensional polytope, so that they are far away from each other and from the boundary of the polytope. In many real-world scenarios however, the points arrive and depart at different times, and decisions must be made without knowing future events. Therefore we study, for the first time in the literature, the online dispersion problem in Euclidean space. There are two natural objectives when time is involved: the all-time worst-case (ATWC) problem tries to maximize the minimum distance that ever appears at any time; and the cumulative distance (CD) problem tries to maximize the integral of the minimum distance throughout the whole time interval. Interestingly, the online problems are highly non-trivial even on a segment. For cumulative distance, this remains the case even when the problem is time-dependent but offline, with all the arriving and departure times given in advance. For the online ATWC problem on a segment, we construct a deterministic polynomial-time algorithm which is (2ln2+epsilon)-competitive, where epsilon>0 can be arbitrarily small and the algorithm\u27s running time is polynomial in 1/epsilon. We show this algorithm is actually optimal. For the same problem in a square, we provide a 1.591-competitive algorithm and a 1.183 lower-bound. Furthermore, for arbitrary k-dimensional polytopes with k>=2, we provide a 2/(1-epsilon)-competitive algorithm and a 7/6 lower-bound. All our lower-bounds come from the structure of the online problems and hold even when computational complexity is not a concern. Interestingly, for the offline CD problem in arbitrary k-dimensional polytopes, we provide a polynomial-time black-box reduction to the online ATWC problem, and the resulting competitive ratio increases by a factor of at most 2. Our techniques also apply to online dispersion problems with different boundary conditions

Note On Certain Inequalities for Neuman Means

In this paper, we give the explicit formulas for the Neuman means $N_{AH}$, $N_{HA}$, $N_{AC}$ and $N_{CA}$, and present the best possible upper and lower bounds for theses means in terms of the combinations of harmonic mean $H$, arithmetic mean $A$ and contraharmonic mean $C$.Comment: 9 page

Electrosprayed PLGA smart containers for active anti-corrosion coating on magnesium alloy Amlite

A novel self-healing system, consisting of poly(lactic-co-glycolic) acid (PLGA) porous particles loaded with a corrosion inhibitor, i.e. benzotriazole (BTA), has been successfully achieved via direct electro-spray deposition and subsequent epoxy spraying upon magnesium (Mg) alloy AMlite. The two-step process greatly simplified the multi-step fabrication of smart coatings reported previously. The PLGA particles demonstrate rapid response to both water and pH increase incurred by corrosion of Mg, ensuring instant and ongoing release of BTA to self-heal the protective functionality and retard further corrosion. Furthermore, nanopores in the PLGA–BTA microparticles, formed by the fast evaporation of dichloromethane during the electrospray process, also contribute to the fast release of BTA. Using Mg alloy AMlite as a model substrate which requires corrosion protection, potentiodynamic polarisation characterisation and scratch testing were adopted to reveal the anti-corrosion capability of the active coating