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

Comparisons and Applications of Four Independent Numerical Approaches for Linear Gyrokinetic Drift Modes

To help reveal the complete picture of linear kinetic drift modes, four independent numerical approaches, based on integral equation, Euler initial value simulation, Euler matrix eigenvalue solution and Lagrangian particle simulation, respectively, are used to solve the linear gyrokinetic electrostatic drift modes equation in Z-pinch with slab simplification and in tokamak with ballooning space coordinate. We identify that these approaches can yield the same solution with the difference smaller than 1\%, and the discrepancies mainly come from the numerical convergence, which is the first detailed benchmark of four independent numerical approaches for gyrokinetic linear drift modes. Using these approaches, we find that the entropy mode and interchange mode are on the same branch in Z-pinch, and the entropy mode can have both electron and ion branches. And, at strong gradient, more than one eigenstate of the ion temperature gradient mode (ITG) can be unstable and the most unstable one can be on non-ground eigenstates. The propagation of ITGs from ion to electron diamagnetic direction at strong gradient is also observed, which implies that the propagation direction is not a decisive criterion for the experimental diagnosis of turbulent mode at the edge plasmas.Comment: 12 pages, 10 figures, accept by Physics of Plasma

Experimental investigation of piloted flameholders

Four configurations of piloted flameholders were tested. The range of flame stabilization, flame propagation, pressure oscillation during ignition, and pressure drop of the configurations were determined. Some tests showed a very strong effect of inlet flow velocity profile and flameholder geometry on flame stabilization. These tests led to the following conclusions. (1) The use of a piloted flameholder in the turbofan augmentor may minimize the peak pressure rise during ignition. At the present experimental conditions, delta P/P asterisk over 2 is less than 10 percent; therefore, the use of a piloted flameholder is a good method to realize soft ignition. (2) The geometry of the piloted flameholder and the amount of fuel injected into the flameholder have a strong effect on the pressure oscillation during ignition of the fuel-air mixture in the secondary zone. (3) Compared with the V-gutter flameholder with holes in its wall, the V-gutter flameholder without holes not only has advantages such as simple structure and good rigidity but offers a wide combustion stability limit and a high capability of igniting the fuel-air mixture of the secondary zone