Triangles capturing many lattice points

We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices $(0,0)$, $(x,0)$, and $(0,y)$ and fixed area, which one encloses the most lattice points from $\mathbb{Z}_{>0}^2$? Moreover, does its shape necessarily converge to the isosceles triangle $(x=y)$ as the area becomes large? Laugesen and Liu suggested that, in contrast to similar problems, there might not be a limiting shape. We prove that the limiting set is indeed nontrivial and contains infinitely many elements. We also show that there exist `bad' areas where no triangle is particularly good at capturing lattice points and show that there exists an infinite set of slopes $y/x$ such that any associated triangle captures more lattice points than any other fixed triangle for infinitely many (and arbitrarily large) areas; this set of slopes is a fractal subset of $[1/3, 3]$ and has Minkowski dimension at most $3/4$.Comment: 23 pages, 9 figure

The generation of noise by the fluctuations in gas temperature into a turbine

An actuator disc analysis is used to calculate the pressure fluctuations produced by the convection of temperature fluctuations (entropy waves) into one or more rows of blades. The perturbations in pressure and temperature must be small, but the mean flow deflection and acceleration are generally large. The calculations indicate that the small temperature fluctuations produced by combustion chambers are sufficient to produce large amounts of acoustic power. Although designed primarily to calculate the effect of entropy waves, the method is more general and is able to predict the pressure and vorticity waves generated by upstream or downstream going pressure waves or by vorticity waves impinging on blade rows

Infinite Excess Entropy Processes with Countable-State Generators

We present two examples of finite-alphabet, infinite excess entropy processes generated by invariant hidden Markov models (HMMs) with countable state sets. The first, simpler example is not ergodic, but the second is. It appears these are the first constructions of processes of this type. Previous examples of infinite excess entropy processes over finite alphabets admit only invariant HMM presentations with uncountable state sets.Comment: 13 pages, 3 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/ieepcsg.ht