The staircase method: integrals for periodic reductions of integrable lattice equations

We show, in full generality, that the staircase method provides integrals for mappings, and correspondences, obtained as traveling wave reductions of (systems of) integrable partial difference equations. We apply the staircase method to a variety of equations, including the Korteweg-De Vries equation, the five-point Bruschi-Calogero-Droghei equation, the QD-algorithm, and the Boussinesq system. We show that, in all these cases, if the staircase method provides r integrals for an n-dimensional mapping, with 2r<n, then one can introduce q<= 2r variables, which reduce the dimension of the mapping from n to q. These dimension-reducing variables are obtained as joint invariants of k-symmetries of the mappings. Our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable. We also study reductions on other quad-graphs than the regular 2D lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.Comment: 40 pages, 23 Figure

Involutivity of integrals for sine-Gordon, modified KdV and potential KdV maps

Closed form expressions in terms of multi-sums of products have been given in \cite{Tranclosedform, KRQ} of integrals of sine-Gordon, modified Korteweg-de Vries and potential Korteweg-de Vries maps obtained as so-called $(p,-1)$-traveling wave reductions of the corresponding partial difference equations. We prove the involutivity of these integrals with respect to recently found symplectic structures for those maps. The proof is based on explicit formulae for the Poisson brackets between multi-sums of products.Comment: 24 page

Flotation tests on a table sludge from the Golden Rod Milling Company at Tar River, Oklahoma

The purpose of this experimental work is to give to the operators of the Joplin District some facts concerning the possibilities of flotation in their district. An attempt will be made to show that high-grade concentrates can be made economically; also that the usual gravity concentration. It is believed that the oil and reagent used will not put a prohibitive cost burden upon the operators. It is, of course, realized that flotation concentrates of any zinc tenor bear a penalty because of the fineness of the product --Introduction, page 3

Are Giant Planets Forming Around HR 4796A?

We have obtained FUSE and HST STIS spectra of HR 4796A, a nearby 8 Myr old main sequence star that possesses a dusty circumstellar disk whose inclination has been constrained from high resolution near-infrared observations to be ~17 deg from edge-on. We searched for circumstellar absorption in the ground states of C II at 1036.3 A, O I at 1039.2 A, Zn II at 2026.1 A, Lyman series H2, and CO (A-X) and failed to detect any of these species. We place upper limits on the column densities and infer upper limits on the gas masses assuming that the gas is in hydrostatic equilibrium, is well-mixed, and has a temperature, Tgas ~ 65 K. Our measurements suggest that this system possesses very little molecular gas. Therefore, we infer an upper limit for the gas:dust ratio (<4.0) assuming that the gas is atomic. We measure less gas in this system than is required to form the envelope of Jupiter.Comment: 10 pages, 3 figures (including 1 color figure), accepted for publication in Ap