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

Effects of different land use on soil hydraulic properties

Non-Peer ReviewedAn understanding of hydraulic properties of surface soils is needed for sound soil management because it determines the partition of rainfall and snow melt into runoff or soil water storage. The purpose of this study was to evaluate the effects of three land uses (native grassland, brome grassland and cultivated land) on soil physical properties. For each land use, water infiltration rates were measured with a tension infiltrometer at 3, 7, 15, and 22 cm water tensions. Total porosity and macro porosity were determined and hydraulic properties were estimated. The highest total porosity and macro porosity were observed in native grassland while cultivation had significantly reduced macro porosity. At 15 and 22 cm tensions native grassland had significantly lower infiltration rates than cultivated fields while brome grassland had intermediate infiltration rates. Surface soil hydraulic properties differ markedly among land uses, with grasslands having higher saturated hydraulic conductivity. The observations indicate that cultivated soils have lower macroporosity than grassland soils and, therefore, lower infiltration of rain and increased potential for runoff. In contrast, cultivated soils are able to absorb more rainfall and snow melt under unsaturated conditions. Land use changes may alter the water balance of the area by changing the amount of surface runoff and therefore, any changes in existing land use must be done cautiously

Arboreal birds do not avoid scattered trees in West Africa

Arboreal birds tend to remain in woody vegetation and avoid crossing open areas. Therefore, few tree-dwelling birds are to be expected in scattered trees. We tested this expectation with field data collected in the deserts, savannas and open agricultural parklands of West Africa where woody cover in 1,327 stratified random study sites varied between 0.2 and 29%. We found no evidence that scattered trees were avoided. Instead, bird density in trees was independent of trees occurring clumped or singly. The presence of birds in an individual tree was related to tree species and tree-related variables, but not to woody cover or species composition of the surrounding woody vegetation. We hypothesise that scattered trees are not avoided because (1) travel time between trees is too short to have a negative impact on foraging time, (2) predation risk of arboreal passerines is very low (bird-hunting raptors are scarce in the deserts and savannas of West Africa and mostly prey on ground-feeding, not arboreal, birds), and (3) the probability of being chased away by other arboreal birds is less when trees are more scattered. Scattered trees are ecologically important since hundreds of millions of migratory woodland birds, of which several species are in decline, spend the northern winter in Africa in open, often human-modified, landscapes where trees are well spaced

The chemical potential of the electron gas on a one dimensional lattice

The chemical potential of the electron gas on a one-dimensional lattice is determined within the discrete Hubbard model. The result will have applications in studies of transport properties of quasi one-dimensional organic conductors such as the Bechgaard salts.Comment: 4 pages,plain TeX,presented at the 9 National Congress of Yugoslav Physicists,held in May 1995.,and published in the proceedings.The author can be contacted at: [email protected]

Integrable and superintegrable systems associated with multi-sums of products

We construct and study certain Liouville integrable, superintegrable, and non-commutative integrable systems, which are associated with multi-sums of products.Comment: 26 pages, submitted to Proceedings of the royal society