A list of all integrable 2D homogeneous polynomial potentials with a polynomial integral of order at most 4 in the momenta

We searched integrable 2D homogeneous polynomial potential with a polynomial first integral by using the so-called direct method of searching for first integrals. We proved that there exist no polynomial first integrals which are genuinely cubic or quartic in the momenta if the degree of homogeneous polynomial potentials is greater than 4.Comment: 22 pages, no figures, to appear in J. Phys. A: Math. Ge

Integrable systems without the Painlev\'e property

We examine whether the Painlev\'e property is a necessary condition for the integrability of nonlinear ordinary differential equations. We show that for a large class of linearisable systems this is not the case. In the discrete domain, we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non-Painlev\'e continuous linearisable systems. We find that while these discrete systems are themselves linearisable, they possess nonconfined singularities

Sampling a Littoral Fish Assemblage: Comparison of Small-Mesh Fyke Netting and Boat Electrofishing

We compared small-mesh (4-mm) fyke netting and boat electrofishing for sampling a littoral fish assemblage in Muskegon Lake, Michigan. We hypothesized that fyke netting selects for small-bodied fishes and electrofishing selects for large-bodied fishes. Three sites were sampled during May (2004 and 2005), July (2005 only), and September (2004 and 2005). We found that the species composition of captured fish differed considerably between fyke netting and electrofishing based on nonmetric multidimensional scaling (NMDS). Species strongly associated with fyke netting (based on NMDS and relative abundance) included the brook silverside Labidesthes sicculus, banded killifish Fundulus diaphanus, round goby Neogobius melanostomus, mimic shiner Notropis volucellus, and bluntnose minnow Pimephales notatus, whereas species associated with electrofishing included the Chinook salmon Oncorhynchus tshawytscha, catostomids (Moxostoma spp. and Catostomus spp.), freshwater drum Aplodinotus grunniens, walleye Sander vitreus, gizzard shad Dorosoma cepedianum, and common carp Cyprinus carpio. The total length of fish captured by electrofishing was 12.8 cm (95% confidence interval ¼ 5.5– 17.2 cm) greater than that of fish captured by fyke netting. Size selectivity of the gears contributed to differences in species composition of the fish captured, supporting our initial hypothesis. Thus, small-mesh fyke nets and boat electrofishers provided complementary information on a littoral fish assemblage. Our results support use of multiple gear types in monitoring and research surveys of fish assemblages. Copyright by the American Fisheries Society 2007, Originally published in the North American Journal of Fisheries Management 27: 825-831, 2007

The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies

The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This symmetry enables formulae normally given implicitly in terms of residues, such as conserved charges and fluxes, to be calculated explicitly. As a corollary of these results the equivalence of the Benney and Toda hierarchies is established. It is further shown that such quantities may be expressed in terms of generalized hypergeometric functions, the simplest example involving Legendre polynomials. These results are then extended to systems derived from a rational Lax function and a logarithmic function. Various reductions are also studied.Comment: 29 pages, LaTe

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Backlund transformations for the equations considered. We are also able to derive Backlund transformations onto other ODEs in the Painleve classification.Comment: To appear in Nonlinearity (22 pages

On the physical meaning of Fermi coordinates

(Some Latex problems should be removed in this version) Fermi coordinates (FC) are supposed to be the natural extension of Cartesian coordinates for an arbitrary moving observer in curved space-time. Since their construction cannot be done on the whole space and even not in the whole past of the observer we examine which construction principles are responsible for this effect and how they may be modified. One proposal for a modification is made and applied to the observer with constant acceleration in the two and four dimensional Minkowski space. The two dimensional case has some surprising similarities to Kruskal space which generalize those found by Rindler for the outer region of Kruskal space and the Rindler wedge. In perturbational approaches the modification leads also to different predictions for certain physical systems. As an example we consider atomic interferometry and derive the deviation of the acceleration-induced phase shift from the standard result in Fermi coordinates.Comment: 11 pages, KONS-RGKU-94/02 (Latex

Structural Properties of Planar Graphs of Urban Street Patterns

Recent theoretical and empirical studies have focused on the structural properties of complex relational networks in social, biological and technological systems. Here we study the basic properties of twenty 1-square-mile samples of street patterns of different world cities. Samples are represented by spatial (planar) graphs, i.e. valued graphs defined by metric rather than topologic distance and where street intersections are turned into nodes and streets into edges. We study the distribution of nodes in the 2-dimensional plane. We then evaluate the local properties of the graphs by measuring the meshedness coefficient and counting short cycles (of three, four and five edges), and the global properties by measuring global efficiency and cost. As normalization graphs, we consider both minimal spanning trees (MST) and greedy triangulations (GT) induced by the same spatial distribution of nodes. The results indicate that most of the cities have evolved into networks as efficienct as GT, although their cost is closer to the one of a tree. An analysis based on relative efficiency and cost is able to characterize different classes of cities.Comment: 7 pages, 3 figures, 3 table

Gluing construction of initial data with Kerr-de Sitter ends

We construct initial data sets which satisfy the vacuum constraint equa- tions of General Relativity with positive cosmologigal constant. More pre- silely, we deform initial data with ends asymptotic to Schwarzschild-de Sitter to obtain non-trivial initial data with exactly Kerr-de Sitter ends. The method is inspired from Corvino's gluing method. We obtain here a extension of a previous result for the time-symmetric case by Chru\'sciel and Pollack.Comment: 27 pages, 3 figure

Multiple-Time Higher-Order Perturbation Analysis of the Regularized Long-Wavelength Equation

By considering the long-wave limit of the regularized long wave (RLW) equation, we study its multiple-time higher-order evolution equations. As a first result, the equations of the Korteweg-de Vries hierarchy are shown to play a crucial role in providing a secularity-free perturbation theory in the specific case of a solitary-wave solution. Then, as a consequence, we show that the related perturbative series can be summed and gives exactly the solitary-wave solution of the RLW equation. Finally, some comments and considerations are made on the N-soliton solution, as well as on the limitations of applicability of the multiple scale method in obtaining uniform perturbative series.Comment: 15 pages, RevTex, no figures (to appear in Phys. Rev. E

Balancing Minimum Spanning and Shortest Path Trees

This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and epsilon > 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+epsilon times the shortest-path distance, and yet the total weight of the tree is at most 1+2/epsilon times the weight of a minimum spanning tree. This is the best tradeoff possible. The paper also describes a fast parallel implementation.Comment: conference version: ACM-SIAM Symposium on Discrete Algorithms (1993