The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to (1+1)-dimensional completely integrable soliton equations of differential-difference type.Comment: 47 page

A characteristic particle method for traffic flow simulations on highway networks

A characteristic particle method for the simulation of first order macroscopic traffic models on road networks is presented. The approach is based on the method "particleclaw", which solves scalar one dimensional hyperbolic conservations laws exactly, except for a small error right around shocks. The method is generalized to nonlinear network flows, where particle approximations on the edges are suitably coupled together at the network nodes. It is demonstrated in numerical examples that the resulting particle method can approximate traffic jams accurately, while only devoting a few degrees of freedom to each edge of the network.Comment: 15 pages, 5 figures. Accepted to the proceedings of the Sixth International Workshop Meshfree Methods for PDE 201

On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa-Holm system

The Camassa-Holm equation and its two-component Camassa-Holm system generalization both experience wave breaking in finite time. To analyze this, and to obtain solutions past wave breaking, it is common to reformulate the original equation given in Eulerian coordinates, into a system of ordinary differential equations in Lagrangian coordinates. It is of considerable interest to study the stability of solutions and how this is manifested in Eulerian and Lagrangian variables. We identify criteria of convergence, such that convergence in Eulerian coordinates is equivalent to convergence in Lagrangian coordinates. In addition, we show how one can approximate global conservative solutions of the scalar Camassa-Holm equation by smooth solutions of the two-component Camassa-Holm system that do not experience wave breaking

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments

The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion

The paper focuses on discrete-type approximations of solutions to non-homogeneous stochastic differential equations (SDEs) involving fractional Brownian motion (fBm). We prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by fBm with Hurst index $H>1/2$ can be estimated by $O(\delta^{2H-1})$ ($\delta$ is the diameter of partition). For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is $O(\delta^H)$.Comment: 21 pages, (incorrect) weak convergence result removed, to appear in Stochastic

Peatlands and the carbon cycle: from local processes to global implications - a synthesis

Peatlands cover only 3% of the Earth's land surface but boreal and subarctic peatlands store about 15-30% of the world's soil carbon ( C) as peat. Despite their potential for large positive feedbacks to the climate system through sequestration and emission of greenhouse gases, peatlands are not explicitly included in global climate models and therefore in predictions of future climate change. In April 2007 a symposium was held in Wageningen, the Netherlands, to advance our understanding of peatland C cycling. This paper synthesizes the main findings of the symposium, focusing on (i) small-scale processes, (ii) C fluxes at the landscape scale, and (iii) peatlands in the context of climate change. The main drivers controlling most are related to some aspects of hydrology. Despite high spatial and annual variability in Net Ecosystem Exchange ( NEE), the differences in cumulative annual NEE are more a function of broad scale geographic location and physical setting than internal factors, suggesting the existence of strong feedbacks. In contrast, trace gas emissions seem mainly controlled by local factors. Key uncertainties remain concerning the existence of perturbation thresholds, the relative strengths of the CO2 and CH4 feedback, the links among peatland surface climate, hydrology, ecosystem structure and function, and trace gas biogeochemistry as well as the similarity of process rates across peatland types and climatic zones. Progress on these research areas can only be realized by stronger co-operation between disciplines that address different spatial and temporal scales

Selecting and preparing seed corn.

The following circular has been prepared in answer to the many letters which are being received daily asking for information regarding the purchasing and testing of seed corn and its preparation for planting. When we consider that more than nine million acres, considerably over one-fourth of the entire area of the state, will be planted to corn the coming season and that it will require more than 1,300,000 bushels of seed to plant this area; and when we realize that the character of the seed, its vitality, breeding, purity, adaptability to the soil and climate and uniformity in both size and shape of kernels, all exercise a great influence on the future yield, the great importance of paying the closest attention to the testing and the preparation of the seed corn for the planter cannot be over estimated

The Evolution of the Field and Cluster Morphology-Density Relation for Mass-Selected Samples of Galaxies

The Sloan Digital Sky Survey (SDSS) and photometric/spectroscopic surveys in the GOODS-South field (the Chandra Deep Field-South, CDFS) are used to construct volume-limited, stellar mass-selected samples of galaxies at redshifts 0<z<1. The CDFS sample at 0.6<z<1.0 contains 207 galaxies complete down to M=4x10^10 Msol (for a ``diet'' Salpeter IMF), corresponding to a luminosity limit for red galaxies of M_B=-20.1. The SDSS sample at 0.020<z<0.045 contains 2003 galaxies down to the same mass limit, which corresponds to M_B=-19.3 for red galaxies. Morphologies are determined with an automated method, using the Sersic parameter n and a measure of the residual from the model fits, called ``bumpiness'', to distinguish different morphologies. These classifications are verified with visual classifications. In agreement with previous studies, 65-70% of the galaxies are located on the red sequence, both at z~0.03 and at z~0.8. Similarly, 65-70% of the galaxies have n>2.5. The fraction of E+S0 galaxies is 43+/-3%$ at z~0.03 and 48+/-7% at z~0.8, i.e., it has not changed significantly since z~0.8. When combined with recent results for cluster galaxies in the same redshift range, we find that the morphology-density relation for galaxies more massive than 0.5M* has remained constant since at least z~0.8. This implies that galaxies evolve in mass, morphology and density such that the morphology-density relation does not change. In particular, the decline of star formation activity and the accompanying increase in the stellar mass density of red galaxies since z~1 must happen without large changes in the early-type galaxy fraction in a given environment.Comment: 16 pages, 13 figures, 2 tables. Updated to match journal version. Will appear in ApJ (vol. 670, p. 206

The Galaxy Population of Cluster RXJ0848+4453 at z=1.27

We present a study of the galaxy population in the cluster RXJ0848+4453 at z=1.27, using deep HST NICMOS and WFPC2 images. We morphologically classify all galaxies to K_s=20.6 that are covered by the HST imaging, and determine photometric redshifts using deep ground based BRIzJK_s photometry. Of 22 likely cluster members with morphological classifications, eleven (50%) are classified as early-type galaxies, nine (41%) as spiral galaxies, and two (9%) as ``merger/peculiar''. At HST resolution the second brightest cluster galaxy is resolved into a spectacular merger between three red galaxies of similar luminosity, separated from each other by ~6 kpc, with an integrated magnitude K=17.6 (~3 L* at z=1.27). The two most luminous early-type galaxies also show evidence for recent or ongoing interactions. Mergers and interactions between galaxies are possible because RXJ0848+4453 is not yet relaxed. The fraction of early-type galaxies in our sample is similar to that in clusters at 0.5<z<1, and consistent with a gradual decrease of the number of early-type galaxies in clusters from z=0 to z=1.3. We find evidence that the color-magnitude relation of the early-type galaxies is less steep than in the nearby Coma cluster. This may indicate that the brightest early-type galaxies have young stellar populations at z=1.27, but is also consistent with predictions of single age ``monolithic'' models with a galactic wind. The scatter in the color-magnitude relation is ~0.04 in rest frame U-V, similar to that in clusters at 0<z<1. Taken together, these results show that luminous early-type galaxies exist in clusters at z~1.3, but that their number density may be smaller than in the local Universe. Additional observations are needed to determine whether the brightest early-type galaxies harbor young stellar populations.Comment: Accepted for publication in The Astrophysical Journal Letter