Integrability of Dirac reduced bi-Hamiltonian equations

First, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three hierarchies of bi-Hamiltonian PDE's, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies.Comment: 15 pages. Corrected some typos and added missing equations in Section 5 for g=sl_n, n>

Adaptive walks in a gene network model of morphogenesis: insights into the Cambrian explosion

The emergence of complex patterns of organization close to the Cambrian boundary is known to have happened over a (geologically) short period of time. It involved the rapid diversification of body plans and stands as one of the major transitions in evolution. How it took place is a controversial issue. Here we explore this problem by considering a simple model of pattern formation in multicellular organisms. By modeling gene network-based morphogenesis and its evolution through adaptive walks, we explore the question of how combinatorial explosions might have been actually involved in the Cambrian event. Here we show that a small amount of genetic complexity including both gene regulation and cell-cell signaling allows one to generate an extraordinary repertoire of stable spatial patterns of gene expression compatible with observed anteroposterior patterns in early development of metazoans. The consequences for the understanding of the tempo and mode of the Cambrian event are outlined.Comment: to appear in International Journal of Developmental Biology, special issue on Evo-Devo (2003

Testing hypotheses of the cause of peripheral thinning of the Greenland Ice Sheet: is land-terminating ice thinning at anomalously high rates?

Recent observations have shown that the periphery of the Greenland ice sheet (GrIS) is thinning rapidly and that this thinning is greatest around marine-terminating outlet glaciers. Several theories have been proposed which provide a link between climate and ice thinning. We present surface elevation change (<i>dh/dt</i>) data from NASA's Program for Arctic Regional Climate Assessment (PARCA) laser altimetry surveys for fourteen and eleven of the largest outlet glaciers in Southern Greenland from 1993 to 1998 and 1998 to 2006 respectively to test the applicability of these theories to the GrIS. <br><br> Initially, outlet glacier <i>dh/dt</i> data are compared with data from concurrent surveys over inland ice (slow flowing ice that is not obviously draining into an outlet glacier) to confirm the effect of ice flow on surface thinning rates. Land-terminating and marine-terminating outlet glacier <i>dh/dt</i> data are then compared from 1993 to 1998 and from 1998 to 2006. Finally, ablation anomalies (the difference between the "normal" ablation rate from 1970 to 2000 and the ablation rate in the time period of interest) calculated with a positive degree day model are compared to both marine-terminating and land-terminating outlet glacier <i>dh/dt</i> data. <br><br> Our results support earlier conclusions that certain marine-terminating outlet glaciers have thinned much more than land-terminating outlet glaciers during both time periods. Furthermore we show that these differences are not limited to the largest, fastest-flowing outlet glaciers – almost all marine-terminating outlet glaciers are thinning more than land-terminating outlet glaciers. There was a four fold increase in mean marine-terminating outlet glacier thinning rates below 1000 m elevation between the periods 1993 to 1998 and 1998 to 2006, while thinning rates of land-terminating outlet glaciers remained statistically unchanged. This suggests that a change in a controlling mechanism specific to the thinning rates of marine-terminating outlet glaciers occurred in the late 1990s and that this change did not affect thinning rates of land-terminating outlet glaciers. <br><br> Thinning rates of land-terminating outlet glaciers are statistically the same as ablation anomalies, while thinning rates of marine-terminating outlet glaciers are not. Thinning of land-terminating outlet glaciers therefore seems to be a response to changes in local mass balance (principally increases in air temperature) while thinning of marine-terminating outlet glaciers is principally controlled by ice dynamics. The mechanism by which this dynamic thinning occurs is still not clear although its association with marine-terminating outlet glaciers suggests perturbations at marine termini (calving) as the likely cause

On classification of Poisson vertex algebras

We describe a conjectural classification of Poisson vertex algebras of CFT type and of Poisson vertex algebras in one differential variable (= scalar Hamiltonian operators)

Quantitative analysis of Clausius inequality

In the context of driven diffusive systems, for thermodynamic transformations over a large but finite time window, we derive an expansion of the energy balance. In particular, we characterize the transformations which minimize the energy dissipation and describe the optimal correction to the quasi-static limit. Surprisingly, in the case of transformations between homogeneous equilibrium states of an ideal gas, the optimal transformation is a sequence of inhomogeneous equilibrium states.Comment: arXiv admin note: text overlap with arXiv:1404.646

Minimum dissipation principle in stationary non equilibrium states

We generalize to non equilibrium states Onsager's minimum dissipation principle. We also interpret this principle and some previous results in terms of optimal control theory. Entropy production plays the role of the cost necessary to drive the system to a prescribed macroscopic configuration