Lower bounds for nodal sets of eigenfunctions

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.Comment: To appear in Communications in Mathematical Physics; revised to include two additional references and update bibliographic informatio

Tamarisk biocontrol using Tamarisk Beetles: Potential consequences for riparian birds in the southwestern United Stains

The tamarisk beetle (Diorhabda spp.), a non-native biocontrol agent, has been introduced to eradicate tamarisk (Tamarix spp.), a genus of non-native tree that has become a dominant component of riparian woodlands in the southwestern United States. Tamarisk beetles have the potential to spread widely and defoliate large expanses of tamarisk habitat, but the effects of such a widespread loss of riparian vegetation on birds remains unknown. We reviewed literature on the effects of other defoliating insects on birds to investigate the potential for tamarisk beetles to affect birds positively or negatively by changing food abundance and vegetation structure. We then combined data on the temporal patterns of tamarisk defoliation by beetles with nest productivity of a well studied riparian obligate, the Southwestern Willow Flycatcher (Empidonax traillii extimus), to simulate the potential demographic consequences of beetle defoliation on breeding riparian birds in both the short and long term. Our results highlight that the effects of tamarisk biocontrol on birds will likely vary by species and population, depending upon its sensitivity to seasonal defoliation by beetles and net loss of riparian habitat due to tamarisk mortality. Species with restricted distributions that include areas dominated by tamarisk may be negatively affected both in the short and long term. The rate of regeneration and/or restoration of native cottonwoods (Populus spp.)and willows (Salix spp.) relative to the rate of tamarisk loss will be critical in determining the long-term effect of this large-scale ecological experiment

Age, habitat, and yearly variation in the diet of a generalist insectivore, the Southwestern Willow Flycatcher

The article presents a study which examines the age, habitat, and yearly variation in the diet of a generalist insectivore, the southwestern Willow Flycatcher at Roosevelt Lake in central Arizona. It focuses on the five prey categories which were accounted for 70% of the proportional abundance in flycatcher diet, including Hymenoptera, Diptera, Cicadellidae, Coleoptera, and Formicide. It states that there were no detected differences in diet between sexes of adults, but adults and nestling samples differed, with higher proportions of Hymenoptera in adult samples and more Diptera in nestling samples. It discusses the comparison between flycatcher diet in habitat patches dominated by native cottonwood and willow, exotic salt cedar, or a mix of the three species

Global existence problem in T3T^3-Gowdy symmetric IIB superstring cosmology

We show global existence theorems for Gowdy symmetric spacetimes with type IIB stringy matter. The areal and constant mean curvature time coordinates are used. Before coming to that, it is shown that a wave map describes the evolution of this system

Generalized and weighted Strichartz estimates

In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schr\"odinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension 2 and 3.Comment: Final version, to appear in the Communications on Pure and Applied Analysis. 33 pages. 2 more references adde

Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we previously used to study analogous problems on $S^1$, and on $\mathbb{R}$.Comment: 38 page

Localness of energy cascade in hydrodynamic turbulence, II. Sharp spectral filter

We investigate the scale-locality of subgrid-scale (SGS) energy flux and inter-band energy transfers defined by the sharp spectral filter. We show by rigorous bounds, physical arguments and numerical simulations that the spectral SGS flux is dominated by local triadic interactions in an extended turbulent inertial-range. Inter-band energy transfers are also shown to be dominated by local triads if the spectral bands have constant width on a logarithmic scale. We disprove in particular an alternative picture of ``local transfer by nonlocal triads,'' with the advecting wavenumber mode at the energy peak. Although such triads have the largest transfer rates of all {\it individual} wavenumber triads, we show rigorously that, due to their restricted number, they make an asymptotically negligible contribution to energy flux and log-banded energy transfers at high wavenumbers in the inertial-range. We show that it is only the aggregate effect of a geometrically increasing number of local wavenumber triads which can sustain an energy cascade to small scales. Furthermore, non-local triads are argued to contribute even less to the space-average energy flux than is implied by our rigorous bounds, because of additional cancellations from scale-decorrelation effects. We can thus recover the -4/3 scaling of nonlocal contributions to spectral energy flux predicted by Kraichnan's ALHDIA and TFM closures. We support our results with numerical data from a $512^3$ pseudospectral simulation of isotropic turbulence with phase-shift dealiasing. We conclude that the sharp spectral filter has a firm theoretical basis for use in large-eddy simulation (LES) modeling of turbulent flows.Comment: 42 pages, 9 figure