Semiclassical bounds for spectra of biharmonic operators

We provide complementary semiclassical bounds for the Riesz means $R_1(z)$ of the eigenvalues of various biharmonic operators, with a second term in the expected power of $z$. The method we discuss makes use of the averaged variational principle (AVP), and yields two-sided bounds for individual eigenvalues, which are semiclassically sharp. The AVP also yields comparisons with Riesz means of different operators, in particular Laplacians

On sums of eigenvalues of elliptic operators on manifolds

We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kr{\"o}ger 's bound for Neumann spectra of Laplacians on Euclidean domains [12]. Among the operators we consider are the Laplace-Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces (here extending a result of Strichartz [21] with a simplified proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schr{\"o}dinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.Comment: in Journal of Spectral Theory, 201

The One-Dimensional Schrödinger-Newton Equations

We prove an existence and uniqueness result for ground states of one-dimensional Schrödinger-Newton equation

Comparative Demography and Dietary Resource Partitioning of Two Wild Ranging Asiatic Equid Populations

Tooth wear signatures allow inference on the dietary traits of herbivorous ungulates. Comparing dietary regimes of taxonomically closely related populations further allows inference on habitat structure and food availability. The mesowear method of tooth wear evaluation has opened up a pathway to reconstruct subtle differences in dietary behavior and resource partitioning based on skeletal material as the only source of information. Eighty cheek dentitions of Asian wild asses (Equus hemionus) from the Southern Gobi (Mongolia) and 61 dentitions of African free ranging donkeys (Equus asinus) from the Emirate Sharjah (United Arab Emirates) were investigated for their mesowear signatures. It is tested if sexes and age classes of individuals are different in their mesowear signatures. Cluster analyses and principal components analyses are applied in order to test hypotheses using 27 ungulate species with known diets as a reference. The wild asses from the Gobi are found to classify as typical grazers within this spectrum and have a more abrasion-dominated signature then the donkey population from the United Arab Emirates (UAE). The diet available to the latter population is thus considered to be more heterogeneous. This indicates that the donkey habitat in the UAE is a more diverse mosaic of feeding resources compared to the wild ass habitat of the Mongolian desert. In both populations there are more intraspecific differences in the diet between sexes then there are interspecific differences between the two African zebra species E. burchelii and E. grevyi. The dietary signal is further interpreted as to reflect the social structure of the animals as associated with specific environmental conditions. Between the two sexes of Asian wild asses and UAE donkeys, one would expect a more pronounced segregation in the diet when male territories are small and poor in resources at the same time. This condition would best characterise the habitat inhabited by the UAE donkey population. The data suggest that the male territories of this population are comparatively small and thus provide highly abrasive forage only