Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian

We consider the problem of minimising the $n^{th}-$eigenvalue of the Robin Laplacian in $\mathbb{R}^{N}$. Although for $n=1,2$ and a positive boundary parameter $\alpha$ it is known that the minimisers do not depend on $\alpha$, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on $\alpha$. We derive a Wolf-Keller type result for this problem and show that optimal eigenvalues grow at most with $n^{1/N}$, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive eigenvalues does go to zero as $n$ goes to infinity. Numerical results then support the conjecture that for each $n$ there exists a positive value of $\alpha_{n}$ such that the $n^{\rm th}$ eigenvalue is minimised by $n$ disks for all $0<\alpha<\alpha_{n}$ and, combined with analytic estimates, that this value is expected to grow with $n^{1/N}$

Summation formula inequalities for eigenvalues of the perturbed harmonic oscillator

We derive explicit inequalities for sums of eigenvalues of one-dimensional Schr\"{o}dinger operators on the whole line. In the case of the perturbed harmonic oscillator, these bounds converge to the corresponding trace formula in the limit as the number of eigenvalues covers the whole spectrum.Comment: 15 pages, to appear in Osaka J. Mat

3C273 variability at 7 mm: Evidences of shocks and precession in the jet

We report 4 years of observations of 3C273 at 7 mm obtained with the Itapetinga Radiotelescope, in Brazil, between 2009 and 2013. We detected a flare in 2010 March, when the flux density increased by 50% and reached 35 Jy. After the flare, the flux density started to decrease and reached values lower than 10 Jy. We suggest that the 7 mm flare is the radio counterpart of the $\gamma$-ray flare observed by Fermi/LAT in 2009 September, in which the flux density at high energies reached a factor of fifty of its average value. A delay of 170 days between the radio and $\gamma$-ray flares was revealed using the Discrete Correlation Function (DCF) that can be interpreted in the context of a shock model, in which each flare corresponds to the formation of a compact superluminal component that expands and becomes optically thin at radio frequencies at latter epochs. The difference in flare intensity between frequencies and at a different times, is explained as a consequence of an increase in the Doppler factor $\delta$, as predicted by the 16 year precession model proposed by Abraham & Romero, which has a large effect on boosting at high frequencies while does not affect too much the observed optically thick radio emission. We discuss other observable effects of the variation in $\delta$, as the increase in the formation rate of superluminal components, the variations in the time delay between flares and the periodic behaviour of the radio light curve that we found compatible with changes in the Doppler factor.Comment: 9 pages, 8 figure

Economics of Scope, Agglomeration and Location of the Multinational Firm

A classification of the locational patterns of firms (with economies of scope present) is outlined, in order to cast light on the location of the multiplant, multinational firm. This is driven by three forces: economies of scope that follow from the co-location of different activities; transportation costs of the final good; and factor mobility costs. It is concluded that the single-plant firm prevails when spatial economies of scope are strong in relation to either transportation or mobility costs. The vertical (horizontal) multiplant firm emerges when mobility costs of factors (transportation costs on the final good) are high. These two kinds of spatial costs, which are mutually exclusive, favor the formation of multinational firms.Economies of scope; Economies of agglomeration; Multiplant firms; Multinational firms.