A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behaviour of the $s$-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain $\Omega$ with smooth boundary $\partial\Omega$. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on $\partial\Omega$. It will be shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order $p$ for which $p>(dim\Omega-1)/3$. Moreover, we also give a simple sufficient condition for the resolvent difference of two generalized Robin Laplacians to belong to a Schatten-von Neumann class of arbitrary small order. Our results extend and complement classical theorems due to M.Sh.Birman on Schatten-von Neumann properties of the resolvent differences of Dirichlet, Neumann and self-adjoint Robin Laplacians

Le problème de l'évaluation de l'effort militaire soviétique, 1988-1990

The problem of evaluating the Soviet military effort, 1989-1990. Since estimating Soviet military expenditure is, as we all know, a difficult art, the author himself, in his conclusion, states that his article raises more questions then it answers. He does, however, endeavour to clear the ground, and to define the real problems. He shows the futility of direct comparison of American and Soviet GNP, and the respective proportions of military expenditure in the GNP, and reminds the reader that the idea of military expenditure can be understood in either a limited sense (expenditure exclusively relating to the military) or in a wider sense (global military expenditure). There is no adequate method of expressing the military potential of a country by a single index. By the same token, the idea of parity between military powers has little relevance. Certain items of expenditure are not comparable in the USA and the USSR. Annual expenditure is always compared, when what should be accounted for are outgoings in previous years, not forgetting the problems of currency convertibility and differences in price systems, etc. Nevertheless, the author is prepared to venture an estimate, in figures whose approximate nature he is careful to stress. He thinks that expenditure on arms production fell in 1989-1990, touches upon the problems of converting the arms industry to civil industry, the costs and the speed involved, and ends with a summary table, qualified with many reservations. He estimates that total military expenditure, in 1989, was in no way less then 200 billion roubles, and that its share in the GNP exceeded (? greatly) 20 %. Of these 200 billion roubles, arms and military supplies (including R&D and investments) amounted to at least 160 billion, the remaining 40 billion being used for upkeep, instruction and health care of personnel, pensions, construction for military purposes, information, transport, etc.L'évaluation des dépenses militaires soviétiques étant comme chacun sait un art difficile, l'auteur déclare lui-même en conclusion que son article pose plus de questions qu'il n'apporte de réponses. Il tente néanmoins de déblayer le terrain et de cerner les vrais problèmes. Il démontre l'inanité d'une comparaison directe des PNB américain et soviétique et de la part des dépenses militaires respectives dans le PNB et rappelle que la notion de dépenses militaires peut être comprise en son sens restreint (dépenses concernant exclusivement la puissance militaire) ou large (dépenses militaires globales). Il n'existe pas de méthode adéquate pour exprimer en un seul indice la puissance militaire d'un pays. De même la notion de parité des puissances militaires n'est pas très pertinente. Certains éléments des dépenses ne sont pas comparables aux États-Unis et en U.R.S.S. On compare toujours les dépenses annuelles alors qu'il faudrait tenir compte des dépenses effectuées pendant les années antérieures. Sans oublier les problèmes de conversion des monnaies, de système des prix, etc. L'auteur se risque néanmoins à une évaluation chiffrée dont il souligne bien l'aspect approximatif. Il pense que les dépenses pour la production d'armements ont diminué en 1989-1990, évoque la difficile reconversion de l'industrie militaire en industrie civile, son coût et son rythme et finit par dresser un tableau récapitulatif, tout en l'accompagnant de multiples réserves. Il estime que les dépenses militaires totales n'étaient, en 1989, en aucun cas inférieure à 200 milliards de roubles et que leur part dans le PNB dépassait (fortement ?) 20 %. Sur ces 200 milliards de roubles, les armes et le matériel militaire (y compris la recherche-développement et les investissements) se montent au minimum à 160 milliards de roubles, les quelque 40 milliards restants étant destinés à l'entretien, l'instruction et la santé des personnels, aux pensions, à la construction à des fins militaires, au renseignement, aux transports, etc.Birman Igor. Le problème de l'évaluation de l'effort militaire soviétique, 1988-1990. In: Revue d'études comparatives Est-Ouest, vol. 22, 1991, n°4. pp. 5-20

Observation of WWWWWW Production in pppp Collisions at s\sqrt s =13 TeV with the ATLAS Detector

International audienceThis Letter reports the observation of $WWW$ production and a measurement of its cross section using 139 fb$^{-1}$ of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. Events with two same-sign leptons (electrons or muons) and at least two jets, as well as events with three charged leptons, are selected. A multivariate technique is then used to discriminate between signal and background events. Events from $WWW$ production are observed with a significance of 8.0 standard deviations, where the expectation is 5.4 standard deviations. The inclusive $WWW$ production cross section is measured to be $820 \pm 100\,\text{(stat)} \pm 80\,\text{(syst)}$ fb, approximately 2.6 standard deviations from the predicted cross section of $511 \pm 18$ fb calculated at next-to-leading-order QCD and leading-order electroweak accuracy