Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation

It is known that the eigenvalues λn(n = 1, 2, ...) numbered in decreasing order and taking the multiplicity of the self-adjoint Sturm-Liouville operator with a completely continuous inverse operator L^{−1} have the following property (*) λn → 0, when n → ∞, moreover, than the faster convergence to zero so the operator L^{−1} is best approximated by finite rank operators. The following question: - Is it possible for a given nonlinear operator to indicate a decreasing numerical sequence characterized by the property (*)? naturally arises for nonlinear operators. In this paper, we study the above question for the nonlinear Sturm-Liouville operator. To solve the above problem the theorem on the maximum regularity of the solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space L2(R) (R = (−∞, ∞)) is proved. Twosided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear SturmLiouville equation are also obtained. As is known, the obtained estimates of Kolmogorov widths give the opportunity to choose approximation apparatus that guarantees the minimum possible error.

Attractor competition enriches cortical dynamics during awakening from anesthesia

Cortical slow oscillations (≲ 1 Hz) are a hallmark of slow-wave sleep and deep anesthesia across animal species. They arise from spatiotemporal patterns of activity with low degree of complexity, eventually increasing as wakefulness is approached and cognitive functions emerge. The arousal process is then an open window on the widely unknown mechanisms underlying the emergence of the dynamical richness of awake cortical networks. Here, we investigated the changes in the network dynamics as anesthesia fades out and wakefulness is approached in layer 5 neuronal assemblies of the rat visual cortex. Far from being a continuum, this transition displays both gradual and abrupt activity changes. Starting from deep anesthesia, slow oscillations increase their frequency eventually expressing maximum regularity. This stage is followed by the abrupt onset of an infra-slow (~ 0.2 Hz) alternation between sleep-like oscillations and activated states. A population rate model reproduces this transition driven by an increased excitability that brings it to periodically cross a critical point. We conclude that dynamical richness emerges as a competition between two metastable attractor states whose existence is here experimentally confirmed.

A study on soft rough semigroups and corresponding decision making applications

In this paper, we study a kind of soft rough semigroups according to Shabir’s idea. We define the upper and lower approximations of a subset of a semigroup. According to Zhan’s idea over hemirings, we also define a kind of new C-soft sets and CC-soft sets over semigroups. In view of this theory, we investigate the soft rough ideals (prime ideals, bi-ideals, interior ideals, quasi-ideals, regular semigroups). Finally, we give two decision making methods: one is for looking a best a parameter which is to the nearest semigroup, the other is to choose a parameter which keeps the maximum regularity of regular semigroups.

The Organisation of Air Routes for Night Flying

The operation of scheduled night flying presents two tasks to aviation:—(1) The technical task of taking off from and landing on an aerodrome.(2) The operation of the flight from one aerodrome to the other.The first task presents no completely new problem, as sufficient experience was gained during the war. All that was necessary was to provide facilities which guaranteed regular traffic.The second task was purely one of navigation. It had to be solved in accordance with the strict demands of maximum safety and maximum regularity of the service. The only means of navigation then available, the compass, was not sufficient for these requirements, and at night it was not possible to count on keeping on the course by observing objects on the ground.