Emergent Quantum Euler Equation and Bose–Einstein Condensates

In this paper, proceeding from the recently developed way of deriving the quantum-mechanical equations from the classical ones, the complete system of hydrodynamical equations, including the quantum Euler equation, is derived for a perfect fluid and an imperfect fluid with pairwise interaction between the particles. For the Bose–Einstein condensate of the latter one the Bogolyubov spectrum of elementary excitations is easily reproduced in the acoustic approximation.     

Multiple-valued steady states and their stability in a saturable-absorber laser

We present the results of a theoretical consideration of steady states in a saturable-absorber laser for different frequencies of transition in the active medium, the absorber, and the cavity. The conditions for the coexistence of several (up to five) stationary values of intensity are obtained. The stability of such states is analytically studied, and the possible regimes of generation and switching for the parameters corresponding to solid-state lasers with a saturable absorber are shown. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 67, No. 3, pp. 322–326, May–June, 2000.     

Compact Composition Operators and Deddens Algebras

We consider the Deddens algebras associated to compact composition operators on the Hardy space \(H^2\) on the unit disk. When the compact composition operator corresponds to a function \(\varphi \) that satisfies \(\varphi (0)=0\) and \(\varphi '(0)\ne 0\), we show that the lattice of invariant subspaces of this algebra is \(\{0\}\cup \{z^n H^2: n=0,1,2,\ldots \}\). As a consequence, for this class of operators the associated Deddens algebra is weakly dense in the algebra of lower triangular matrices.     

Periodic optimal control for parabolic Volterra-Lotka type equations

This paper considers the optimal harvesting control of a biological species, whose growth is governed by the parabolic diffusive Volterra-Lotka equation. We prove that such equation with L^∞ periodic coefficients has an unique positive periodic solution. We show the existence and uniqueness of an optimal control, and under certain conditions, we characterize the optimal control in terms of a parabolic optimality system. A monotone sequence which converges to the optimal control is constructed.