On critical points of the objective functional for maximization of qubit observables

We study unconstrained control of a two-level quantum system and analyse critical points of the objective functional which represents quantum average of system observable at some final time $T$. In Proc. Steklov Inst. Math. 285, 233-240 (2014) it was shown that all maxima and minima of the objective functional are global if $T\ge \pi$ (in suitable units). In the present work we show that all maxima and minima are global as soon as $T\ge \pi/2$. Hence we reduce by the factor of two the minimal time for which traps, i.e., local but not global maxima or minima of the objective functional, do not exist

On convergence of solutions of fractal Burgers equation toward rarefaction waves

In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation $u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0$ with $\alpha\in (1,2)$ is studied. It is shown that if the nondecreasing initial datum approaches the constant states $u_\pm$ ($u_-<u_+$) as $x\to \pm\infty$, respectively, then the corresponding solution converges toward the rarefaction wave, {\it i.e.} the unique entropy solution of the Riemann problem for the nonviscous Burgers equation.Comment: 15 page

Dust extinction and absorption: the challenge of porous grains

In many models of dusty objects in space the grains are assumed to be composite or fluffy. However, the computation of the optical properties of such particles is still a very difficult problem. We analyze how the increase of grain porosity influences basic features of cosmic dust -- interstellar extinction, dust temperature, infrared bands and millimeter opacity. Porous grains can reproduce the flat extinction across the 3 - 8 \mkm wavelength range measured for several lines of sight by {\it ISO} and {\it Spitzer}. Porous grains are generally cooler than compact grains. At the same time, the temperature of very porous grains becomes slightly larger in the case of the EMT-Mie calculations in comparison with the results found from the layered-sphere model. The layered-sphere model predicts a broadening of infrared bands and a shift of the peak position to larger wavelengths as porosity grows. In the case of the EMT-Mie model variations of the feature profile are less significant. It is also shown that the millimeter mass absorption coefficients grow as porosity increases with a faster growth occurring for particles with Rayleigh/non-Rayleigh inclusions. As a result, for very porous particles the coefficients given by two models can differ by a factor of about 3. It is found that an increase of porosity leads to an increase of extinction cross sections at some wavelengths and a decrease at others depending on the grain model. However, this behaviour is sufficient to reproduce the extinction curve in the direction of the star $\sigma$ Sco using current solar abundances. In the case of the star $\zeta$ Oph our model requires larger amounts of carbon and iron in the dust-phase than is available.Comment: Astronomy and Astrophysics (accepted; 11 pages, 11 figures