Oscillations of certain second order nonlinear differential equations

A second order nonlinear differential equation

Electrophysical characteristics and stability of solid apatite-like electrolytes La x Si6O12 + 1.5x and La x Ge6O12 + 1.5x

Oxygen-ion conduction in apatite-like compounds based on silicates and germanates of lanthanum La _x A₆O_{12 + 1.5x} (A = Si, Ge; x = 9.11–10.22) is studied. The compounds are shown to be purely ionic conductors at 600–900°C and partial oxygen pressures 10^?16 to 10⁵ Pa. The electroconductivity of the best conducting specimens of La _x A₆O_{12 + 1.5x} (A = Si, Ge; x = 9.77–10) exceeds that of electrolyte YSZ at moderate temperatures. The electroconductivity of lanthanum germanate is substantially greater than that of lanthanum silicate, specifically, 7.85 × 10^?2 and 2.35 × 10^?2 S cm^?1, respectively, at 800°C. An inflection is discovered at ～750°C in the temperature dependences of electroconductivity of La _x Ge₆O_{12 + 1.5x} (x = 9.77–10.22). A dilatometric examination points to a second-kind phase transition that may be due to the oxygen sublattice disordering. The behavior of apatite-like electrolytes La _x A₆O_{12 + 1.5x} (A = Si, Ge) during long exploitation periods in the interval of working temperatures of electrochemical devices is studied for the first time ever. The electrolytes’ aging at 800°C in air for 1000 h was investigated by the electroconductivity method. The electroconductivity of lanthanum germanates decayed with time by 5% and that of lanthanum silicates, by 9.5%. The steady-state values of electroconductivity of all compounds studied is reached after 600–700 h. The compounds studied form a class of materials that hold some promise as solid electrolytes for medium-temperature fuel cells and other electrochemical devices.     

Evaluation of mean and variance integrals without integration

The mean and variance of some continuous distributions, in particular the exponentially decreasing probability distribution and the normal distribution, are considered. Since they involve integration by parts, many students do not feel comfortable. In this note, a technique is demonstrated for deriving mean and variance through differential calculus. The general nature of the technique has potential for wider applications.     

Motzkin numbers out of Random Domino Automaton

Motzkin numbers are derived from a special case of Random Domino Automaton – recently proposed a slowly driven system being a stochastic toy model of earthquakes. It is also a generalisation of 1D Drossel–Schwabl forest-fire model. A solution of the set of equations describing stationary state of Random Domino Automaton in inverse-power case is presented. A link with Motzkin numbers allows to present explicit form of asymptotic behaviour of the automaton.     

Open problems in pluripotential theory

We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold setting.     

An approach for solving a fuzzy multiobjective programming problem

In this paper we present a new approach, based on the Nearest Interval Approximation Operator, for dealing with a multiobjective programming problem with fuzzy-valued objective functions.     

Interval,ellipsoidal, and super-ellipsoidal calculi for experimental and theoretical treatment of uncertainty: Which one ought to be preferred?

In this study we compare three calculi listed in the title for analysis of structures involving uncertainty. The main idea is based on the consideration that the maximum structural response predicted by the preferred theory ought to be minimal, and the minimum structural response predicted by the preferred theory ought to be maximal, to constitute a lower overestimation. We present analytic results that allow one to calculate the structural response via the interval, ellipsoidal or super-ellipsoidal calculus. We provide several examples of truss structures and illustrate that in different situations, depending on the available data, one of these calculi ought to be preferred. Conclusion is made on the preferable approach to be the super-ellipsoidal calculus.