In a one-parameter model for evolution of random trees strong law of large numbers and central limit theorem are proved for the number of vertices with low degree. The proof is based on elementary martingale theory.
A process of evolving random graphs is considered where vertices are added to the graph one by one, and edges connecting the
new vertex to the old ones are drawn independently, each with probability depending linearly on the degree of the endpoint.
In the paper the asymptotic degree distribution and the order of the maxdegree are determined.
For everyk≥1 consider the waiting time until each pattern of lengthk over a fixed alphabet of sizen appears at least once in an infinite sequence of independent, uniformly distributed random letters. Lettingn→∞ we determine the limiting finite dimensional joint distributions of these waiting times after suitable normalization and
provide an estimate for the rate of convergence. It will turn out that these waiting times are getting independent.
In a string ofn independent coin tosses we consider the difference between the lengths of the longest blocks of consecutive heads resp. tails. A complete characterization of the a.s. limit properties of this quantity is proved.
Sufficient conditions of covariance type are presented for weighted averages of random variables with arbitrary dependence
structure to converge to 0, both for logarithmic and general weighting. As an application, an a.s. local limit theorem of
Csáki, Földes and Révész is revisited and slightly improved.
When an opportunistic predator is looking for a given type of prey and encounters another one from different species, it tries to utilize this random opportunity. We characterize the optimal levels of this opportunism in the framework of stochastic models for the two prey-one predator case. We consider the spatial dispersal of preys and the optimal diet choice of predator as well. We show that when both preys have no handling time, the total opportunism provides maximal gain of energy for the predator. When handling times differ with prey, we find a conditional optimal behavior: for small density of both prey species the predator prefers the more valuable one and is entirely opportunistic. However, when the density of the more valuable prey is higher than that of the other species, then the predator prefers the first one and intentionally neglects the other. Furthermore, when the density of the less valuable prey is high and that of the other one is small, then predator will look for the less valuable prey and is therefore totally opportunistic. We demonstrate that prey preference is remunerative whenever the advantage of a proper prey preference is larger than the average cost of missed prey preference. We also propose a dynamics which explicitly contains two sides of shared predation: apparent mutualism and apparent competition, and we give conditions when the rare prey goes extinct.
The heat capacity of 9.70 and 11.35 mol% yttria stabilized zirconia ((ZrO2)1–x(Y2O3)x; x=0.0970, 0.1135) was measured by adiabatic calorimetry between 13 and 300 K, and some thermodynamic functions were calculated and given in a table. A large excess heat capacity extending from the lowest temperature to room temperature with a broad maximum at about 75 K was found in comparison with the heat capacity calculated from those of pure zirconia and yttria on the basis of simple additivity rule. The shape of the excess heat capacity is very similar to the Schottky anomaly, which may be attributed to a softening of lattice vibration. The amount of the excess heat capacity decreased with increasing yttria doping, while the maximum temperature did not vary. The relationships among the excess heat capacity, defect structure and interatomic force constants, and also the role of oxygen vacancy were discussed.
Y2O3 has a crystal structure of c-type rare-earth oxide. Y2O3 does not show an oxide ionic conductivity. On the other hand, CeO2 based oxide is one of the most interesting of the fluorite oxides since the ionic conductivity of it is higher than that of yttria-stabilized zirconia. However, CeO2 based oxides are partially reduced and develop electronic conductivity under reduced atmosphere.In this study, the effective index for the improvement of ionic conductivity in Y2O3 and CeO2 systems was defined using ionic radii from the viewpoint of crystallography. The utility of this effective index on some electrical properties was investigated.