Dependence and phase changes in random m‐ary search trees

We study the joint asymptotic behavior of the space requirement and the total path length (either summing over all root‐key distances or over all root‐node distances) in random m‐ary search trees. The covariance turns out to exhibit a change of asymptotic behavior: it is essentially linear when , but becomes of higher order when . Surprisingly, the corresponding asymptotic correlation coefficient tends to zero when , but is periodically oscillating for larger m, and we also prove asymptotic independence when . Such a less anticipated phenomenon is not exceptional and our results can be extended in two directions: one for more general shape parameters, and the other for other classes of random log‐trees such as fringe‐balanced binary search trees and quadtrees. The methods of proof combine asymptotic transfer for the underlying recurrence relations with the contraction method. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 353–379, 2017     

Miniemulsion polymerization of styrene using alkyl methacrylates as the reactive cosurfactant

 Stable styrene miniemulsions were prepared by using alkyl methacrylates as the reactive cosurfactant. Like conventional cosurfactants (e.g., cetyl alcohol (CA) and hexadecane (HD)), alkyl methacrylates (e.g., dodecyl methacrylate (DMA) and stearyl methacrylate (SMA)) may act as a cosurfactant in stabilizing the homogenized miniemulsions. Furthermore, the methacrylate group may be chemically incorporated into latex particles in subsequent miniemulsion polymerization. The data of the monomer droplet size, creaming rate and phase separation of monomer as a function of time were used to evaluate the shelf-life of miniemulsions stabilized by sodium dodecyl sulfate in combination with various cosurfactants. Polystyrene latex particles were produced via both monomer droplet nucleation and homogeneous nucleation in the miniemulsion polymerization using CA or DMA as the cosurfactant, with the result of a quite broad particle size distribution. On the other hand, the miniemulsion polymerization with HD or SMA showed a predominant monomer droplet nucleation. The resultant particle size distribution was relatively narrow. In miniemulsion polymerization, the less hydrophobic DMA is similar to CA, whereas the more hydrophobic SMA is similar to HD. Received: 19 November 1996 Accepted: 20 February 1997     

Giant nonlinear optical properties of bismuth thin films grown by pulsed laser deposition

Thin films of Bi were grown by pulsed laser deposition on glass substrates at room temperature. The thickness and roughness of the films were characterized by grazing-incidence x-ray reflectivity, and the complex refractive indices were measured in the range from 1.5 to 4 eV by spectroscopic ellipsometry. We performed Z-scan measurements to study the third-order optical nonlinearity of the films. It was found that the Bi films exhibited an unusually large nonlinear refractive coefficient, n(I)~1.24x10(-1) cm(2)/kW and nonlinear absorption coefficient, alpha(I)~-3.97 cm/W , at low laser intensity, ~60 kW/cm(2) . This anomaly is believed to have an origin related to melting of the Bi films at the focus spot by the laser beam.     

Magnetic anisotropy in (, .44, .55, .67, and .78) alloy film grown on and MgO(001) by molecular beam epitaxy

Fe_xPd_1−x (x=.30, .44, .55, .67, and .78) films were directly grown on SrTiO₃(001) and MgO(001) by molecular beam epitaxy at 500 C. The thickness of all films is 50 nm. X-ray diffraction shows epitaxial quality and face-center- tetragonal (00l), (002), and (003) peaks indicating an FePd L1₀ order state for films of x=.30, .44 and .55. X-ray diffraction only shows (002) peaks with a relatively weak intensity for the film of x=.67 and no (00l) peak is observed, but a broad body-center-cubic Fe(002) is identified for the film of x=.78. Magnetic hysteresis curves are carried out by a vibrating sample magnetometer (VSM) with an applied field within 12 kOe. Magnetization of both in-plane and perpendicular-to- the-plane measurements show a linear increase of the magnetization saturation from 560 emu/cm³ to 1250 emu/cm³ as x increasing from .30 to .78. For the results of the in-plane measurements, remanence (M_r), however, shows a minimum while the anisotropy field (H_k) shows a maximum for the film with x=.44 indicating the optimal content ratio of Fe/Pd for perpendicular anisotropy in the present alloy films. Further, negative remanence is observed in the hysteresis curves where the field is perpendicular to the film of x=.78 This may indicate that the L1₀order state still affects the magnetic anisotropy for high Fe content films even though the film has a body-center-cubic structure.