Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations

Numerical schemes for initial value problems of stochastic differential equations (SDEs) are considered so as to derive the order conditions of ROW-type schemes in the weak sense. Rooted tree analysis, the well-known useful technique for the counterpart of the ordinary differential equation case, is extended to be applicable to the SDE case. In our analysis, the roots are bi-colored corresponding to the ordinary and stochastic differential terms, whereas the vertices have four kinds of label corresponding to the terms derived from the ROW-schemes. The analysis brings a transparent way for the weak order conditions of the scheme. An example is given for illustration.     

Formation of high-density GaAs hexagonal nano-wire networks by selective MBE growth on pre-patterned (0 0 1) substrates

Attempts were made to grow high-density GaAs hexagonal nano-wire networks on (0 0 1) patterned substrates by selective molecular beam epitaxy. To form a hexagon, - and 5 1 0-directions were combined. By the growth of straight wire arrays in each direction, the growth mode, conditions and mechanism were investigated. The wire width was shown to be determined for both directions by the facet boundary planes resulting from the growth rate difference on different facets. By optimizing growth conditions, highly uniform and smoothly connected hexagonal nano-wire networks with a density of 3×10⁸ cm⁻² were successfully formed.     

Structural analysis of polyelectrolyte film absorbing metal ion by SAXS utilizing with X-ray anomalous dispersion effect

A distribution of Cu ions in polyelectrolyte film (Nafion) is directly observed with a small-angle X-ray scattering (SAXS) method utilizing an X-ray anomalous dispersion effect. A partial structure factor of the Cu ions, GAA(q), can be derived from the SAXS profiles obtained by scanning the incident X-ray energy around the Cu K absorption edge. GAA(q) has two peaks, indicating that the Cu ions hierarchically distribute in Nafion film. In addition, a standard SAXS also shows that Nafion film has a hierarchical structure. These results mean that the Cu ions locate in the domain where the hydrophilic bases aggregate.     

Electronic properties of Si and Ge atoms doped in clusters: In(n)Si(m) and In(n)Ge(m)

Electronic properties of silicon and germanium atom doped indium clusters, In(n)Si(m) and In(n)Ge(m), were investigated by photoionization spectroscopy of the neutrals and photoelectron spectroscopy of the anions. Size dependence of ionization energy and electron affinity for In(n)Si(1) and In(n)Ge(1) exhibit pronounced even-odd alternation at cluster sizes of n = 10-16, as compared to those for pure In(n) clusters. This result shows that symmetry lowering with the doped atom of Si or Ge results in undegeneration of electronic states in the 1d shell formed by monovalent In atoms.     

Encapsulation with the Protrusion of Cruciform 9,10‐Bis(arylethynyl)anthracene Derivatives in a Self‐Assembled Boronic Ester Cavitand Capsule: Photochemical and Photophysical Properties

The self‐assembled boronic ester cavitand capsule 3 quantitatively and tightly encapsulates 2,6‐diacetoxy‐9,10‐bis(arylethynyl)anthracene derivatives 4 a – 4 c as highly fluorescent cruciform guests to form complexes 4 a @ 3 , 4 b @ 3 , and 4 c @( 3 )₂. The structural features of capsule 3 , which possesses two polar bowl‐shaped aromatic cavity ends and four large equatorial windows connected by dynamic boronic ester bonds, made it possible to encapsulate cruciform 4 with protection of the reactive anthracene core inside the capsule and with two protruding arylethynyl groups, the π‐conjugated arms of compound 4 , through two of the equatorial windows of the capsule. Thus, complexes 4 a @ 3 , 4 b @ 3 , and 4 c @( 3 )₂ show greater resistance to photochemical reactions in solution and fluorescence quenching in the powder state compared to free guests 4 . In addition to the improved photostability, restriction of the free rotation of the arylethynyl groups of guests 4 upon encapsulation results in sharpening of the UV/Vis absorption peaks with a red‐shift and a significant increase in some of the two‐photon‐absorption peaks of complexes 4 a @ 3 , 4 b @ 3 , and 4 c @( 3 )₂ compared with free guests 4 .     

Synthesis and characterization of Ti(IV)-substituted calcium hydroxyapatite particles by forced hydrolysis of Ca(OH)2-Na5P3O10-TiCl4 mixed solution

Ti(IV)-substituted calcium hydroxyapatite (TiHap) particles were prepared by aging Ca(OH)₂, TiCl₄, and sodium triphosphate (sodium tripolyphosphate, Natpp: Na₅P₃O₁₀) mixed solution at 100 °C for 18 h. The ellipsoidal secondary TiHap particles with ca. 100～150 nm in length composing by aggregation of small ellipsoidal primary particles with ca. 20 nm in length were produced at atomic ratio of Ti/(Ca+Ti) [X_Ti]≦0.2. The in situ IR spectra of these TiHap particles exhibited very small bulk OH^? band at 3,570 cm^?1. This result indicated that the TiHap particles were formed by aggregation of fine primary particles and OH^? ions along with c-axis in the primary particles were disordered. The TiHap particles with Ca/P atomic ratio larger than theoretical value of 1.67 did not exhibit surface P–OH groups at 3,659 and 3,682 cm^?1. The diffuse reflectance UV spectra of TiHap particles revealed that these particles have a UV absorption property, especially fabricated at X_Ti?=?0.1. The particles prepared at X_Ti?=?0.6 and 0.8 were amorphous and nanoparticles with 5～10 nm in diameter, but those precipitated at X_Ti?=?1.0 were poorly crystallized anataze-type TiO₂ nanoparticles.     

The Influence of Defects on the Propagation Light within Polymeric Optical Waveguides Studied by Polarized Near-Field Scanning Optical Microscopy

We have developed a polarization-preserving near-field scanning optical microscopy (NSOM) optical fiber probe and with it observed the influence of defects and weak stresses on a propagation light within polymeric optical waveguides. To characterize the influence, we intentionally printed an indentation in the vicinity of the waveguide and then evaluated the resulting influence using polarized guide-collection-mode NSOM images taken around the indentation. When transverse magnetic polarized light enters a waveguide, the light intensity becomes greater on the near side of the indentation than on the far side, as measured by a linearly polarized component perpendicular to the direction of light propagation. The most probable cause of this phenomenon is microdefects generated by the printing of the indentation. The polarized NSOM technique is useful in searching for small defects or stresses within integrated photonic devices.     

A stochastic linear–quadratic problem with Lévy processes and its application to finance

We study a Linear–Quadratic Regulation (LQR) problem with Lévy processes and establish the closeness property of the solution of the multi-dimensional Backward Stochastic Riccati Differential Equation (BSRDE) with Lévy processes. In particular, we consider multi-dimensional and one-dimensional BSRDEs with Teugel’s martingales which are more general processes driven by Lévy processes. We show the existence and uniqueness of solutions to the one-dimensional regular and singular BSRDEs with Lévy processes by means of the closeness property of the BSRDE and obtain the optimal control for the non-homogeneous case. An application of the backward stochastic differential equation approach to a financial (portfolio selection) problem with full and partial observation cases is provided.