Some functional equations originating from number theory

We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.     

A new VOF‐based numerical scheme for the simulation of fluid flow with free surface. Part II: application to the cavity filling and sloshing problems

Finite element analysis of fluid flow with moving free surface has been performed in 2‐D and 3‐D. The new VOF‐based numerical algorithm that has been proposed by the present authors (Int. J. Numer. Meth. Fluids, submitted) was applied to several 2‐D and 3‐D free surface flow problems. The proposed free surface tracking scheme is based on two numerical tools; the orientation vector to represent the free surface orientation in each cell and the baby‐cell to determine the fluid volume flux at each cell boundary. The proposed numerical algorithm has been applied to 2‐D and 3‐D cavity filling and sloshing problems in order to demonstrate the versatility and effectiveness of the scheme. The proposed numerical algorithm resolved successfully the free surfaces interacting with each other. The simulated results demonstrated applicability of the proposed numerical algorithm to the practical problems of large free surface motion. It has been also demonstrated that the proposed free surface tracking scheme can be easily implemented in any irregular non‐uniform grid systems and can be extended to 3‐D free surface flow problems without additional efforts. Copyright © 2003 John Wiley & Sons, Ltd.     

SEQUENTIAL QUADRATIC PROGRAMMING METHODS FOR OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

A Kind of direct methods is presented for the solution of optimal control problems with state constraints.These methods are sequential quadratic programming methods.At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and Linear approximations to constraints is solved to get a search direction for a merit function.The merit function is formulated by augmenting the Lagrangian funetion with a penalty term.A line search is carried out along the search direction to determine a step length such that the merit function is decreased.The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadrade programming methods.     

Evaluation of compatibility in polymer blend systems by simultaneous DSC-FTIR measurement

The concept of crystallization dynamics method evaluating the miscibility of binary blend system including crystalline component was proposed. Three characteristic rates, nucleation, crystal growth rates (N*, G*) and growth rate of conformation (G _c*) were used to evaluate the miscibility of PVDF/at-PMMA and PVDF/iso-PMMA by the simultaneous DSC-FTIR. N*, G* and G _c* depended remarkably on both temperature and blend fraction (ϕ_PMMA) for PVDF/at-PMMA system, which indicated the miscible system. PVDF/iso-PMMA showed small ϕ_PMMA dependency of N*, G* and G _c*, was estimated the immiscible system. The ΔT/T _m⁰ values, corresponding to Gibbs energy required to attend the constant G* and G _c*, evaluated from G* and G _c* showed the good linear relationships with different slope. The experimental results suggested that the concentration fluctuation existed in PVDF/iso-PMMA system.     

Characteristics of Ti0.97Co0.03O2:Sb0.01 dilute magnetic semiconducting thin films deposited at 500 °C by pulsed laser deposition

Epitaxial Ti_0.97Co_0.03O₂:Sb_0.01(TCO:Sb) films were deposited on R-Al₂O₃ (1 1 0 2) substrates at 500 °C in various deposition pressures by pulsed laser deposition. The solubility of cobalt within the films increases with decreasing deposition pressure at a deposition temperature of 500 °C. The TCO:Sb films deposited at 5×10⁻⁶ Torr exhibit a p-type anomalous Hall effect having a hole concentration of 6.1×10²²/cm³ at 300 K. On the other hand, films deposited at 4×10⁻⁴ Torr exhibits an n-type anomalous Hall effect having an electron concentration of about 1.1×10²¹/cm³. p- or n-type DMS characteristics depends on the change of the structure of TCO:Sb films and the solubility of Co is possible by controlling the deposition pressure.     

Badge-type diffusive sampler using 3-methyl-2-benzothiazolinone hydrazone for measuring formaldehyde in indoor air.

The evaluation of a badge-type diffusive sampler for measuring formaldehyde using 3-methyl-2-benzothiazolinone hydrazone (MBTH) was investigated. On average, the formaldehyde concentration in blanks was reduced by approximately 31% by cleaning procedures. The cleaning techniques did not significantly differ in effectiveness. The maximum sampling rate was 22.4 +/- 3.5 mL min(-1) at MBTH concentrations of 0.05%. The formaldehyde concentration in blanks did not appreciably increase over a period of about 1 month at room temperature, and was 0.36 +/- 0.03 microg, with a relative standard deviation of 8%. The diffusive sampler had good precision and accuracy for measuring formaldehyde in indoor environments. For a 24-h exposure time, the limits of detection and quantification calculated with the field blanks were 9.7 and 13.8 ppb, respectively. The minimum exposure times were calculated based on the measured and calculated limits of quantification, the sampling rate, and the atmospheric formaldehyde concentration. The capacity of the diffusive sampler with 0.5% MBTH was 3 ppm h(-1), approximately 1.5-times the capacity when the MBTH concentrations were 0.05%.