On the solutions of the linear integral equations of Volterra type

Some boundaries about the solution of the linear Volterra integral equations of the form f(t)=1?K*f were obtained as |f(t)|?1, |f(t)|?2 and |f(t)|?4 in (J. Math. Anal. Appl. 1978; 64 :381–397; Int. J. Math. Math. Sci. 1982; 5 (1):123–131). The boundary of the solution function of an equation in this type was found as |f(t)|?2ⁿ in (Integr. Equ. Oper. Theory 2002; 43 :466–479), where t∈[0, ∞) and n is a natural number such that n?2. In (Math. Comp. 2006; 75 :1175–1199), it is shown that the boundary of the solution function of an equation in the same form can also be derived as that of (Integr. Equ. Oper. Theory 2002; 43 :466–479) under different conditions than those of (Integr. Equ. Oper. Theory 2002; 43 :466–479). In the present paper, the sufficient conditions for the boundedness of functions f, f′, f′′, …, f⁽ⁿ⁺³⁾, (n∈?) defined on the infinite interval [0, ∞) are given by our method, where f is the solution of the equation f(t)=1?K*f. Copyright © 2007 John Wiley & Sons, Ltd.     

Electrochemical and optical properties of novel donor‐acceptor thiophene‐perylene‐thiophene polymers

In this study, donor‐acceptor type thiophene‐perylene‐thiophene monomers were synthesized and polymerized by both oxidative polymerization using FeCl₃ as catalyst and the electrochemical process. UV–vis, FTIR, ¹H NMR, and elemental analysis techniques were used for structural characterization. Thermal behaviors of these compounds were determined by using TGA system. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) energy levels and electrochemical and optical band gap values were calculated by using the results of cyclic voltammetry and UV–vis measurements, respectively. The number–average molecular weight (M_n), weight–average molecular weight (M_w), and polydispersity index (PDI) values of synthesized polymers were determined by size exclusion chromatography. Conductivity measurements of these polymers were carried out by electrometer by using a four‐point probe technique. The conductivity was observed to be increased by iodine doping. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 1974–1989, 2008     

In situ generated rhodium‐based catalyst for addition of phenylboronic acid to aldehydes

New 1,3‐dialkylperhydrobenzimidazolinium and 1,3‐dialkylimidazolinium salts ( 2,4 ) as NHC precursors have been synthesized and characterized. These salts in combination with [RhCl(COD)]₂ provided active catalysts for the addition of phenylboronic acid to aldehydes under mild conditions. The in situ prepared three‐component system [RhCl(COD)]₂/imidazolinium salts ( 2,4 ) and KOBu^t catalyse the addition of phenylboronic acid to sterically hindered aldehydes affording the corresponding arylated secondary alcohols in good yields. © 2005 Wiley Periodicals, Inc. Heteroatom Chem 16:461–465, 2005; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20132     

Component testing of repairable systems in multistage missions

We consider the component testing problem of a device that is designed to perform a mission consisting of a sequence of stages. Once a stage is over, the device is overhauled to replace all failed components before the next stage starts to improve mission reliability. The components fail exponentially where the failure rate depends on the stage of the mission. The reliability structure of the device involves a series connection of redundant subsystems. The optimal component testing problem is formulated as a semi-infinite linear programme. We present an algorithmic procedure to compute optimal test times based on the column generation technique, and illustrate it with numerical results.     

Positive integer solutions of the diophantine equation x 2 − L n xy + (−1) n y 2 = ±5 r

In this paper, we consider the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r and determine the values of n for which the equation has positive integer solutions x and y. Moreover, we give all positive integer solutions of the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r when the equation has positive integer solutions.     

Electrokinetic and electrorheological properties of poly(vinyl chloride)/polyindole conducting composites

In this study, poly(vinyl chloride) (PVC), polyindole (PIN), and PVC/PIN conducting composites having five different compositions were used. Particle sizes, densities, dielectric constants, and sedimentation ratios of the materials were determined. The zeta‐potentials of the samples were measured in aqueous and nonaqueous (silicone oil [SO]) media. The dispersions prepared in SO were subjected to external electric field strength, and their electrorheological properties were investigated. Then the effects of dispersed particle volume fraction, shear rate, external electric field strength, frequency, and temperature onto electrorheological activities of the dispersions were examined. Further, creep and creep‐recovery tests were applied to the PIN/SO and PVC/PIN/SO dispersions, and reversible viscoelastic deformations were observed. Finally, the vibration damping capacity of PVC (66%)/PIN (34%)/SO dispersion system was tested by using an automobile shock absorber. Copyright © 2011 John Wiley & Sons, Ltd.