Counting derangements, involutions and unimodal elements in the wreath product Cr ? Sn

We count derangements, involutions and unimodal elements in the wreath product C _r ≀ S _n by the numbers of excedances, fixed points and 2-cycles. Properties of the generating functions, including combinatorial formulas, recurrence relations and real-rootedness are studied. The results obtained specialize to those on the symmetric group S _n and on the hyperoctahedral group B _n when r = 1, 2, respectively.     

The commutation relation <Emphasis Type="Italic">xy</Emphasis>=<Emphasis Type="Italic">qyx</Emphasis>+<Emphasis Type="Italic">hf</Emphasis>(<Emphasis Type="Italic">y</Emphasis>) and Newton’s binomial formula

Let x and y be two variables satisfying the commutation relation xy=qyx+hf(y), where f(y) is a polynomial. In this paper, using Young diagrams and generating functions techniques, we study the binomial formula (x+y) ⁿ and we present an identity for x ^m y. The connection to Operator Calculus is discussed and several special cases are treated explicitly.     

On integer and fractional parts of powers of Salem numbers

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers , when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αⁿ are divisible by N for infinitely many n if and only if N≦ 2d − 3. Received: 27 April 2005     

Approximate Controllability of Fractional Neutral Stochastic Evolution Equations with Nonlocal Conditions

In this paper, the approximate controllability of neutral stochastic fractional differential equations involving nonlocal initial conditions is studied. By using Sadovskii’s fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of semilinear fractional stochastic differential equations with nonlocal conditions under the assumption that the corresponding linear system is approximately controllable. Finally, an application to a fractional partial stochastic differential equation with nonlocal initial condition is provided to illustrate the obtained theory.     

Oxidizing agent concentration effect on metal-assisted electroless etching mechanism in HF-oxidizing agent-H2O solutions

The mechanism of metal-assisted electroless etching of silicon in HF-oxidizing agent-H₂O etching system as a function of oxidizing agent concentration was studied. Three types of oxidizing agent were experimented namely Na₂S₂O₈, K₂Cr₂O₇ and KMnO₄. Their concentrations were varied from 0.05 M to 0.3 M. The layers formed on silicon were investigated by scanning electron microscopy (SEM), X-ray diffraction (XRD) and energy-dispersive X-ray (EDX). It is shown that an insoluble solid-phase film (K₂SiF₆) form on silicon surface when concentration of K₂Cr₂O₇ or KMnO₄ increases in chemical solutions. On other hand, when Na₂S₂O₈ concentration increases, the surface roughness decreases without any chemical complex formation.     

An alternative approach for the estimation of equivalent temperature in gas chromatography

The temperature at which the isothermal retention time (tRiso) is equal to the temperature-programmed retention time (tRTPGC) or the isothermal retention index (Iiso) is equal to the temperature-programmed retention index (ITPGC) is defined as the equivalent temperature (Teq). The Teq of one-, two-, three-, and four-step temperature-programmed gas chromatography (TPGC) of unsaturated fatty acid methyl esters (FAMEs) from Chinese mustard seed oil is calculated. All of the tR values obtained at the Teq (tRTeq) are very close to the tRTPGC. The highest difference for each chromatogram is less than 0.5%. The slight deviation may partly arrive from the difficulty in setting the desired carrier gas flow rate. Also, good agreement among equivalent chain length values determined by different methods is observed, including the graphical method at the Teq. Therefore, it is speculated that the proposed method may facilitate the GC identification of FAMEs as well as other organic compounds in TPGC by using the available isothermal retention index database.