The factorization approximation for the Master Equation

We examine the validity of the application of the Factorization Approximation to derive the Master Equation for a microscopic system coupled to a reservoir. We developed a formal perturbation expansion for the time evolution of the system reduced density matrix. We employed a diagrammatic schemes to produce each term of the perturbation series. The diagrams in the time domain provide a distinct criteria to distinguish the diagrams which survive the Factorization Approximation. The Feynmann-like diagrams in the energy domain, originated from the Resolvent method, are used for execution of diagram summations to estimate their overall contributions. We demonstrated that for a two level atomic system, interacting with a thermal reservoir, the summation over the diagrams which survived the Factorization Approximation, yields the proper time evolution of the system, in agreement with the solution of the Master Equation. The summation of the diagrams which are excluded by applying the Factorization Approximation are characterized by a dimensionless parameter: Γ/ω₀, where ω₀ is the frequency of the transition line, and Γ is the line width. The Factorization Approximation is thus rigorously justified when this expansion parameter is very small.     

The entropies of renewal systems

Renewal systems are symbolic dynamical systems originally introduced by Adler. IfW is a finite set of words over a finite alphabetA, then the renewal system generated byW is the subshiftX _W ⊂A ^Z formed by bi-infinite concatenations of words fromW. Motivated by Adler’s question of whether every irreducible shift of finite type is conjugate to a renewal system, we prove that for every shift of finite type there is a renewal system having the same entropy. We also show that every shift of finite type can be approximated from above by renewal systems, and that by placing finite-type constraints on possible concatenations, we obtain all sofic systems. The authors were supported in part by NFS grants DMS-8706284, DMS-8814159 and DMS-8820716.     

On centroid branches of trees from certain families

We obtain various results involving the centroid branches of randomly chosen trees from simply generated families.     

Some asymptotic results useful in enumeration problems

Suppose one is given two related generating functionsa(x) = a _n x ⁿ andb(x) = b _n x ⁿ , often it is of interest to determine the limiting behaviour of the quantitiesa _n /b _n We survey some earlier results of this nature and give some new ones     

XRD study of the dehydration and rehydration behaviour of Al-pillared smectites differing in source of charge

The swelling properties of Al-pillared clays, obtained from five different smectites, were studied using X-ray diffraction. These clays, the dioctahedral beidellite and montmorillonite and the trioctahedral saponite, hectorite and laponite differ in source of isomorphic substitution and represent a series of decreasing basicity along the siloxane plane. An Al oxyhydroxy cation was inserted between the layers to form the respective pillared clays and these clays were heated incrementally to 600°C. The XRD peaks at each stage of heating were recorded as well as the same samples subsequently wetted. Basal spacings of each clay at each stage of dehydration d rehydration indicated that the swelling of tetrahedrally substituted saponite and beidellite was indeed restricted, compared with the other three clays. This was attributed to greater basicity of the oxygen plane of beidellite and saponite due to tetrahedral substitution of Si by Al, resulting in an increase in the strength of hydrogen bonds between either water or the interlayer polyhydroxy cation and the clay.The data from the XRD analyses helped in addition, to clarify the thermal transformations of the Keggin ion itself. According to the changes in thed-spacings of the pillared clays it was concluded that the Keggin ion lost its structural water at 200°C and dehydroxylated in a range beginning at 350°C. Between 500 to 600°C this polymer cation, which is thought to form the Al₂O₃ oxide, did not rehydrate.Dedicated to Prof. Menachem Steinberg on the occasion of his 65th birthdayThe authors wish to thank Laporte Industries, Inc., U.K. for the laponite sample.     

Imprinting of chiral molecular recognition sites in thin TiO2 films associated with field-effect transistors: novel functionalized devices for chiroselective and chirospecific analyses

(R)- or (S)-2-Methylferrocene carboxylic acids, (R)-1 or (S)-1, (R)- or (S)-2-phenylbutanoic acid, (R)-2 or (S)-2, and (R)- or (S)-2-propanoic acid, (R)-3 or (S)-3, can be imprinted in thin TiO2 films on the gate surface of ion-sensitive field-effect transistor (ISFET) devices. The imprinting is performed by hydrolyzing the respective carboxylate TiIV butoxide complex on the gate surface, followed by washing off the acid from the resulting TiO2 film. The imprinted sites reveal chiroselectivity only towards the sensing of the imprinted enantiomer. The chiral recognition sites reveal not only chiroselectivity but also chirospecificity and, for example, the (R)-2-imprinted film is active in the sensing of (R)-2, but insensitive towards the sensing of (R)2-phenylpropanoic acid, (R)-3, which exhibits a similar chirality. Similarly, the (R)-3-imprinted film is inactive in the analysis of (R)-2. The chiroselectivity and chirospecificity of the resulting imprinted films are attributed to the need to align and fit the respective substrates in precise molecular contours generated in the cross-linked TiO2 films upon the imprinting process.     

The outer-distance of nodes in random trees

The outer-distance of a nodeu in a rooted treeT _n is the height of the subtree determined byu and all nodesv such thatu is on the path joiningv and the root ofT. We show that the expected outer-distance of nodes of treesT _n in certain families is asymptotic toB logn where the constantB depends on .