The curvature invariant for a class of homogeneous operators

For an operator T in the class B-n(), introduced by Cowen and Douglas, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle E-T determine the unitary equivalence class of the operator T. In a subsequent paper, the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant derivatives are necessary to determine the unitary equivalence class of the operator T is an element of B-n(). Here we show that some of the covariant derivatives are necessary. Our examples consist of homogeneous operators in B-n(). For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives at any point w in the unit disc are determined from the simultaneous unitary equivalence class at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples. One of our main results is that the curvature along with its covariant derivative of order (0, 1) at 0 determines the equivalence class of generic homogeneous Hermitian holomorphic vector bundles over the unit disc.

Active nucleoprotein filaments of single-stranded binding protein and recA protein on single-stranded DNA have a regular repeating structure

When E. coli single-stranded DNA binding protein (SSB) coats single-stranded DNA (ssDNA) in the presence of 1 mM MgCl2 it inhibits the subsequent binding of recA protein, whereas SSB binding to ssDNA in 12 mM MgCl2 promotes the binding of recA protein. These two conditions correspond respectively to those which produce 'smooth' and 'beaded' forms of ssDNA-SSB filaments. By gel filtration and immunoprecipitation we observed active nucleoprotein filaments of recA protein and SSB on ssDNA that contained on average 1 monomer of recA protein per 4 nucleotides and 1 monomer of SSB per 20-22 nucleotides. Filaments in such a mixture, when digested with micrococcal nuclease produced a regular repeating pattern, approximately every 70-80 nucleotides, that differed from the pattern observed when only recA protein was bound to the ssDNA. We conclude that the beaded ssDNA-SSB nucleoprotein filament readily binds recA protein and forms an intermediate that is active in the formation of joint molecules and can retain substantially all of the SSB that was originally bound.

Wigner distributions for finite-state systems without redundant phase-point operators

We set up Wigner distributions for N-state quantum systems following a Dirac-inspired approach. In contrast to much of the work in this study, requiring a 2N x 2N phase space, particularly when N is even, our approach is uniformly based on an N x N phase-space grid and thereby avoids the necessity of having to invoke a `quadrupled' phase space and hence the attendant redundance. Both N odd and even cases are analysed in detail and it is found that there are striking differences between the two. While the N odd case permits full implementation of the marginal property, the even case does so only in a restricted sense. This has the consequence that in the even case one is led to several equally good definitions of the Wigner distributions as opposed to the odd case where the choice turns out to be unique.

d-Regular Graphs of Acyclic Chromatic Index at Least d+2

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). It was conjectured by Alon, Suclakov and Zaks (and earlier by Fiamcik) that a'(G) <= Delta+2, where Delta = Delta(G) denotes the maximum degree of the graph. Alon et al. also raised the question whether the complete graphs of even order are the only regular graphs which require Delta+2 colors to be acyclically edge colored. In this article, using a simple counting argument we observe not only that this is not true, but in fact all d-regular graphs with 2n vertices and d>n, requires at least d+2 colors. We also show that a'(K-n,K-n) >= n+2, when n is odd using a more non-trivial argument. (Here K-n,K-n denotes the complete bipartite graph with n vertices on each side.) This lower bound for Kn,n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that d >= 5, n >= 2d+3 and dn even, there exist d-regular graphs which require at least d+2-colors to be acyclically edge colored. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 63: 226-230, 2010.

Extension of Darken's Quadratic Formalism to Dilute Multicomponent Solutions

Darken's quadratic formalism is extended to multicomponent solutions. Equations are developed for the representation of the integral and partial excess free energies, entropies and enthalpies in dilute multicomponent solutions. Quadratic formalism applied to multicomponent solutions is thermodynamically consistent. The formalism is compared with the conventional second order Maclaurin series or interaction parameter representation and the relations between them are derived. Advantages of the quadratic formalism are discussed.

Modelling of symmetric laminates under extension

A higher-order theory of laminated composites under in-plane loads is developed. The displacement field is expanded in terms of the thickness co-ordinate, satisfying the zero shear stress condition at the surfaces of the laminate. Using the principle of virtual displacement, the governing equations and boundary conditions are established. Numerical results for interlaminar stresses arising in the case of symmetric laminates under uniform extension have been obtained and are compared with similar results available in the literature.

Spectral Invariance of SG Pseudo-Differential Operators on L-P(R-n)

We prove the spectral invariance of SG pseudo-differential operators on L-P(R-n), 1 < p < infinity, by using the equivalence of ellipticity and Fredholmness of SG pseudo-differential operators on L-p(R-n), 1 < p < infinity. A key ingredient in the proof is the spectral invariance of SC pseudo-differential operators on L-2(R-n).

Perturbation of critical solution temperatures by impurity doping

An expression is developed for the variation of the critical solution temperature of a binary liquid system when a third component (dopant) is added, using an extension of the regular solution theory. The model can be used for UCST, LCST and for closed loop systems and has the correct features in the limiting cases.

Descriptions of operators in quantum mechanics

The problem of expressing a general dynamical variable in quantum mechanics as a function of a primitive set of operators is studied from several points of view. In the context of the Heisenberg commutation relation, the Weyl representation for operators and a new Fourier-Mellin representation are related to the Heisenberg group and the groupSL(2,R) respectively. The description of unitary transformations via generating functions is analysed in detail. The relation between functions and ordered functions of noncommuting operators is discussed, and results closely paralleling classical results are obtained.

Wigner distribution for angle coordinates in quantum mechanics

The method of Wigner distribution functions, and the Weyl correspondence between quantum and classical variables, are extended from the usual kind of canonically conjugate position and momentum operators to the case of an angle and angular momentum operator pair. The sense in which one has a description of quantum mechanics using classical phase‐space language is much clarified by this extension.

A classification of homogeneous operators in the Cowen-Douglas class

An explicit construction of all the homogeneous holomorphic Hermitian vector bundles over the unit disc D is given. It is shown that every such vector bundle is a direct sum of irreducible ones. Among these irreducible homogeneous holomorphic Hermitian vector bundles over D, the ones corresponding to operators in the Cowen-Douglas class B-n(D) are identified. The classification of homogeneous operators in B-n(D) is completed using an explicit realization of these operators. We also show how the homogeneous operators in B-n(D) split into similarity classes. (C) 2011 Elsevier Inc. All rights reserved.

Metastable extension of the fluorite phase field in Y2O3---ZrO2 and its effect on grain growth

Pure Y2O3 and Y2O3---ZrO2 solid solutions have been prepared by melt atomization and by pyrolysis of nitrate solutions. Extended solubility is readily achieved in both techniques for the entire composition range investigated: melts with 0–30% ZrO2 and precursors with 0–50% ZrO2. However, solidification of under cooled droplets yields almost exclusively single phase powders with the structure of cubic yttria (D53). In contrast, the pyrolysis route leads to a sequence of metastable microstructures beginning with a nanocrystalline disordered fluorite-based (C1) solid solution. Further heating leads to the evolution of much larger (micron size) flake crystals with a {001} texture, concurrent with partial ordering of the oxygen ions to the sites occupied in the D53 structure. The driving force for ordering and the rate of grain growth decrease with increasing ZrO2 addition. Abrupt heating to high temperatures or electron irradiation can induce ordering without substantial grain growth. There is no significant reduction in porosity during the recrystallization, which with the other observations suggests that grain growth is driven by the free energy available for the ordering transformation from fluorite to the yttria structure. This route offers opportunities for single crystal thin film development at relatively low processing temperatures.

Metastable extension of the fluorite phase field in Y2O3---ZrO2 and its effect on grain growth

Pure Y2O3 and Y2O3-ZrO2 solid solutions have been prepared by melt atomization and by pyrolysis of nitrate solutions. Extended solubility is readily achieved in both techniques for the entire composition range investigated: melts with 0-30% ZrO2 and precursors with 0-50% ZrO2. However, solidification of under cooled droplets yields almost exclusively single phase powders with the structure of cubic yttria (D5(3)). In contrast, the pyrolysis route leads to a sequence of metastable microstructures beginning with a nanocrystalline disordered fluorite-based (C1) solid solution. Further heating leads to the evolution of much larger (micron size) flake crystals with a {001} texture, concurrent with partial ordering of the oxygen ions to the sites occupied in the D5(3) structure. The driving force for ordering and the rate of grain growth decrease with increasing ZrO2 addition. Abrupt heating to high temperatures or electron irradiation can induce ordering without substantial grain growth. There is no significant reduction in porosity during the recrystallization, which with the other observations suggests that grain growth is driven by the free energy available for the ordering transformation from fluorite to the yttria structure. This route offers opportunities for single crystal thin film development at relatively low processing temperatures.

Perturbative corrections for staggered four-fermion operators

We present results for one-loop matching coefficients between continuum four-fermion operators, defined in the Naive Dimensional Regularization scheme, and staggered fermion operators of various types. We calculate diagrams involving gluon exchange between quark fines, and ''penguin'' diagrams containing quark loops. For the former we use Landau-gauge operators, with and without O(a) improvement, and including the tadpole improvement suggested by Lepage and Mackenzie. For the latter we use gauge-invariant operators. Combined with existing results for two-loop anomalous dimension matrices and one-loop matching coefficients, our results allow a lattice calculation of the amplitudes for KKBAR mixing and K --> pipi decays with all corrections of O(g2) included. We also discuss the mixing of DELTAS = 1 operators with lower dimension operators, and show that, with staggered fermions, only a single lower dimension operator need be removed by non-perturbative subtraction.