A Review of Results on Axially Symmetric Navier-Stokes Equations,with Addendum by X. Pan and Q. Zhang

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail.In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.     

q-Hypergeometric Series and Macdonald Functions

We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type ₄₃, for which we obtain linear transformation formulas in terms of the Jacobi theta function and the q-Gamma function. The transformation formulas are then used to give the duality formula and a new formula for two-row Macdonald functions in terms of the vertex operators. The Jack polynomials are also treated accordingly.     

Time-dependent fluctuations in infrared vsCH2 frequencies in acyl chain membranes of Acholeplasma laidlawii cells

We measured FT-IR spectra of intact Acholeplasma laidlawii cells grown at 37 °C on palmitic acid (C16:0) or on binary palmitic acid-d31/oleic acid (C16:0-d31/C18:1(9)) at an initial mole ratio of 2:3, which have been previously reported to produce significant fluctuations in CH₂ symmetric stretching (ν_sCH₂) and CD₂ asymmetric stretching (ν_aCD₂) frequencies (Biochim. Biophys. Acta 1279 (1996) 49). Time courses for acyl chain ν_sCH₂ and ν_aCD₂ frequencies determined from fourth derivative spectra are presented. Fluctuations were detected with the C16:0 enriched cells at temperatures above 40 °C as well as with the cells enriched in 2:3 C16:0-d₃₁/C18:1(9). These observations at temperatures above 40 °C for the C16:0 enriched cells were not in agreement with the conclusion in the previous work by Moore et al. Our results have suggested that the 2850 cm⁻¹ ν_sCH₂ band comprises two components arising from trans and gauche conformations, and that the fluctuations in ν_sCH₂ frequency are caused by random temporal changes in the relative intensities of these two components.     

The computation of the representation matrices of the generators of the unitary group

A method is presented for the efficient computation of the representation matrices of the unitary group, U(n) in the Gelfand—Tsetlin basis (corresponding to the usual spin-symmetry adapted basis for an N electron CI). The present scheme is conceptually and computationally attractive in that it is formulated directly in terms of Weyl tableaux and also that it permits simultaneous basis vector generation and matrix element evaluation. In addition the basis vectors are ordered so that subsequent restriction to the three dimensional rotation group is facilitated. An illustrative example is also presented.Taken in part from a thesis submitted to the University of London in partial fulfilment of the requirements for the degree of PhD.     

Investigation of the limitations of saturated fractional factorial experimental designs,with confounding effects for an HPLC ruggedness test

Summary Recently much attention has been directed to the validation of analytical methods. The level of validation will depend on the method application. If a method is developed for general use, for instance by different analysts, instruments or laboratories, then the effect of these changes need evaluation. This is normally referred to as ruggedness testing, where the limitations of the method to changes in the specified conditions are examined.This paper examines the use of Plackett-Burman fractional factorial experiment designs for ruggedness testing in high performance liquid chromatography (HPLC). The ability of these designs to identify confounding effects is applied to the stability indicating analysis of Salbutamol and its major degradation product, AH4045. The results show that confounding effects, proving extremely pertinent to HPLC, can be identified.The assay proved rugged to most tested changes although not to the use of column packing material from different manufacturers.     

Bayes estimation with spherically symmetric,convex loss

Summary It is desired to estimate a parameter with the loss function of the formL(θ, a)=W(‖θ−a‖), where is convex, differentiable, and non-decreasing. With this structure a characterization of Bayes estimators is given. Also it is noted that if the sample space, , for the observation,X, is a complete separable metric space then a Bayes estimator exists.     

Locally s-distance transitive graphs and pairwise transitive designs

The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs with a star quotient is equivalent to the study of a particular family of designs with strong symmetry properties that we call nicely affine and pairwise transitive. We show that a group acting regularly on the points of such a design must be abelian and give general construction for this case.     

The maximum number of columns in supersaturated designs with

Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows () and a maximum in absolute value correlation between any two columns (). In particular, they proved that for (mod ) and . However, the only known SSD satisfying this upper bound is when . By utilizing a computer search, we prove that for , and . These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the lower bound for SSDs with rows and columns, for , and . Finally, we show that a skew‐type Hadamard matrix of order can be used to construct an SSD with rows and columns that proves . Hence, we establish for and for all (mod ) such that . Our result also implies that when is a prime power and (mod ). We conjecture that for all and (mod ), where is the maximum number of equiangular lines in with pairwise angle .