Dual Variational Principles and Pad?-type Approximants

Families of approximants for <?, f> are derived from dualvariational functionals associated with the linear equation(1+yL)?=f. Expansions in both ascending and descending powersof y are considered, and the approximants are identified eitheras one- or two-point Pad? approximants, or as approximants ofa closely related type. Compact formulae are obtained for theapproximants, and their duality and bounding properties areexhibited. Attention is paid to the special situations occurringwhen the non-negative self-adjoint operator L (i) has a zeroeigenvalue, (ii) is bounded.     

Hardy spaces on Lie groups of polynomial growth

We give several characterizations of Hardy spaces associated with complex, second-order,subelliptic operators on Lie groups with polynomial growth.     

Use of Atomic Force Microscopy to Determine the in vitro Growth of Hydroxyapatite Crystals from Mammalian Skeletal Tissues

The control of hydroxyapatite crystal deposition and growth has long been thought to be mediated via the proteins of the extracellular matrix^[1]. However, the precise mechanisms remain obscure. Dental enamel is the most extreme example of mammalian biomineralisation and provides an excellent model for studies of biological hydroxyapatite crystal growth. During its development, enamel passes through a number of discrete stages which have been defined both chemically and histologically. Elucidation of these mechanisms would be an important step in the design of novel therapeutics and treatment of osteopathologies. Unfortunately, despite a great deal of in vitro work using indirect measurements of crystal growth (for example by depletion of ions from solution) there have been no models to date capable of measuring crystal growth directly and in real time. Atomic force microscopy (AFM) has provided unprecedented opportunities for the imaging of biological specimens, generating quantitative data in three dimensions with molecular resolution.It is ideally suited for biological imaging as specimens do not need to be dehydrated, fixed, stained or coated and, most importantly, imaging can be carried out under fluids, under the biological conditions of choice^[2-3].     

Pad?-approximant Bounds and Approximate Solutions for Kirkwood-Riseman Integral Equations

Two-point Pad? approximants are used to calculate tight upperand lower bounds on the quantity <?, f> associated withKirkwood-Riseman integral equations (1+yL)?=f, which arise inthe diffusion theory of flexible macromolecules. The self-adjointoperator L is an integral operator on –1 x 1, with weaklysingular kernel |x – x'|^–?, and the two specificcases (i) f = 1, (ii) f = x² are studied. In case (i) directbounds on <?, 1> are obtained; this quantity is inverselyproportional to the translational diffusion constant. In case(ii) bounds on <?, 1 > are found by a new technique involvingcombinations of bounds for the three cases f = 1, f = x² andf = bx²?b^–1. Various types of Pade and related approximantsare compared, using the information <f, Lⁿf>, n = –2,–1, 0, 1, 2, 3 and (an upper bound on L) for severalvalues of the positive parameter y. Pad?-approximant-generating trial vectors are investigated anda convergence theorem is established. The vector consistingof an optimum linear combination of L^–1f, f and Lf isfound to be an accurate approximation to a numerical solutionin case (ii), for all values of y and x. Specific analyticalexpressions are derived for the approximate solutions.     

Signature Analysis Extended: A System Approach to Machinery Dynamics

Signature analysis has a definite function and role to play in understanding machinery dynamics and maintaining the health of operating systems. It does not, however, provide all the answers. Experimental modal analysis can be used to extend the reach of signature analysis by providing a more fundamental understanding of machinery dynamics. Combined experimental and analytical techniques for system analysis can take a further step in understanding and designing better machinery. These techniques allow the engineer to see how various components or subsystems interact, even before total system hardware is available. They also can provide valuable load information so that better, more efficient component design may be done.
The broad application of system analysis technology is just beginning to take place. Developments in hardware and user-oriented software have taken some of the "black-magic" out of the technology and have made it something that can be used as a routine tool. Many areas are open for development and are actively being pursued. A number of particular interest are:⁵ coupling of equipment dynamics with dynamic fluid forces created in attached piping; continuation of development of techniques for flexible rotor balancing using a system model, modal data base, and future instrumentation developments;⁶ additional refinement techniques for coupling of foundation dynamics with equipment dynamics; additional insight in the use of computerized condition monitoring systems by identifying the optimal locations for instrumentation and insight into the proper monitoring system diagnostics.