Implicit function theorems and lagrange multipliers

Several implicit function theorems are proved, for systems of inequalities as well as equalities, assuming weaker differentiability than continuous Fréchet. These results give sufficient hypotheses for the Kuhn-Tucker conditions to hold, in a constrained minimization problem, with cone constraints and in any dimension. A condition is given for an inequality constraint system to be equivalently replaced by another, for which the function involved has surjective derivative. This allows some known implicit function theorems to be applied to inequality systems.     

Symmetric dual fractional programming

A pair of symmetric dual fractional programming problems is formulated and appropriate duality theorems are established.

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Zusammenfassung Es wird ein symmetrisches Paar primaler und dualer Quotientenprogramme eingeführt, für das Dualitätsbeziehungen hergeleitet werden.

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Research by this author was carried out while he was a visiting research fellow at the University of Melbourne.     

An algorithm for minimax

An algorithm is proposed for computing an unconstrained minimax, based on differential equations with suitable stabilization terms. Methods for accelerating the convergence are discussed. For computing a constrained minimax, the augmented Lagrangian algorithm of Powell, Hestenes and Rockafellar is generalized to minimax, assuming the unconstrained minimax algorithm as a subroutine. An estimate of the convergence rate is obtained.     

Characterisation of InAs/GaAs short period superlattices using column ratio mapping in aberration-corrected scanning transmission electron microscopy

The image processing technique of column ratio mapping was applied to aberration-corrected high angle annular dark field (HAADF) images of short period MBE (molecular beam epitaxy) grown InAs/GaAs superlattices. This method allowed the Indium distribution to be mapped and a more detailed assessment of interfacial quality to be made. Frozen-phonon multislice simulations were also employed to provide a better understanding of the experimental column ratio values. It was established that ultra-thin InAs/GaAs layers can be grown sufficiently well by MBE. This is despite the fact that the Indium segregated over 3-4 monolayers. Furthermore, the effect of the growth temperature on the quality of the layers was also investigated. It was demonstrated that the higher growth temperature resulted in a better quality superlattice structure.     

A combinatorial selective labeling method for the assignment of backbone amide NMR resonances

A combinatorial selective labeling (CSL) method is presented for the assignment of backbone amide NMR resonances, which has a particular application in the identification of protein-ligand interaction sites. The method builds on the dual amino acid selective labeling technique. In the CSL method a number of different samples are produced, each with a different pattern of labeled amino acids. By analyzing peak intensities in HSQC and 2D HNCO spectra of these samples, a large number of combinations of amino acid pairs can be simultaneously assigned. We demonstrate the method on the 27 kDa protein GFP. The samples can be produced rapidly and cost-effectively in a commercially available in vitro translation system. The method greatly simplifies the process of backbone assignment and would be very straightforward to automate.     

Specific detection of Clostridium botulinum types A,B, E,and F using the polymerase chain reaction

Clostridium botulinum organisms generally produce 1 of 4 neurotoxin types (A, B, E, and F) associated with human illness. Neurotoxin type determination is important in identification of the bacterium. A polymerase chain reaction (PCR) method was developed to identify 24 h botulinal cultures as potential types A, B, E, and F neurotoxin producers as well as other clostridial species which also produce neurotoxins. Components of the PCR and amplification conditions were adjusted for optimal amplification of toxin gene target regions to enable simultaneous testing for types A, B, E, and F in separate tubes using a single thermal cycler. Each primer set was specific for its corresponding toxin type. A DNA extraction procedure was also included to remove inhibitory substances that may affect amplification. This procedure is rapid, sensitive, and specific for identification of toxigenic C. botulinum.     

Weak minimization and duality

Sufficient and necessary optimality conditions are given for weakly minimized optimization problems in terms of a vector valued Lagrangian. Lagrangian and Wolfe type duals are constructed and duality established using an ordering that accords with the definition of a weak minimum. The results for differentiable problems continue to hold under weakened convexity assumptions and for problems which quasiminimize rather than minimize.