Some results in asymptotic field point theory

In 1944, Levinson ([22]) introduced the concept of dissipativeness for a map T in a finite-dimensional space which leads to the existence of a fixed point of some iterate T ⁿ for n large, rather than a fixed point of T. Browder ([3]) gave an asymptotic field point theorem which proved that T itself had a field point. Although Browder’s result was a big step, it was not suitable for hyperbolic PDEs and neutral functional differential equations because, in those cases, the map T is not compact. For α-contraction maps the result was extended by Nussbaum ([25]) and Hale and Lopes ([13]) using different methods. In this paper, we review these ideas and some more recent applications. Dedicated to Felix Browder on the occasion of his 80th birthday     

Intralaboratory validation of cell-free translation assay for detecting ricin toxin biological activity

A cell-free translation (CFT) assay for determining ricin biological activity was validated. The statistical data from the validation study showed a high level of precision within and between runs of the assay. The assay was specific for determining ricin biological activity in food-based matrixes and discriminated ricin from other ribosome-inactivating proteins. The mean bias (relative error) between measured ricin concentrations of 3 validation samples and their nominal concentrations was 1.1, 6.6, and 20.3%, while the coefficient of variation (CV) was 14.1, 7.7, and 13.5%, respectively, demonstrating good precision, accuracy, and linearity. The CVs of ricin concentrations in 2 ricin-containing samples calculated from a dilution series were <5 and <12%, respectively, demonstrating very good parallelism. The analyte stability of ricin-containing samples stored for 1 month either at 4 or -20 degrees C, the stability of ricin stock solutions, and the results of assays executed by different analysts and using different luminometers were evaluated. The statistical validation data confirmed that the 4-parameter logistic equation, y = (a - d)/[1 + (x/c)b] + d, provided an accurate representation of a sigmoidal relationship between the measured response and the observed ricin concentration for the CFT assay.     

High density,high performance optical data storage via volume holography: Viability at last?

The long held promise of using volume holography to deliver high performance optical storage is reviewed. The problems, which limited the development for many years, are accessed. Finally, we describe a series of innovations, which may make the technology viable at last.     

A new stereocontrolled synthetic route to (-)-echinosporin from D-glucose via Padwa allenylsulfone [3 + 2]-anionic cycloadditive elimination

A new formal total synthesis of (-)-echinosporin has been developed based upon the Padwa [3 + 2]-cycloadditive elimination reaction of allenylsulfone 4 with the D-glucose-derived enone 14 which provides cycloadduct 12.     

On uniformity of exponential dichotomies for delay equations

It is shown that there cannot exist a uniform exponential dichotomy for any linear delay equation with a positive finite delay.     

Smoothness of bounded solutions of nonlinear evolution equations

It is shown that in many cases globally defined, bounded solutions of evolution equations are as smooth (in time) as the corresponding operator, even if a general solution of the initial-value problem is much less smooth; i.e., initial values for bounded solutions are selected in such a way that optimal smoothness is attained. In particular, solutions which bifurcate from certain steady states, such as periodic orbits, almost-periodic orbits and also homo- and heteroclinic orbits, have this property. As examples, a neutral functional differential equation, a slightly damped non-linear wave equation, and a heat equation are considered. In the latter case the space variable is included into the discussion of smoothness. Finally, generalized Hopf bifurcation in infinite dimensions is considered. Here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized.     

Eigenvalue distributions of large Hermitian matrices; Wigner's semi-circle law and a theorem of Kac,Murdock, and Szegö

Wigner's semi-circle law describes the eigenvalue distribution of certain large random Hermitian matrices. A new proof is given for the case of Gaussian matrices, that involves reducing a random matrix to tridiagonal form by a method that is well known as a technique for numerical computation of eigenvalues. The result is a generalized Toeplitz matrix whose eigenvalue distribution can be found using a theorem of Kac, Murdock, and Szegö. A new and more elementary proof of the latter is also given. The arguments use only direct L² estimates, rather than the transform methods or moment calculations employed previously.