Accelerating the convergence of the method of alternating projections

The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. It achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the individual subspaces which make up the intersection. The main practical drawback of this algorithm, at least for some applications, is that the method is slowly convergent. In this paper, we consider a general class of iterative methods which includes the MAP as a special case. For such methods, we study an ``accelerated' version of this algorithm that was considered earlier by Gubin, Polyak, and Raik (1967) and by Gearhart and Koshy (1989). We show that the accelerated algorithm converges faster than the MAP in the case of two subspaces, but is, in general, not faster than the MAP for more than two subspaces! However, for a ``symmetric' version of the MAP, the accelerated algorithm always converges faster for any number of subspaces. Our proof seems to require the use of the Spectral Theorem for selfadjoint mappings.

     

Joint local quasinilpotence and common invariant subspaces

In this article we obtain some positive results about the existence of a common nontrivial invariant subspace forN-tuples of not necessarily commuting operators on Banach spaces with a Schauder basis. The concept of joint quasinilpotence plays a basic role. Our results complement recent work by Kosiek [6] and Ptak [8].     

Stability of Decoherence-Free Subspaces under Stochastic Phase Fluctuations

We study the stability of decoherence-free subspaces under stochastic phase fluctuations by analytically and numerically evaluating the fidelity of the corresponding decoherence-free subspace bases with stochastic phase fluctuations under the evolution of environment. The environment is modeled by a bath of oscillators with infinite degrees of freedom and the register-bath coupling is chosen to be a general dissipation-decoherence form. It is found that the decoherence-free subspaces take on good stability in the case of small dissipation and small phase fluctuations.     

Conjugate residual methods for almost symmetric linear systems

This paper concerns the use of conjugate residual methods for the solution of nonsymmetric linear systems arising in applications to differential equations. We focus on an application derived from a seismic inverse problem. The linear system is a small perturbation to a symmetric positive-definite system, the nonsymmetries arising from discretization errors in the solution of certain boundary-value problems. We state and prove a new error bound for a class of generalized conjugate residual methods; we show that, in some cases, the perturbed symmetric problem can be solved with an error bound similar to the one for the conjugate residual method applied to the symmetric problem. We also discuss several applications for special distributions of eigenvalues.This work was supported in part by the National Science Foundation, Grants DMS-84-03148 and DCR-81-16779, and by the Office of Naval Research, Contract N00014-85-K-0725.     

On quasispectral maximal subspaces of a class of Volterra-type operators

The concept of quasispectral maximal subspaces for quasinilpotent (but not nilpotent) operators was introduced by M. Omladi\v{c} in 1984. As an application a class of quasinilpotent operators on -spaces, close to the Volterra kernel operator, was studied. In the present Banach function space setting we determine all quasispectral maximal subspaces of analogues of such operators and prove that these subspaces are all the invariant bands. An example is given showing that (in general) they are not all the closed, invariant ideals of the operator.

     

Multiplication and Compact-friendly Operators

During the last few years the authors have studied extensively the invariant subspace problem of positive operators; see [6] for a survey of this investigation. In [4] the authors introduced the class of compact-friendly operators and proved for them a general theorem on the existence of invariant subspaces. It was then asked if every positive operator is compact-friendly. In this note, we present an example of a positive operator which is not compact-friendly but which, nevertheless, has a non-trivial closed invariant subspace.In the process of presenting this example, we also characterize the multiplication operators that commute with non-zero finite-rank operators. We show, among other things, that a multiplication operator M commutes with a non-zero finite-rank operator if and only the multiplier function is constant on some non-empty open set.     

Precolumn derivatization of reducing carbohydrates with 4-(3-Methyl-5-oxo-2-pyrazolin-1-yl) benzoic acid. Study of reaction, high-performance liquid chromatographic separation and quantitative performance of method

Summary A simple method for quantitative determination of carbohydrates by reversed-phase, high-performance liquid chromatography after pre-column derivatization and UV detection has been developed. Reducing sugars are condensed with 4-(3-methyl-5-oxo-2-pyrazolin-l-yl) benzoic acid (PMPA) to yield UV adducts absorbing at 271 nm, which are resolved under typical reversed-phase conditions. After derivatization, excess PMPA is easily removed from the reaction mixture by precipitation with mineral acids at pH<4. The influence of experimental conditions on reaction yield, as well as chromatographic separation of derivatives, were investigated. The quantitative performance was evaluated by means of a protocol comprising replicate measurements at several analyte levels. The calibration curves obtained for 8 sugars showed excellent linearity over 10–5000 pmol. Limits of detection and quantification for several monosaccharides were ca. 10 and 50 picomoles, respectively. Optimized conditions were successfully used for quantitative determination of monosaccharides released after hydrolysis from fetuin, mucin, α₁-acid glycoprotein, ovalbumin and transferrin.     

Note on a Density Question for Neural Networks

We prove a uniform density result for three-layer feedforward sigma-pi neural networks of hyperbolic type.