Abstract. In this paper, we consider ill-posed problems which discretize to linear least squares problems with matrices K of high dimensions. The algorithm proposed uses K only as an operator and does not need to explicitly store or modify it. A method related to one of Lanczos is used to project the problem onto a subspace for which K is bidiagonal. It is then an easy matter to solve the projected problem by standard regularization techniques. These ideas are illustrated with some integral equations of the first kind with convolution kernels, and sample numerical results are given.

A detailed computational study is presented for the radial–axial modes— both leaky and nonleaky—in an infinitely clad isotropic rods. The complex phase velocities of leaky modes are located using an application of the argument principle. Particle orbits are determined, and leaky modes are shown to have an asymptotic leakage angle away from the interface. By using the homotopic methods of varying densities and elastic constants, clad-rod modes are compared with those in a bare rod. The topology of the clad-rod mode dispersion diagram differs qualitatively from that of a bare rod, even when the cladding has negligible density, with no velocity cutoffs and with wave mode knitting. Comparison is also given with modes occurring in a cladding without a rod present (a tunnel) and for a planar interface. Most leaky modes can be correlated with rod modes; only a limited number of tunnel modes exist. Energy flow contours within modes are also calculated. The local energy velocity, which generalizes group velocity, can vary considerably in the radial direction for bare rod modes. For leaky modes the contours are quite complex due to the cylindrical geometry, giving rise to apparent shift in wave position across the interface.

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£^t j 3t j These Proceedings contain research papers, discussions thereon, and panel discussions for the conference on "Fundamental Aspects of Dislocation Theory," held under the auspices of the Institute for Materials Research, April 21-25, 1969, at the laboratories of the National Bureau of Standards, Gaithersburg, Maryland. Approximately 75 contributed papers and two panel discussions are included. Topics covered are Dis-

The current flow for a local electrolytic cell with rectangular anode and cathode is calculated as a function of a polarization parameter, electrolyte thickness and conductivity, anode and cathode size, and zero current potential difference. The total current is obtained from the numerical computation (with error analysis) of a series solution of Laplace's equation assuming a linear polarization relation at the electrode-electrolyte interface. For high values of the polarization parameter and sufficiently thick electrolytes an asymptotic formula obtains for the current approximately proportional to the anodic fraction, as was found by Waber et al. However, in general the cell geometry greatly influences the amount of current flow, especially for anode shapes near to linear rather than near to square. This effect can be greater than an order of magnitude. The calculated results are compared with experimental data obtained by Ogburn and Schlissel and are shown to be in good agreement. Corrosion currents in randomly pitted surfaces may be estimated from the above results.