Cauchy and the spectral theory of matrices

It is well-known that Cauchy (1829) provided the first general proof that the eigenvalues of a symmetric matrix are real. Furthermore, Cauchy's paper initiated the developments that resulted in the creation of a substantial spectral theory of matrices by the early 1870's. The following essay relates Cauchy's work and its historical significance to the consideration of eigenvalue problems during the 18th century.     

Capital Replacement Under Variable Discounting

This paper presents a theoretical model for dealing with uncertainity in capital investment. Utilizing the problem of recurring capital replacement, an optimum strategy is derived under variable discounting. A linear approximation to the variable discount function is given as an illustration.     

Geometric Quantization of Vector Bundles and the Correspondence with Deformation Quantization

I repeat my definition for quantization of a vector bundle. For the cases of the Toeplitz and geometric quantizations of a compact K?hler manifold, I give a construction for quantizing any smooth vector bundle, which depends functorially on a choice of connection on the bundle. Using this, the classification of formal deformation quantizations, and the formal, algebraic index theorem, I give a simple proof as to which formal deformation quantization (modulo isomorphism) is derived from a given geometric quantization. Received: 16 November 1998 / Accepted: 29 June 2000     

The influence of core geometry on the crystallography of silicon optical fiber

Crystalline semiconductor core optical fibers have received growing attention as greater understanding of the underlying materials science, coupled with advances in fiber processing and fabrication, have expanded the quality and portfolio of available materials. In a continued effort to better understand the nature of the crystal formation this work studies the role of the cross-sectional geometry on the resultant core crystallography with respect to the fiber axis. More specifically, a molten-core approach was used to fabricate silicon optical fibers clad in silica tubes of either circular or square inner cross-sections. In both geometric cases, the silicon core was found to possess regions of single crystallinity where specific crystal orientations persisted along a fiber length of about 4–5 mm prior to transitioning through polycrystalline regions. However, the rotation and tilting angular combination needed to align a given crystallographic axis with the fiber axis was more constant over the single crystalline region in the case of the square-core fiber while more significant variations were observed in the round-core case. This work begins to elucidate some of the microstructural features, not present in conventional glass optical fibers, that could be important for future low-loss single crystalline semiconductor optical fibers.     

Energy-dissipation mechanisms associated with rapid fracture of concrete

A hybrid experimental-numerical procedure, involving moiré interferometry and dynamic finite-element analysis, was used to analyze rapid crack growth in an impact loaded three-point-bend concrete specimen with an offset straight precrack. The dissipated energy rates in the fracture process zone (FPZ), which trails the rapidly extending crack, and in the frontal FPZ ahead of the crack tip, the kinetic-energy rate and energy-release rate were computed. The results showed that while the trailing FPZ was the dominant energy dissipation mechanism, much of the released energy was converted to kinetic energy in the fracturing concrete specimen. Paper was presented at the 1993 SEM Spring Conference on Experimental Mechanics held in Dearborn, MI on June 11.