Inf-convolution and regularization of convex functions on Riemannian manifolds of nonpositive curvature

We show how an operation of inf-convolution can be used to approximate convex functions with $C^{1}$ smooth convex functions on Riemannian manifolds with nonpositive curvature (in a manner that not only is explicit but also preserves some other properties of the original functions, such as ordering, symmetries, infima and sets of minimizers), and we give some applications.Comment: 17 page

Viscosity solutions to second order partial differential equations on Riemannian manifolds

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$. Finest results (with hypothesis that require the function $F$ to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable $x$) are obtained under the assumption that $M$ has nonnegative sectional curvature, while, if one additionally requires $F$ to depend on $d^{2}u$ in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.Comment: Final version: the domain of F in the equation F=0 has been changed in order to get more generality and simplicity in the definitions and assumptions, and several important misprints have been correcte

Nonsmooth Morse-Sard theorems

We prove that every function $f:\mathbb{R}^n\to \mathbb{R}$ satisfies that the image of the set of critical points at which the function $f$ has Taylor expansions of order $n-1$ and non-empty subdifferentials of order $n$ is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential $\partial_{P}$, we see that for every lower semicontinuous function $f:\mathbb{R}^2\to\mathbb{R}$ the set $f(\{x\in\mathbb{R}^2 : 0\in\partial_{P}f(x)\})$ is $\mathcal{L}^{1}$-null.Comment: Final version. The main result has been strengthened thanks to the suggestions of a refere

Nanometer-scale placement in electron-beam lithography

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2000.Includes bibliographical references (p. 259-268).Electron-beam lithography is capable of high-resolution lithographic pattern generation (down to 10 nm or below). However, for conventional e-beam lithography, pattern-placement accuracy is inferior to resolution. Despite significant efforts to improve pattern placement, a limit is being approached. The placement capability of conventional e-beam tools is insufficient to fabricate narrow-band optical filters and lasers, which require sub-micrometer-pitch gratings with a high degree of spatial coherence. Moreover, it is widely recognized that placement accuracy will not be sufficient for future semiconductor device generations, with minimum feature sizes below 100 nm. In electron-beam lithography, an electromagnetic deflection system is used in conjunction with a laser-interferometer-controlled stage to generate high-resolution patterns over large areas. Placement errors arise because the laser interferometer monitors the stage position, but the e-beam can independently drift relative to the stage. Moreover, the laser interferometer can itself drift during exposure. To overcome this fundamental limitation, the method of spatial phase-locked electron-beam lithography has been proposed. The beam position is referenced to a high-fidelity grid, exposed by interference lithography, on the substrate surface. In this method, pattern-placement performance depends upon the accuracy of the reference grid and the precision with which patterns can be locked to the grid. The grid must be well characterized to serve as a reliable fiducial. This document describes work done to characterize grids generated by interference lithography. A theoretical model was developed to describe the spatial-phase progression of interferometric gratings and grids. The accuracy of the interference lithography apparatus was found to be limited by substrate mounting errors and uncertainty in setting the geometrical parameters that determine the angle of interference. Experimental measurements were performed, which agreed well with the theoretical predictions. A segmented-grid spatial-phase locking system was implemented on a vector-scan e-beam tool to correct field placement errors, in order to fabricate high-quality Bragg reflectors for optical filters and distributed-feedback lasers. Before this work, Bragg reflectors of adequate fidelity had not been fabricated by e-beam lithography. The phase coherence of the gratings fabricated with the segmented-grid method was characterized by measuring the displacement between adjacent fields. From these measurements, field-placement errors of ~ 20 nm (mean + 3 sigma) were estimated. The segmented grid method was used to pattern Bragg gratings, which were used in the fabrication of integrated optical filters. The devices demonstrated excellent performance.by Juan Ferrera.Ph.D

Intercalation and dynamics of hydrated Fe2+ in the vermiculites from Santa Olalla and Ojén

Although the intercalation of Fe3+ into layered phyllosicilicates-especially into smectites-attracted much attention in the past two decades, the information about Fe2+ loaded phyllosilicates is sparse. Here we present an investigation of the Fe2+ exchanged vermiculites from Santa Olalla and Ojén (Andalusia, Spain) by means of Mössbauer spectroscopy. The room temperature Mössbauer spectra are very similar to those of the starting compounds (Na forms) except for a decrease of the contribution of structural Fe3+ and a concomitant increase of the contribution of Fe2+ sites, indicating an internal redox process. The extent of this redox reaction is different for the two vermiculites. Thus, the intercalated Fe2+ acts as an electron mediator from the external medium to the structural Fe3+ ions. A new component attributable to intercalated Fe2+ is practically invisible in the room temperature Mössbauer spectra, but increases strongly and continuously during cooling to 4.2 K, where it is the dominant feature of the Mössbauer patterns. At 4.2 K, its quadruple splitting amounts to 3.31 mm/s, which is in excellent agreement with the quadrupole slitting of Fe2+ coordinated to six water molecules in a highly symmetric octahedral arrangement. The strong decrease of the Mössbauer-Lamb factor of this component with increasing temperature indicates a weak bonding of the Fe 2+ in the interlayer space

BIOMECHANICAL ANALYSIS APPLIED TO THE EVALUATION OF POSTURE IN LAPAROSCOPIC SURGERY

Laparoscopic techniques allow less-invasive treatment of common surgical problems in the field of Sports Medicine and Orthopaedics. Laparoscopic instruments have been reported as being uncomfortable, leading to musculoskeletal disorders (pain, stiffness and numbness) during and after the work, due to the forced posture that surgeons have to adopt for long periods (Berguer, et al. 1999; 2001). These muscular disorders are work related (WRMSDs) because of the remote visualization of the surgical field and the deficient design of the laparoscopic instruments. Even if several studies have been developed to obtain information in this field, there is still lack of research findings concerning the surgical procedures developed in the "real world" of the operating room conditions. The main aim of this study in progress, is to establish objective criteria concerning the design and use of laparoscopic instruments using the Ergonomics methodology. Besides, research findings could support the training process of the surgeons. To this purpose there is a need for developing a feasible methodology to detect the presence of risk factors on the surgeons' health