Nanobump microresonator

We introduce a whispering gallery-mode (WGM) nanobump microresonator (NBMR) and develop its theory. This microresonator is formed by an asymmetric nanoscale-high deformation of the translationally symmetric optical fiber surface, which is employed in fabrication of surface nanoscale axial photonics (SNAP) structures. It is shown that an NBMR causes strong localization of WGMs near a closed ray (geodesic) at the fiber surface, provided that this ray is stable. Our theory explains and describes the experimentally observed localization of WGMs by NBMRs and is useful for the design and fabrication of SNAP devices.

Algorithms for super-resolution of images based on Sparse Representation and Manifolds

lmage super-resolution is defined as a class of techniques that enhance the spatial resolution of images. Super-resolution methods can be subdivided in single and multi image methods. This thesis focuses on developing algorithms based on mathematical theories for single image super­ resolution problems. lndeed, in arder to estimate an output image, we adopta mixed approach: i.e., we use both a dictionary of patches with sparsity constraints (typical of learning-based methods) and regularization terms (typical of reconstruction-based methods). Although the existing methods already per- form well, they do not take into account the geometry of the data to: regularize the solution, cluster data samples (samples are often clustered using algorithms with the Euclidean distance as a dissimilarity metric), learn dictionaries (they are often learned using PCA or K-SVD). Thus, state-of-the-art methods still suffer from shortcomings. In this work, we proposed three new methods to overcome these deficiencies. First, we developed SE-ASDS (a structure tensor based regularization term) in arder to improve the sharpness of edges. SE-ASDS achieves much better results than many state-of-the- art algorithms. Then, we proposed AGNN and GOC algorithms for determining a local subset of training samples from which a good local model can be computed for recon- structing a given input test sample, where we take into account the underlying geometry of the data. AGNN and GOC methods outperform spectral clustering, soft clustering, and geodesic distance based subset selection in most settings. Next, we proposed aSOB strategy which takes into account the geometry of the data and the dictionary size. The aSOB strategy outperforms both PCA and PGA methods. Finally, we combine all our methods in a unique algorithm, named G2SR. Our proposed G2SR algorithm shows better visual and quantitative results when compared to the results of state-of-the-art methods.

Geodesically reversible Finsler 2-spheres of constant curvature

A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodesically reversible Finsler metric of constant flag curvature on the 2-sphere is necessarily projectively flat. As a corollary, using a previous result of the author, it is shown that a reversible Finsler metric of constant flag curvature on the 2-sphere is necessarily a Riemannian metric of constant Gauss curvature, thus settling a long- standing problem in Finsler geometry.

On surfaces with prescribed shape operator

The problem of immersing a simply connected surface with a prescribed shape operator is discussed. I show that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved are analyzed and it is found that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable and another depending essentially on two arbitrary functions of one variable. The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed.

The Herglotz variational problem on spheres and its optimal control approach

The main goal of this paper is to extend the generalized variational problem of Herglotz type to the more general context of the Euclidean sphere S^n. Motivated by classical results on Euclidean spaces, we derive the generalized Euler-Lagrange equation for the corresponding variational problem defined on the Riemannian manifold S^n. Moreover, the problem is formulated from an optimal control point of view and it is proved that the Euler-Lagrange equation can be obtained from the Hamiltonian equations. It is also highlighted the geodesic problem on spheres as a particular case of the generalized Herglotz problem.

Some Inverse Problems in Analysis and Geometry

Thesis (Ph.D.)--University of Washington, 2016-08

Cornered Asymptotically Hyperbolic Metrics

Thesis (Ph.D.)--University of Washington, 2016-08

Monitoreo de movimientos de terreno en el sector del cerro Tamuga del cantón Paute mediante técnicas DGPS

vulnerabilidad a deslizamientos ubicado en el Cerro Tamuga del cantón Paute, provincia del Azuay, la metodología empleada consiste en utilizar la técnica DGPS (Differential Global Positioning System), la misma que incluye el uso simultaneo de dos o más receptores, el método de medida empleado para las observaciones DGPS es el estático rápido con un tiempo de medida de diez minutos para cada hito, los resultados fueron comparados con mediciones realizadas con estación total, para lo que se aplicó el método de medida y cálculo de triangulación; que consiste en observar desde dos bases diferentes al mismo hito para realizar la triangulación y procesamiento de los datos. Durante la etapa de muestreo se realizó 20 campañas de medición con técnicas DGPS, monitoreando un total de 14 hitos, con técnicas convencionales (Topográficas) se realizó 7 campañas y se monitoreó 14 hitos. De estos datos se obtiene la diferencia entre la última y la primera medición tanto para valores de X, Y y Z, y por tanto se obtiene la variación de precisión para los dos métodos de medición (DGPS y Estación Total). Con los resultados (∆X, ∆Y, ∆Z) se realiza el análisis de la direccionalidad de los vectores de desplazamiento mediante la diferencia entre el promedio de todas las mediciones con el primer punto medido. Los resultados DGPS presentan menor variabilidad de los datos, por lo que se sugiere emplear esta técnica en la medición de desplazamiento en extensiones grandes. Con relación al caso de estudio del Cerro Tamuga, se determinó que mediante las mediciones con DGPS, éste no presenta movimientos, pero se deben continuar las campañas de monitoreo para analizar la situación a largo plazo.

Gravitational Waves: completeness and connectedness by geodesics

Only few months ago some physicists gave the official announcement that gravitational waves exist, but, from a geometrical point of view, they have always been ``real objects'' and their properties have been widely investigated. The aim of this talk is introducing generalized plane waves and discussing some of their properties such as geodesic connectedness and geodesic completeness.

Contribuição da Geodésia ao monitoramento costeiro do Litoral Setentrional do Rio Grande do Norte, área de influência da indústria petrolífera

The general objective of this thesis has been seasonal monitoring (quarterly time scale) of coastal and estuarine areas of a section of the Northern Coast of Rio Grande do Norte, Brazil, environmentally sensitive and with intense sediment erosion in the oil activities to underpin the implementation of projects for containment of erosion and mitigate the impacts of coastal dynamics. In order to achieve the general objective, the work was done systematically in three stages which consisted the specific objectives. The first stage was the implementation of geodetic reference infrastructure for carrying out the geodetic survey of the study area. This process included the implementation of RGLS (Northern Coast of the RN GPS Network), consisting of stations with geodetic coordinates and orthometric heights of precision; positioning of Benchmarks and evaluation of the gravimetric geoid available, for use in GPS altimetry of precision; and development of software for GPS altimetry of precision. The second stage was the development and improvement of methodologies for collection, processing, representation, integration and analysis of CoastLine (CL) and Digital Elevation Models (DEM) obtained by geodetic positioning techniques. As part of this stage have been made since, the choice of equipment and positioning methods to be used, depending on the required precision and structure implanted, and the definition of the LC indicator and of the geodesic references best suited, to coastal monitoring of precision. The third step was the seasonal geodesic monitoring of the study area. It was defined the execution times of the geodetic surveys by analyzing the pattern of sediment dynamics of the study area; the performing of surveys in order to calculate and locate areas and volumes of erosion and accretion (sandy and volumetric sedimentary balance) occurred on CL and on the beaches and islands surfaces throughout the year, and study of correlations between the measured variations (in area and volume) between each survey and the action of the coastal dynamic agents. The results allowed an integrated study of spatial and temporal interrelationships of the causes and consequences of intensive coastal processes operating in the area, especially to the measurement of variability of erosion, transport, balance and supply sedimentary over the annual cycle of construction and destruction of beaches. In the analysis of the results, it was possible to identify the causes and consequences of severe coastal erosion occurred on beaches exposed, to analyze the recovery of beaches and the accretion occurring in tidal inlets and estuaries. From the optics of seasonal variations in the CL, human interventions to erosion contention have been proposed with the aim of restoring the previous situation of the beaches in the process of erosion.

Efficient algorithms for beacon routing in polygons

In a paper by Biro et al. [7], a novel twist on guarding in art galleries is introduced. A beacon is a fixed point with an attraction pull that can move points within the polygon. Points move greedily to monotonically decrease their Euclidean distance to the beacon by moving straight towards the beacon or sliding on the edges of the polygon. The beacon attracts a point if the point eventually reaches the beacon. Unlike most variations of the art gallery problem, the beacon attraction has the intriguing property of being asymmetric, leading to separate definitions of attraction region and inverse attraction region. The attraction region of a beacon is the set of points that it attracts. For a given point in the polygon, the inverse attraction region is the set of beacon locations that can attract the point. We first study the characteristics of beacon attraction. We consider the quality of a "successful" beacon attraction and provide an upper bound of $\sqrt{2}$ on the ratio between the length of the beacon trajectory and the length of the geodesic distance in a simple polygon. In addition, we provide an example of a polygon with holes in which this ratio is unbounded. Next we consider the problem of computing the shortest beacon watchtower in a polygonal terrain and present an $O(n \log n)$ time algorithm to solve this problem. In doing this, we introduce $O(n \log n)$ time algorithms to compute the beacon kernel and the inverse beacon kernel in a monotone polygon. We also prove that $\Omega(n \log n)$ time is a lower bound for computing the beacon kernel of a monotone polygon. Finally, we study the inverse attraction region of a point in a simple polygon. We present algorithms to efficiently compute the inverse attraction region of a point for simple, monotone, and terrain polygons with respective time complexities $O(n^2)$, $O(n \log n)$ and $O(n)$. We show that the inverse attraction region of a point in a simple polygon has linear complexity and the problem of computing the inverse attraction region has a lower bound of $\Omega(n \log n)$ in monotone polygons and consequently in simple polygons.

Black holes as quantum bound states

We present a new quantum description for the Oppenheimer-Snyder model of gravitational collapse of a ball of dust. Starting from the geodesic equation for dust in spherical symmetry, we introduce a time-independent Schrödinger equation for the radius of the ball. The resulting spectrum is similar to that of the Hydrogen atom and Newtonian gravity. However, the non-linearity of General Relativity implies that the ground state is characterised by a principal quantum number proportional to the square of the ADM mass of the dust. For a ball with ADM mass much larger than the Planck scale, the collapse is therefore expected to end in a macroscopically large core and the singularity predicted by General Relativity is avoided. Mathematical properties of the spectrum are investigated and the ground state is found to have support essentially inside the gravitational radius, which makes it a quantum model for the matter core of Black Holes. In fact, the scaling of the ADM mass with the principal quantum number agrees with the Bekenstein area law and the corpuscular model of Black Holes. Finally, the uncertainty on the size of the ground state is interpreted within the framework of an Uncertainty Principle.