The Minimum Number of Points Taking Part in k-Sets in Sets of Unaligned Points

En este trabajo se da un ejemplo de un conjunto de n puntos situados en posición general, en el que se alcanza el mínimo número de puntos que pueden formar parte de algún k-set para todo k con 1menor que=kmenor quen/2. Se generaliza también, a puntos en posición no general, el resultado de Erdõs et al., 1973, sobre el mínimo número de puntos que pueden formar parte de algún k-set. The study of k- sets is a very relevant topic in the research area of computational geometry. The study of the maximum and minimum number of k-sets in sets of points of the plane in general position, specifically, has been developed at great length in the literature. With respect to the maximum number of k-sets, lower bounds for this maximum have been provided by Erdõs et al., Edelsbrunner and Welzl, and later by Toth. Dey also stated an upper bound for this maximum number of k-sets. With respect to the minimum number of k-set, this has been stated by Erdos el al. and, independently, by Lovasz et al. In this paper the authors give an example of a set of n points in the plane in general position (no three collinear), in which the minimum number of points that can take part in, at least, a k-set is attained for every k with 1 ≤ k < n/2. The authors also extend Erdos’s result about the minimum number of points in general position which can take part in a k-set to a set of n points not necessarily in general position. That is why this work complements the classic works we have mentioned before.

Perfect imaging of three object points with only two analytic lens surfaces in two dimensions

In this work, a new two-dimensional analytic optics design method is presented that enables the coupling of three ray sets with two lens profiles. This method is particularly promising for optical systems designed for wide field of view and with clearly separated optical surfaces. However, this coupling can only be achieved if different ray sets will use different portions of the second lens profile. Based on a very basic example of a single thick lens, the Simultaneous Multiple Surfaces design method in two dimensions (SMS2D) will help to provide a better understanding of the practical implications on the design process by an increased lens thickness and a wider field of view. Fermat?s principle is used to deduce a set of functional differential equations fully describing the entire optical system. The transformation of these functional differential equations into an algebraic linear system of equations allows the successive calculation of the Taylor series coefficients up to an arbitrary order. The evaluation of the solution space reveals the wide range of possible lens configurations covered by this analytic design method. Ray tracing analysis for calculated 20th order Taylor polynomials demonstrate excellent performance and the versatility of this new analytical optics design concept.

Meromorphic continuation of functions and arbitrary distribution of interpolation points

We characterize the region of meromorphic continuation of an analytic function ff in terms of the geometric rate of convergence on a compact set of sequences of multi-point rational interpolants of ff. The rational approximants have a bounded number of poles and the distribution of interpolation points is arbitrary.

Image analysis by moment invariants using a set of step-like basis functions

Moment invariants have been thoroughly studied and repeatedly proposed as one of the most powerful tools for 2D shape identification. In this paper a set of such descriptors is proposed, being the basis functions discontinuous in a finite number of points. The goal of using discontinuous functions is to avoid the Gibbs phenomenon, and therefore to yield a better approximation capability for discontinuous signals, as images. Moreover, the proposed set of moments allows the definition of rotation invariants, being this the other main design concern. Translation and scale invariance are achieved by means of standard image normalization. Tests are conducted to evaluate the behavior of these descriptors in noisy environments, where images are corrupted with Gaussian noise up to different SNR values. Results are compared to those obtained using Zernike moments, showing that the proposed descriptor has the same performance in image retrieval tasks in noisy environments, but demanding much less computational power for every stage in the query chain.

Discontinuity Set Extractor

Disponible en Github: https://github.com/adririquelme/DSE

New York, N.Y. and vicinity, 1961 : Paterson, N.J.-N.Y. (8 Sheet Set) (Raster Image)

This layer is a georeferenced raster image of the United States Geological Survey 7.5 minute topographic sheet map entitled: New York and vicinity : Paterson, N.J.-N.Y., 1955. It is part of an 8 sheet map set covering the metropolitan New York City area. It was published in 1961. Scale 1:24,000. The source map was prepared by the Geological Survey from 1:24,000-scale maps of Hackensack, Paterson, Orange, and Weehawken 1955 7.5 minute quadrangles. The Orange quadrangle was previously compiled by the Army Map Service. Culture revised by the Geological Survey. Hydrography compiled from USC&GS charts 287 (1954), 745 (1956), and 746 (1956). The image inside the map neatline is georeferenced to the surface of the earth and fit to the Universal Transverse Mercator (UTM) Zone 18N NAD27 projection. All map collar and inset information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, index maps, legends, or other information associated with the principal map. USGS maps are typical topographic maps portraying both natural and manmade features. They show and name works of nature, such as mountains, valleys, lakes, rivers, vegetation, etc. They also identify the principal works of humans, such as roads, railroads, boundaries, transmission lines, major buildings, etc. Relief is shown with standard contour intervals of 10 and 20 feet; depths are shown with contours and soundings. Please pay close attention to map collar information on projections, spheroid, sources, dates, and keys to grid numbering and other numbers which appear inside the neatline. This layer is part of a selection of digitally scanned and georeferenced historic maps from The Harvard Map Collection as part of the Imaging the Urban Environment project. Maps selected for this project represent major urban areas and cities of the world, at various time periods. These maps typically portray both natural and manmade features at a large scale. The selection represents a range of regions, originators, ground condition dates, scales, and purposes.

New York, N.Y. and vicinity, 1961 : Harlem, N.Y.-N.J. (8 Sheet Set) (Raster Image)

This layer is a georeferenced raster image of the United States Geological Survey 7.5 minute topographic sheet map entitled: New York and vicinity : Harlem, N.Y.-N.J., 1956. It is part of an 8 sheet map set covering the metropolitan New York City area. It was published in 1961. Scale 1:24,000. The source map was compiled from 1:24,000-scale maps of Mount Vernon 1956, Yonkers 1956, Central Park 1956, and Flushing 1955 7.5 minute quadrangles. Hydrography compiled from USC&GS charts 222 (1955), 223 (1954), 748 (1955), 226, 274, 745, 746, and 747 (1956). The image inside the map neatline is georeferenced to the surface of the earth and fit to the Universal Transverse Mercator (UTM) Zone 18N NAD27 projection. All map collar and inset information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, index maps, legends, or other information associated with the principal map. USGS maps are typical topographic maps portraying both natural and manmade features. They show and name works of nature, such as mountains, valleys, lakes, rivers, vegetation, etc. They also identify the principal works of humans, such as roads, railroads, boundaries, transmission lines, major buildings, etc. Relief is shown with standard contour intervals of 10 and 20 feet; depths are shown with contours and soundings. Please pay close attention to map collar information on projections, spheroid, sources, dates, and keys to grid numbering and other numbers which appear inside the neatline. This layer is part of a selection of digitally scanned and georeferenced historic maps from The Harvard Map Collection as part of the Imaging the Urban Environment project. Maps selected for this project represent major urban areas and cities of the world, at various time periods. These maps typically portray both natural and manmade features at a large scale. The selection represents a range of regions, originators, ground condition dates, scales, and purposes.