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This is a post-peer-review, pre-copyedit version of an article published in Journal of Global Optimization. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10898-021-01020-3, Let O be a set of k orientations in the plane, and let P be a simple polygon in the plane. Given two points p, q inside P, we say that p O-sees q if there is an O-staircase contained in P that connects p and q. The O-Kernel of the polygon P, denoted by O-Kernel(P), is the subset of points of P which O-see all the other points in P. This work initiates the study of the computation and maintenance of O-Kernel(P) as we rotate the set O by an angle ¿, denoted by O-Kernel¿(P). In particular, we consider the case when the set O is formed by either one or two orthogonal orientations, O={0°} or O={0°,90°}. For these cases and P being a simple polygon, we design efficient algorithms for computing the O-Kernel¿(P) while ¿ varies in [-p2,p2), obtaining: (i) the intervals of angle ¿ where O-Kernel¿(P) is not empty, (ii) a value of angle ¿ where O-Kernel¿(P) optimizes area or perimeter. Further, we show how the algorithms can be improved when P is a simple orthogonal polygon. In addition, our results are extended to the case of a set O={a1,…,ak}., This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922, Peer Reviewed, Postprint (author's final draft)

In this article, a new concept for SBAS integrity validation is presented. The proposed concept is a modification of the well known Stanford diagram [2], where a 2D histogram shows the relationship of position errors against protection levels for a set of measurements using an all in view satellite selection. The new method consists on two diagrams: the Worse-Safety Index diagram and the “All-Geometries” diagram, known here as the Stanford-ESA and the All-Stanford-ESA, respectively. The first consist on taking, at each sample time and given location, the worst possible satellite geometrical combination (out of all possible combinations) from a SBAS integrity margin viewpoint. In the second, all possible geometries are displayed and, in case of MIs, the geometries associated to each epoch are leveled with different symbols and colors. It allows, to easily identify the different clusters and to assess the time correlation of the events. Real measurement results are presented here showing that the EGNOS integrity margins remain safe under this very exigent criterion, a certainly very positive result. It is suggested here to use the Stanford-ESA Integrity concept, for routine performance monitoring and to support and complement the safety case of the EGNOS system with real experimental data., Peer Reviewed, Postprint (published version)

In the present thesis we elaborate the theory of universally prestarlike functions. We study the generating functions of normalised completely monotone sequences and we obtain a suitable characterization of them. Then, using Pick functions and the Hadamard convolution as our main tools, we are studying the geometric properties (convexity and starlikeness of the image sets of discs or half planes) of functions holomorphic in the slit plane ¢\ [1,∞) and their connections with completely monotone sequences and their generating functions. We characterize those functions that map discs and half planes which contain 0 in their interior to convex or starlike (with respect to the origin) sets as the universally prestarlike functions of specific orders. After we find suitable integral representations for these functions, we find necessary or sufficient conditions on the measures in them to define functions with specific geometric properties. In the end, we apply the results in Gaussian hypergeometric functions and polylogarithms.

In the present thesis we elaborate the theory of universally prestarlike functions. We study the generating functions of normalised completely monotone sequences and we obtain a suitable characterization of them. Then, using Pick functions and the Hadamard convolution as our main tools, we are studying the geometric properties (convexity and starlikeness of the image sets of discs or half planes) of functions holomorphic in the slit plane ¢\ [1,∞) and their connections with completely monotone sequences and their generating functions. We characterize those functions that map discs and half planes which contain 0 in their interior to convex or starlike (with respect to the origin) sets as the universally prestarlike functions of specific orders. After we find suitable integral representations for these functions, we find necessary or sufficient conditions on the measures in them to define functions with specific geometric properties. In the end, we apply the results in Gaussian hypergeometric functions and polylogarithms.

In this paper we consider the complexification of the Arnold standard family of circle maps given by $\widetilde F_{\alpha,\ep}(u)=u{\rm e}^{{\rm i}\alpha} {\rm e}^{\frac{\ep}{2}(u-\frac{1}{u})}$, with $\alpha=\alpha(\ep)$ chosen so that $\widetilde F_{\alpha(\ep),\ep}$ restricted to the unit circle has a prefixed rotation number $\theta$ belonging to the set of Brjuno numbers. In this case, it is known that $\widetilde F_{\alpha(\ep),\ep}$ is analytically linearizable if $\ep$ is small enough, and so, it has a Herman ring $\widetilde U_{\ep}$ around the unit circle. Using Yoccoz's estimates, one has that \emph{the size} $\widetilde R_\ep$ of $\widetilde U_{\ep}$ (so that $\widetilde U_{\ep}$ is conformally equivalent to $\{u\in\bc:\mbox{ } 1/\widetilde R_\ep < |u| < \widetilde R_\ep\}$) goes to infinity as $\ep\to 0$, but one may ask for its asymptotic behavior. We prove that $\widetilde R_\ep=\frac{2}{\ep}(R_0+{\cal O}(\ep\log\ep))$, where $R_0$ is the conformal radius of the Siegel disk of the complex semistandard map $G(z)=z{\rm e}^{{\rm i}\omega}{\rm e}^z$, where $\omega= 2\pi\theta$. In the proof we use a very explicit quasiconformal surgery construction to relate $\widetilde F_{\alpha(\ep),\ep}$ and $G$, and hyperbolic geometry to obtain the quantitative result.

by Dong Xinhan., "March 2002.", Thesis (Ph.D.)--Chinese University of Hong Kong, 2002., Includes bibliographical references (p. 113-117)., Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web., Mode of access: World Wide Web., s in English and Chinese., http://library.cuhk.edu.hk/record=b6073401, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

by Dong Xinhan., "March 2002.", Thesis (Ph.D.)--Chinese University of Hong Kong, 2002., Includes bibliographical references (p. 113-117)., Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web., Mode of access: World Wide Web., s in English and Chinese., http://library.cuhk.edu.hk/record=b6073401, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)