Novel approaches for constructing persistent Delaunay triangulations by applying different equations and different methods

“Delaunay triangulation and data structures are an essential field of study and research in computer science, for this reason, the correct choices, and an adequate design are essential for the development of algorithms for the efficient storage and/or retrieval of information. However, most structures are usually ephemeral, which means keeping all versions, in different copies, of the same data structure is expensive. The problem arises of developing data structures that are capable of maintaining different versions of themselves, minimizing the cost of memory, and keeping the performance of operations as close as possible to the original structure. Therefore, this research aims to aims to examine the feasibility concepts of Spatio-temporal structures such as persistence, to design a Delaunay triangulation algorithm so that it is possible to make queries and modifications at a certain time t, minimizing spatial and temporal complexity. Four new persistent data structures for Delaunay triangulation (Bowyer-Watson, Walk, Hybrid, and Graph) were proposed and developed. The results of using random images and vertex databases with different data (DAG and CGAL), proved that the data structure in its partial version is better than the other data structures that do not have persistence. Also, the full version data structures show an advance in the state of the technique. All the results will allow the algorithms to minimize the cost of memory”--Abstract, page iii

Computational Geometric and Algebraic Topology

Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity. At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field

Proceedings of the Workshop on Algorithmic Aspects of Advanced Programming Languages: WAAAPL'99: Paris, France, September 30, 1999

The first Workshop on Algorithmic Aspects of Advanced Programming Languages was held on September 30, 1999, in Paris, France, in conjunction with the PLI'99 conferences and workshops. The choice of programming languages has a huge effect on the algorithms and data structures that are to be implemented in that language. Traditionally, algorithms and data structures have been studied in the context of imperative languages. This workshop considers the algorithmic implications of choosing an advanced functional or logic programming language instead. A total of eight papers were selected for presentation at the workshop, together with an invited lecture by Robert Harper. We would like to thank Dider Remv, general chair of PLI'99, for his assistance in organizing this workshop

Distributed and Collaborative Synthetic Environments

Fast graphics workstations and increased computing power, together with improved interface technologies, have created new and diverse possibilities for developing and interacting with synthetic environments. A synthetic environment system is generally characterized by input/output devices that constitute the interface between the human senses and the synthetic environment generated by the computer; and a computation system running a real-time simulation of the environment. A basic need of a synthetic environment system is that of giving the user a plausible reproduction of the visual aspect of the objects with which he is interacting. The goal of our Shastra research project is to provide a substrate of geometric data structures and algorithms which allow the distributed construction and modification of the environment, efficient querying of objects attributes, collaborative interaction with the environment, fast computation of collision detection and visibility information for efficient dynamic simulation and real-time scene display. In particular, we address the following issues: (1) A geometric framework for modeling and visualizing synthetic environments and interacting with them. We highlight the functions required for the geometric engine of a synthetic environment system. (2) A distribution and collaboration substrate that supports construction, modification, and interaction with synthetic environments on networked desktop machines

Abstracts of Ph.D. theses in mathematics

summary:Stolín, Radek: Teaching of financial and insurance mathematics at secondary and higher professional schools -- coinsurance in bonus-malus systems in automobile insurance. Olejníèková, Jana: Scientific work of Bohumil Bydžovský. Flašková, Jana: Ultrafilters and small sets. Pražák, Pavel: Differential equations and their applications in economics. Bartl, David: Theorems of the alternative and linear programming in infinite-dimensional spaces. Smetana, Petr: Some problems of the private health insurance. Kosinka, Jiøí: Algorithms for Minkowski Pythagorean hodograph curves. Kopa, Miloš: Utility functions in portfolio optimization. Orsáková, Martina: M-estimation in nonlinear regression for longitudal data. Šiman, Miroslav: On portmanteau tests of randomness. Purmová, Lucie: Continuous population models for single species. Koubková, Alena: Sequential change-point analysis. Omelka, Marek: Second order properties of some M-estimators and R-estimators. Barto, Libor: Full embeddings and their modifications. Pecinová, Eliška: Ladislav Svante Rieger (1916--1963). Nguyen, Duc Huy: On existence and regularity of solutions to perturbed systems of Stokes type. Prachaø, Aleš: Analysis of the discontinuous Galerkin method for elliptic problems. Franek, Peter: Several Dirac operators in parabolic geometry. Hladík, Milan: Explicit description of supporting and separating hyperplanes of two convex polyhedral sets depending on parameters. Janeèek, Martin: Valuation techniques of life insurance liabilities. Ranocha, Pavel: Stationary distribution of time series. Bímová, Daniela: Kinematic geometry in $n$-dimensional Euclidean space

Optimal topological simplification of discrete functions on surfaces

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.Comment: 27 pages, 8 figure