Optimal Point Placement for Mesh Smoothing

We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using generalized linear programming. We also give efficient algorithms for some mesh smoothing problems that do not fit into the generalized linear programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This is the final version, and will appear in a special issue of J. Algorithms for papers from SODA '9

Subdivisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter

In this work we study subdivisions of k-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called k-partitions, consisting of k curves meeting in an interior vertex, we prove that the so-called standard k-partition (given by k equiangular inradius segments) is minimizing for any k 2 N, k > 3. For general subdivisions, we show that the previous result only holds for k 6 6. We also study the optimal set for this problem, obtaining that for each k 2 N, k > 3, it consists of the intersection of the unit circle with the corresponding regular k-gon of certain area. Finally, we also discuss the problem for planar convex sets and large values of k, and conjecture the optimal k-subdivision in this case.Ministerio de Educación y Ciencia MTM2010-21206-C02-01Ministerio de Economía e Innovación MTM2013-48371-C2-1-PJunta de Andalucía FQM-325Junta de Andalucía P09-FQM-508

Experimental and computational analyses reveal that environmental restrictions shape HIV-1 spread in 3D cultures

Here, using an integrative experimental and computational approach, Imle et al. show how cell motility and density affect HIV cell-associated transmission in a three-dimensional tissue-like culture system of CD4+ T cells and collagen, and how different collagen matrices restrict infection by cell-free virions

Alexandrov geometry: preliminary version no. 1

This is a preliminary version of our book. It goes up to the definition of dimension, which is about 30% of the material we plan to include. If you use it as a reference, do not forget to include the version number since the numbering will be changed.Comment: 238 pages, 35 figure

A concise guide to existing and emerging vehicle routing problem variants

Vehicle routing problems have been the focus of extensive research over the past sixty years, driven by their economic importance and their theoretical interest. The diversity of applications has motivated the study of a myriad of problem variants with different attributes. In this article, we provide a concise overview of existing and emerging problem variants. Models are typically refined along three lines: considering more relevant objectives and performance metrics, integrating vehicle routing evaluations with other tactical decisions, and capturing fine-grained yet essential aspects of modern supply chains. We organize the main problem attributes within this structured framework. We discuss recent research directions and pinpoint current shortcomings, recent successes, and emerging challenges

New geometric techniques for linear programming and graph partitioning

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (leaves 79-82).In this thesis, we advance a collection of new geometric techniques for the analysis of combinatorial algorithms. Using these techniques, we resolve several longstanding questions in the theory of linear programming, polytope theory, spectral graph theory, and graph partitioning. The thesis consists of two main parts. In the first part, which is joint work with Daniel Spielman, we present the first randomized polynomial-time simplex algorithm for linear programming, answering a question that has been open for over fifty years. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input. To do this, we begin by reducing the input linear program to a special form in which we merely need to certify boundedness of the linear program. As boundedness does not depend upon the right-hand-side vector, we run a modified version of the shadow-vertex simplex method in which we start with a random right-hand-side vector and then modify this vector during the course of the algorithm. This allows us to avoid bounding the diameter of the original polytope.(cont.) Our analysis rests on a geometric statement of independent interest: given a polytope ... in isotropic position, if one makes a polynomially small perturbation to b then the number of edges of the projection of the perturbed polytope onto a random 2-dimensional subspace is expected to be polynomial. In the second part of the thesis, we address two long-open questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [25] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case? We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O( g/n) and vertex bisectors of size O(Vng) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng.(cont.) While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces. While the questions addressed in the two parts of the thesis are quite different, we show that a common methodology runs through their solutions. We believe that this methodology provides a powerful approach to the analysis of algorithms that will prove useful in a variety of broader contexts.by Jonathan A. Kelner.Ph.D

Geometristen muotojen reaaliaikainen tunnistus

Kynä- ja kosketuskäyttöliittymät vaativat toimiakseen tehokasta ja tarkkaa hahmontunnistusta. Tässä työssä esitellään reaaliaikaisen hahmontunnistuksen käsitteistöä, yleisiä menetelmiä ja aikaisempaa tutkimusta. Lyhyesti käsitellään eri tutkimusryhmien esittämiä hahmontunnistusjärjestelmiä. Lisäksi esitellään geometrisiin piirteisiin perustuva hahmontunnistusjärjestelmä. Työ antaa yksityiskohtaiset kuvaukset piirtoviivan esiprosessointi- ja piirteenirrotusalgoritmeista sekä hahmoluokittelumenetelmästä. Lisäksi kuvaillaan hahmontunnistusheuristiikka kahdelle yksinkertaiselle muodolle (nuoli ja tähti). Joukko koehenkilöitä käytti työssä toteutettua graa_sta käyttöliittymää, minkä tuloksena saatiin realistiset tulokset järjestelmän laskennallisesta suorituskyvystä ja tarkkuudesta: toteutettu järjestelmä on laskennallisesti nopea mutta tunnistustarkkuus monitulkintainen. Lopuksi pohditaan valitun lähestymistavan ongelmia ja rajoitteita.Effective sketch recognition is the basis for pen and touch-based human-computer interfaces. In this thesis the concepts, common methods and earlier work in the research area of online symbol recognition are presented. A set of shape recognition approaches proposed in the past by various research teams are briefly introduced. An online shape recognizer using global geometric features is described. The preprocessing and feature extraction algorithms as well as the shape classification method are described in detail. Recognition heuristics for two simple shapes (arrow and star) are suggested. A graphical user interface was implemented and a group of subjects employed to obtain realistic results of the computational performance and recognition accuracy of the system: the implemented system performs fast but the results on the recognition accuracy were ambiguous. Finally, the problems and restrictions of the approach are discussed