7 research outputs found

    Induced Ramsey-type results and binary predicates for point sets

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    Let kk and pp be positive integers and let QQ be a finite point set in general position in the plane. We say that QQ is (k,p)(k,p)-Ramsey if there is a finite point set PP such that for every kk-coloring cc of (Pp)\binom{P}{p} there is a subset QQ' of PP such that QQ' and QQ have the same order type and (Qp)\binom{Q'}{p} is monochromatic in cc. Ne\v{s}et\v{r}il and Valtr proved that for every kNk \in \mathbb{N}, all point sets are (k,1)(k,1)-Ramsey. They also proved that for every k2k \ge 2 and p2p \ge 2, there are point sets that are not (k,p)(k,p)-Ramsey. As our main result, we introduce a new family of (k,2)(k,2)-Ramsey point sets, extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new result to show that for every kk there is a point set PP such that no function Γ\Gamma that maps ordered pairs of distinct points from PP to a set of size kk can satisfy the following "local consistency" property: if Γ\Gamma attains the same values on two ordered triples of points from PP, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.Comment: 22 pages, 3 figures, final version, minor correction

    Integrated modelling for 3D GIS

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    A three dimensional (3D) model facilitates the study of the real world objects it represents. A geoinformation system (GIS) should exploit the 3D model in a digital form as a basis for answering questions pertaining to aspects of the real world. With respect to the earth sciences, different kinds of objects of reality can be realized. These objects are components of the reality under study. At the present state-of-the-art different realizations are usually situated in separate systems or subsystems. This separation results in redundancy and uncertainty when different components sharing some common aspects are combined. Relationships between different kinds of objects, or between components of an object, cannot be represented adequately. This thesis aims at the integration of those components sharing some common aspects in one 3D model. This integration brings related components together, minimizes redundancy and uncertainty. Since the model should permit not only the representation of known aspects of reality, but also the derivation of information from the existing representation, the design of the model is constrained so as to afford these capabilities. The tessellation of space by the network of simplest geometry, the simplicial network, is proposed as a solution. The known aspects of the reality can be embedded in the simplicial network without degrading their quality. The model provides finite spatial units useful for the representation of objects. Relationships between objects can also be expressed through components of these spatial units which at the same time facilitate various computations and the derivation of information implicitly available in the model. Since the simplicial network is based on concepts in geoinformation science and in mathematics, its design can be generalized for n-dimensions. The networks of different dimension are said to be compatible, which enables the incorporation of a simplicial network of a lower dimension into another simplicial network of a higher dimension.The complexity of the 3D model fulfilling the requirements listed calls for a suitable construction method. The thesis presents a simple way to construct the model. The raster technique is used for the formation of the simplicial network embedding the representation of the known aspects of reality as constraints. The prototype implementation in a software package, ISNAP, demonstrates the simplicial network's construction and use. The simplicial network can facilitate spatial and non spatial queries, computations, and 2D and 3D visualizations. The experimental tests using different kinds of data sets show that the simplicial network can be used to represent real world objects in different dimensionalities. Operations traditionally requiring different systems and spatial models can be carried out in one system using one model as a basis. This possibility makes the GIS more powerful and easy to use

    Subject Index Volumes 1–200

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    From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices

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    A set P = H cup {w} of n+1 points in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on P, it suffices to know the so-called frequency vector of P. While there are roughly 2^n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2^{n/2}. We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, w-embracing triangles, and many more. Based on these formulas, the corresponding numbers of graphs can be computed efficiently. Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of simplices spanned by H that contain w. While the concept of frequency vectors does not generalize easily, we show how to apply similar methods in higher dimensions. The result is an O(n^{d-1}) time algorithm for computing the simplicial depth of a point w in a set H of n d-dimensional points, improving on the previously best bound of O(n^d log n). Configurations equivalent to wheel sets have already been used by Perles for counting the faces of high-dimensional polytopes with few vertices via the Gale dual. Based on that we can compute the number of facets of the convex hull of n=d+k points in general position in R^d in time O(n^max(omega,k-2)) where omega = 2.373, even though the asymptotic number of facets may be as large as n^k.ISSN:1868-896
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