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Extra resources for A statistical measure of tissue heterogeneity with application to 3D PET sarcoma data (2003)(en)(16s

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The arguments we are now going to use follow the general lines of the proofs of [14, Theorem 5] and [11, Theorem 4]. For a complete proof of Theorem 1 we refer to the full version of this paper. For the moment let us fix the integer parameters L and Observe that the closed formula is valid on the database D. Therefore the constructible set is nonempty. Let and be the Zariski-closures of and in and respectively, and let be the morphism of affine varieties defined for by Observe that and are absolutely irreducible affine varieties and that the image of is It turns out that is a finite, bijective and birational morphism of affine varieties.

We use a vector to denote a point in space. , if it has the form where for all Moving Objects and Their Equations of Motion Definition 1. A trajectory is a piecewise linear function from be the set of all trajectories. to 43 Let We also introduce the derivative variables Although we will consider them as an ordinary variable, the semantics of a derivative variable is the derivative of (as a function of time) with respect to the time variable. A differential constraint over the variables has the following general form: where is a multi-variate polynomial with integer coefficients in the variables and Example 2.

F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985. 29. P. Revesz. Introduction to Constraint Databases. Springer, New York, 2002. 30. P. Revesz, R. Chen, P. Kanjamala, Y. Li, Y. Liu, and Y. Wang. The MLPQ/GIS constraint database system. In Proc. ACM SIGMOD International Conference on Management of Data, 2000. 31. P. Revesz and L. Li. Constraint-based visualization of spatial interpolation data. In Proc. of the Sixth International Conference on Information Visualization, pages 563–569, London, England, 2002.

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A statistical measure of tissue heterogeneity with application to 3D PET sarcoma data (2003)(en)(16s by O'sullivan F., Roy S.

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