Projection of polyhedral sets is a fundamental operation in both geometry and symbolic computation. In most cases, however, it is not practically feasible to generate projections as the size of the output can be exponential in the size of the input. Even when the size of the output is manageable, we still face two serious problems: overwhelming redundancy and degeneracy. Here, we address these problems from a practical point of view. We discuss three algorithms based on algebraic and geometric techniques and we compare their performance in order to assess the feasibility of these approaches.
Options are contracts whose value is contingent upon the value of underlying assets. We will focus on the most common types of options-those on common stocks. A call option gives owners the right to buy (exercise the option) a fixed number of shares at a fixed (exercise/strike) price until a certain (maturity/expiration) date. Conversely, aput option gives owners the right to sell shares at a fixed price. Option contracts are usually purchased in lots of 100 shares. Individuals
The filtering method considered in this paper is based on approximation of a spatial object in d-dimensional space by the minimal convex polyhedron that encloses the object and whose facets are normal to preselected axes. These axes are not necessarily the standard coordinate axes and, furthermore, their number is not determined by the dimension of the space.' We optimize filtering by selecting optimal such axes based on analysis of stored objects or a sample thereof. The number of axes selected represents a trade-off between access time and storage overhead, as more axes usually lead to better filtering but require more overhead to store the associated access structures.We address the problem of minimizing the number of axes required to achieve a predefine quality of filtering and the reverse problem of optimizing the quality of filtering when the number of axes is fixed. In both cases we also show how to find an optimal collection of axes. In order to solve these problems, we introduce and study the key notion of separability y classification, whick, is a general tool potentially useful in many applications of a computational geometry flavor.The approach is best suited to applications in which the spatial data is relatively static, some directions are more dominant than others, and the dimension of the space is not high; maps are a prime example.
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