By L. Bostock, F.S. Chandler
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Additional resources for Applied Mathematics: v. 1
We can, therefore, conclude from these examples that the Toy system does not ensure that every shape constructed by the system is physically valid. However, the notion of physical validity of a shape is rather intuitive and to a large extent application dependent. To specify the concept of “valid shapes” required for some particular application, we feel the need of a set of explicit axioms. Depending on the requirements of the intended application, the notion of physically validity of a shape could be reformulated as a set of mathematical conditions that a valid shape must satisfy, and this set of conditions has to be included in the system as the set A.
Unfortunately, relatively few formal tools are available at present to judge quantitatively the relation between conciseness and computational complexity. In particular, there is a serious need for a theory in the direction of space–complexity tradeoffs. 11 Efﬁcacy in the Context of Applications We have emphasized several times that every shape description scheme should be application oriented, and that the efficacy of a scheme must be judged in the context of its application. Any application task T can be viewed as a two-part procedure.
A) One approach is to restrict the definition of the shape operator. ) (b) Another approach is to extend the definitions of operands and products in such way that both of them can be treated as the same type. ) Consider, for example, a description scheme where the primitive shapes are twodimensional polygons and the set intersection operator (∩) is the shape operator. The product shapes are also expected to be only two-dimensional polygons. 15(a) we show a situation in which the product shape is not homogeneously two-dimensional; it consists of a two-dimensional polygon together with a one-dimensional line segment as a dangling portion.
Applied Mathematics: v. 1 by L. Bostock, F.S. Chandler