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However, larger elements have other undesirable effects, which will be discussed in appropriate places. Like the array size determination, the element size determination is a compromise between cost and performance. In the first part of the chapter we discuss the array size and the element size and their effects on a shaped beam in terms of ripple heights inside the coverage region. Guidelines for selecting these two important parameters are presented. Woodward s superposition method for pattern synthesis is presented next. In the second part, mathematical foundations of three optimization algorithms that are commonly used for beam shaping are presented. They include the gradient search algorithm, conjugate match algorithm, and successive projection algorithm. Design examples of one- and two-dimensional shaped beam are shown to demonstrate the effectiveness of the algorithms. Shaped-beam synthesis using phase-only optimizations is also presented. Effects of amplitude tapers are discussed. The genetic algorithm and its operating principle are briefly mentioned. A general guideline for the array size determination is presented at the end.

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The isotropic (sub)group of an element of E is de ned by Ge : Ge = g G | g e = e . Let H be a subgroup of G; the orbit of e under the action of H is: H e= g e|g H . The generalization of this action to subsets of E is obtained by: g.E = g e | e E . Usually, the group of symmetries is obtained from permutation relations de ned over a set of atomic propositions (i.e. labels of states). In a well-formed net, this can be derived from the group of admissible permutations de ned for the color classes. Consider for instance that E is the set AP of atomic propositions of the net; then a proposition [p(u) = 1] means that the place p contains a token of color u and possibly other colored tokens. Let G be the group of admissible permutations over the elements of C, and let g be an element of G which permutates u and v, so we have g.[p(u) = 1] = [p(g(u)) = 1] = [p(v) = 1]. Let {[p(u) = 1], [p(v) = 1]} be a subset of atomic propositions, then the isotropic subgroup of this subset is characterized by the subgroup of permutations which leave the subset {u, v} globally invariant. We are now able to formally de ne a symmetric Kripke structure and, within it, the symmetric states. DEFINITION 14.7. Let SK be a Kripke structure and G a group acting on AP ; SK is said to be symmetric (w.r.t. G) iff: Each state of SK has a symmetric state in SK with regard to each element of G: s S, g G, s S, (s ) = g (s)

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Semmler, W and Gong, G (1996) Estimating Parameters of Real Business Cycle Models Journal of Economic Behavior and Organization 30, 301 25 Ser ing, R J (1980) Approximation Theorems of Mathematical Statistics New York: John Wiley & Sons Shao, J and Tu, D (1995) The Jackknife and Bootstrap New York: Springer-Verlag Sieber, J E (1991) Sharing Social Science Data In Sharing Social Science Data, J E Sieber (ed) Newbury Park, CA: Sage Publications, pp 1 19 Sierpinski, W (1960) Sur un probl` me concernant les nombres Elemente der Mathematik e 15, 73 74 Simon, S D and LeSage, J P (1988) Benchmarking Numerical Accuracy of Statistical Algorithms Computational Statistics and Data Analysis 7, 197 209 Simon, S D and LeSage, J P (1990) Assessing the Accuracy of ANOVA Calculations in Statistical Software Computational Statistics and Data Analysis 8, 325 32 Sirkin, R M (1999).

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life cycle (PLC)-stage terms, and on other project characteristics like complexity and novelty The possibilities for varying approaches are so numerous a general treatment is not feasible Stepping back from a comprehensive approach could involve limiting the extent of application, making the RMP less formal, restricting its focus, or reducing the scope of analysis in a given context Limiting the extent of application could involve employing an RMP only on particular kinds of projects or only at speci c, selected stages of the PLC The implications of this will be clear from the discussion in 14 The degree of formality sought in using a given RMP framework can be a key in uence in achieving an effective and ef cient approach At one extreme a purely informal, intuitive approach could be adopted.

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