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A chiral invariant background field method is introduced in order to study the structure of higher order terms in chiral perturbation theory. Explicit one-loop calculations are presented together with the construction of the most general O(p6) chiral lagrangian relevant for anomalous processes. The non-renormalization of the coefficient of the Wess-Zumino term in both two and four dimensions is discussed in general. (0 1991 Academic PIN. I~C.

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The fundamental problem of distance geometry, F P DG , involves the characterization and study of sets of points based only on given values of (some of) the distances between pairs of points. This problem has a wide range of applications in various areas of mathematics, physics, chemistry, and engineering. Euclidean Distance Matrices, EDM , play an important role in F P DG . They use the squared distances and provide elegant and powerful convex relaxations for F P DG . These EDM problems are closely related to graph realization, GRL ; and graph rigidity, GRD , plays an important role. Moreover, by relaxing the embedding dimension restriction, EDM problems *

A bar framework G(p) in r-dimensional Euclidean space is a graph G = (V, E) on the vertices 1, 2, . . . , n, where each vertex i is located at point p i in R r . Given a framework G(p) in R r , a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained from G(p) by a rigid motion, such that ||qThis problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework G(q) is restricted to be in the same r-dimensional space or not. The stress matrix S of a bar framework G(p) plays a key role in these and other related problems.In this paper, we show that semidefinite programming (SDP) can be effectively used to address the universal rigidity problem. In particular, we use the notion of non-degeneracy of SDP to obtain a sufficient condition for universal rigidity, and to re-derive the known sufficient condition for generic universal rigidity. We present new results concerning positive semidefinite stress matrices and we use a semidefinite version of Farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix S. *