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An upper bound on the number of algebraically independent invariants in an enveloping algebra U under the action of a Lie algebra G0 of derivations is obtained. We are able to determine the exact number of invariants for the case [G0,G0]=G0. This generalizes previous results about Casimir invariants.

This paper contains a general description of the theory of invariants under the adjoint action of a given finite‐dimensional complex Lie algebra G, with special emphasis on polynomial and rational invariants. The familiar ’’Casimir’’ invariants are identified with the polynomial invariants in the enveloping algebra U (G). More general structures (quotient fields) are required in order to investigate rational invariants. Some useful criteria for G having only polynomial or rational invariants are given. Moreover, in most of the physically relevant Lie algebras the exact computation of the maximal number of algebraically independent invariants turns out to be very easy. It reduces to finding the rank of a finite matrix. We apply the general method to some typical examples.

This paper extends to nonlinear evolution equations of odd order the analysis of existence and structure of the polynomial conserved densities. The results for low order densities are similar to the case of even order. The situation for densities with high order derivatives is now radically different. An asymptotic algorithm is presented for the search of such densities, which are shown to be quadratic in the highest derivatives. The very existence of just one high order conserved density is shown to severely restrict the evolution equation, and in the third order case it leads, with some minor additional hypothesis, to the KdV family.

An asymptotic algorithm presented in a previous paper is applied to investigate the possible structures of evolution equations ut=uM+K(u,...,uM−1), M=3,5, which could be compatible with the existence of a conserved density ρ0(u), depending only on u, and with the existence as well of conserved densities with arbitrarily high‐order derivatives. For M=3 it is shown that the Calogero–Degasperis–Fokas equation is essentially the only nonpolynomial equation of that type. For the case ut=D[u4+Q], with Q(u,...,u3) a polynomial, we find a very narrow class of admissible structures for Q, typified by the few particular examples known up to date. Actually, there is in this case an essentially unique structure, modulo a Miura transformation.

The adjoint action of a semisimple complex Lie algebra G singles out some singular directions with special properties. The directions of breaking of the hadronic internal symmetry groups, e.g., SU(3) or SU(3) × SU(3), by the electromagnetic, semileptonic, and nonleptonic weak interactions, are of this type. They are Hermitian (Q,Y) or nilpotent (Cabibbo direction) elements in the corresponding complexified Lie algebra. Their mathematical characterization is provided by the adjoint action of G. In particular, they satisfy a generalized extremal property for every G−invariant polynomial.

We use the Weyl quantization W in a general context valid for any finite‐dimensional Lie algebra G, to derive an explicit formula for (P1,P2) ≡W−1([W (P1),W (P2)]), P1,P2 polynomials. In the particular case of the Heisenberg Lie algebra, this formula reduces to the familiar Moyal bracket.

The existence of closed connected subgroups X of SU(2j+1) such that Dj[SU(2)] lim ≠

The design of a three element piezoelectric microparticle impact sensing diaphragm is described which is sensitive to the detection of momentum propagated by the bending wave. The design achieves a sensitivity of .03 microdyn/sec and optimizes the detection of the direct-path pulse from impact relative to secondary reflections and interference from discontinuities. Measurement of the relative arrival times and the maximum amplitudes of the outputs from the three piezoelectric sensors leads to the determination of the impact position and the normally resolved impact momentum exchange. Coincidence of the signals and a partial redundancy of data leads to a very high noise discrimination.