By Aldo Ghizzetti (auth.), Aldo Ghizzetti (eds.)

A. Ghizzetti: a) Lezioni sui procedimenti di quasilinearizzazione e applicazioni. b) Nozioni fondamentali sulle equazioni alle differenze e sulle frazioni continue.- P. Wynn: 4 lectures at the numerical program of persisted fractions.- W. Gautschi: energy and weak spot of three-term recurrence relation.- F.L. Bauer: Use of persisted fractions and algorithms regarding them.

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4]) for arbitrary rings. Set d = dU(g) . LEMMA. Let M be a finitely generated U(g) module. (i) d(M) = ma;x{d(N),d(M/N)} foranysubmoduleN ofM. Moreover if all three integers coincide, then e(M) = e(N) + e(M/N). (ii) Let M = Ml ;2 M2 ;2 ... , be an infinite decreasing chain of submodules of M. One has d(MdMi-d < d(M) for all i suHiciently large. An immediate consequence of this result is that for a finitely generated module M over the Weyl algebra An (over a field of characteristic zero) one has d(M) = n ===:} M has finite length.

Commutativity means that grm+n(a, b) = 0, Va E An, bEAm. This gives rise to a Poisson bracket structure on gr A which is defined on homogeneous elements by setting Call an ideal of gr A involutive if it is stable under Poisson bracket. It is obvious that gr I is involutive for any left ideal of A; but this is not so useful. V. Guillemin, D. Quillen and S. Sternberg [23] pointed out that one should ask if its radical Jgr I is involutive and proved this in some cases. This used a trace argument necessitating our characteristic zero hypothesis.

Consider i as a module for the adjoint action of g. 9) there exists A E g* such that the generalized weight subspace p. of i is non-zero. Choose a E p.. Since ,YI,Y2 commute, are ad 9 eigenvectors and act locally ad-nilpotently on A we can assume that a commutes with Y1 , Y2. L E g* is at most 1 dimensional over K (and in this case a monomial in Y1, Y2) . LAo. Thus writing b = az it is clear that a, b are both divisable by the monomial in Y1, Y2 spanning A>.. Consequently we can fur~her assume A = O.