Showing papers in "American Journal of Mathematics in 1971"

Generation of semi-groups of nonlinear transformations on general banach spaces,

Abstract: : The authors continue a discussion of a problem posed by Hille (1951) in a paper titled, 'On the Generation of Semigroups and the Theory of Conjugate Functions.'

Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci

Abstract: 0. Introduction. Let R be a commutative Noetherian ring with identity and let I be a proper ideal of R. A classical result of Krull is that if I can be generated by r elements then the rank or altitude of I (the greatest rank of any minimal prime of I) is at most r. If, moreover, the grade of I (the length of the longest R-sequence contained in I) is r, then I enjoys certain special properties summarized in the term "perfect" as used by iRees [30, p. 32]: I is perfect if the homological (or projective) dimension of R/I as an R-module is equal to the grade of I. The associated primes of a perfect ideal I all have the same grade as I, that is, perfect ideals are grade unmixed. If R is Cohen-Macaulay, the grade of any ideal is equal to its little rank of height (the least rank of any minimal prime) ; in particular, the notions of grade and rank coincide on -primes, and perfect ideals are rank unmixed. Moreover, if I is perfect in a Cohen-Macaulay ring R, R/I is again (Cohen-Macaulay. Macaulay's famous theorem that in a polynomial ring over a field a rank r ideal which can be generated by r elements is rank unmixed [36, p. 203] is then a consequence of two facts: a polynomial ring over a field is CohenMacaulay, and a grade r ideal generated by r elements is perfect. This is the classical example of a perfect ideal. Good discussions of the subject. are available: see [9], [24, ? 25], [30], [18, Ch. 3], and [36, Appendix 6]. The Noetherian restriction on R is, for certain purposes, unnecessary in the discussion of perfect ideals, if one adopts a suitable definition of grade. This idea is worked out in [1]. Suppose that R is (locally) regular, and I is an ideal of R such that R/I is not the direct product of two rings in a nontrivial way. Then I is perfect if and only if R/I is Cohen-Macaulay. In particular, this is the situation when R is a polynomial ring over a field and I is homogeneous. It is very natural, then, to hunt for perfect ideals. Relatively few classes are known, but several authors [4, 6, 8, 33] have established the perfection

Generators, relations and coverings of chevalley groups over commutative rings.

Abstract: A universal central extension of a group G is a central extension, G, of G satisfying the additional conditions (i) G = [G, G], and (ii) every central extension of G is split. If, by analogy with topology, we call a group G connected if H1(G, Z) = 0 and simply connected if H1(G, Z) H(G, Z) = 0, the above conditions are equivalent to saying G is simply connected. In this language, a universal central extension of G is called a universal covering. Now suppose b is a reduced irreducible root system, A is a commutative ring with unit, and E (cD, A) is the elementary subgroup of the points in A of a Chevalley-Demazure group scheme with root system (. Define the Steinberg group, St ((, A), by generators and relations as in [19]. Then the main concern of this paper is to determine under what conditions E ((, A) is connected and St ((, A) is simply connected. When A is a field, these questions were posed and resolved in a well-known paper of Steinberg [19]. For the groups of type Al, 1 ? 2, they have been treated by Kervaire [10] and Steinberg [20]. The results of this paper were announced in [16]. Connectivity is discussed in Section 4. For groups of rank > 2, the main result is most suggestively stated "E ((, A) and St ((F, A)) are connected if and only if E ((, A/m) is connected for every maximal ideal m C A." Thus these groups are connected unless ( C= or G2 and A has a residue field with. two elements. The above statement is also true for groupsof rank 1, provided A is semi-local. In fact, a stronger result holds in the rank 1 case: E (A1, A) and St (A,, A) are connected if the ideal generated by {u2 _1, u C A*} is all of A. Although this implies that A has no residue field with 2 or 3 elements, I do not know whether these are, in general, equivalent conditions. Despite the relationship with the case of fields in the formulation. of these theorems, their proofs do not resemble the proofs for fields. They are based instead on careful exploitation of commutator relations in the groups of rank 2. To decide when St ((, A) is simply connected, it remains to determine when every central extension of St ((, A) is split. The answer, roughly

Uniqueness and Differential Characterization of Approximations from Manifolds of Functions

Abstract: For example such conditions arise in the unisolvent and locally unisolvent theories ( [10], [22] see [17, 8. 3] for a brief comparison of nonlinear results by Tornheim, Motzkin Rice, Meinardus and Schwedt). In Section II we characterize local best approximations from a manifold in terms of best approximations from the corresponding tangent spaces. The characterization, which is basic to our argument, requires, roughly speaking, that approximations from the tangent space satisfy a strong uniqueness condition. In Section I we characterize approximations from linear spaces which satisfy a strong uniqueness condition. We find, for example, that all approximations from a Chebyshev subspace of L1 satisfy a strong uniqueness condition.

Some Applications of Representative Functions to Solvmanifolds

Abstract: The Auslander-Tolimieri method rests on "the splitting of G," a construction they define for certain torsion-free solvable groups and whose definition involves some arbitrary choices. Now in fact, their "splitting" can be defined invariantly in terms of a functor A (G) introduced in 1957 by Hochschild and the author for arbitrary groups G (cf. [2a]). The group A (G) is defined as the group of automorphisms of the G-algebra of representative functions on G. A (G) is a projective limit of algebraic groups and in general is infinite dimensional. However its unipotent radical U(G) which we call the unipotent hull of G (cf. [2b]) turns out to be finite dimensional in a number of important cases. In a paper " Representative functions on discrete groups" (cf. [4c]), which will be referred to hereafter as RFDG, I had studied the map A (C) -4 A (G) where C is a discrete uniform subgroup of a solvable analytic group G for the purpose of showing that C is arithmetic in G. One of the key observations in RFDG is: although C does not determine G, their unipotent hulls

On the total curvature of immersed manifolds, i: an inequality of fenchel-borsuk-willmore.

Abstract: where dV denotes the volume element of M2. The equality holds when and only when M2 is a sphere in E3. The result (A) was proved by Fenchel [5] in 1929, and (B) by Willmore [6] in 1968. The results (A) and (B) were generalized to closed curves and surfaces in higher dimensional e-Lelidean space by Borsuk [1] and Chen [3], respectively. In this paper we give some generalizations of (A) and (B). We consider a closed manifold Mn of dimension n and an immersion x: Mn > En+N of Mn into euclidean space of dimension n + N. Let B, (x) be the bundle of unit normal vectors of x (Mn) so that a point of B, (x) is a pair (p,e) where e is a unit normal vector at x(p). Then B,(x) is a bundle of (N -1)-dimensional spheres over Mn and is a (smooth) manifold of dimension n + N1. Let dV be the volume element of Mn. There is a differential form doof degree N 1 on B, (x) such that its restriction to a fibre is the volume element

The Dual of a Parabolic Subgroup and a Degenerate Principal Series of Sp(n, C)

Abstract: Introduction. Although the work of Harish-Chandra yields, the Plancherel formula for complex semi-simple Lie groups, the (unitary) representation theory of such groups remains incomplete in the sense that the dual object has not yet been completely determined, except in the case of SL(2, C)and possibly SL (n, C) for n> 2. The suggested constituents of the dual object fall into various series of representations which one refers to as pritncipal series or complementary series.