Showing papers in "Transactions of the American Mathematical Society in 1965"

Categoricity in power

Abstract: Introduction. A theory, 1, (formalized in the first order predicate calculus) is categorical in power K if it has exactly one isomorphism type of models of power K. This notion was introduced by Los [ 9] and Vaught [ 16] in 1954. At that time they pointed out that a theory (e.g., the theory of dense linearly ordered sets without end points) may be categorical in power N0 and fail to be categorical in any higher power. Conversely, a theory may be categorical in every uncountable power and fail to be categorical in power N0 (e.g., the theory of algebraically closed fields of characteristic 0). Los' then raised the following question. Is a theory categorical in one uncountable power necessarily categorical in every uncountable power? The principal result of this paper is an affirmative answer to that question. We actually prove a stronger result, namely: If a theory is categorical in some uncountable power then every uncountable model of that theory is saturated. (Terminology used in the Introduction will be defined in the body of the paper; roughly speaking, a model is saturated, or universalhomogeneous, if it contains an element of every possible elementary type relative to its subsystems of strictly smaller power.) It is known(2) that a theory can have (up to isomorphism) at most one saturated model in each power. It is interesting to note that our results depend essentially on an analogue of the usual analysis of topological spaces in terms of their derived spaces and the Cantor-Bendixson theorem. The paper is divided into five sections. In ?1 terminology and some meta-mathematical results are summarized. In particular, for each theory, 1, there is described a theory, *, which has essentially the same models as z but is "neater" to work with. In ?2 is defined a topological space, S(A), corresponding to each subsystem, A, of a model of a theory, 1; the points of S(A) being the "isomorphism types" of elements with respect to A. With each monomorphism (= isomorphic imbedding), f: A -+ B, is associated a "dual" continuous map, f*: S(B) -+ S(A). Then there is defined for each S(A) a decreasing sequence ISa(A) I of subspaces which is analogous to (but different from)

Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines

Abstract: Introduction. In the following all semigroups are of finite order. One semigroup S, is said to divide another semigroup S2, written SlIS2, if S, is a homomorphic image of a subsemigroup of S2. The semidirect product of S2 by Sl, with connecting homomorphism Y, is written S2 X y Sl. See Definition 1.6. A semigroup S is called irreducible if for all finite semigroups S2 and Si and all connecting homomorphisms Y, S I (S2 X Y SJ) implies S I S2 or S I S1. It is shown that S is irreducible if and only if either:

Complete Riemannian manifolds and some vector fields

Abstract: Introduction and theorems. In this paper we shall always deal with connected Riemannian manifolds with positive definite metric, and suppose that manifolds and quantities are differentiable of class C'. Let M be an n-dimensional Riemannian manifold with metric tensor field (1) g,,. We call a nonconstant scalar field p in M a concircular scalar field, or simply a concircularfield, if it satisfies the equation

Twisting spun knots

Abstract: 1. Introduction. In [5] Mazur constructed a homotopy 4-sphere which looked like one of the strongest candidates for a counterexample to the 4-dimensional Poincaré Conjecture. In this paper we show that Mazur's example is in fact a true 4-sphere after all. This raises the odds in favour of the 4-dimensional Poincaré Conjecture. The proof involves a smooth knot of S2 in S4 with unusual properties.

The Space of Minimal Prime Ideals of a Commutative Ring

Abstract: Introduction. Of the various spaces of ideal of rings that have been studied (see [1 ], for example) we are focusing attention on the space of minimal prime ideals because of its special role in the case of rings of continuous functions. For the simplest manifestation of this role, consider the compact space N* obtained from the discrete space of positive integers N by adjoining a single point, o. In the ring C(N*) of all continuous real-valued functions on N*, the maximal ideals are very easy to describe. They are in one-one correspondence with the points of N*, and are given by

On continuous rings and self injective rings

Abstract: 0. Throughout this paper we assume that every ring has a unit element. A module is called injective if it is a direct summand of every extension module. A ring is said to be left self injective if it is iniective as the left module over itself. The main results we shall show in the present paper are the following: Let S be a left self injective ring. Then S/N(S) is also left self injective, where N(S) denotes the Jacobson radical of S. Any system of orthogonal idempotents of S/N(S) can be lifted to a system of orthogonal idempotents of S. This theorem about orthogonal idempotents can be proved under a somewhat weaker assumption than the left self injectivity of S. In fact, it is enough to suppose that S is a ring satisfying the following two conditions: 0.1. CONDITION. For any left ideal A there is an idempotent e such that Se is an essential extension of A.

Quasi-convexity and lower semi-continuity of multiple variational integrals of any order

Abstract: Here x = (x', , x'), u = (u .., utm), Q is a bounded domain and the integrand f(x, p, **, p) is a continuous function of its arguments. In 1952 Morrey studied the case I = 1 and introduced the concept of quasiconvexity (see [3]). Extending this concept to the cases I > 1, we say that an integrand f(pl) is quasi-convex if each polynomial of degree ? I minimizes the integral, fnf(D'u(x))dx, among all functions whose derivatives of order I 1 satisfy a Lipschitz condition on Q (we denote this function space by 4' (Q)) and assume the same Dirichlet data on a0 as the polynomial. The reason for the term quasi-convexity becomes clear when one sees that convexity implies quasiconvexity and quasi-convexity in turn implies the Legendre condition (at least for smooth integrands) which contains within it various convexities. Hence, quasi-convexity is a condition which falls between convexity and a weaker kind of convexity. In Theorems 1 and 2 of ?2,1 extend Morrey's results by showing that the necessary and sufficient condition for lower semi-continuity of I(u; Q) in f under uniform convergence of derivatives of order ? I 1 and uniform boundedness of derivatives of order 1, is that f(x, ps,.,p') be quasi-convex in p1 for each fixed value of the variables (x, p, ... p1). The proof is a straightforward extension of Morrey's for the case 1 = 1. However, it contains the added feature that the necessity is derived asssuming only that the admissible functions satisfy fixed Dirichlet boundary conditions. I then go on to consider lower semi-continuity under weak convergence in a space 4/ r(Q) (1 < r < oo ), the space of functions with strong derivatives up to the order I which are in Yr(Q). Two cases are considered, though they do not require separate treatment: first, the case where the admissible functions satisfy a fixed Dirichlet boundary condition, and second, the case of no boundary condition.

Integral equations associated with Hankel convolutions

Abstract: 1. Objectives. We denote by V[al, .., a,,] the number of variations of sign of the sequence a,, * , a,, of real numbers. Hence, for example, V[-1,0,I]=1, V[1,0,1]= 0, V[O,0,0]= -1. If f is a real function defined for 0 < x < cx, then V[f(x)] = l.u.b. V[f(xl), * ,f(x,,)], taken over all sets 0 X < X2< .< Let H be a real-valued function of L1(co, co), q a real-valued, continuous function of L0(c, co) and let

Integro-differential operators on vector bundles

Abstract: Introduction. This article considers a fairly general class of operators on sections of a vector bundle over a compact manifold, including the \"smooth\" differential operators and singular integral operators. The members of this class share many of the properties of differential operators, particularly the elliptic ones. Two general advantages have motivated the development. First, it leads to transparent proofs of the familiar results for elliptic equations, on regularity, the Fredholm alternative, and eigenfunction expansions; and for a larger class than the differential operators. These proofs are not new; rather some of the techniques used in the case of differential operators appear here as general properties of the class of integro-differential operators considered. A second advantage of the larger system, not extensively exploited in this article, is topological. Homotopies (in the class of smooth functions) of the characteristic polynomial of a differential operator can be \"lifted\" to homotopies of the operator itself in the class of integro-differential operators considered, but not (generally) in the class of differential operators. This is an important help in treating some questions raised by Gelfand [6] ; some of the questions concerning the index have now been answered by Atiyah and Singer [1]. To find the notation and main results, one can read §1—§3 (except for proofs), §6, and the definitions and statements of theorems and corollaries from the remaining sections. The paper is organized as follows. §1 describes the well-known function spaces on /{\"that are involved, as well as certain operators on them. §2 describes the singular integral operators and their symbols. §3 extends this collection to one that contains the differential operators on R\", as well as the inverses of the invertible elliptic operators. The symbol c(A) of an operator A is defined, and the behavior of a under composition of operators is discussed. §4 considers the behavior of a under coordinate changes. §5 gives some necessary lemmas from functional analysis. §6 establishes the notation for vector bundles, and the analogs for bundles of the function spaces of §1. §7 defines the singular integral operators on sections of a vector bundle E over a compact manifold X, and their symbols. If A is a singular integral operator from sections of one bundle E

Hyponormal operators and spectral density

Abstract: Introduction. We say an operator T on a Hilbert space H is hyponormal if Tx || > || T*x || for xeH, or equivalently T*T-TT* > 0. In this paperwe will first examine some general properties of hyponormal operators. Then we restrict our interest to hyponormal operators with "thin" spectra. The importance of the topological nature of the spectrum is evident in our main result (Theorem 4) which states that a hyponormal operator whose spectrum lies on a smooth Jordan arc is normal. We continue with a general discussion of a certain growth condition on the resolvent which obtains for hyponormal operators. We conclude with a counterexample to a relation between hyponormal and subnormal operators. The reader is advised that additional facts about hyponormal operators may be found in [l1]. We shall denote the spectrum and the resolvent set of an operator by o(T) and p(T), respectively. The spectral radius R,,(T) sup {j z : z E a(T)}. The numerical range = closure {z: z Tx.x) 11 x = 1} is designated by W(T). Throughout the paper the underlyingvector space is always a separable Hilbert space H.

On Klein’s combination theorem. IV

Abstract: This paper contains an expansion of the combination theorems to cover the following problems. New rank 1 parabolic subgroups are produced, while, as in previous versions, all elliptic and parabolic elements are tracked. A proof is given that the combined group is analytically finite if and only if the original groups are; in the analytically finite case, we also give a formula for the hyperbolic area of the combined group (i.e., the hyperbolic area of the set of discontinuity on the 2-sphere modulo G) in terms of the hyperbolic areas of the original groups. There is also a new variation on the first combination theorem in which the common subgroup has finite index in one of the two groups

Multi-valued monotone nonlinear mappings and duality mappings in Banach spaces

Abstract: It is our object in the present paper to generalize to the multi-valued case the results obtained in a number of recent papers by the author and G. J. Minty for single-valued mappings T (cf. [2]-[14]). The first results for multi-valued mappings for X a Hubert space have been obtained in an unpublished paper of Minty [15]. The methods of [15] are not directly extendable to more general spaces, but our discussion of the finite-dimensional case (Lemma 2.1) has been very much influenced by the manuscript of [15] which Minty has recently transmitted to the author. (The basic result of [15] is stated at the end of §2 below.) Our results for general multi-valued monotone mappings have an interesting specific application given in §3 below to the generalization of a theorem of Beurling and Livingston [1] on duality mappings in Banach spaces. In a previous paper [12], we showed that for strictly convex reflexive spaces, this theorem could be obtained from results on single-valued monotone mappings. In §3 below we give a generalization of this theorem to general reflexive Banach spaces which runs as follows: Let X be a reflexive Banach space, ej)(r) a non-negative nondecreasing function on P1 with ej)(0) =0. The duality map Tof X with respect to c¡> is defined by

The integrability problem for $G$-structures

Abstract: Introduction. One of the main problems of local differential geometry is to determine when a given differentiable structure (for example, a Riemannian structure, a projective structure or an almost complex structure) is integrable. There are really two problems involved in this: (1) the problem of finding a consistent system of differential equations whose unknowns yield a solution to the problem; and (2) the problem of solving these equations. It goes without saying that the second problem is more or less trivial if we are considering analytic structures. On the other hand for C*-structures it is extremely difficult, since the equations that occur in (1) are generally not elliptic. In the paper we will only deal with the first problem which we will refer to as the problem of formal flatness or formal integrability. Also, for simplicity, we will restrict ourselves to studying G-structures, though most of the results of this paper can be extended to more general kinds of pseudogroup structures (such as projective structures). We will also formulate the problem in slightly more intuitive terms: In ?2 we will define what we mean by a mapping of one G-structure into another which is structure preserving to kth order at some point, a notion which is almost self-evident. We will try to find conditions on a given G-structure, E -4 M, such that every point of M admits a mapping into a neighborhood of an integrable G-structure which is structure preserving to order k at that point, for arbitrarily large k. The obstructions to constructing such mappings turn out to be tensors of type Hk2(g), defined on E, where the Hki(g) are the bigraded homology groups of a certain chain complex due to Spencer (cf. ?1). Therefore, the character of the integrability problem will be determined by the cohomology sequence Hk( g). We will also show in this paper that for G-structures of finite type formal integrability in the sense above implies integrability. For analytic Gstructures we will give a direct proof of the equivalence of these two notions (not using the Cartan-Kaehler theorem) in a paper to follow this one. Finally we mention that the following are known for CG-structures:

Transformation groups on compact symmetric spaces

Abstract: The symmetric spaces constitute the most important class of Riemannian manifolds; some of them have been the standard spaces in various branches of geometry, and many authors threw light upon their deep properties. Still there seems to be no thorough study of general transformation groups L (other than the isometry groups H, but containing G) of compact(2) symmetric spaces M, which we call geometric transformation groups of M in this introduction. Its need will be patent if it will reveal interrelations of symmetric spaces, and if L will be the automorphism group of some geometric structure of M, more or less closely related with the Riemannian structure of M, by which M is geometrically distinguished from the other symmetric spaces. Let us observe a few examples. Let M be the sphere as a symmetric space. M has the projective [respectively, conformal] structure; it can be thought of as the set of all geodesics [respectively, the function which gives the angles between two tangent vectors at the same points] of M. This can be defined, of course, for any Riemannian manifold, but the automorphism group L, or the projective [respectively, conformal] transformation group, differs from the isometry group G for the sphere M, and by this fact, M is distinguished from all other symmetric spaces (except the real projective space which is locally isometric with M), as asserted by E. Cartan [Oeuvres completes, Partie I, Vol. II, Gauthier-Villars, Paris, 1952, p. 659]. (See [7], [8] for the proof.) M is the standard space in the projective [respectively, conformal] differential geometry. And, 'a la F. Klein, this group L on M gives rise to the (real) projective [respectively, conformal or Moebius] geometry. Next, to observe another example, we select a compact hermitian symmetric space for M. The structure is the complex structure connected with the Riemannian metric in a certain way. The automorphism group L is the holomorphic transformation group. L is a complex Lie group whose complex structure essentially determines that of M.