Showing papers in "Transactions of the American Mathematical Society in 1974"

On vector measures

Abstract: The four sections of this paper treat four different but somewhat related topics in the theory of vector measures. In §1 necessary and sufficient conditions for a Banach space X to have the property that bounded additive X-valued maps on o-algebras be strongly bounded are presented, namely, X can contain no copy of /„. The next two sections treat the Jordan decomposition for measures with values in Z.|-spaces on c0(r) spaces and criteria for integrability of scalar functions with respect to vector measures. In particular, a different proof is presented for a result of D. R. Lewis to the effect that scalar integrability implies integrability is equivalent to noncontainment of c0. The final section concerns the Radon-Nikodym theorem for vector measures. A generalization of a result due to E. Leonard and K. Sundaresan is given, namely, if a Banach space X has an equivalent very smooth norm (in particular, a Fréchet differentiable normithenitsdualhas the Radon-Nikodym property. Consequently, a C(H) space which is a Grothendieck space (weak-star and weak-sequential convergence in dual coincide) cannot be renormed smoothly. Several open questions are mentioned throughout the paper. The present paper contains results on various aspects of the general theory of vector-valued measures. It proceeds in four sections which are unrelated to each other except for their general relationship to the topic of the title. A brief outline of the results of each section is presented below—a more complete discussion of the sections is delayed (largely because of their disconnected nature) until the sections themselves. §1 is concerned with the theory of strongly bounded vector measures. The main result of this section (Theorem 1.1) provides criteria for a Banach space X to possess the property that every X-valued bounded additive map with values in X be strongly bounded. This theorem sharpens the classical Pettis theorem on weakly countably additive set functions and allows a sharpening of several other related results. §2 is concerned with the Jordan decomposition of vector measures with values in a Banach lattice. The results of this section are necessarily meager: not much is possible. Our most precise results are in case the range space is an abstract £space or c0. A few remarks are also made concerning the range of certain vector measures. §3 deals with the integrability of certain scalar functions with respect to a vector measure. Utilizing the series representation of a scalar function and its integral, a result of D. R. Lewis is generalized. Also, a criterion for integrability Received by the editors February 5, 1973 and, in revised form, May 25, 1973. AMS (MOS) subject classifications (1970). Primary 46B05.

Weighted norm inequalities for fractional integrals

Abstract: The principal problem considered is the determination of all nonnegative functions, V(x), such that IIhTf(x)V(x)14 < ClIf(x)V(x)lIp where the functions are defined on R', 0 < y < n, 1 < p < n/y, l/q = l/p -yn, C is a constant independent of f and Tyf (x) = S f(x y) Ly|f dy. The main result is that V(x) is such a function if and only if (-fQ [V(X)]qdx (jff [vx) P'd)1 (IQIJQVX]^ (QI J [V(X)]-p, ) where Q is any n dimensional cube, IQI denotes the measure of Q, p' = p/(p 1) and K is a constant independent of Q. Substitute results for the cases p = 1 and q = oo and a weighted version of the Sobolev imbedding theorem are also proved.

Existence and stability for partial functional differential equations

Abstract: Publisher Summary This chapter discusses the existence and stability for a class of partial functional differential equations. As a model for this class, one may take the equation wt(x, t) = wxx (x, t) + ƒ(t, w(x, t − r)), 0 ≤ x ≤ Π, t ≥ 0, w(0, t) = w(Π, t) = 0, t ≥ 0, w(x, t) = Φ (x, t), 0 ≤ x ≤ Π, −r ≤ t ≤ 0, where ƒ is a linear or nonlinear scalar-valued function, r a positive real number, and Φ a given initial function. The second derivative term corresponds to a strongly continuous semigroup of linear operators on a Banach space of functions determined by the boundary conditions. Accordingly, the approach relies primarily on semigroup methods and the treatment of equation as an ordinary functional differential equation in a Banach space. The chapter also describes semigroup and infinitesimal generator in the autonomous case.

Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients

Abstract: Variational boundary value problems for quasilinear elliptic systems in divergence form are studied in the case where the nonlinearities are nonpolynomial. Monotonicity methods are used to derive several existence theorems which generalize the basic results of Browder and Leray-Lions. Some features of the mappings of monotone type which arise here are that they act in nonreflexive Banach spaces, that they are unbounded and not everywhere defined, and that their inverse is also unbounded and not everywhere defined. © 1974 American Mathematical Society.

On totally real submanifolds

Abstract: Complex analytic submanifolds and totally real submanifolds are two typical classes among all submanifolds of an almost Hermitian mani- fold. In this paper, some characterizations of totally real submanifolds are given. Moreover some classifications of totally real submanifolds in complex space forms are obtained.

Explicit class field theory for rational function fields

Abstract: Developing an idea of Carlitz, I show how one can describe explicitly the maximal abelian extension of the rational function field over F, (the finite field of q elements) and the action of the idèle class group via the reciprocity law homomorphism. The theory is closely analogous to the classical theory of cyclotomic extensions of the rational numbers. The class field theory of the rational numbers Q is \"explicit\" in the sense that one can write down a sequence of polynomials whose roots generate the maximal abelian extension of Q, and one can describe concretely how a given Q-idele class operates on each of these roots via the reciprocity law homomorphism (see [1, Chapter 7]). A similar program can be carried out for imaginary quadratic fields using the theory of elliptic curves (see [4, Chapter 13]). These results are quite old, having originally been conceived by Kronecker in the late 19th century. More recently, Lubin and Täte [5] have given such an explicit description of the class field theory for any local field using the theory of formal groups. All of these results use the same basic procedure: A ring of \"integers\" in the ground field is made to act on part of the algebraic closure of that field, and the maximal abelian extension is gotten essentially by adjoining the torsion points of that action. For example, one obtains the maximal abelian extension of Q by adjoining the torsion points of Z acting by exponentiation on the multiplicative group of the field of algebraic numbers. This paper contains a similar explicit description for the class field theory of a rational function field (over a finite field of constants). The main idea comes from a paper of Carlitz [2], the aim of which was to develop an analog of the cyclotomic polynomial for the ring of polynomials over a finite field. In brief, this Carlitz cyclotomic theory goes as follows: Let k be the field of rational functions over the finite field F, of q elements. Of the q3 q generators of k over Fq pick one, say T, and consider the polynomial subring RT = Fq[T] of k. Carlitz makes RT act as a ring of endomorphisms on the additive group of kK, the algebraic closure of k. For M G RT, the action of M is given by a separable polynomial Received by the editors March 10, 1972. AMS (MOS) subject classifications (1970). Primary 12A65, 12A35.

Generation of analytic semigroups by strongly elliptic operators

Abstract: Strongly elliptic operators are shown to generate analytic semigroups of evolution operators in the topology of uniform convergence, when realized under general boundary conditions on (possibly) unbounded domains. An application to the existence and regularity of solutions to parabolic initial-boundary value problems is indicated. Introduction. Extending the results of a previous paper [19], we propose to prove a theorem on the generation of analytic semigroups by strongly elliptic operators A of order 2m in the topology of uniform convergence, under more general boundary conditions. As before, there is a direct application to parabolic initial-boundary value problems: we give an existence and uniqueness theorem for such problems, using the Kato-Tanabe theory for temporally inhomogeneous evolution equations du/dt + A(t)u = / The topology of uniform convergence in the space variables yields classical solutions of this parabolic problem which are analytic in time at each space point; furthermore, the initial values are assumed in the pointwise continuous sense. Since we treat general boundary conditions, our parabolic problem is roughly comparable to the one discussed in Arima [4], where existence and uniqueness are treated (using a fundamental solution of the parabolic problem), but not analyticity. The semigroup generation theorem for strongly elliptic operators is by now well established in the Lp spaces. The L2 case was treated by Browder in [5]; shortly thereafter Agmon, in [1], gave a method of proving the basic a priori estimate \\\\u\\\\ < (M/\\z\\)\\\\iA + z)u\\\\, |arg _| < \\m + e, (E) in Lp, and this method has been filled out with existence theorems and used by several authors, including Friedman [8], Higuchi [9], Lau [14], and Freeman and Schechter [7]. The work of Lau offers an attractive combination of full development and general hypotheses, so we shall borrow Lp results from [14]. We recall that the Kato-Tanabe theorem applied in the Lp topology gives solutions which are Received by the editors June 25, 1979. AMS (MOS) subject classifications (1970). Primary 35K35, 47D05. 'This work was supported in part by the U. S. Department of Energy under contract EY-76-C-02-0016. Accordingly, the U. S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U. S. Government purposes. © 1980 American Mathematical Society 0002-9947/80/0000-02 1 8/$04.00 299 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

The Riemann problem for general 2×2 conservation laws

Abstract: The Riemann Problem for a system of hyperbolic conservation laws of form ut + f(u, v) = 0, (1) ' vt + g(u, v)x = 0 with arbitrary initial constant states (2) (u0(*). «o<*» í(«/. v¡), \(.ur, vr), x<0, x > 0, is considered. We assume that fv < 0, gu < 0. Let l¡ (r¡) be the left (right) eigenvectors of dF ■ d(f, g) for eigenvalues \j < \2. Instead of assuming the usual convexity condition d\¡(r¡) i* 0,1 « 1, 2, we assume that d\¡(r¡) = 0 on disjoint union of 1-dim manifolds in the (u, v) plane. Oleinik's condition (E) for single equation is extended to system (1); again call this new condition (E). Our condition (E) implies Lax's shock inequalities and, in case d\¡(r¡) ¥= 0, the two are equivalent. We then prove that there exists a unique solution to the Riemann Problem (1) and (2) in the class of shocks, rarefaction waves and contact discontinuities which satisfies condition (E). Introduction. We consider the system ut + fiu, v)r = 0, (o.i) *Jy )x t>o,-~ 0. Presented to the Society, January 26, 1972; received by the editors October 16, 1973. AMS (MOS) subject classifications (1970). Primary 35L65, 35F25.

Differentiability of solutions to hyperbolic initial-boundary value problems

Abstract: This paper establishes conditions for the differentiability of solutions to mixed problems for first order hyperbolic systems of the form (3/3» — 2 Aja/dxj — B)u = F on [0, r] X ß, Mu g on [0,7\"] x 3Í2, u(0, x) f(x), x e Q. Assuming that X1 a priori inequalities are known for this equation, it is shown that if F e H'([0, T] x Q), g e H'*l'2([0, T] X 3ñ),/ e H'(Q) satisfy the natural compatibility conditions associated with this equation, then the solution is of class C from [0, T] to H'~f(Q), 0 < p < s. These results are applied to mixed problems with distribution initial data and to quasi-linear mixed problems.

On dynamical systems with the specification property

Abstract: A continuous transfornation T of a compact metric space X satisfies the specification property if one can approximate distinct pieces of orbits by single periodic orbits with a certain uniformity. There are many examples of such transformations which have recently been studied in ergodic theory and statistical mechanics. This paper investigates the relation between Tinvariant measures and the frequencies of T-orbits. In particular, it is shown that every invariant measure (and even every closed connected subset of such measures) has generic points, but that the set of all generic points is of first category in X. This generalizes number theoretic results concerning decimal expansions and normal numbers.

Symplectic homogeneous spaces

Abstract: It is proved in this paper that for a given simply connected Lie group G with Lie algebra g, every left-invariant closed 2-form induces a symplectic homogeneous space. This fact generalizes the results in [7] and [12] that if H1l(g)= H2(g) =0, then the most general symplectic homogeneous space covers an orbit in the dual of the Lie algebra S. A one-to-one correspondence can be established between the orbit space of equivalent classes of 2-cocycles of a given Lie algebra and the set of equivalent classes of simply connected symplectic homogeneous spaces of the Lie group. Lie groups with left-invariant symplectic structure cannot be semisimple; hence such groups of dimension four have to be solvable, and connected unimodular groups with left-invariant symplectic structure are solvable [4]. 1. Symplectic manifolds. Let M be a 2n-dimensional connected differentiable manifold. A symplectic structure on M is defined by a closed differential 2form c) which is everywhere of maximal rank. Such a form is called a symplectic form of the symplectic structure defined on M. On a symplectic manifold M, a one-to-one map from the space of vector fields X(M) onto the space of linear differential forms D1(M) can be defined as follows. If X is a vector field, the map x "i(X)w (where i(X)w denotes the interior product of X with wo) is a bijective map from X(M) onto DI(M). In fact, at each point x c M, this map from TX(M) onto 7* (M) is given by the nonsingular bilinear form o) A classical theorem attributed to Darboux [11] states that for an n-dimensional manifold M with a closed 2-form w) of rank exactly p everywhere there can be introduced about every point a system of coordinates x1,*. *, xn-P, y * yP, in terms of which the local representation of cw becomes Received by the editors April 23, 1973 and, in revised form, July 10, 1973. AMS (MOS) subject classifications (1970). Primary 53C30; Secondary 57F15, 22E25.

Coproducts and some universal ring constructions

Abstract: Let R be an algebra over a field k, and P, Q be two nonzero finitely generated projective R-modules. By adjoining further generators and relations to R, one can obtain an extension S of R having a universal isomorphism of modules, i: P(R S -Q (R S. We here study this and several similar constuctions, including (given a single finitely generated projective R-module P) the extension S of R having a universal idempotent module-endomorphism e: P 0 S P 0 S, and (given a positive integer n) the k-algebra S with a universal k-algebra homomorphism of R into its n X n matrix ring, f: R mn(S). A trick involving matrix rings allows us to reduce the study of each of these constructions to that of a coproduct of rings over a semisimple ring Ro (= k X k X k, k X k, and k respectively in the above cases), and hence to apply the theory of such coproducts. As in that theory, we find that the homological properties of the construction are extremely good: The global dimension of S is the same as that of R unless this is 0, in which case it can increase to 1, and the semigroup of isomorphism classes of finitely generated projective modules is changed only in the obvious fashion; e.g., in the first case mentioned, by the adjunction of the relation [PI = [QJ. These results allow one to construct a large number of unusual examples. We discuss the problem of obtaining similar results for some related constructions: the adjunction to R of a universal inverse to a given homomorphism of finitely generated projective modules, f: P Q, and the formation of the factor-ring R/Tp by the trace ideal of a given finitely generated projective R-module P (in other words, setting P = 0). The idea for a category-theoretic generalization of the ideas of the paper

Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers)

Abstract: COMPUTABILITY AND ITS RELATION TO THE GENERAL PURPOSE ANALOG COMPUTER (SOME CONNECTIONS BETWEEN LOGIC, DIFFERENTIAL EQUATIONS AND ANALOG COMPUTERS) BY MARIAN BOYKAN POUR-EL(l) ABSTRACT. Our aim is to study computability from the viewpoint of the analog computer. We present a mathematical definition of an analog generable function of a real variable. This definition is formulated in terms of a simultaneous set of nonlinear differential equations possessing a \"domain of generation.\" (The latter concept is explained in the text.) Our definition includes functions generated by existing general-purpose analog computers. Using it we prove two theorems which provide a characterization of analog generable functions in terms of solutions of algebraic differential polynomials. The characterization has two consequences. First we show that there are entire functions which are computable (in the sense of recursive analysis) but which cannot be generated by any analog computer in any interval—e.g. l/r(x) and 2^-i (x°/n^')). Second we note that the class of analog generable functions is very large: it includes special functions which arise as solutions to algebraic differential polynomials. Although not all computable functions are analog generable, a kind of converse holds. For entire functions,/(x) = X\"o b,x', the theorem takes the following form. If f(x) is analog generable on some closed, bounded interval then there is a finite number of bk such that, on every closed bounded interval, f(x) is computable relative to these bk. A somewhat similar theorem holds if/is not entire. Although the results are stated and proved for functions of a real variable, they hold with minor modifications for functions of a complex variable. Our aim is to study computability from the viewpoint of the analog computer. We present a mathematical definition of an analog generable function of a real variable. This definition is formulated in terms of a simultaneous set of nonlinear differential equations possessing a \"domain of generation.\" (The latter concept is explained in the text.) Our definition includes functions generated by existing general-purpose analog computers. Using it we prove two theorems which provide a characterization of analog generable functions in terms of solutions of algebraic differential polynomials. The characterization has two consequences. First we show that there are entire functions which are computable (in the sense of recursive analysis) but which cannot be generated by any analog computer in any interval—e.g. l/r(x) and 2^-i (x°/n^')). Second we note that the class of analog generable functions is very large: it includes special functions which arise as solutions to algebraic differential polynomials. Although not all computable functions are analog generable, a kind of converse holds. For entire functions,/(x) = X\"o b,x', the theorem takes the following form. If f(x) is analog generable on some closed, bounded interval then there is a finite number of bk such that, on every closed bounded interval, f(x) is computable relative to these bk. A somewhat similar theorem holds if/is not entire. Although the results are stated and proved for functions of a real variable, they hold with minor modifications for functions of a complex variable. Introduction. This work represents a chapter in the development of a mathematical theory of the analog computer. As stated in the abstract, the definition of analog generable function which we present is expressed in terms of a simultaneous set of nonlinear differential equations. We will see that it includes functions generated by existing general purpose analog computers—i.e., the electronic analog computer and the mechanical differential analyzer. Our definition with its \"domain of generation\" appears to differ considerably from the approach taken in [17]. We have found our approach necessary for reasons stated in footnotes 4 and 12. Received by the editors April 12, 1971 and, in revised form, June 20, 1972. AMS (MOS) subject classifications (1970). Primary 02F50, 02F99; Secondary 26A42, 33A15, 34A10.

Fixed point iterations using infinite matrices

Abstract: Let E be a closed, bounded, convex subset of a Banach space X, f: E → E. Consider the iteration scheme defined by , where A is a regular weighted mean matrix. For particular spaces X and functions f this iterative scheme converges to a fixed point of f. [5] extends or generalizes related work of Browder and Petryshyn, Dotson, Franks and Marzec, Johnson, Kannan, Mann, and Reinermann. This paper continues investigations begun in [6].

Canonical forms and principal systems for general disconjugate equations

Abstract: It is shown that the disconjugate equation (1) Lx = (1/,,)(d/dt) (l//,) **... (d/dt)(l//A)(d/dt)(x/,lo) = 0, a 0, and (2) A, Ee C(a,b), can be written in essentially unique canonical forms so that fb /,dt = X0 (fa Jidt = 00) for 1 < i < n 1. From this it follows easily that (1) has solutions xi, ..., x. which are positive in (a, b) near b(a) and satisfy lim,.b-xi(t)/xj(t) = (lim,.ax&(t)/xj(t) = oo) for 1 ? i < j < n. Necessary and sufficient conditions are given for (1) to have solutions yl, ...,y,, such that lim, .b-y,(t)/yJ(t) = lim,.a+yj(t)/yi(t) = 0 for 1 < i

Equisingular deformations of plane algebroid curves

Abstract: We construct a formal versal equisingular deformation of a plane algebroid curve (in characteristic zero), and show it is smoothly embedded in the whole deformation space of the singularity. Closer analysis relates equisingular deformations of the curve to locally trivial deformations of a certain (nonreduced) projective curve. Finally, we prove that algebraic r1 of the complement of a plane algebroid curve remains constant during formal equisingular deformation. Introduction. In a series of papers ([10], [11], [121), Zariski has studied the concept of equisingularity of plane algebroid curves. Two curves are equisingular if one can simultaneously resolve their singularities; this equivalence relation is weaker than analytic equivalence, but stronger than equimultiplicity. Using topological techniques, Zariski proves that two equisingular curves over C have locally the same topological embedding in C2; in particular, the characteristic pairs of their branches are the same, whence they yield knots of the same knot type in R3 (cf. [4]). Utilizing techniques developed by M. Schlessinger [61, we study infinitesimal equisingular families of curves. Our deformation theory takes place over the category C of artin local C-algebras. Recall that if f e C[[X, Y]] is reduced, and if g1, *.*. * gm e C[[X, Y]] induce a basis of the artin ring C[[X, Y]/(jf fx' fy), then the formal family f + t1g1 + * .*+ tmgm C[[X, Y, t1,., till induces a formal versal (or semiuniversal) deformation of the singularity defined by (f ). Thus, in a weak sense, the family represents the functor on C of infinitesimal deformations of the singularity. To define equisingular deformation, we emulate Zariski's original definition. Recall that every plane algebroid singularity can be reduced to a number of ordinary double points by a finite number of quadratic transforms. We say a deformation Received by the editors December 19, 1972. AMS (MOS) subject classifications (1970). Primary 14D15, 14H20; Secondary 14B10, 32C40.

Linear operators and vector measures

Abstract: This paper is a continuation of a study of operators on function spaces initiated by the authors in [2] and [3]. The operators are studied in terms of their representing measures. In w 2 a characterization of compact operators is obtained using recent results of Brooks [1]. In w 3 factorization and characterization theorems for p-dominated and absolutely p-summing maps are given. The factorization results are motivated by Pietsch's work [5]. The general setting is as follows. Each of E and F is a B-space (= Banach space), H is a compact Hausdorff space and C(H, E) is the B-space (sup norm) of all continuous E-valued defined on H. We shall be interested in operators (= continuous linear transformations) L: C(H,E)--~F and representing measures m: Z---~B(E, F**), where B(E, F**) is the B-space of all operators from E into the bidual of E and E is the Borel o--algebra of subsets of H. A finitely additive set function m: Z--~B(E, F**) is called a representing measure if m has finite semivariation and Imz[ (= to ta l variation of the adjoint measure) is a regular Borel measure for each z~F~ (=closed unit ball in F*). The Riesz Representation Theorem in this setting asserts that to each operator L: C(H, E)---~F there corresponds a unique representing measure m: E--~B(E, F**) so that L(f)=~nfdm and IILII =~(H), where ffz denotes the semivariation; this association is denoted by L,,--.m. The reader may consult Brooks and Lewis [3] for a detailed discussion of this setting. In particular, CA will be the characteristic function of a set A, S(Z) will denote the scalar valued simple functions over Z, and U(Z) will be the uniform closure of S(E); the spaces SE(2~ ) and Ue(2~) are defined analogously for E-valued functions. The reader should note that if m~-~,L, then m(A)x=L**(r ). Also, we shall say that a representing measure m is strongly bounded (s-bounded) if rh(Ai)---~0 for a disjoint sequence (Ai)c S. The first named author acknowledges support of the National Science Foundation during the preparation of this paper.

Modules over coproducts of rings

Abstract: Let Ro be a skew field, or more generally, a finite product of full matrix rings over skew fields. Let (RX)XEA be a family of faithful Rorings (associative unitary rings containing RO) and let R denote the coproduct ("free product") of the RX as R0-rings. An easy way to obtain an R-module M is to choose for each X E A U {0} an Rx-module Mx, and put M = MA OR; R. Such an M will be called a "standard" R-module. (Note that these include the free R-modules.) We obtain results on the structure of standard R-modules and homomorphisms between them, and hence on the homological properties of R. In particular: (1) Every submodule of a standard module is isomorphic to a standard

Elementary divisor rings and finitely presented modules

Abstract: Throughout, rings are commutative with unit and modules are unital. We prove that R is an elementary divisor ring if and only if every finitely presented module over R is a direct sum of cyclic modules, thus providing a converse to a theorem of Kaplansky and answering a question of Warfield. We show that every Bezout ring with a finite number of minimal prime ideals is Hermite. So, in particular, semilocal Bezout rings are Hermite answering affirmatively a question of Henriksen. We show that every semihereditary Bezout ring is Hermite. Semilocal adequate rings are characterized and a partial converse to a theorem of Henriksen is established.

A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings

Abstract: We define and study the properties of a topological degree for ultimately compact, multivalued vector fields defined on the closures of open subsets of certain locally convex topological vector spaces. In addition to compact mappings, the class of ultimately compact mappings includes condensing mappings, generalized condensing mappings, perturbations of compact mappings by certain Lipschitz-type mappings, and others. Using this degree we obtain fixed point theorems and mapping theorems.

Equivariant endomorphisms of the space of convex bodies

Abstract: We consider maps of the set of convex bodies in d-dimensional Euclidean space into itself which are linear with respect to Minkowski addition, continuous with respect to Hausdorff metric, and which commute with rigid motions. Examples constructed by means of different methods show that there are various nontrivial maps of this type. The main object of the paper is to find some reasonable additional assumptions which suffice to single out certain special maps, namely suitable combinations of dilatations and reflections, and of rotations if d = 2. For instance, we determine all maps which, besides having the properties mentioned above, commute with affine maps, or are surjective, or preserve the volume. The method of proof consists in an application of spherical harmonics, together with some convexity arguments.