The Picard group of noncommutative rings, in particular of orders
A.
Fröhlich
1-45
Abstract: The structure of the Picard group of not necessarily commutative rings, and specifically of orders, and its relation to the automorphism group are studied, mainly with arithmetic applications in mind.
Higher derivations and field extensions
R. L.
Davis
47-52
Abstract: Let $K$ be a field having prime characteristic $ p$. The following conditions on a subfield $k$ of $K$ are equivalent: (i) ${ \cap _n}{K^{{p^n}}}(k) = k$ and $K/k$ is separable. (ii) $ k$ is the field of constants of an infinite higher derivation defined in $ K$. (iii) $k$ is the field of constants of a set of infinite higher derivations defined in $ K$. If $K/k$ is separably generated and $ k$ is algebraically closed in $K$, then $k$ is the field of constants of an infinite higher derivation in $K$. If $K/k$ is finitely generated then $ k$ is the field of constants of an infinite higher derivation in $K$ if and only if $K/k$ is regular.
Optimal arcs and the minimum value function in problems of Lagrange
R. Tyrrell
Rockafellar
53-83
Abstract: Existence theorems are proved for basic problems of Lagrange in the calculus of variations and optimal control theory, in particular problems for arcs with both endpoints fixed. Emphasis is placed on deriving continuity and growth properties of the minimum value of the integral as a function of the endpoints of the arc and the interval of integration. Control regions are not required to be bounded. Some results are also obtained for problems of Bolza. Conjugate convex functions and duality are used extensively in the development, but the problems themselves are not assumed to be especially ``convex". Constraints are incorporated by the device of allowing the Lagrangian function to be extended-real-valued. This necessitates a new approach to the question of what technical conditions of regularity should be imposed that will not only work, but will also be flexible and general enough to meet the diverse applications. One of the underlying purposes of the paper is to present an answer to this question.
Dual spaces of groups with precompact conjugacy classes
John R.
Liukkonen
85-108
Abstract: We show that a second countable locally compact type I group with a compact invariant neighborhood of the identity is CCR, and has a Hausdorff dual if and only if its conjugacy classes are precompact. We obtain sharper results if the group is almost connected or has a fundamental system of invariant neighborhoods of the identity. Along the way we show that for a locally compact abelian group $ A$ and a group $ B$ of topological group automorphisms of $A, A$ has small $B$ invariant neighborhoods at 1 if and only if $\hat A$ has precompact orbits under the dual actions of $B$.
The structure of Dedekind cardinals
Erik
Ellentuck
109-125
Abstract: Semantic criteria are given for provability in set theory without the axiom of choice of positive sentences about the Dedekind cardinals. These criteria suggest that Dedekind cardinals (as well as general cardinals) have an internal structure.
Maximal regular right ideal space of a primitive ring. II
Kwangil
Koh;
Hang
Luh
127-141
Abstract: If $R$ is a ring, let $X(R)$ be the set of maximal regular right ideals of $R$. For each nonempty subset $E$ of $R$, define the hull of $ E$ to be the set $ \{ I \epsilon\, X(R)\vert E \subseteq I\}$ and the support of $ E$ to be the complement of the hull of $E$. Topologize $X(R)$ by taking the supports of right ideals of $ R$ as a subbase. If $ R$ is a right primitive ring, then $X(R)$ is homeomorphic to an open subset of a compact space $X({R^\char93 })$ of a right primitive ring ${R^\char93 }$, and $X(R)$ is a discrete space if and only if $ X(R)$ is a compact Hausdorff space if and only if either $R$ is a finite ring or a division ring. Call a closed subset $F$ of $X(R)$ a line if $F$ is the hull of $I \cap J$ for some two distinct elements $ I$ and $J$ in $X(R)$. If $R$ is a semisimple ring, then every line contains an infinite number of points if and only if either $ R$ is a division ring or $ R$ is a dense ring of linear transformations of a vector space of dimension two or more over an infinite division ring such that every pair of simple (right) $R$-modules are isomorphic.
Inversion formulae for the probability measures on Banach spaces
G.
Gharagoz Hamedani;
V.
Mandrekar
143-169
Abstract: Let $B$ be a real separable Banach space, and let $\mu$ be a probability measure on $\mathcal{B}(B)$, the Borel sets of $B$. The characteristic functional (Fourier transform) $\phi$ of $\mu$, defined by $\phi (y) = \int_B {\exp \{ i(y,x)\} d\mu (x)\;}$ for $y \in {B^\ast }$ (the topological dual of $B$), uniquely determines $\mu$. In order to determine $\mu$ on $ \mathcal{B}(B)$, it suffices to obtain the value of $\int_B {G(s)d\mu (s)}$ for every real-valued bounded continuous function $G$ on $B$. Hence an inversion formula for $\mu$ in terms of $\phi$ is obtained if one can uniquely determine the value of $\int_B {G(s)d\mu (s)}$ for all real-valued bounded continuous functions $G$ on $B$ in terms of $\phi$ and $G$. The main efforts of this paper will be to prove such inversion formulae of various types. For the Orlicz space $ {E_\alpha }$ of real sequences we establish inversion formulae (Main Theorem II) which properly generalize the work of L. Gross and derive as a corollary the extension of the Main Theorem of L. Gross to ${E_\alpha }$ spaces (Corollary 2.2.12). In Part I we prove a theorem (Main Theorem I) which is Banach space generalization of the Main Theorem of L. Gross by reinterpreting his necessary and sufficient conditions in terms of convergence of Gaussian measures. Finally, in Part III we assume our Banach space to have a shrinking Schauder basis to prove inversion formulae (Main Theorem III) which express the measure directly in terms of $\phi$ and $G$ without the use of extension of $\phi$ as required in the Main Theorems I and II. Furthermore this can be achieved without using the Lévy Continuity Theorem and we hope that one can use this theorem to obtain a direct proof of the Lévy Continuity Theorem.
Orbits of families of vector fields and integrability of distributions
Héctor J.
Sussmann
171-188
Abstract: Let $D$ be an arbitrary set of ${C^\infty }$ vector fields on the ${C^\infty }$ manifold $M$. It is shown that the orbits of $D$ are $ {C^\infty }$ submanifolds of $M$, and that, moreover, they are the maximal integral submanifolds of a certain $ {C^\infty }$ distribution $ {P_D}$. (In general, the dimension of ${P_D}(m)$ will not be the same for all $ m \in M$.) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow's theorem to the maximal integral submanifolds of the smallest distribution $\Delta$ such that every vector field $X$ in the Lie algebra generated by $ D$ belongs to $ \Delta$ (i.e. $X(m) \in \Delta (m)$ for every $m \in M$). Their work therefore requires the additional assumption that $\Delta$ be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of $\Delta$ is not assumed in proving the first main result. It turns out that $\Delta$ is integrable if and only if $\Delta = {P_D}$, and this fact makes it possible to derive a characterization of integrability and Chow's theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.
Hermitian vector bundles and value distribution for Schubert cycles
Michael J.
Cowen
189-228
Abstract: R. Bott and S. S. Chern used the theory of characteristic differential forms of a holomorphic hermitian vector bundle to study the distribution of zeroes of a holomorphic section. In this paper their methods are extended to study how often a holomorphic mapping into a Grassmann manifold hits Schubert cycles of fixed type.
Extension of Loewner's capacity theorem
Raimo
Näkki
229-236
Abstract: Analogues of a well-known theorem of Loewner concerning conformal capacity of a space ring are given in the case of an arbitrary domain.
Constructing isotopes on noncompact $3$-manifolds
Marianne S.
Brown
237-263
Abstract: We consider the question ``When are two homeomorphisms of a noncompact $3$-manifold onto itself isotopic?'' Roughly, the answer is when they are homotopic to each othet. More precisely, this paper deals with the question for irreducible $3$-manifolds which either have an infinite hierarchy or have such a hierarchy after the removal of a compact set. Manifolds having the first property are called end-irreducible; the others are called eventually endirreducible. There are two results fot each type of manifold depending on whether the homotopy between the two homeomorphisms sends the boundary of the manifold into itself or not.
Variational problems within the class of solutions of a partial differential equation
Robert
Delver
265-289
Abstract: The subject of this paper is the optimization of a multiple integral over a domain $G$ of a function, containing as arguments the independent variables, the unknown function and its partial derivatives up to order $l$, within the class of all sufficiently smooth solutions in $G$ of a given partial differential equation of order greater than or equal to $2l$. Necessary conditions in the form of a boundary value problem are derived. A physical application occurs in the control with boundary and initial conditions of a process in $G$ that is described by a specific partial differential equation.
On Lagrangian groups
J. F.
Humphreys;
D. L.
Johnson
291-300
Abstract: We study the class $\mathcal{L}$ of Lagrangian groups, that is, of finite groups $G$ possessing a subgroup of index $n$ for each factor $n$ of $\vert G\vert$. These groups and their analogues were considered by McLain in [4] and the object of the present work is to extend the results in this article. We study the classes $(G) = \{ H\vert G \times H \in \mathcal{L}\}$ and also the closure of $ \mathcal{L}$ under wreath products. We also consider the two classes $\mathfrak{X}$ and $ \mathfrak{Y}$ introduced in [2] and [4] respectively.
On symmetric orders and separable algebras
T. V.
Fossum
301-314
Abstract: Let $K$ be an algebraic number field, and let $\Lambda$ be an $R$-order in a separable $K$-algebra $A$, where $R$ is a Dedekind domain with quotient field $ K$; let $\Delta$ denote the center of $ \Lambda$. A left $ \Lambda$-lattice is a finitely generated left $\Lambda$-module which is torsion free as an $ R$-module. For left $ \Lambda$-modules $ M$ and $N, \operatorname{Ext} _\Lambda ^1(M,N)$ is a module over $\Delta$. In this paper we examine ideals of $ \Delta$ which are the annihilators of $\operatorname{Ext} _\Lambda ^1(M,\_)$ for certain classes of left $\Lambda$-lattices $M$ related to the central idempotents of $ A$, and we compute these ideals explicitly if $\Lambda$ is a symmetric $R$-algebra. For a group algebra, these ideals determine the defect of a block. We then compare these annihilator ideals with another set of ideals of $ \Delta$ which are closely related to the homological different of $\Lambda$, and which in a sense measure deviation from separability. Finally we show that, for $\Lambda$ to be separable over $ R$, it is necessary and sufficient that $\Lambda$ is a symmetric $R$-algebra, $\Delta$ is separable over $R$, and the center of each localization of $\Lambda$ at the maximal ideals of $R$ maps onto the center of its residue class algebra.
A characteristic zero non-Noetherian factorial ring of dimension three
John
David
315-325
Abstract: This paper shows the previously unknown existence of a finite dimensional non-Noetherian factorial ring in characteristic zero. The example, called ``$J$", contains a field of characteristic zero and is contained in a pure transcendental extension of degree three of that field. $J$ is seen to be an ascending union of polynomial rings and degree functions are introduced on each of the polynomial rings. These are the basic facts that enable it to be seen that two extensions of $ J$ are Krull. One of these extensions is a simple one and the other is a localization of $J$ at a prime ideal $P$. In the case of the latter extension, it is necessary to show that the intersection of the powers of $P$ is zero. As $J$ is the intersection of these two extensions, a theorem of Nagata is all that is needed to show then that $ J$ is factorial. It is easily proved that $J$ is non-Noetherian once it is known to be factorial.
$P$-commutative Banach $\sp{\ast} $-algebras
Wayne
Tiller
327-336
Abstract: Let $A$ be a complex $^ \ast $-algebra. If $ f$ is a positive functional on $A$, let ${I_f} = \{ x \in A:f(x^ \ast x) = 0\}$ be the corresponding left ideal of $A$. Set $ P = \cap {I_f}$, where the intersection is over all positive functionals on $ A$. Then $A$ is called $P$-commutative if $ xy - yx \in P$ for all $ x,y \in A$. Every commutative $^ \ast$-algebra is $P$-commutative and examples are given of noncommutative $^ \ast$-algebras which are $ P$-commutative. Many results are obtained for $P$-commutative Banach $^ \ast$-algebras which extend results known for commutative Banach $^ \ast$-algebras. Among them are the following: If $ {A^2} = A$, then every positive functional on $A$ is continuous. If $A$ has an approximate identity, then a nonzero positive functional on $A$ is a pure state if and only if it is multiplicative. If $A$ is symmetric, then the spectral radius in $ A$ is a continuous algebra seminorm.
Scattering theory for hyperbolic systems with coefficients of Gevrey type
William L.
Goodhue
337-346
Abstract: Using the techniques developed by P. D. Lax and R. S. Phillips, qualitative results on the location of the poles of the scattering matrix for symmetric, hyperbolic systems are obtained. The restrictions placed on the system are that the coefficient matrices be of Gevrey type and that the bicharacteristic rays tend to infinity.
Almost spherical convex hypersurfaces
John Douglas
Moore
347-358
Abstract: Let $M$ be a smooth compact hypersurface with positive sectional curvatures in $n$-dimensional euclidean space. This paper gives a sufficient condition for $M$ to lie in the spherical shell bounded by concentric spheres of radius $1 - \epsilon$ and $1 + \epsilon$. This condition is satisfied, in the case where $n = 3$, if the Gaussian curvature or the mean curvature of $M$ is sufficiently pointwise close to one.
$C\sp{2}$-preserving strongly continuous Markovian semigroups
W. M.
Priestley
359-365
Abstract: Let $X$ be a compact $ {C^2}$-manifold. Let $\{ {P^t}\}$ be a Markovian semigroup on $ C(X)$. The semigroup's infinitesimal generator $A$, with domain $ \mathcal{D}$, is defined by $Af = {\lim _{t \to 0}}{t^{ - 1}}({P^t}f - f)$, whenever the limit exists in $\vert\vert\;\vert\vert$. Theorem. Assume that $\{ {P^t}\}$ preserves $ {C^2}$-functions and that the restriction of $ \{ {P^t}\}$ to $ {C^2}(X) \subset \mathcal{D}$ and $A$ is a bounded operator from $ C(X),\vert\vert\;\vert\vert$. From the conclusion is obtained a representation of $Af \cdot (x)$ as an integrodifferential operator on ${C^2}(X)$. The representation reduces to that obtained by Hunt [Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264-293] in case $X$ is a Lie group and ${P^t}$ commutes with translations. Actually, a stronger result is proved having the above theorem among its corollaries.
Quasiconformal mappings and sets of finite perimeter
James C.
Kelly
367-387
Abstract: Let $D$ be a domain in ${R^n},n \geqslant 2,f$ a quasiconformal mapping on $ D$. We give a definition of bounding surface of codimension one lying in $ D$, and show that, given a system $\Sigma$ of such surfaces, the image of the restriction of $f$ to ``almost every'' surface is again a surface. Moreover, on these surfaces, $f$ takes $ {H^{n - 1}}$ (Hausdorff $ (n - 1)$-dimensional) null sets to ${H^{n - 1}}$ null sets. ``Almost every'' surface is given a precise meaning via the concept of the module of a system of measures, a generalization of the concept of extremal length.
Homeomorphisms with polyhedral irregular sets. I
P. F.
Duvall;
L. S.
Husch
389-406
Abstract: Homeomorphisms on open manifolds with polyhedral irregular sets are studied. For high dimensions, necessary and sufficient conditions for certain codimension three irregular sets to be tame are given. Several examples of homeomorphisms with wild irregular sets are given.
A Laurent expansion for solutions to elliptic equations
Reese
Harvey;
John C.
Polking
407-413
Abstract: Let $P(\xi )$ be a homogeneous elliptic polynomial of degree $m$. Let $E$ be a fundamental solution for the partial differential operator $P(D)$. Suppose $\Omega$ is a neighborhood of 0 in ${{\mathbf{R}}^n}$. Suppose $f \in {C^\infty }(\Omega \sim \{ 0\} )$ satisfies $P(D)f = 0$ in $ \Omega \sim \{ 0\}$. It is shown that there is a differential operator $ H(D)$ (perhaps of infinite order) and a function $g \in {C^\infty }(\Omega )$ satisfying $ P(D)g = 0$ in $ \Omega$, such that $f = H(D)E + g$ in $\Omega \sim \{ 0\}$. This analog of the Laurent expansion for $f$ is made unique by requiring that the Cauchy principal value of $H(D)E$ be equal to $H(D)E$.
Solutions of partial differential equations with support on leaves of associated foliations
E. C.
Zachmanoglou
415-421
Abstract: Suppose that the linear partial differential operator $P(x,D)$ has analytic coefficients and that it can be written in the form $P(x,D) = R(x,D)S(x,D)$ where $S(x,D)$ is a polynomial in the homogeneous first order operators ${A_1}(x,D), \cdots ,{A_r}(x,D)$. Then in a neighborhood of any point ${x^0}$ at which the principal part of $ S(x,D)$ does not vanish identically, there is a solution of $P(x,D)u = 0$ with support the leaf through $ {x^0}$ of the foliation induced by the Lie algebra generated by ${A_1}(x,D), \cdots ,{A_r}(x,D)$. This result yields necessary conditions for hypoellipticity and uniqueness in the Cauchy problem. An application to second order degenerate elliptic operators is also given.
Obstructions to embedding $n$-manifolds in $(2n-1)$-manifolds
J. W.
Maxwell
423-435
Abstract: Suppose $f:({M^n},\partial {M^n}) \to ({Q^{2n - 1}},\partial {Q^{2n - 1}})$ is a proper PL map between PL manifolds $ {M^n}$ and ${Q^{2n - 1}}$ of dimension $n$ and $2n - 1$ respectively, $M$ compact. J. F. P. Hudson has shown that associated with each such map $f$ that is an embedding on $\partial M$ is an element $\bar \alpha (f)$ in $ {H_1}(M;{Z_2})$ when $ n$ is odd and an element $\bar \beta (f)$ in ${H_1}(M;Z)$ when $n$ is even. These elements are invariant under a homotopy relative to $ \partial M$. We show that, under slight additional assumptions on $ M,Q$ and $f,f$ is homotopic to an embedding if and only if $ \bar \alpha (f) = 0$ for $ n$ odd and $\bar \beta (f) = 0$ for $n$ even. This result is used to give a sufficient condition for extending an embedding $ f:\partial {M^n} \to \partial {B^{2n - 1}}$ ( $ {B^{2n - 1}}$ denotes $ (2n - 1)$-dimensional ball) to an embedding $ F:({M^n},\partial {M^n}) \to ({B^{2n - 1}},\partial {B^{2n - 1}})$.
On the Arens products and certain Banach algebras
Pak Ken
Wong
437-448
Abstract: In this paper, we study several problems in Banach algebras concerned with the Arens products.
Multipliers and linear functionals for the class $N\sp{+}$
Niro
Yanagihara
449-461
Abstract: Multipliers for the classes ${H^p}$ are studied recently by several authors, see Duren's book, Theory of ${H^p}$ spaces, Academic Press, New York, 1970. Here we consider corresponding problems for the class ${N^ + }$ of holomorphic functions in the unit disk such that $\displaystyle \mathop {\lim }\limits_{r \to 1} \int_0^{2\pi } {{{\log }^ + }} \... ...= \int_0^{2\pi } {{{\log }^ + }\vert f({e^{i\theta }})\vert} d\theta < \infty .$ Our results are: 1. ${N^ + }$ is an $F$-space in the sense of Banach with the distance function $\displaystyle \rho (f,g) = \frac{1}{{2\pi }}\int_0^{2\pi } {\log (1 + \vert f({e^{i\theta }}) - g({e^{i\theta }})\vert)} d\theta .$ 2. A complex sequence $\Lambda = \{ {\lambda _n}\}$ is a multiplier for ${N^ + }$ into ${H^q}$ for a fixed $ q,0 < q < \infty$, if and only if ${\lambda _n} = O(\exp [ - c\sqrt n ])$ for a positive constant $c$. 3. A continuous linear functional $ \phi$ on the space $ {N^ + }$ is represented by a holomorphic function $g(z) = \Sigma {b_n}{z^n}$ which satisfies $ {b_n} = O(\exp [ - c\sqrt n ])$ for a positive constant $c$. Conversely, such a function $ g(z) = \Sigma {b_n}{z^n}$ defines a continuous linear functional on the space $ {N^ + }$.
Weak compactness in locally convex spaces
D. G.
Tacon
463-474
Abstract: The notion of weak compactness plays a central role in the theory of locally convex topological vector spaces. However, in the statement of many theorems, completeness of the space, or at least quasi-completeness of the space in the Mackey topology is an important assumption. In this paper we extend the concept of weak compactness in a general way and obtain a number of useful particular cases. If we replace weak compactness by these generalized notions we can drop the completeness assumption from the statement of many theorems; for example, we generalize the classical theorems of Eberlein and Kreĭn. We then consider generalizations of semireflexivity and reflexivity and characterize these properties in terms of our previous ideas as well as in terms of known concepts. In most of the proofs we use techniques of nonstandard analysis.
Closed hulls in infinite symmetric groups
Franklin
Haimo
475-484
Abstract: Let $\operatorname{Sym} M$ be the symmetric group of an infinite set $M$. What is the smallest subgroup of $\operatorname{Sym} M$ containing a given element if the subgroup is subject to the further condition that it is also the automorphism group of some finitary algebra on $M$? The structures of such closed hulls are related to the disjoint-cycle decompositions of the given elements. If the closed hull is not just the cyclic subgroup on the given element then it is nonminimal as a closed hull and is represented as a subdirect product of finite cyclic groups as well as by a quotient group of a group of infinite sequences. We determine the conditions under which it has a nontrivial primary component for a given prime $p$ and show that such components must be bounded abelian groups.
Heegaard splittings of homology $3$-spheres
Dean A.
Neumann
485-495
Abstract: We investigate properties of Heegaard splittings of closed $ 3$-manifolds which are known for simply-connected manifolds and which might provide the basis for a general test for simple-connectivity. Our results are negative: each property considered is shown to hold in a wider class of manifolds.
Barycenters, pinnacle points, and denting points
Surjit Singh
Khurana
497-503
Abstract: Some properties of probability measures, on closed convex bounded sets in locally convex spaces, having barycenters are obtained. Also some geometric and measure-theoretic characterizations of pinnacle points are given, and a result about denting points is proved.
Addendum to: ``Differential-boundary operators''
Allan M.
Krall
505
Abstract: The proof of a lemma and the statement of another were omitted from an earlier paper. This corrects that omission.
Addendum to: ``Modular representations of metabelian groups'' (Trans. Amer. Math. Soc. {\bf 169}(1972), 389--399)
B. G.
Basmaji
507-508