Plurisubharmonic functions and convexity properties for general function algebras
C. E.
Rickart
1-24
Abstract: A ``natural system'' consists of a Hausdorff space $\Sigma$ plus an algebra $\mathfrak{A}$ of complex-valued continuous functions on $\Sigma$ (which contains the constants and determines the topology in $\Sigma$) such that every continuous homomorphism of $\mathfrak{A}$ onto $ {\mathbf{C}}$ is given by an evaluation at a point of $\Sigma$ (compact-open topology in $\mathfrak{A}$). The prototype of a natural system is $ [{{\mathbf{C}}^n},\mathfrak{P}]$, where $ \mathfrak{P}$ is the algebra of polynomials on $ {{\mathbf{C}}^n}$. In earlier papers (Pacific J. Math. 18 and Canad. J. Math. 20), the author studied $ \mathfrak{A}$-holomorphic functions, which are generalizations of ordinary holomorphic functions in ${{\mathbf{C}}^n}$, and associated concepts of $\mathfrak{A}$-analytic variety and $ \mathfrak{A}$-holomorphic convexity in $\Sigma$. In the present paper, a class of extended real-valued functions, called $ \mathfrak{A}$-subharmonic functions, is introduced which generalizes the ordinary plurisubharmonic functions in ${{\mathbf{C}}^n}$. These functions enjoy many of the properties associated with plurisubharmonic functions. Furthermore, in terms of the $ \mathfrak{A}$-subharmonic functions, a number of convexity properties of ${{\mathbf{C}}^n}$ associated with plurisubharmonic functions can be generalized. For example, if $ G$ is an open $ \mathfrak{A}$-holomorphically convex subset of $\Sigma$ and $K$ is a compact subset of $G$, then the convex hull of $K$ with respect to the continuous $ \mathfrak{A}$-subharmonic functions on $G$ is equal to its hull with respect to the $ \mathfrak{A}$-holomorphic functions on $G$.
A characterization of $M$-spaces in the class of separable simplex spaces
Alan
Gleit
25-33
Abstract: We show that a separable simplex space is an $M$-space iff the arbitrary intersection of closed ideals is always an ideal.
Ascent, descent, and commuting perturbations
M. A.
Kaashoek;
D. C.
Lay
35-47
Abstract: In the present paper we investigate the stability of the ascent and descent of a linear operator $T$ when $T$ is subjected to a perturbation by a linear operator $C$ which commutes with $T$. The domains and ranges of $T$ and $C$ lie in some linear space $X$. The results are used to characterize the Browder essential spectrum of $T$. We conclude with a number of remarks concerning the notion of commutativity used in the present paper.
A comparison principle for terminal value problems in ordinary differential equations
Thomas G.
Hallam
49-57
Abstract: A comparison principle for a terminal value problem of an ordinary differential equation is formulated. Basic related topics such as the existence of maximal and minimal solutions of terminal value problems are investigated. The close relationship between the existence of the extremal solutions of a terminal value problem and the concept of asymptotic equilibrium of the differential equation is explored. Several applications of the terminal comparison principle are given.
On the ideal structure of Banach algebras
William E.
Dietrich
59-74
Abstract: For Banach algebras $ A$ in a class which includes all group and function algebras, we show that the family of ideals of $A$ with the same hull is typically quite large, containing ascending and descending chains of arbitrary length through any ideal in the family, and that typically a closed ideal of $A$ whose hull meets the Šilov boundary of $ A$ cannot be countably generated algebraically.
A sheaf-theoretic duality theory for cylindric algebras
Stephen D.
Comer
75-87
Abstract: Stone's duality between Boolean algebras and Boolean spaces is extended to a dual equivalence between the category of all $ \alpha$-dimensional cylindric algebras and a certain category of sheaves of such algebras. The dual spaces of important types of algebras are characterized and applications are given to the study of direct and subdirect decompositions of cylindric algebras.
Holomorphic maps into complex projective space omitting hyperplanes
Mark L.
Green
89-103
Abstract: Using methods akin to those of E. Borel and R. Nevanlinna, a generalization of Picard's theorem to several variables is proved by reduction to a lemma on linear relations among exponentials of entire functions. More specifically, it is shown that a holomorphic map from ${{\mathbf{C}}^m}$ to $ {{\mathbf{P}}_n}$ omitting $n + 2$ distinct hyperplanes has image lying in a hyperplane. If the map omits $n + 2$ or more hyperplanes in general position, the image will lie in a linear subspace of low dimension, being forced to be constant if the map omits $2n + 1$ hyperplanes in general position. The limits found for the dimension of the image are shown to be sharp.
Close-to-convex multivalent functions with respect to weakly starlike functions
David
Styer
105-112
Abstract: It is the object of this article to define close-to-convex multivalent functions in terms of weakly starlike multivalent functions. Six classes are defined, and shown to be equal. These generalize the class of close-to-convex functions developed by Livingston in the article, $p$-valent close-to-convex functions, Trans. Amer. Math. Soc. 115 (1965), 161-179.
Harmonic analysis on $F$-spaces with a basis
J.
Kuelbs;
V.
Mandrekar
113-152
Abstract: We establish Bochner's theorem and the Levy continuity theorem in the case that the underlying space is a real $F$-space with a basis, and then examine the infinitely divisible probability measures on a class of such spaces.
Wall manifolds with involution
R. J.
Rowlett
153-162
Abstract: Consider smooth manifolds $W$ with involution $t$ and a Wall structure described by a map $f:W \to {S^1}$ such that $ft = f$. For such objects we define cobordism theories ${\text{W}}_\ast ^I$ (in case $ W$ is closed, $ t$ unrestricted), ${\text{W}}_ \ast ^F$ (for $W$ closed, $t$ fixed-point free), and ${\text{W}}_ \ast ^{{\text{rel}}}$ ($ W$ with boundary, $ t$ free on $W$). We prove that there is an exact sequence $\displaystyle 0 \to {\text{W}}_ \ast ^I \to {\text{W}}_ \ast ^{{\text{rel}}} \to {\text{W}}_ \ast ^F \to 0.$ As a corollary, $ {\text{W}}_ \ast ^I$ imbeds in the cobordism of unoriented manifolds with involution. We also describe how ${\text{W}}_ \ast ^I$ determines the $ 2$-torsion in the cobordism of oriented manifolds with involution.
An algebra of distributions on an open interval
Harris S.
Shultz
163-181
Abstract: Let $(a,b)$ be any open subinterval of the reals which contains the origin and let $\mathfrak{B}$ denote the family of all distributions on $(a,b)$ which are regular in some interval $( \in ,0)$, where $ \in < 0$. Then $\mathfrak{B}$ is a commutative algebra: Multiplication is defined so that, when restricted to those distributions on $(a,b)$ whose supports are contained in $[0,b)$, it is ordinary convolution. Also, $\mathfrak{B}$ can be injected into an algebra of operators; this family of operators is a sequentially complete locally convex space. Since it preserves multiplication, this injection serves as a generalization (there are no growth restrictions) of the two-sided Laplace transformation.
A notion of capacity which characterizes removable singularities
Reese
Harvey;
John C.
Polking
183-195
Abstract: In this paper the authors define a capacity for a given linear partial differential operator acting on a Banach space of distributions. This notion has as special cases Newtonian capacity, analytic capacity, and AC capacity. It is shown that the sets of capacity zero are precisely those sets which are removable sets for the corresponding homogeneous equation. Simple properties of the capacity are derived and special cases examined.
Radon-Nikodym theorems for vector valued measures
Joseph
Kupka
197-217
Abstract: Let $\mu$ be a nonnegative measure, and let $ m$ be a measure having values in a real or complex vector space $V$. This paper presents a comprehensive treatment of the question: When is $m$ the indefinite integral with respect to $ \mu$ of a $V$ valued function $f?$ Previous results are generalized, and two new types of Radon-Nikodym derivative, the ``type $\rho$'' function and the ``strongly $\Gamma$ integrable'' function, are introduced. A derivative of type $\rho$ may be obtained in every previous Radon-Nikodym theorem known to the author, and a preliminary result is presented which gives necessary and sufficient conditions for the measure $m$ to be the indefinite integral of a type $\rho$ function. The treatment is elementary throughout, and in particular will include the first elementary proof of the Radon-Nikodym theorem of Phillips.
On antiflexible algebras
David J.
Rodabaugh
219-235
Abstract: In this paper we begin a classification of simple and semisimple totally antiflexible algebras (finite-dimensional) over splitting fields of char. $\ne 2,3$. For such an algebra $A$, let $P$ be the largest associative ideal in $ {A^ + }$ and let $ {N^ + }$ be the radical of $ P$. We determine all simple and semisimple totally antiflexible algebras in which $N \cdot N = 0$. Defining $ A$ to be of type $ (m,n)$ if ${N^ + }$ is nilpotent of class $ m$ with $\dim A = n$, we then characterize all simple nodal totally anti-flexible algebras (over fields of char. $\ne 2,3$) of types $(n,n)$ and $(n - 1,n)$ and give preliminary results for certain other types.
M\"untz-Szasz type approximation and the angular growth of lacunary integral functions
J. M.
Anderson
237-248
Abstract: We consider analogues of the Müntz-Szasz theorem, as in [15] and [4], for functions regular in an angle. This yields necessary and sufficient conditions for the existence of integral functions which are bounded in an angle and have gaps of a very regular nature in their power series expansion. In the case when the gaps are not so regular, similar results hold for formal power series which converge in the angle concerned.
A characterization of badly approximable functions
S. J.
Poreda
249-256
Abstract: Complex valued functions, continuous on a closed Jordan curve in the plane and having the property that all of their polynomials of best uniform approximation on that curve are identically zero are characterized in terms of their mapping properties on that curve.
Torsion differentials and deformation
D. S.
Rim
257-278
Abstract: Let $S$-scheme $X$ be a Schlessinger deformation of a curve $ {X_0}$ defined over a field $k$. In §§1 and 2, the dimension of the parameter space $S$, the relative differentials of $ X$ over $S$, and the fibres with singularity were studied, in case when ${X_0}$ is locally complete-intersection. In §3 we show that if $k$-scheme ${X_0}$ is a specialization of a smooth $ k$-scheme, then the punctured spectrum $\operatorname{Spex} ({O_{{X_{0,x}}}})$ has to be connected for every point $ x \in {X_0}$ such that $ \dim {O_{{X_{0,x}}}} \geqslant 2$. In turn we construct a rigid singularity on a surface. In the last section a few conjectures amplifying those of P. Deligne are made.
A method of symmetrization and applications. II
Dov
Aharonov;
W. E.
Kirwan
279-291
Abstract: In this paper we make use of a new method of symmetrization introduced in [1] to study various covering properties of univalent functions. More specifically, we introduce a generalization of a classical problem raised by Fekete and give a partial solution.
On the finitely generated subgroups of an amalgamated product of two groups
R. G.
Burns
293-306
Abstract: Sufficient conditions are found for the free product $G$ of two groups $A$ and $B$ with an amalgamated subgroup $U$ to have the properties (1) that the intersection of each pair of finitely generated subgroups of $G$ is again finitely generated, and (2) that every finitely generated subgroup containing a nontrivial subnormal subgroup of $G$ has finite index in $G$. The known results that Fuchsian groups and free products (under the obvious conditions on the factors) have properties (1) and (2) follow as instances of the main result.
Spaces of set-valued functions
David N.
O’Steen
307-315
Abstract: If $X$ and $Y$ are topological spaces, the set of all continuous functions from $X$ into $CY$, the space of nonempty, compact subsets of $ Y$ with the finite topology, contains a copy (with singleton sets substituted for points) of ${Y^X}$, the continuous point-valued functions from $ X$ into $Y$. It is shown that ${Y^X}$ is homeomorphic to this copy contained in ${(CY)^X}$ (where all function spaces are assumed to have the compact-open topology) and that, if $ X$ or $Y$ is $ {T_2},{(CY)^X}$ is homoemorphic to a subspace of $ {(CY)^{CX}}$. Further, if $ Y$ is ${T_2}$, then these images of $ {Y^X}$ and ${(CY)^X}$ are closed in ${(CY)^X}$ and $ {(CY)^{CX}}$ respectively. Finally, it is shown that, under certain conditions, some elements of ${X^Y}$ may be considered as elements of $ {(CY)^X}$ and that the induced $1$-$1$ function between the subspaces is open.
Complex structures on Riemann surfaces
Garo
Kiremidjian
317-336
Abstract: Let $X$ be a Riemann surface (compact or noncompact) with the property that the length of every closed geodesic is bounded away from zero. Then we show that sufficiently small complex structures on $ X$ can be described without making use of Schwarzian derivatives or the theory of quasiconformal mappings. Instead, we use methods developed in Kuranishi's work on the existence of locally complete families of deformations of compact complex manifolds. We introduce norms $\vert\quad {\vert _k}$ ($k$ a positive integer) on the space of ${C^\infty }(0,p)$-forms with values in the tangent bundle on $X$, which are similar to the usual Sobolev $\vert\vert\quad \vert{\vert _k}$-norms. (In the compact case $\vert\quad {\vert _k}$ is equivalent to $\vert\vert\quad \vert{\vert _k}$.) Then we prove that certain properties of $\vert\vert\quad \vert{\vert _k}$, crucial for Kuranishi's approach, are also satisfied by $\vert\quad {\vert _k}$.
Characteristic classes of stable bundles of rank $2$ over an algebraic curve
P. E.
Newstead
337-345
Abstract: Let $X$ be a complete nonsingular algebraic curve over $ {\mathbf{C}}$ and $ L$ a line bundle of degree 1 over $X$. It is well known that the isomorphism classes of stable bundles of rank 2 and determinant $L$ over $X$ form a nonsingular projective variety $ S(X)$. The Betti numbers of $S(X)$ are also known. In this paper we define certain distinguished cohomology classes of $S(X)$ and prove that these classes generate the rational cohomology ring. We also obtain expressions for the Chern character and Pontrjagin classes of $S(X)$ in terms of these generators.
Endomorphism rings of reduced torsion-free modules over complete discrete valuation rings
Wolfgang
Liebert
347-363
Abstract: The purpose of this paper is to find necessary and sufficient conditions that an abstract ring be isomorphic to the ring of all endomorphisms of a reduced torsion-free module over a (possibly noncommutative) complete discrete valuation ring.
Lattice-ordered injective hulls
Stuart A.
Steinberg
365-388
Abstract: It is well known that the injective hull of a lattice-ordered group ($ l$-group) $M$ can be given a lattice order in a unique way so that it becomes an $l$-group extension of $M$. This is not the case for an arbitrary $ f$-module over a partially ordered ring (po-ring). The fact that it is the case for any $l$-group is used extensively to get deep theorems in the theory of $l$-groups. For instance, it is used in the proof of the Hahn-embedding theorem and in the characterization of ${\aleph _a}$-injective $ l$-groups. In this paper we give a necessary and sufficient condition on the injective hull of a torsion-free $f$-module $M$ (over a directed essentially positive po-ring) for it to be made into an $f$-module extension of $M$ (in a unique way). An $f$-module is called an $i - f$-module if its injective hull can be made into an $f$-module extension. The class of torsion-free $ i - f$-modules is closed under the formation of products, sums, and Hahn products of strict $f$-modules. Also, an $l$-submodule and a torsion-free homomorphic image of a torsion-free $i - f$-module are $i - f$-modules. Let $R$ be an $f$-ring with zero right singular ideal whose Boolean algebra of polars is atomic. We show that $ R$ is a $qf$-ring (i.e., ${R_R}$ is an $i - f$-module) if and only if each torsion-free $ R - f$-module is an $ i - f$-module. There are no injectives in the category of torsion-free $ R - f$-modules, but there are ${\aleph _a}$-injectives. These may be characterized as the $f$-modules that are injective $R$-modules and ${\aleph _a}$-injective $l$-groups. In addition, each torsion-free $ f$-module over $ R$ can be embedded in a Hahn product of $l$-simple $Q(R) - f$-modules. We note, too, that a totally ordered domain has an $i - f$-module if and only if it is a right Ore domain.
Modular representations of metabelian groups
B. G.
Basmaji
389-399
Abstract: The irreducible modular representations, the blocks, and the defect groups of finite metabelian groups are determined. Also the dimensions of the principal indecomposable modules are computed.
Transverse cellular mappings of polyhedra
Ethan
Akin
401-438
Abstract: We generalize Marshall Cohen's notion of transverse cellular map to the polyhedral category. They are described by the following: Proposition. Let $f:K \to L$ be a proper simplicial map of locally finite simplicial complexes. The following are equivalent: (1) The dual cells of the map are all cones. (2) The dual cells of the map are homogeneously collapsible in $K$. (3) The inclusion of $L$ into the mapping cylinder of $ f$ is collared. (4) The mapping cylinder triad $({C_f},K,L)$ is homeomorphic to the product triad $(K \times I;K \times 1,K \times 0)$ rel $K = K \times 1$. Condition (2) is slightly weaker than ${f^{ - 1}}$(point) is homogeneously collapsible in $K$. Condition (4) when stated more precisely implies $f$ is homotopic to a homeomorphism. Furthermore, the homeomorphism so defined is unique up to concordance. The two major applications are first, to develop the proper theory of ``attaching one polyhedron to another by a map of a subpolyhedron of the former into the latter". Second, we classify when two maps from $ X$ to $Y$ have homeomorphic mapping cylinder triads. This property turns out to be equivalent to the equivalence relation generated by the relation $ f \sim g$, where $f,g:X \to Y$ means $f = gr$ for $r:X \to X$ some transverse cellular map.
Diffusion and Brownian motion on infinite-dimensional manifolds
Hui Hsiung
Kuo
439-459
Abstract: The purpose of this paper is to construct certain diffusion processes, in particular a Brownian motion, on a suitable kind of infinite-dimensional manifold. This manifold is a Banach manifold modelled on an abstract Wiener space. Roughly speaking, each tangent space ${T_x}$ is equipped with a norm and a densely defined inner product $g(x)$. Local diffusions are constructed first by solving stochastic differential equations. Then these local diffusions are pieced together in a certain way to get a global diffusion. The Brownian motion is completely determined by $g$ and its transition probabilities are proved to be invariant under ${d_g}$-isometries. Here ${d_g}$ is the almost-metric (in the sense that two points may have infinite distance) associated with $g$. The generalized Beltrami-Laplace operator is defined by means of the Brownian motion and will shed light on the study of potential theory over such a manifold.
Hypersurfaces of nonnegative curvature in a Hilbert space
Leo
Jonker
461-474
Abstract: In this paper we prove the following generalizations of known theorems about hypersurfaces in ${{\mathbf{R}}^n}$: Let $M$ be a hypersurface in a Hilbert space. (1) If on $ M$ the sectional curvature $K(\sigma )$ is nonnegative for every $ 2$-plane section $ \sigma$ and if $K(\sigma ) > 0$ for at least one $\sigma$, then $M$ is the boundary of a convex body. (2) If $K(\sigma ) = 0$ for all $\sigma$, then $M$ is a hypercylinder. The main tool in these theorems is Smale's infinite dimensional Sard's theorem.
The structure of pseudocomplemented distributive lattices. III. Injective and absolute subretracts
G.
Grätzer;
H.
Lakser
475-487
Abstract: Absolute subretracts are characterized in the classes $ {\mathcal{B}_n},n \leqslant \omega$. This is applied to describe the injectives in $ {\mathcal{B}_1}$ (due to R. Balbes and G. Grätzer) and ${\mathcal{B}_2}$.
Abelian subgroups of finite $p$-groups
Susan Claire
Dancs
489-493
Abstract: Information is obtained about the order of maximal abelian subgroups of central powers and crown products of finite $ p$-groups. This is used to construct groups with ``small'' maximal abelian subgroups.
A non-Noetherian factorial ring
John
David
495-502
Abstract: This paper supplies a counterexample to the conjecture that factorial implies Noetherian in finite Krull dimension. The example is the integral closure of a three-dimensional Noetherian ring, and is the union of Noetherian domains, which are proven to be factorial by means of derivation techniques.
A note on the geometric means of entire functions of several complex variables
P. K.
Kamthan
503-508
Abstract: Let $f({z_1}, \cdots ,{z_n})$ be an entire function of $n( \geqslant 2)$ complex variables. Recently Agarwal [Trans. Amer. Math. Soc. 151 (1970), 651-657] has obtained certain results involving geometric mean values of $ f$. In this paper we have constructed examples to contradict some of the results of Agarwal and have thereafter given improvements and modifications of his results.