Finite groups with Sylow 2-subgroups of class two. I
Robert
Gilman;
Daniel
Gorenstein
1-101
Abstract: In this paper we classify finite simple groups whose Sylow 2-subgroups have nilpotency class two.
Finite groups with Sylow 2-subgroups of class two. II
Robert
Gilman;
Daniel
Gorenstein
103-126
Abstract: In this paper we classify finite simple groups whose Sylow 2-subgroups have nilpotence class two.
Convergence and divergence of series conjugate to a convergent multiple Fourier series
J. Marshall
Ash;
Lawrence
Gluck
127-142
Abstract: In this note we consider to what extent the classical theorems of Plessner and Kuttner comparing the set of convergence of a trigonometric series with that of the conjugate trigonometric series can be generalized to higher dimensions. We show that if a function belongs to ${L^p},p > 1$, of the $2$-torus, then the convergence (= unrestricted rectangular convergence) of the Fourier series on a set implies its three conjugate functions converge almost everywhere on that set. That this theorem approaches the best possible may be seen from two examples which show that the dimension may not be increased to 3, nor the required power of integrability be decreased to 1. We also construct a continuous function having a boundedly divergent Fourier series of power series type and an a.e. circularly convergent double Fourier series whose $y$-conjugate diverges circularly a.e. Our $ {L^p}$ result depends on a theorem of L. Gogöladze (our proof is included for the reader's convenience), work of J. M. Ash and G. Welland on $(C,1,0)$ summability, and on a result deducing the boundedness of certain partial linear means from convergence of those partial means. The construction of the counterexamples utilizes examples given by C. Fefferman, J. Marcinkiewicz, A. Zygmund, D. Menšov, and the present authors' earlier work.
Generalization of right alternative rings
Irvin Roy
Hentzel;
Giulia Maria
Piacentini Cattaneo
143-161
Abstract: We study nonassociative rings $R$ satisfying the conditions (1) $ (ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b$ for all $a,b,c,d \in R$, and (2) $ (x,x,x) = 0$ for all $x \in R$. We furthermore assume weakly characteristic not 2 and weakly characteristic not 3. As both (1) and (2) are consequences of the right alternative law, our rings are generalizations of right alternative rings. We show that rings satisfying (1) and (2) which are simple and have an idempotent $\ne 0, \ne 1$, are right alternative rings. We show by example that $(x,e,e)$ may be nonzero. In general, $(a,b,c) + (b,c,a) + (c,a,b) = 0$ for all $a,b,c \in R$. We generate the Peirce decomposition. If $R'$ has no trivial ideals contained in its center, the table for the multiplication of the summands is associative, and the nucleus of $R'$ contains ${R'_{10}} + {R'_{01}}$. Without the assumption on ideals, the table for the multiplication need not be associative; however, if the multiplication is defined in the most obvious way to force an associative table, the new ring will still satisfy (1), (2), (3).
The zeros of holomorphic functions in strictly pseudoconvex domains
Lawrence
Gruman
163-174
Abstract: We determine a sufficient condition on a positive divisor in certain strictly pseudoconvex domains in ${{\mathbf{C}}^n}$ such that there exists a function in the Nevanlinna class which determines the divisor.
Families of holomorphic maps into Riemann surfaces
Theodore J.
Barth
175-187
Abstract: In analogy with the Hartogs theorem that separate analyticity of a function implies analyticity, it is shown that a separately normal family of holomorphic maps from a polydisk into a Riemann surface is a normal family. This contrasts with examples of discontinuous separately analytic maps from a bidisk into the Riemann sphere. The proof uses a theorem on pseudoconvexity of normality domains, which is proved via the following convergence criterion: a sequence $ \{ {f_j}\}$ of holomorphic maps from a complex manifold into a Riemann surface converges to a nonconstant holomorphic map if and only if the sequence $\{ f_j^{ - 1}\}$ of set-valued maps, defined on the Riemann surface, converges to a suitable set-valued map. Extending Osgood's theorem, it is also shown that a separately analytic map (resp. a separately normal family of holomorphic maps) from a polydisk into a hyperbolic complex space is analytic (resp. normal).
A generalized topological measure theory
R. B.
Kirk;
J. A.
Crenshaw
189-217
Abstract: The theory of measures in a topological space, as developed by V. S. Varadarajan for the algebra ${C^b}$ of bounded continuous functions on a completely regular topological space, is extended to the context of an arbitrary uniformly closed algebra $ A$ of bounded real-valued functions. Necessary and sufficient conditions are given for ${A^ \ast }$ to be represented in the natural way by a space of regular finitely-additive set functions. The concepts of additivity and tightness for these set functions are considered and some remarks about weak convergence are made.
Boundary behavior of the Carath\'eodory and Kobayashi metrics on strongly pseudoconvex domains in $C\sp{n}$ with smooth boundary
Ian
Graham
219-240
Abstract: The Carathéodory and Kobayashi distance functions on a bounded domain $G$ in $ {{\mathbf{C}}^n}$ have related infinitesimal forms. These are the Carathéodory and Kobayashi metrics. They are denoted by $ F(z,\xi )$ (length of the tangent vector $\xi$ at the point $z$). They are defined in terms of holomorphic mappings, from $G$ to the unit disk for the Carathéodory metric, and from the unit disk to $G$ for the Kobayashi metric. We consider the boundary behavior of these metrics on strongly pseudoconvex domains in $ {{\mathbf{C}}^n}$ with $ {C^2}$ boundary. $ \xi$ is fixed and $ z$ is allowed to approach a boundary point ${z_0}$. The quantity $F(z,\xi )d(z,\partial G)$ is shown to have a finite limit. In addition, if $\xi$ belongs to the complex tangent space to the boundary at ${z_0}$, then this first limit is zero, and $ {(F(z,\xi ))^2}d(z,\partial G)$ has a (nontangential) limit in which the Levi form appears. We prove an approximation theorem for bounded holomorphic functions which uses peak functions in a novel way. The proof was suggested by N. Kerzman. This theorem is used here in studying the boundary behavior of the Carathéodory metric.
Fiber preserving equivalence
Richard T.
Miller
241-268
Abstract: We give a theory of fibered regular neighborhoods based on a remarkable property of simplicial fibered projections. All the usual properties of regular neighborhoods are retained. Using Millett's fibered general position, together with the regular neighborhoods, we prove THEOREM. The simplicial space of codimension 4 PL embeddings of a complex into a PL manifold is locally contractible at each point of the space of topological embeddings.
Cone complexes and PL transversality
Clint
McCrory
269-291
Abstract: A definition of PL transversality is given, using the orderreversing duality on partially ordered sets. David Stone's theory of stratified polyhedra is thereby simplified; in particular, the symmetry of blocktransversality is proved. Also, polyhedra satisfying Poincaré duality are characterized geometrically.
Finitary imbeddings of certain generalized sample spaces
Marie A.
Gaudard;
Robert J.
Weaver
293-307
Abstract: A generalized sample space each of whose subspaces has as its logic an orthomodular poset is called an HD sample space. In this paper it is shown that any HD sample space may be imbedded in a natural way in a generalized sample space which is HD and at the same time admits a full set of dispersion free weight functions.
Approximate isometries on finite dimensional Banach spaces
Richard D.
Bourgin
309-328
Abstract: A map $ T:{{\mathbf{E}}_1} \to {{\mathbf{E}}_2}$ ( $ {{\mathbf{E}}_1},{{\mathbf{E}}_2}$ Banach spaces) is an $\epsilon$-isometry if $ \vert\;\vert\vert T(X) - T(Y)\vert\vert - \vert\vert X - Y\vert\vert\;\vert \leqslant \epsilon$ whenever $X,Y \in {{\mathbf{E}}_1}$. The problem of uniformly approximating such maps by isometries was first raised by Hyers and Ulam in 1945 and subsequently studied for special infinite dimensional Banach spaces. This question is here broached for the class of finite dimensional Banach spaces. The only positive homogeneous candidate isometry $ U$ approximating a given $ \epsilon$-isometry $ T$ is defined by the formal limit $U(X) = {\lim _{r \to \infty }}{r^{ - 1}}T(rX)$. It is shown that, whenever $T:{\mathbf{E}} \to {\mathbf{E}}$ is a surjective $\epsilon$-isometry and $ {\mathbf{E}}$ is a finite dimensional Banach space for which the set of extreme points of the unit ball is totally disconnected, then this limit exists. When ${\mathbf{E}} = \ell _1^k( = k$ - dimensional$\; {\ell _1})$ a uniform bound of uniform approximation is obtained for surjective $\epsilon $-isometries by isometries; this bound varies linearly in $\epsilon$ and with ${k^3}$.
The subclass algebra associated with a finite group and subgroup
John
Karlof
329-341
Abstract: Let $G$ be a finite group and let $H$ be a subgroup of $G$. If $g \in G$, then the set ${E_g} = \{ hg{h^{ - 1}}\vert h \in H\}$ is the subclass of $G$ containing $g$ and $ {\Sigma _{x \in {E_g}}}x$ is the subclass sum containing $g$. The algebra over the field of complex numbers generated by these subclass sums is called the subclass algebra (denoted by $S$) associated with $ G$ and $H$. The irreducible modules of $ S$ are demonstrated, and results about Schur algebras are used to develop formulas relating the irreducible characters of $S$ to the irreducible characters of $ G$ and $H$.
Quantum logic and the locally convex spaces
W. John
Wilbur
343-360
Abstract: An important theorem of Kakutani and Mackey characterizes an infinite dimensional real (complex) Hilbert space as an infinite dimensional real (complex) Banach space whose lattice of closed subspaces admits an orthocomplementation. This result, also valid for quaternionic spaces, has proved useful as a justification for the unique role of Hilbert space in quantum theory. With a like application in mind, we present in the present paper a number of characterizations of real and complex Hilbert space in the class of locally convex spaces. One of these is an extension of the Kakutani-Mackey result from the infinite dimensional Banach spaces to the class of all infinite dimensional complete Mackey spaces. The implications for the foundations of quantum theory are discussed.
$I$-rings
W. K.
Nicholson
361-373
Abstract: A ring $ R$, possibly with no identity, is called an ${I_0}$-ring if each one-sided ideal not contained in the Jacobson radical $J(R)$ contains a nonzero idempotent. If, in addition, idempotents can be lifted modulo $J(R),R$ is called an $I$-ring. A survey of when these properties are inherited by related rings is given. Maximal idempotents are examined and conditions when $ {I_0}$-rings have an identity are given. It is shown that, in an ${I_0}$-ring $R$, primitive idempotents are local and primitive idempotents in $R/J(R)$ can always be lifted. This yields some characterizations of ${I_0}$-rings $R$ such that $R/J(R)$ is primitive with nonzero socle. A ring $ R$ (possibly with no identity) is called semiperfect if $R/J(R)$ is semisimple artinian and idempotents can be lifted modulo $J(R)$. These rings are characterized in several new ways: among them as ${I_0}$-rings with no infinite orthogonal family of idempotents, and as ${I_0}$-rings $R$ with $R/J(R)$ semisimple artinian. Several other properties are derived. The connection between $ {I_0}$-rings and the notion of a regular module is explored. The rings $ R$ which have a regular module $M$ such that $J(R) = \operatorname{ann} (M)$ are studied. In particular they are ${I_0}$-rings. In addition, it is shown that, over an ${I_0}$-ring, the endomorphism ring of a regular module is an ${I_0}$-ring with zero radical.
On the inverse problem of Galois theory
J.
Kovacic
375-390
Abstract: Let $k$ be a field, $F$ a finite subfield and $G$ a connected solvable algebraic matric group defined over $F$. Conditions on $G$ and $k$ are given which ensure the existence of a Galois extension of $k$ with group isomorphic to the $F$-rational points of $G$.
Functions of vanishing mean oscillation
Donald
Sarason
391-405
Abstract: A function of bounded mean oscillation is said to have vanishing mean oscillation if, roughly speaking, its mean oscillation is locally small, in a uniform sense. In the present paper the class of functions of vanishing mean oscillation is characterized in several ways. This class is then applied to answer two questions in analysis, one involving stationary stochastic processes satisfying the strong mixing condition, the other involving the algebra ${H^\infty } + C$.
Erratum to ``Transversally parallelizable foliations of codimension two''
Lawrence
Conlon
406