Higher products
Gerald J.
Porter
315-345
Analytic sheaves of local cohomology
Yum-tong
Siu
347-366
A density theorem with an application to gap power series
K. G.
Binmore
367-384
Abstract: Let N be a set of positive integers and let $\displaystyle F(z) = \sum {{A_n}{z^n}}$ be an entire function for which ${A_n} = 0(n \notin N)$. It is reasonable to expect that, if D denotes the density of the set N in some sense, then $F(z)$ will behave somewhat similarly in every angle of opening greater than $2\pi D$. For functions of finite order, the appropriate density seems to be the Pólya maximum density $ \mathcal{P}$. In this paper we introduce a new density $ \mathcal{D}$ which is perhaps the appropriate density for the consideration of functions of unrestricted growth. It is shown that, if $ \vert I\vert > 2\pi \mathcal{D}$, then $\displaystyle \log M(r) \sim \log M(r,I)$ outside a small exceptional set. Here $M(r)$ denotes the maximum modulus of $ F(z)$ on the circle $ \vert z\vert = r$ and $ M(r,I)$ that of $F(r{e^{i\theta }})$ for values of $\theta$ in the closed interval I. The method used is closely connected with the question of approximating to functions on an interval by means of linear combinations of the exponentials ${e^{ixn}}(n \in N)$.
Representation theorems for complemented algebras
Freda E.
Alexander
385-398
Strongly separable pairings of rings
Robert S.
Cunningham
399-416
Abstract: The theory of adjoint functors has been used by Morita to develop a theory of Frobenius and quasi-Frobenius extensions subsuming the work of Kasch, Müller, Nakayama, and others. We use adjoint functors to define a pairing of the two rings and develop a theory of relative projective and injective modules for pairings generalizing that of Hochschild for extensions. The main purpose of this paper is to define ``strongly separable pairings'' generalizing strongly separable (i.e. finitely generated projective separable) algebras. We show that such pairings have very close connections to category equivalences, so that it is natural to investigate those properties shared by two rings which admit a strongly separable pairing. We show that most ``categorical'' properties are so shared.
The pseudo-circle is not homogeneous
James T.
Rogers
417-428
Recursive functions modulo ${\rm CO}-r$-maximal sets
Manuel
Lerman
429-444
Abstract: Define the equivalence relation ${ \sim _A}$ on the set of recursive functions of one variable by $f\sim_A g$ if and only if $ f(x) = g(x)$ for all but finitely many $x \in \bar A$, where $ \bar A$ is an r-cohesive set, to obtain the structure $\mathcal{R}/\bar A$. Then the recursive functions modulo such an equivalence relation form a semiring with no zero divisors. It is shown that if A is r-maximal, then the structure obtained above is not a nonstandard model for arithmetic, a result due to Feferman, Scott, and Tennenbaum. Furthermore, if A and B are maximal sets, then a necessary and sufficient condition for $\mathcal{R}/\bar A$ and $\mathcal{R}/\bar B$ to be elementarily equivalent is obtained. It is also shown that many different elementary theories can be obtained for $\mathcal{R}/\bar A$ by proper choice of $ \bar A$.
Compact imbedding theorems for quasibounded domains
Robert A.
Adams
445-459
Convolutions with kernels having singularities on a sphere
Robert S.
Strichartz
461-471
Abstract: We prove that convolution with $(1 - \vert x{\vert^2})_ + ^{ - \alpha }$ and related convolutions are bounded from ${L^p}$ to ${L^q}$ for certain values of p and q. There is a unique choice of p which maximizes the measure of smoothing $1/p - 1/q$, in contrast with fractional integration where $1/p - 1/q$ is constant. We apply the results to obtain a priori estimates for solutions of the wave equation in which we sacrifice one derivative but gain more smoothing than in Sobolev's inequality.
Nice homology coalgebras
A. K.
Bousfield
473-489
Sur le rel\`evement des representations modulaires d'un groupe fini
François
Aribaud
491-499
Mesures associ\'ees aux fonctionnelles additives de Markov. I
D.
Revuz
501-531
Abstract: With each additive functional of Markov processes we associate a measure and characterize, under duality hypotheses, those which correspond to $\sigma$-finite measures. This enables us to weaken the hypotheses of Meyer's theorem on representation of potentials of measures as potentials of additive functional. We characterize also the measures which are associated with continuous additive functionals. This leads us to show that for each finite continuous additive functional of the process there exists a finite continuous additive functional of the dual process such that the corresponding time-changed processes are in duality. Similar results are also stated for subprocesses which generalize results by Hunt and Blumenthal and Getoor.
Finite nilpotent characteristic nonverbal groups
Orin
Chein
533-548
Abstract: In this paper, we study nilpotent groups which are quotient groups of finitely generated free groups with respect to characteristic but nonverbal subgroups. We show that there are no abelian groups of the type in question. We also show that all such groups of nilpotence class 2 or 3 are finite and have minimal sets of two generators. In fact, formal presentations for all such groups are given. The direct product of two finite CNV groups (as the groups in question will be called) which have minimal sets of generators of the same size is shown to again be a CNV group, provided that the orders of the original two groups are relatively prime. Conversely, if a finite CNV group is a direct product of groups of relatively prime orders, then at least one of these direct factors is a CNV group. Several other related results are also obtained.
Injective and projective Heyting algebras
Raymond
Balbes;
Alfred
Horn
549-559
A class of decompositions of $E\sp{n}$ which are factors of $E\sp{n+1}$
John L.
Bailey
561-575
Characterizations of $C\sp{\ast} $-algebras. II
T. W.
Palmer
577-588
Nest generated intersection rings in Tychonoff spaces
A. K.
Steiner;
E. F.
Steiner
589-601
A topologically strongly mixing symbolic minimal set
K. E.
Petersen
603-612
Abstract: Recent papers by the author, Keynes and Robertson, and others have shown that weakly mixing minimal flows are objects of considerable interest, but examples of such flows, other than the horocycle flows, have been scarce. We give here a ``machinal'' construction of a bilateral sequence with entries from 0, 1 whose orbit closure is topologically strongly mixing and minimal. We prove in addition that the flow we obtain has entropy zero, is uniquely ergodic, and fails to be measure-theoretically strongly mixing.
Some remarks on self-dual locally compact Abelian groups
Lawrence
Corwin
613-622
Abstract: The main results of this paper are the construction of some new self-dual locally compact Abelian groups and the proof of a structure theorem for a certain class of such groups. The construction is based on an investigation of when the extension of a compact Abelian group by its dual yields a self-dual group. It turns out that such extensions can be described algebraically ; the structure theorem follows from an analysis of the algebraic description.
Corrections to: ``On sequential convergence''
R. M.
Dudley
623-624
Errata to: ``Vector cross products on manifolds''
Alfred
Gray
625