The rank $3$ permutation representations of the finite classical groups
William M.
Kantor;
Robert A.
Liebler
1-71
Abstract: The permutation representations in the title are all determined, and no surprises are found to occur.
Type structure complexity and decidability
T. S.
Millar
73-81
Abstract: We prove that for every countable homogeneous model $\mathcal{A}$ such that the set of recursive types of $\operatorname{Th} (\mathcal{A})$ is $ \sum _2^0$, $\mathcal{A}$ is decidable iff the set of types realized in $\mathcal{A}$ is a $\sum _2^0$ set of recursive types. As a corollary to a lemma, we show that if a complete theory $ T$ has a recursively saturated model that is decidable in the degree of $ T$, then $T$ has a prime model.
Trees, Gleason spaces, and coabsolutes of $\beta {\bf N}\sim {\bf N}$
Scott W.
Williams
83-100
Abstract: For a regular Hausdorff space $X$, let $ \mathcal{E}(X)$ denote its absolute, and call two spaces $X$ and $Y$ coabsolute ( $ \mathcal{G}$-absolute) when $ \mathcal{E}(X)$ and $\mathcal{E}(Y)$ ( $\beta \mathcal{E}(X)$ and $\beta \mathcal{E}(Y)$) are homeomorphic. We prove $ X$ is $\mathcal{G}$-absolute with a linearly ordered space iff the POSET of proper regular-open sets of $ X$ has a cofinal tree; a Moore space is $ \mathcal{G}$-absolute with a linearly ordered space iff it has a dense metrizable subspace; a dyadic space is $\mathcal{G}$-absolute with a linearly ordered space iff it is separable and metrizable; if $ X$ is a locally compact noncompact metric space, then $\beta X \sim X$ is coabsolute with a compact linearly ordered space having a dense set of $ P$-points; CH implies but is not implied by "if $X$ is a locally compact noncompact space of $ \pi$-weight at most ${2^\omega }$ and with a compatible complete uniformity, then $\beta X \sim X$ and $\beta N \sim N$ are coabsolute."
Universal bundles for deformations of asymmetric structures
Paul R.
Dippolito
101-115
Abstract: Riemannian parallel transport of frames and Cartan connections can be considered as special cases of a type of structure in which the connection form on the principal bundle can transform by an arbitrary extension of the adjoint representation of the fiber group; there always exist universal deformations for asymmetric objects in any of these geometries.
The Hodge theory of flat vector bundles on a complex torus
Jerome William
Hoffman
117-131
Abstract: We study the Hodge spectral sequence of a local system on a compact, complex torus by means of the theory of harmonic integrals. It is shown that, in some cases, Baker's theorems concerning linear forms in the logarithms of algebraic numbers may be applied to obtain vanishing theorems in cohomology. This is applied to the study of Betti and Hodge numbers of compact analytic threefolds which are analogues of hyperelliptic surfaces. Among other things, it is shown that, in contrast to the two-dimensional case, some of these varieties are nonalgebraic.
Transfer and the spectral sequence of a fibration
Carlos
Prieto
133-142
Abstract: The purpose of this paper is to show that transfers for fibrations behave well with respect to spectral sequences which are induced by filtrations of the base space. In particular, for the spectral sequence of a fibration (induced by the skeletal filtration of the base space), one obtains the expected effect on the ${E_2}$-terms: We prove that the transfer in the $ {E_2}$-terms is determined by the transfer of the fiber (considered trivially as a fibration over a point). As an application, results of Atiyah on the $K$-theory of classifying spaces are transcribed to generalized cohomology theories.
Torsion-groups of abelian coverings of links
John P.
Mayberry;
Kunio
Murasugi
143-173
Abstract: If $M$ is an abelian branched covering of $ {S^3}$ along a link $ L$, the order of $ {H_1}(M)$ can be expressed in terms of (i) the Alexander polynomials of $ L$ and of its sublinks, and (ii) a "redundancy" function characteristic of the monodromy-group. In 1954, the first author thus generalized a result of Fox (for $L$ a knot, in which case the monodromy-group is cyclic and the redundancy trivial); we now prove earlier conjectures and give a simple interpretation of the redundancy. Cyclic coverings of links are discussed as simple special cases. We also prove that the Poincaré conjecture is valid for the above-specified family of $3$-manifolds $M$. We state related results for unbranched coverings.
Permutation-partition pairs. II. Bounds on the genus of the amalgamation of graphs
Saul
Stahl
175-182
Abstract: Bounds are derived on the extent to which the parameter $\mu (P,\,\prod )$ can fail to be additive over disjoint permutations. This is done by associating an Eulerian digraph to each such pair and relating the maximum orbiticity $ \mu (P,\,\prod )$ to the decompositions of this digraph's arc set into arc disjoint cycles. These bounds are then applied to obtain information about the genus of the amalgamation of graphs.
The minimum modulus of certain small entire functions
P. C.
Fenton
183-195
Abstract: Suppose that $ f(z)$ is an entire function satisfying $\displaystyle \mathop {\min }\limits_{\vert z\vert = r} \vert f(z)\vert \leqslant C(\sigma )\mathop {\max }\limits_{\vert z\vert = r} \vert f(z)\vert,$ for $r \geqslant {\rho _0} > 0$, where $\sigma > 0$ and $\displaystyle C(\sigma ) = \prod\limits_{k = 1}^\infty {{{\left\{ {\frac{{1 - \... ...- (2k - 1) / 4\sigma )}} {{1 + \exp ( - (2k - 1) / 4\sigma )}}} \right\}}^2}.}$ It is shown that $\displaystyle \mathop {\underline {\lim } }\limits_{r \to \infty } \frac{{{{\ma... ...)}^2}}} {{\log r}} \geqslant - 2\sigma \log (\max ({\rho _0},\vert{a_1}\vert)),$ where $ {a_1}$ is the first nonzero zero of $f$.
Siegel domains and representations of Jordan algebras
O. S.
Rothaus
197-213
Abstract: In the analysis of infinitesimal automorphisms of arbitrary Siegel domains, a certain class of nonsemisimple Jordan algebras occurs. The description of all the infinitesimal automorphisms of the domain may be based on a study of representations of the associated Jordan algebra satisfying a certain "strange identity". In this paper, all the possibilities for the Jordan algebra and representations satisfying the identity are given.
Tangent $2$-fields on even-dimensional nonorientable manifolds
Benedict J.
Pollina
215-224
Abstract: This paper uses the Postnikov decomposition of a nonsimple fibration to describe the obstructions to a tangent $2$-field on an even-dimensional nonorientable manifold.
The sums of powers theorem for commuting block maps
Frank
Rhodes
225-236
Abstract: A block map is a map $ f:{\{ 0,\,1\} ^n} \to \{ 0,\,1\}$ for some $n \geqslant 1$. A block map $ f$ induces an endomorphism ${f_\infty }$ of the full $2$-shift $ (X,\,\sigma )$. Composition of block maps is defined in such a way that ${(f \circ g)_\infty } = {f_\infty } \circ {g_\infty }$. In this paper some recent results concerning the set $ \{ g\vert g \circ f = f \circ g\}$ for certain types of block maps $f$ are extended.
Tauberian conditions for $L\sp{1}$-convergence of Fourier series
Časlav V.
Stanojević
237-244
Abstract: It is proved that Fourier series with asymptotically even coefficients and satisfying $ {\lim _{\lambda \to 1}}\lim {\sup _{n \to \infty }}\sum _{j = n}^{[\lambda n]}{j^{p - 1}}\vert\Delta \hat f(j){\vert^p} = 0$, for some $1 < p \leqslant 2$, converge in ${L^1}$-norm if and only if $ \vert\vert\hat f(n){E_n} + \hat f( - n){E_{ - n}}\vert\vert = o(1)$, where ${E_n}(t) = \sum _{k = 0}^n{e^{ikt}}$. Recent results of Stanojević [1], Bojanic and Stanojević [2], and Goldberg and Stanojević [3] are special cases of some corollaries to the main theorem.
Growth of solutions of linear differential equations at a logarithmic singularity
A.
Adolphson;
B.
Dwork;
S.
Sperber
245-252
Abstract: We consider differential equations $Y' = AY$ with a regular singular point at the origin, where $A$ is an $n \times n$ matrix whose entries are $ p$-adic meromorphic functions. If the solution matrix at the origin is of the form $Y = P\exp (\theta \log x)$, where $P$ is an $n \times n$ matrix of meromorphic functions and $\theta$ is an $ n \times n$ constant matrix whose Jordan normal form consists of a single block, then we prove that the entries of $P$ have logarithmic growth of order $ n - 1$.
Analogues of the Denjoy-Young-Saks theorem
C. L.
Belna;
G. T.
Cargo;
M. J.
Evans;
P. D.
Humke
253-260
Abstract: In this paper, an analogue of the Denjoy-Young-Saks theorem concerning the almost everywhere classification of the Dini derivates of an arbitrary real function is established in both the case where the exceptional set is of first category and the case where it is $ \sigma$-porous. Examples are given to indicate the sharpness of these results.
On the construction and distribution of a local martingale with a given absolute value
Edwin
Perkins
261-281
Abstract: A local martingale is constructed on an appropriate Loeb space whose absolute value equals a given nonnegative local submartingale. Nonstandard analysis is used to reduce the problem to the discrete time setting where the original construction of D. Gilat is fairly simple. This approach has the advantage of allowing explicit computations. In particular, the distribution of the local martingale is described in terms of the Doob-Meyer decomposition of the original local submartingale.
Derivations into the integral closure
Richard
Draper;
Klaus
Fischer
283-298
Abstract: Let $A$ be a reduced analytical $ k$-algebra of dimension $ r$ and $A'$ its integral closure in the full ring of quotients of $A$. We investigate the condition on $A$ that there exist $r$ elements ${x_1}, \ldots ,{x_r}$ in $A$ and $k$-derivations $ {d_1}, \ldots ,{d_r}$ from $ A$ into $A'$ so that $ {d_i}({x_j})$ is the $r \times r$ identity matrix and so that ${d_1}, \ldots ,{d_r}$ freely generate
Algebras generated by a subnormal operator
Robert F.
Olin;
James E.
Thomson
299-311
Abstract: We use the notion of generalized Toeplitz operators to obtain some basic results concerning the $ {C^{\ast}}$-algebra generated by a subnormal operator. We apply these results to problems concerning the intersection of ${C^{\ast}}(S)$ with rationally closed algebras generated by $S$. In particular, we prove that ${C^{\ast}}(S) \cap \mathcal{W}(S) = \{ f(S):f \in R({\sigma _{\mathcal{W}(S)}}(S))\}$. The spectral inclusion property for generalized Toeplitz operators with symbols in ${P^\infty }(\mu ) + C(\sigma (N))$ is also considered.
Classification of oriented equivariant spherical fibrations
Stefan
Waner
313-324
Abstract: Classifying spaces for oriented equivariant spherical fibrations are constructed, and the notion of an equivariant $ SF$-fibration is introduced. It is shown that equivariant $SF$-fibrations are naturally oriented in $ RO(G)$-graded equivariant singular cohomology.
Self-maps of projective spaces
C. A.
McGibbon
325-346
Abstract: The classical projective $n$-spaces (real, complex, and quaternionic) are studied in terms of their self maps, from a homotopy point of view. Self maps of iterated suspensions of these spaces are also considered. The goal in both cases is to classify, up to homology, all such maps. This goal is achieved in the stable case. Some partial results are obtained in the unstable case. The results from both cases are used to compute the genus groups and the stable genus groups of the classical projective spaces. Applications to other spaces are also given.
Corrections and additions to: ``Some properties of measure and category''
Arnold W.
Miller
347-348