Applications of a theorem of L\'evy to Boolean terms and algebras
Jonathan
Stavi
1-36
Abstract: The paper begins with a short proof of the Gaifman-Hales theorem and the solution of a problem of Gaifman about the depth and length of Boolean terms. The main results are refinements of the following theorem: Let $\kappa$ be regular, ${\aleph _1} \leq \kappa \leq \infty$. A $ < \kappa$-complete Boolean algebra on $ {\aleph _0}$ generators, which are restricted by just one countably long equation, is either atomic with $ \leq {\aleph _0}$ atoms or isomorphic to the free $< \kappa$-complete Boolean algebra on ${\aleph _0}$ generators. The main tools are a Skolem-Löwenheim type theorem of Azriel Lévy and a coding of Borel sets and Borel-measurable functions by Boolean terms.
Location of the zeros of a polynomial relative to certain disks
R. C.
Riddell
37-45
Abstract: The zeros of the complex polynomial $P(z) = {z^n} + \Sigma {\alpha _i}{z^{n - 1}}$ are studied under the assumption that some $\vert{\alpha _k}\vert$ is large in comparison with the other $\vert{\alpha _i}\vert$. It is shown under certain conditions that $P(z)$ has $n - k$ zeros in $ \vert z\vert \leq {m_ - }$ and $k$ zeros in $ \vert z\vert \geq {m_ + }$, where ${m_ - } < {m_ + } \leq \vert{\alpha _k}{\vert^{1/k}}$; and under suitably strengthened conditions, one of the $k$ zeros of larger modulus is shown to lie in each of the $k$ disks $\vert z - {( - {\alpha _k})^{1/k}}\vert \leq R$, where ${m_ - } + R < \vert{\alpha _k}{\vert^{1/k}}$.
Hall-Higman type theorems. II
T. R.
Berger
47-69
Abstract: This paper continues the investigations of this series. Suppose $ {\mathbf{K}} =$ GF${\text{(}}r{\text{)}}$ is a field for a prime $ r;G$-is a nilpotent; $ V$ is a nonsingular symplectic space with form $g$; and $V$ is a faithful irreducible ${\mathbf{K}}[G]$-module where $G$ fixes the form $g$. This paper describes completely the structure of $G$ and its representation upon $ V$ when $G$ is symplectic primitive. This latter condition is described in §4 and is a primitivity condition.
Stable thickenings in the homotopy category
R. L.
Chazin
71-77
Abstract: This paper extends the result that the set of stable thickenings of a simply-connected complex $K$ are in 1-1 correspondence with $[K,BQ](Q = {\rm O},$PL$,{\text{or TOP)}}$. which holds in the smooth, PL, and topological categories, to the homotopy category.
A construction of Lie algebras by triple systems
W.
Hein
79-95
Abstract: A construction of Lie algebras by means of special unital representations of Jordan algebras on a certain kind of triple systems is given which generalizes the construction due to Freudenthal, Faulkner and Koecher.
Fourier coefficients of Eisenstein series of one complex variable for the special linear group
A.
Terras
97-114
Abstract: The Eisenstein series in question are generalizations of Epstein's zeta function, whose Fourier expansions generalize the formula of Selberg and Chowla (for the binary quadratic form case of Epstein's zeta function). The expansions are also analogous to Siegel's calculation of the Fourier coefficients of Eisenstein series for the symplectic group. The only ingredients not appearing in Siegel's formula are the Bessel functions of matrix argument studied by Herz. These functions generalize the modified Bessel function of the second kind appearing in the Selberg-Chowla formula.
Trees of homotopy types of 2-dimensional ${\rm CW}$ complexes. II
Micheal N.
Dyer;
Allan J.
Sieradski
115-125
Abstract: A $\pi$-complex is a finite, connected $ 2$-dimensional CW complex with fundamental group $\pi$. The tree HT$(\pi )$ of homotopy types of $\pi $-complexes has width $ \leq N$ if there is a root $ Y$ of the tree such that, for any $\pi$-complex $X,X \vee ( \vee _{i = 1}^NS_i^2)$ lies on the stalk generated by $Y$. Let $\pi$ be a finite abelian group with torsion coefficients $ {\tau _1}, \cdots ,{\tau _n}$. The main theorem of this paper asserts that width HT$ (\pi ) \leq n(n - 1)/2$. This generalizes the results of [4].
$S$-operations in representation theory
Evelyn Hutterer
Boorman
127-149
Abstract: For $G$ a group and ${\text{A} ^G}$ the category of $G$-objects in a category A$$, a collection of functors, called ``$S$-operations,'' is introduced under mild restrictions on A$$. With certain assumptions on A$ $ and with $G$ the symmetric group $ {S_k}$, one obtains a unigeneration theorem for the Grothendieck ring formed from the isomorphism classes of objects in ${\text{A} ^{{S_k}}}$. For A = finite-dimensional vector spaces over $C$, the result says that the representation ring $ R({S_k})$ is generated, as a $\lambda$-ring, by the canonical $ k$-dimensional permutation representation. When A = finite sets, the $ S$-operations are called ``$ \beta$-operations,'' and the result says that the Burnside ring $B({S_k})$ is generated by the canonical $ {S_k}$-set if $ \beta$-operations are allowed along with addition and multiplication.
Classification of $3$-manifolds with certain spines
Richard S.
Stevens
151-166
Abstract: Given the group presentation $ \varphi = \left\langle {a,b\backslash {a^m}{b^n},{a^p}{b^q}} \right\rangle$ with $ m,n,p,q \ne 0$, we construct the corresponding $2$-complex $ {K_\varphi }$. We prove the following theorems. THEOREM 7. ${K_\varphi }$ is a spine of a closed orientable $ 3$-manifold if and only if (i) $ \vert m\vert = \vert p\vert = 1$ or $ \vert n\vert = \vert q\vert = 1$, or (ii) $(m,p) = (n,q) = 1$. THEOREM 10. If $ M$ is a closed orientable $ 3$-manifold having ${K_\varphi }$ as a spine and $\lambda = \vert mq - np\vert$ then $ M$ is a lens space ${L_{\lambda ,k}}$ where $(\lambda ,k) = 1$ except when $\lambda = 0$ in which case $M = {S^2} \times {S^1}$.
$p$-factorable operators
C. V.
Hutton
167-180
Abstract: Several classes of operators on Banach spaces, defined by certain summability conditions on the $k$th approximation numbers, are introduced and studied. Characterizations of these operators in terms of tensor-product representations are obtained. The relationship between these operators and other classes of operators introduced by various authors is studied in some detail.
Some theorems on $({\rm CA})$ analytic groups
David
Zerling
181-192
Abstract: An analytic group $ G$ is called $ (CA)$ if the group of inner automorphisms of $G$ is closed in the Lie group of all (bicontinuous) automorphisms of $G$. We show that each non-$(CA)$ analytic group $G$ can be written as a semidirect product of a $(CA)$ analytic group and a vector group. This decomposition yields a natural dense immersion of $ G$ into a $(CA)$ analytic group $ H$, such that each automorphism of $G$ can be extended to an automorphism of $ H$. This immersion and extension property enables us to derive a sufficient condition for the normal part of a semidirect product decomposition of a $(CA)$ analytic group to be $(CA)$.
Smooth $Z\sb{p}$-actions on spheres which leave knots pointwise fixed
D. W.
Sumners
193-203
Abstract: The paper produces, via handlebody construction, a family of counterexamples to the generalized Smith conjecture; that is, for each pair of integers $(n,p)$ with $n \geq 2$ and $p \geq 2$ there are infinitely many knots $({S^{n + 2}},k{S^n})$ which admit smooth semifree ${Z_p}$-actions (fixed on the knotted submanifold $ k{S^n}$ and free on the complement $ ({S^{n + 2}} - k{S^n}))$. This produces previously unknown $ {Z_p}$-actions on $({S^4},k{S^2})$ for $p$ even, the one case not covered by the work of C. H. Giffen. The construction is such that all of the knots produced are equivariantly null-cobordant. Another result is that if a knot admits ${Z_p}$ -actions for all $ p$, then the infinite cyclic cover of the knot complement is acyclic, and thus leads to an unknotting theorem for $ {Z_p}$-actions.
Asymptotic enumeration of partial orders on a finite set
D. J.
Kleitman;
B. L.
Rothschild
205-220
Abstract: By considering special cases, the number ${P_n}$ of partially ordered sets on a set of $ n$ elements is shown to be $ (1 + O(1/n)){Q_n}$, where $ {Q_n}$ is the number of partially ordered sets in one of the special classes. The number ${Q_n}$ can be estimated, and we ultimately obtain $\displaystyle {P_n} = \left( {1 + O\left( {\frac{1}{n}} \right)} \right)\left( ... ...{{2^i} - 1} \right)}^j}{{\left( {{2^j} - 1} \right)}^{n - i - j}}} } } \right).$
Order summability of multiple Fourier series
G. E.
Peterson;
G. V.
Welland
221-246
Abstract: Jurkat and Peyerimhoff have characterized monotone Fouriereffective summability methods as those which are stronger than logarithmic order summability. Here the analogous result for double Fourier series is obtained assuming unrestricted rectangular convergence. It is also shown that there is a class of order summability methods, which are weaker than any Cesàro method, for which the double Fourier series of any $f \in L$ is restrictedly summable almost everywhere. Finally, it is shown that square logarithmic order summability has the localization property for exponentially integrable functions.
Generalized gradients and applications
Frank H.
Clarke
247-262
Abstract: A theory of generalized gradients for a general class of functions is developed, as well as a corresponding theory of normals to arbitrary closed sets. It is shown how these concepts subsume the usual gradients and normals of smooth functions and manifolds, and the subdifferentials and normals of convex analysis. A theorem is proved concerning the differentiability properties of a function of the form $ \max \{ g(x,u):u \in U\}$. This result unifies and extends some theorems of Danskin and others. The results are then applied to obtain a characterization of flow-invariant sets which yields theorems of Bony and Brezis as corollaries.
Hausdorff $m$ regular and rectifiable sets in $n$-space
Pertti
Mattila
263-274
Abstract: The purpose of this paper is to prove the following theorem: If $ E$ is a subset of Euclidean $n$-space and if the $m$-dimensional Hausdorff density of $E$ exists and equals one ${H^m}$ almost everywhere in $E$, then $E$ is countably $({H^m},m)$ rectifiable. Here ${H^m}$ is the $m$-dimensional Hausdorff measure. The proof is a generalization of the proof given by J. M. Marstrand in the special case $ n = 3,m = 2$.
Three local conditions on a graded ring
Jacob
Matijevic
275-284
Abstract: Let $R = {\Sigma _{i \in Z}}{R_i}$ be a commutative graded Noetherian ring with unit and let $A = {\Sigma _{i \in Z}}{A_i}$ be a finitely generated graded $R$ module. We show that if we assume that $ {A_M}$ is a Cohen Macaulay $ {R_M}$ module for each maximal graded ideal $M$ of $R$, then ${A_P}$ is a Cohen Macaulay ${R_P}$ module for each prime ideal $ P$ of $R$. With $A = R$ we show that the same is true with Cohen Macaulay replaced by regular and Gorenstein, respectively.
Splitting isomorphisms of mapping tori
Terry C.
Lawson
285-294
Abstract: Necessary and sufficient conditions involving invertible cobordisms are given for two mapping tori to be isomorphic. These are used to give conditions under which a given isomorphism ${M_f} \to {N_g}$ is pseudoisotopic to an isomorphism which sends $M$ to $N$. An exact sequence for the group of pseudoisotopy classes of automorphisms of $M \times {S^1}$ is derived. The principal tools are an imbedding technique due to C. T. C. Wall as well as arguments involving invertible cobordisms. Applications and examples are given, particularly for manifolds of higher dimension where the $s$-cobordism theorem is applied.
The geometric dimension of some vector bundles over projective spaces
Donald M.
Davis;
Mark E.
Mahowald
295-315
Abstract: We prove that in many cases the geometric dimension of the $ p$-fold Whitney sum $ p{H_k}$ of the Hopf bundle $ {H_k}$ over quaternionic projective space $Q{P^k}$ is the smallest $n$ such that for all $i \leq k$ the reduction of the $i$th symplectic Pontryagin class of $p{H_k}$ to coefficients ${\pi _{4i - 1}}(({\text{R}}{P^\infty }/{\text{R}}{P^{n - 1}})\Lambda bo)$ is zero, where bo is the spectrum for connective KO-theory localized at 2. We immediately obtain new immersions of real projective space $ {\text{R}}{P^{4k + 3}}$ in Euclidean space if the number of 1's in the binary expansion of $k$ is between 5 and 8.
Continua in which all connected subsets are arcwise connected
E. D.
Tymchatyn
317-331
Abstract: Let $X$ be a metric continuum such that every connected subset of $X$ is arcwise connected. Some facts concerning the distribution of local cutpoints of $X$ are obtained. These results are used to prove that $X$ is a regular curve.
Further generalizations of the Nehari inequalities
Duane W.
DeTemple
333-340
Abstract: Inequalities of the Nehari type are obtained for bounded univalent functions on the unit disc, including a form which depends upon the parameters $a$ and $d$, where $d = f(a)$.
Stiefel-Whitney homology classes and bordism
Ethan
Akin
341-359
Abstract: We develop the theory of $\bmod 2$ Stiefel-Whitney homology classes for Euler polyhedra. We then describe a simple method of obtaining p.1. bordism theories. Finally, we define the ungraded bordism theory of Euler spaces and show that it is isomorphic to ordinary total homology.
Some polynomials defined by generating relations
H. M.
Srivastava;
R. G.
Buschman
360-370
Abstract: In an attempt to present a unified treatment of the various polynomial systems introduced from time to time, new generating functions are given for the sets of polynomials $ \{ S_{n,q}^{(\alpha ,\beta )}(\lambda ;x)\}$ and $\{ T_{n,q}^{(\alpha ,\beta )}(\lambda ;x)\}$, defined respectively by (6) and (29) below, and for their natural generalizations in several complex variables. This paper also indicates relevant connections of the results derived here with different classes of generating relations which have appeared recently in the literature.
Exotic singular structures on spheres
Norman
Levitt
371-388
Abstract: It is shown how the category of PL-manifolds may be obtained from the smooth category by an iterative procedure, viz., first form singular smooth manifolds where smooth seven-spheres are allowed as links. Then, in the new category one has obtained, kill all eight-spheres in similar fashion. Repeating this process ad infinitum (but requiring only finitely many stages in each dimension), one obtains the category of PL-manifolds. By taking care that the set of ``singular'' points is always given enough structure, it is seen that this iterative process corresponds to a skeletal filtration of $ BPL \bmod BO$. Also, a geometric interpretation of the Hurewicz map $ {\pi _ \ast }(BPL,BO) \to {H_ \ast }(BPL,BO)$ is inferred.
$\Phi $-like holomorphic functions in ${\bf C}\sp{n}$ and Banach spaces
Kenneth R.
Gurganus
389-406
Abstract: In a recent paper, L. Brickman introduced the concept of $\Phi$-like holomorphic functions as a complete generalization of starlike and spirallike functions of a single complex variable. In the present paper, the author extends this work to locally biholomorphic mappings of several complex variables and then to locally biholomorphic mappings defined in an arbitrary Banach space. Complete characterizations of univalency and starlikeness of locally biholomorphic maps in general Banach spaces are obtained.