A new basis for uniform asymptotic solution of differential equations containing one or several parameters
Gilbert
Stengle
1-43
Some immersion theorems for manifolds
A. Duane
Randall
45-58
Abstract: In this paper we obtain several results on immersing manifolds into Euclidean spaces. For example, a spin manifold $ {M^n}$ immerses in ${R^{2n - 3}}$ for dimension $n \equiv 0\bmod 4$ and n not a power of 2. A spin manifold ${M^n}$ immerses in $ {R^{2n - 4}}$ for $n \equiv 7\bmod 8$ and $n > 7$. Let ${M^n}$ be a 2-connected manifold for $n \equiv 6\bmod 8$ and $n > 6$ such that $ {H_3}(M;Z)$ has no 2-torsion. Then M immerses in ${R^{2n - 5}}$ and embeds in ${R^{2n - 4}}$. The method of proof consists of expressing k-invariants in Postnikov resolutions for the stable normal bundle of a manifold by means of higher order cohomology operations. Properties of the normal bundle are used to evaluate the operations.
Almost locally tame $2$-manifolds in a $3$-manifold
Harvey
Rosen
59-71
Abstract: Several conditions are given which together imply that a 2-manifold M in a 3-manifold is locally tame from one of its complementary domains, U, at all except possibly one point. One of these conditions is that certain arbitrarily small simple closed curves on M can be collared from U. Another condition is that there exists a certain sequence ${M_1},{M_2}, \ldots$ of 2-manifolds in U converging to M with the property that each unknotted, sufficiently small simple closed curve on each $ {M_i}$ is nullhomologous on ${M_i}$. Moreover, if each of these simple closed curves bounds a disk on a member of the sequence, then it is shown that M is tame from $U(M \ne {S^2})$. As a result, if U is the complementary domain of a torus in ${S^3}$ that is wild from U at just one point, then U is not homeomorphic to the complement of a tame knot in ${S^3}$.
Supports of continuous functions
Mark
Mandelker
73-83
Abstract: Gillman and Jerison have shown that when X is a realcompact space, every function in $C(X)$ that belongs to all the free maximal ideals has compact support. A space with the latter property will be called $\mu$-compact. In this paper we give several characterizations of $\mu$-compact spaces and also introduce and study a related class of spaces, the $\psi$-compact spaces; these are spaces X with the property that every function in $ C(X)$ with pseudocompact support has compact support. It is shown that every realcompact space is $\psi$-compact and every $\psi$-compact space is $\mu$-compact. A family $ \mathcal{F}$ of subsets of a space X is said to be stable if every function in $C(X)$ is bounded on some member of $\mathcal{F}$. We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space.
Semigroups on finitely floored spaces
John D.
McCharen
85-89
Abstract: This paper is concerned with certain aspects of acyclicity in a compact connected topological semigroup, and applications to the admissibility of certain multiplications on continua. The principal result asserts that if S is a semigroup on a continuum, finitely floored in dimension 2, then $S = ESE$ implies $S = K$.
Convex hulls of some classical families of univalent functions
L.
Brickman;
T. H.
MacGregor;
D. R.
Wilken
91-107
Abstract: Let S denote the functions that are analytic and univalent in the open unit disk and satisfy $f(0) = 0$ and $f'(0) = 1$. Also, let K, St, ${S_R}$, and C be the subfamilies of S consisting of convex, starlike, real, and close-to-convex mappings, respectively. The closed convex hull of each of these four families is determined as well as the extreme points for each. Moreover, integral formulas are obtained for each hull in terms of the probability measures over suitable sets. The extreme points for each family are particularly simple; for example, the Koebe functions $f(z) = z/{(1 - xz)^2},\vert x\vert = 1$ , are the extreme points of cl co St. These results are applied to discuss linear extremal problems over each of the four families. A typical result is the following: Let J be a ``nontrivial'' continuous linear functional on the functions analytic in the unit disk. The only functions in St. that satisfy $ \operatorname{Re} \,J(f) = \max \;\{ \operatorname{Re} \;J(g):g \in St\}$ are Koebe functions and there are only a finite number of them.
Representations of metabelian groups realizable in the real field
B. G.
Basmaji
109-118
Abstract: A necessary and sufficient condition is found such that all the nonlinear irreducible representations of a metabelian group are realizable in the real field, and all such groups with cyclic commutator subgroups are determined.
Conjugacy separability of certain free products with amalgamation
Peter F.
Stebe
119-129
Abstract: Let G be a group. An element g of G is called conjugacy distinguished or c.d. in G if and only if given any element h of G either h is conjugate to g or there is a homomorphism $ \xi$ from G onto a finite group such that $\xi (h)$ and $\xi (g)$ are not conjugate in $\xi (G)$. Following A. Mostowski, a group G is conjugacy separable or c.s. if and only if every element of G is c.d. in G. In this paper we prove that every element conjugate to a cyclically reduced element of length greater than 1 in the free product of two free groups with a cyclic amalgamated subgroup is c.d. We also prove that a group formed by adding a root of an element to a free group is c.s.
Stationary isotopies of infinite-dimensional spaces
Raymond Y. T.
Wong
131-136
Abstract: Let X denote the Hilbert cube or any separable infinite-dimensional Fréchet space. It has been shown that any two homeomorphisms f, g of X onto itself is isotopic to each other by means of an invertible-isotopy on X. In this paper we generalize the above results to the extent that if f, g are K-coincident on X (that is, $f(x) = g(x)$ for $x \in K$), then the isotopy can be chosen to be K-stationary provided K is compact and has property-Z in X. The main tool of this paper is the Stable Homeomorphism Extension Theorem which generalizes results of Klee and Anderson.
On the inertia group of a product of spheres
Reinhard
Schultz
137-153
Abstract: In this paper it is proved that the smooth connected sum of a product of ordinary spheres with an exotic combinatorial sphere is never diffeomorphic to the original product. This result is extended and compared to certain related examples.
The prime radical in special Jordan rings
T. S.
Erickson;
S.
Montgomery
155-164
Abstract: If R is an associative ring, we consider the special Jordan ring $ {R^ + }$, and when R has an involution, the special Jordan ring S of symmetric elements. We first show that the prime radical of R equals the prime radical of $ {R^ + }$, and that the prime radical of R intersected with S is the prime radical of S. Next we give an elementary characterization, in terms of the associative structure of R, of primeness of S. Finally, we show that a prime ideal of R intersected with S is a prime Jordan ideal of S.
Positive one-relator groups
Gilbert
Baumslag
165-183
Abstract: A group G which can be defined by a single relation in which there are no negative exponents is residually solvable. If G is also torsion-free then it is locally indicable and hence its integral group ring has no zero divisors.
Some iterated logarithm results related to the central limit theorem.
R. J.
Tomkins
185-192
Abstract: An iterated logarithm theorem is presented for sequences of independent, not necessarily bounded, random variables, the distribution of whose partial sums is related to the standard normal distribution in a particular manner. It is shown that if a sequence of independent random variables satisfies the Central Limit Theorem with a sufficiently rapid rate of convergence, then the law of the iterated logarithm holds. In particular, it is demonstrated that these results imply several known iterated logarithm results, including Kolmogorov's celebrated theorem.
Commutators, $C\sp{k}$-classification, and similarity of operators
Shmuel
Kantorovitz
193-218
Abstract: We generalize the results of our recent paper, The $ {C^k}$-classification of certain operators in ${L_p}$. II, to the abstract setting of a pair of operators satisfying the commutation relation $[M,N] = {N^2}$.
Expanding gravitational systems
Donald G.
Saari
219-240
Abstract: In this paper we obtain a classification of motion for Newtonian gravitational systems as time approaches infinity. The basic assumption is that the motion survives long enough to be studied, i.e., the solution exists in the interval $(0,\infty )$. From this classification it is possible to obtain a sketch of the evolving Newtonian universe.
An algebraic classification of noncompact $2$-manifolds
Martin E.
Goldman
241-258
Tensor products of polynomial identity algebras
Elizabeth
Berman
259-271
Abstract: We investigate matrix algebras and tensor products of associative algebras over a commutative ring R with identity, such that the algebra satisfies a polynomial identity with coefficients in R. We call A a P. I. algebra over R if there exists a positive integer n and a polynomial f in n noncommuting variables with coefficients in R, not annihilating A, such that for all ${a_1}, \ldots ,{a_n}$ in A, $f({a_1}, \ldots ,{a_n}) = 0$. We call A a P-algebra if f is homogeneous with at least one coefficient of 1. We define the docile identity, a polynomial identity generalizing commutativity, in that if A satisfies a docile identity, then for all n, ${A_n}$, the set of n-by-n matrices over A, satisfies a standard identity. We similarly define the unitary identity, which generalizes anticommutativity. Claudio Procesi and Lance Small recently proved that if A is a P. I. algebra over a field, then for all n, ${A_n}$ satisfies some power of a standard identity. We generalize this result to P-algebras over commutative rings with identity. It follows that if A is a P-algebra, A satisfies a power of the docile identity.
On the signature of knots and links
Yaichi
Shinohara
273-285
Abstract: In 1965, K. Murasugi introduced an integral matrix M of a link and defined the signature of the link by the signature of $M + M'$. In this paper, we study some basic properties of the signature of links. We also describe the effect produced on the signature of a knot contained in a solid torus by a further knotting of the solid torus.
A strong duality theorem for separable locally compact groups
John
Ernest
287-307
Abstract: We obtain a duality theorem for separable locally compact groups, where the group is regained from the set of factor unitary representations. Loosely stated, the group is isomorphic to the group of nonzero bounded, operator valued maps on the set of factor representations, which preserve unitary equivalence, direct sums, and tensor products. The axiom involving tensor products is formulated in terms of direct integral theory. The topology of G may be regained from the irreducible representations alone. Indeed a sequence $\{ {x_i}\}$ in G, converges to x in G if and and only if $ \pi ({x_i})$ converges strongly to $\pi (x)$ for each irreducible representation $ \pi$ of G. This result supplies the missing topological part of the strong duality theorem of N. Tatsuuma for type I separable locally compact groups (based on irreducible representations). Our result also generalizes this Tatsuuma strong duality theorem to the nontype I case.
Generalized interpolation spaces
Vernon
Williams
309-334
Abstract: In this paper we introduce the notion of ``generalized'' interpolation space, and state and prove a ``generalized'' interpolation theorem. This apparently provides a foundation for an axiomatic treatment of interpolation space theory, for subsequently we show that the ``mean'' interpolation spaces of Lions-Peetre, the ``complex'' interpolation spaces of A. P. Calderón, and the ``complex'' interpolation spaces of M. Schechter are all generalized interpolation spaces. Furthermore, we prove that each of the interpolation theorems established respectively for the above-mentioned interpolation spaces is indeed a special case of our generalized interpolation theorem. In §III of this paper we use the generalized interpolation space concept to state and prove a ``generalized'' duality theorem. The very elegant duality theorems proved by Calderón, Lions-Peetre and Schechter, respectively, are shown to be special cases of our generalized duality theorem. Of special interest here is the isolation by the general theorem of the need in each of the separate theorems for certain ``base'' spaces to be duals of others. At the close of §II of this paper we employ our generalized interpolation theorem ``structure'' to construct new interpolation spaces which are neither complex nor mean spaces.
The structure of pseudocomplemented distributive lattices. I. Subdirect decomposition
H.
Lakser
335-342
Abstract: In this paper all subdirectly irreducible pseudocomplemented distributive lattices are found. This result is used to establish a Stone-like representation theorem conjectured by G. Grätzer and to find all equational subclasses of the class of pseudocomplemented distributive lattices.
The structure of pseudocomplemented distributive lattices. II. Congruence extension and amalgamation
G.
Grätzer;
H.
Lakser
343-358
Abstract: This paper continues the examination of the structure of pseudocomplemented distributive lattices. First, the Congruence Extension Property is proved. This is then applied to examine properties of the equational classes ${\mathcal{B}_n}, - 1 \leqq n \leqq \omega$, which is a complete list of all the equational classes of pseudocomplemented distributive lattices (see Part I). The standard semigroups (i.e., the semigroup generated by the operators H, S, and P) are described. The Amalgamation Property is shown to hold iff $n \leqq 2$ or $n = \omega$. For $3 \leqq n < \omega ,{\mathcal{B}_n}$ does not satisfy the Amalgamation Property; the deviation is measured by a class Amal $ ({\mathcal{B}_n})( \subseteq {\mathcal{B}_n})$. The finite algebras in Amal $ ({\mathcal{B}_n})$ are determined.
Algebras of iterated path integrals and fundamental groups
Kuo-tsai
Chen
359-379
Abstract: A method of iterated integration along paths is used to extend deRham cohomology theory to a homotopy theory on the fundamental group level. For every connected ${C^\infty }$ manifold $ \mathfrak{M}$ with a base point p, we construct an algebra $ {\pi ^1} = {\pi ^1}(\mathfrak{M},p)$ consisting of iterated integrals, whose value along each loop at p depends only on the homotopy class of the loop. Thus ${\pi ^1}$ can be taken as a commutative algebra of functions on the fundamental group ${\pi _1}(\mathfrak{M})$, whose multiplication induces a comultiplication ${\pi ^1} \to {\pi ^1} \otimes {\pi ^1}$, which makes ${\pi ^1}$ a Hopf algebra. The algebra ${\pi ^1}$ relates the fundamental group to analysis of the manifold, and we obtain some analytical conditions which are sufficient to make the fundamental group nonabelian or nonsolvable. We also show that ${\pi ^1}$ depends essentially only on the differentiable homotopy type of the manifold. The second half of the paper is devoted to the study of structures of algebras of iterated path integrals. We prove that such algebras can be constructed algebraically from the following data: (a) the commutative algebra A of $ {C^\infty }$ functions on $\mathfrak{M}$; (b) the A-module M of ${C^\infty }$ 1-forms on $ \mathfrak{M}$; (c) the usual differentiation $d:A \to M$; and (d) the evaluation map at the base point p, $ \varepsilon :A \to K$, K being the real (or complex) number field.
Infinite nodal noncommutative Jordan algebras; differentiably simple algebras
D. R.
Scribner
381-389
Abstract: The first result is that any differentiably simple algebra of the form $A = F1 + R$, for R a proper ideal, 1 the identity element, and F the base field, must be a subalgebra of a (commutative associative) power series algebra over F, and is truncated if the characteristic is not zero. Moreover the algebra A contains the polynomial subalgebra generated by the indeterminates and identity of the power series algebra. This is used to prove that if A is any simple flexible algebra of the form $A = F1 + R$, R an ideal of ${A^ + }$, then ${A^ + }$ is a subalgebra of a power series algebra and multiplication in A is determined by certain elements ${c_{ij}}$ in A as in $\displaystyle fg = f \cdot g + \frac{1}{2}\sum {\frac{{\partial f}}{{\partial {x_i}}} \cdot \frac{{\partial g}}{{\partial {x_j}}} \cdot {c_{ij}},}$ where $ {c_{ij}} = - {c_{ji}}$ and ``$\cdot$'' is the multiplication in $ {A^ + }$. This applies in particular to simple nodal noncommutative Jordan algebras (of characteristic not 2). These results suggest a method of constructing noncommutative Jordan algebras of the given form. We have done this with the restriction that the ${c_{ij}}$ lie in F1. The last result is that if A is a finitely generated simple noncommutative algebra of characteristic 0 of this form, then Der (A) is an infinite simple Lie algebra of a known type.
Summability in amenable semigroups
Peter F.
Mah
391-403
Abstract: A theory of summability is developed in amenable semigroups. We give necessary and (or) sufficient conditions for matrices to be almost regular, almost Schur, strongly regular, and almost strongly regular. In particular, when the amenable semigroup is the additive positive integers, our theorems yield those results of J. P. King, P. Schaefer and G. G. Lorentz for some of the matrices mentioned above.
Theory of random evolutions with applications to partial differential equations
Richard
Griego;
Reuben
Hersh
405-418
Abstract: The selection from a finite number of strongly continuous semigroups by means of a finite-state Markov chain leads to the new notion of a random evolution. Random evolutions are used to obtain probabilistic solutions to abstract systems of differential equations. Applications include one-dimensional first order hyperbolic systems. An important special case leads to consideration of abstract telegraph equations and a generalization of a result of Kac on the classical n-dimensional telegraph equation is obtained and put in a more natural setting. In this connection a singular perturbation theorem for an abstract telegraph equation is proved by means of a novel application of the classical central limit theorem and a representation of the solution for the limiting equation is found in terms of a transformation formula involving the Gaussian distribution.
Harmonic analysis on central topological groups
Siegfried
Grosser;
Martin
Moskowitz
419-454
The Brauer-Wall group of a commutative ring
Charles
Small
455-491
Abstract: Let k be a commutative ring (with 1). We work with k-algebras with a grading $\bmod\;2$, and with graded modules over such algebras. Using graded notions of tensor product, commutativity, and morphisms, we construct an abelian group ${\rm {BW}}(k)$ whose elements are suitable equivalence classes of Azumaya k-algebras. The consruction generalizes, and is patterned on, the definition of the Brauer group $ {\rm {Br}}(k)$ given by Auslander and Goldman. $ {\rm {Br}}(k)$ is in fact a subgroup of $ {\rm {BW}}(k)$, and we describe the quotient as a group of graded quadratic extensions of k.
Characteristic spheres of free differentiable actions of $S\sp{1}$ and $S\sp{3}$ on homotopy spheres
Hsu-tung
Ku;
Mei-chin
Ku
493-504
Spatially induced groups of automorphisms of certain von Neumann algebras
Robert R.
Kallman
505-515
Abstract: This paper gives an affirmative solution, in a large number of cases, to the following problem. Let $ \mathcal{R}$ be a von Neumann algebra on the Hilbert space $\mathcal{H}$, let G be a topological group, and let $a \to \varphi (a)$ be a homomorphism of G into the group of $^ \ast $-automorphisms of $\mathcal{R}$. Does there exist a strongly continuous unitary representation $a \to U(a)$ of G on $\mathcal{H}$ such that each $U(a)$ induces $ \varphi (a)$?