Extending monotone decompositions of 3-manifolds
R. H.
Bing
351-369
On the structure of certain idempotent semigroups
Ahmad
Shafaat
371-378
Abstract: Some general theorems concerning residual finiteness of algebras are given that are applied to show that every idempotent semigroup satisfying $ xyzx = xzyx$ identically is a subcartesian product of certain simple semigroups of order two and three.
On $n$-parameter discrete and continuous semigroups of operators
James A.
Deddens
379-390
Abstract: We prove that n commuting operators on a Hilbert space can be uniquely simultaneously extended to doubly commuting coisometric operators if and only if they satisfy certain positivity conditions, which for the case $n = 1$ state simply that the original operator is a contraction. Our proof establishes the connection between these positivity conditions and the backward translation semigroup on ${l^2}({Z^{ + n}},\mathcal{K})$. A semigroup of operators is unitarily equivalent to backward translation (or a part thereof) on ${l^2}({Z^{ + n}},\mathcal{K})$ if and only if the positivity conditions are satisfied and the individual operators are coisometries (or contractions) whose powers tend strongly to zero. Analogous results are proven in the continuous case ${R^{ + n}}$.
Some fixed point theorems for compact maps and flows in Banach spaces.
W. A.
Horn
391-404
Abstract: Let ${S_0} \subset {S_1} \subset {S_2}$ be convex subsets of the Banach space X, with $ {S_0}$ and ${S_2}$ closed and ${S_1}$ open in ${S_2}$. If f is a compact mapping of $ {S_2}$ into X such that $\cup _{j = 1}^m{f^j}({S_1}) \subset {S_2}$ and ${f^m}({S_1}) \cup {f^{m + 1}}({S_1}) \subset {S_0}$ for some $m > 0$, then f has a fixed point in $ {S_0}$. (This extends a result of F. E. Browder published in 1959.) Also, if $ \{ {T_t}:t \in {R^ + }\}$ is a continuous flow on the Banach space X, $ {S_0} \subset {S_1} \subset {S_2}$ are convex subsets of X with $ {S_0}$ and ${S_2}$ compact and ${S_1}$ open in ${S_2}$, and ${T_{{t_0}}}({S_1}) \subset {S_0}$ for some $ {t_0} > 0$, where ${T_t}({S_1}) \subset {S_2}$ for all $t \leqq {t_0}$, then there exists ${x_0} \in {S_0}$ such that ${T_t}({x_0}) = {x_0}$ for all $t \geqq 0$. Minor extensions of Browder's work on ``nonejective'' and ``nonrepulsive'' fixed points are also given, with similar results for flows.
On properties of subspaces of $l\sb{p},\,0<p<1$
W. J.
Stiles
405-415
Abstract: The material presented in this paper deals with some questions concerning projections, quotient spaces, and linear dimension in $ {l_p}$ spaces, and also includes a remark about weak Schauder bases in $ {l_p}$ spaces and an example of an infinite-dimensional closed subspace of ${l_p}$ which is not isomorphic to ${l_p}$.
A generalization of the Siegel-Walfisz theorem
Larry Joel
Goldstein
417-429
Abstract: The uniform prime number theorem for primes in arithmetic progressions is generalized to the setting of Hecke L-series.
Analogues of Artin's conjecture
Larry Joel
Goldstein
431-442
Abstract: Based on heuristic, a general conjecture is made, which contains Artin's primitive root conjecture as a special case. Special cases of the general conjecture are verified using the generalized Siegel-Walfisz theorem. It is shown that the general conjecture can be considered as an infinite-dimensional analogue of the Tchebotarev density theorem.
On a lemma of Milutin concerning averaging operators in continuous function spaces
Seymour Z.
Ditor
443-452
Abstract: We show that any infinite compact Hausdorff space S is the continuous image of a totally disconnected compact Hausdorff space $S'$, having the same topological weight as S, by a map $\varphi$ which admits a regular linear operator of averaging, i.e., a projection of norm one of $ C(S')$ onto ${\varphi ^ \circ }C(S)$, where $f \in C(S)$ into $f \circ \varphi$. A corollary of this theorem is that if S is an absolute extensor for totally disconnected spaces, the space $S'$ can be taken to be the Cantor space $ {\{ 0,1\} ^\mathfrak{m}}$, where $ \mathfrak{m}$ is the topological weight of S. This generalizes a result due to Milutin and Pełczyński. In addition, we show that for compact metric spaces S and T and any continuous surjection $\varphi :S \to T$, the operator $u:C(S) \to C(T)$ is a regular averaging operator for $\varphi$ if and only if u has a representation $uf(t) = \smallint _0^1f(\theta (t,x))$ for a suitable function $\theta :T \times [0,1] \to S$.
On entropy and generators of measure-preserving transformations
Wolfgang
Krieger
453-464
Abstract: Let T be an ergodic measure-preserving transformation of a Lebesgue measure space with entropy $h(T)$. We prove that T has a generator of size k where ${e^{h(T)}} \leqq k \leqq {e^{h(T)}} + 1$.
Systems of derivations
Frances
Gulick
465-488
Abstract: Let A and B be two complex algebras. A system of derivations of order m from A into B is a set of $m + 1$ linear operators ${D_k}:A \to B(k = 0,1, \ldots ,m)$ such that for $ x,y \in A$ and $k = 0,1,2, \ldots ,m$, $\displaystyle {D_k}(xy) = \sum\limits_{j = 0}^k {\left( {_j^k} \right)} ({D_j}x)({D_{k - j}}y).$ If A is a commutative, regular, semisimple F-algebra with an identity, B the algebra of continuous functions on the closed maximal ideal space of A and $({D_0},{D_1}, \ldots ,{D_m})$ a system of derivations from A into B with ${D_0}$ the Gelfand mapping, then each $ {D_k}$ is continuous. The continuity of the operators in a system of derivations from ${C^n}(U)$ into $C(U)(U \subset R \;{\text{open}})$ is used to obtain a formula for ${D_k}f,f \in {C^n}(U)$, in terms of the ordinary derivatives of f and functions in $C(U)$. Each system of derivations from A into B and each multiplicative seminorm on B determine a multiplicative seminorm on A. Let U be a subset of C and $({D_0},{D_1}, \ldots ,{D_m})$ a system of derivations from the algebra $P(x)$ of polynomials on U into $ C(U)$ with ${D_0}$ the identity operator. Then the system of derivations determines a Hausdorff topology on $P(x)$. If U is open in R and ${D_1}x(t) \ne 0$ for $t \in U(x(t) = t)$, then the completion of $ P(x)$ in this topology is $ {C^m}(U)$. If U is open in C, then the completion of $P(x)$ in this topology is the algebra of functions analytic on U.
A characterization of the Peano derivative
J. Marshall
Ash
489-501
Abstract: For each choice of parameters $\{ {a_i},{b_i}\} ,i = 0,1, \ldots ,n + e$, satisfying certain simple conditions, the expression $\displaystyle \mathop {\lim }\limits_{h \to 0} {h^{ - n}}\sum\limits_{i = 0}^{n + e} {{a_i}f(x + {b_i}h)}$ yields a generalized nth derivative. A function f has an nth Peano derivative at x if and only if all the members of a certain subfamily of these nth derivatives exist at x. The result holds for the corresponding $ {L^p}$ derivatives. A uniformity lemma in the proof (Lemma 2) may be of independent interest. Also, a new generalized second derivative is introduced which differentiates more functions than the ordinary second derivative but fewer than the second Peano derivative.
Representations of twisted group algebras
Robert C.
Busby;
Harvey A.
Smith
503-537
Abstract: We construct a general class of Banach algebras which include as special cases the group algebra of a locally compact group, the group algebra of a group extension (in terms of the subgroup and quotient group), and some other examples, special cases of which have been studied under the name of covariance algebras. We develop the general representation theory and generalize Mackey's theory of induced representations.
Mersions of topological manifolds
David
Gauld
539-560
Abstract: We here generalise the immersion and submersion theorems of Smale, Hirsch, Haefliger and Poenaru, Phillips, Lees, and Lashof, giving a relative version in the case of mersions of topological manifolds. A mersion is a map of manifolds ${M^m} \to {Q^q}$ which in the appropriate local coordinate systems has the form ${R^m} \to {R^q}$ of the standard inclusion or projection of one euclidean space in another. Such a mersion induces a map of tangent bundles satisfying certain properties. In this paper the problem of classifying mersions is reduced to that of classifying such bundle maps.
Exact dynamic systems are tree-like and vice-versa
Kuo-tsai
Chen
561-567
Abstract: This paper gives an analytic characterization of those dynamic systems whose graph of trajectories is a tree.
The tension field of the Gauss map
Ernst A.
Ruh;
Jaak
Vilms
569-573
Abstract: In this paper it is shown that the tension field of the Gauss map can be identified with the covariant derivative of the mean curvature vector field. Since a map with vanishing tension field is called harmonic the following theorem is obtained as a corollary. The Gauss map of a minimal submanifold is harmonic.
Some applications of Waldhausen's results on irreducible surfaces
C. D.
Feustel
575-583
Piecewise linear groups and transformation groups
Herman
Gluck
585-593
Structure spaces of semigroups of continuous functions
K. D.
Magill
595-600
Abstract: In a previous paper, we associated a topological space with each left ideal of a semigroup. Here, we determine this space when the semigroup under consideration is the semigroup of all continuous selfmaps of any space belonging to a fairly extensive class of topological spaces and the left ideal is taken to be the kernel of the semigroup.
Quasi-states on $C\sp{\ast} $-algebras
Johan F.
Aarnes
601-625
Amalgamation of polyadic algebras
James S.
Johnson
627-652
Abstract: The main result of the paper is that for I an infinite set, the class of polyadic I-algebras (with equality) has the strong amalgamation property; i.e., if two polyadic I-algebras have a given common subalgebra they can be embedded in another algebra in such a way that the intersection of the images of the two algebras is the given common subalgebra.
The homotopy type of Fredholm manifolds
Kalyan K.
Mukherjea
653-663
Abstract: Banach manifolds whose tangent bundles admit a reduction to the Fredholm group have been intensively studied in the last few years. Here we show that such a manifold (under appropriate smoothness and separability restrictions) is homotopy equivalent to the union of a nested sequence of closed finite-dimensional submanifolds.
The free envelope of a finitely generated commutative semigroup
Pierre Antoine
Grillet
665-682