The translational hull of a topological semigroup
J. A.
Hildebrant;
J. D.
Lawson;
D. P.
Yeager
251-280
Abstract: This paper is concerned with three aspects of the study of topological versions of the translational hull of a topological semigroup. These include topological properties, applications to the general theory of topological semigroups, and techniques for computing the translational hull. The central result of this paper is that if S is a compact reductive topological semigroup and its translational hull $ \Omega (S)$ is given the topology of continuous convergence (which coincides with the topology of pointwise convergence and the compact-open topology in this case), then $\Omega (S)$ is again a compact topological semigroup. Results pertaining to extensions of bitranslations are given, and applications of these together with the central result to semigroup compactifications and divisibility are presented. Techniques for determining the translational hull of certain types of topological semigroups, along with numerous examples, are set forth in the final section.
The infinitesimal stability of semigroups of expanding maps
Richard
Sacksteder
281-288
Abstract: The concept of ${C^\infty }$ infinitesimal stability for representations of a semigroup by ${C^\infty }$ maps is defined. In the case of expanding linear maps of the torus ${T^d}$ it is shown that certain algebraic conditions assure such stability.
Uniqueness criteria for solutions of singular boundary value problems
D. R.
Dunninger;
Howard A.
Levine
289-301
Abstract: In this paper we consider the equation $u:(0,T) \to D(A) \subset B$ is a Banach space valued function taking values in a dense subdomain $ D(A)$ of the Banach space B. Here A is a closed (possibly unbounded) linear operator on $D(A)$ while k is a real constant. The differential equation is an abstract Euler-Poisson-Darboux equation. We give necessary and sufficient conditions on the point spectrum of A to insure uniqueness of the strong solution $u \equiv 0$ as well as sufficient conditions on the point spectrum to insure uniqueness of weak solutions. u is only required to satisfy (a) $t \to {0^ + }$ if $k > 1$, (b) $t \to {0^ + },0 < k \leqslant 1$, (c) $t \to {0^ + },k < 0$. The operator A need not possess a complete set of eigenvectors nor need one have a backward uniqueness theorem available for (1) for the Cauchy final value problem. Our techniques extend to the n-axially symmetric abstract equation $\displaystyle \sum\limits_{i = 1}^n {[{\partial ^2}u/\partial t_i^2 + ({k_i}/{t_i})\partial u/\partial {t_i}] + Au = 0.}$ ($ 2$) The proofs rest upon an application of the Hahn-Banach Theorem and the consequent separation properties of ${B^\ast}$, the dual of B, as well as the completeness properties of the eigenfunctions of certain Bessel equations associated with (1).
Deformations of formal embeddings of schemes
Miriam P.
Halperin
303-321
Abstract: A family of isolated singularities of k-varieties will be here called equisingular if it can be simultaneously resolved to a family of hypersurfaces embedded in nonsingular spaces which induce only locally trivial deformations of pairs of schemes over local artin k-algebras. The functor of locally trivial deformations of the formal embedding of an exceptional set has a versal object in the sense of Schlessinger. When the exceptional set ${X_0}$ is a collection of nonsingular curves meeting normally in a nonsingular surface X, the moduli correspond to Laufer's moduli of thick curves. When X is a nonsingular scheme of finite type over an algebraically closed field k and ${X_0}$ is a reduced closed subscheme of X, every deformation of $(X,{X_0})$ to $ k[\varepsilon ]$ such that the deformation of ${X_0}$ is locally trivial, is in fact a locally trivial deformation of pairs.
On chain varieties of linear algebras
V. A.
Artamonov
323-338
Abstract: In the present paper we study varieties of linear k-algebras over a commutative associative Noetherian ring k with 1, whose subvarieties form a chain. We describe these varieties in terms of identities in the following cases: residually nilpotent varieties, varieties of alternative, Jordan and $( - 1,1)$-algebras.
Applications of extreme point theory to classes of multivalent functions
David J.
Hallenbeck;
Albert E.
Livingston
339-359
Abstract: Extreme points of the closed convex hulls of several classes of multivalent functions are determined. These are then used to determine the precise bounds on the coefficients of a function majorized by or subordinate to a function in any of the classes. ${L^q}$ means are also discussed and subordination theorems are considered. The classes we consider are generalizations of the univalent starlike, convex and close-to-convex functions in addition to others.
Presentations of $3$-manifolds arising from vector fields
Peter
Percell
361-377
Abstract: A method is given for constructing a smooth, closed, orientable 3-manifold from the information contained in a combinatorial object called an abstract intersection sequence. An abstract intersection sequence of length n is just a cyclic ordering of the set $\{ \pm 1, \ldots , \pm n\} $ plus a map $\nu :\{ 1, \ldots ,n\} \to \{ \pm 1\}$. It is shown that up to diffeomorphism every closed, connected, orientable 3-manifold can be constructed by the method. This is proved by showing that compact, connected, orientable 3-manifolds with boundary the 2-sphere admit vector fields of a certain type. The intersection sequences arise as descriptions of the vector fields.
Reductivity and the automorphism group of locally compact groups
Dong Hoon
Lee
379-389
Abstract: In this paper, we study reductivity of locally compact groups and its effect on the automorphism group and generalize the classical results on the automorphism group of analytic semisimple groups on the one hand and of compact groups on the other.
Competitive processes and comparison differential systems
G. S.
Ladde
391-402
Abstract: Sufficient conditions are given for stability and nonnegativity of solutions of a system of differential equations, in particular, of comparison differential equations. Finally, it has been shown that the comparison differential equations represent the mathematical models for competitive processes in biological, physical and social sciences.
Closed $3$-manifolds with no periodic maps
Frank
Raymond;
Jeffrey L.
Tollefson
403-418
Abstract: Examples of closed, orientable, aspherical 3-manifolds are constructed on which every action of a finite group is trivial.
Dyadic methods in the measure theory of numbers
R. C.
Baker
419-432
Abstract: Some new theorems in metric diophantine approximation are obtained by dyadic methods. We show for example that if ${m_1},{m_2}, \ldots$, are distinct integers with ${m_n} = O({n^p})$ then ${\Sigma _{n \leqslant N}}e({m_n}x) = O({N^{1 - q}})$ except for a set of x of Hausdorff dimension at most $ (p + 4q - 1)/(p + 2q)$; and that for any sequence of intervals ${I_1},{I_2}, \ldots$ in [0, 1) the number of solutions of $\{ {x^n}\} \in {I_n}\;(n \leqslant N)$ is a.e. asymptotic to ${\Sigma _{n \leqslant N}}\vert{I_n}\vert(x > 1)$.
On representations of the group $SU(n,1)$
Hrvoje
Kraljević
433-448
Abstract: A natural bijection is established between the set of equivalence classes of irreducible unitary representations of the group $G = SU(n,1)$, which are not induced from a proper parabolic subgroup, and the set of equivalence classes of irreducible representations of a maximal compact subgroup.
Asymptotically autonomous multivalued differential equations
James P.
Foti
449-452
Abstract: The asymptotic behavior of solutions of the perturbed autonomous multivalued differential equation