On quintic surfaces of general type
Jin Gen
Yang
431-473
Abstract: The study of quintic surfaces is of special interest because $ 5$ is the lowest degree of surfaces of general type. The aim of this paper is to give a classification of the quintic surfaces of general type over the complex number field ${\mathbf{C}}$. We show that if $S$ is an irreducible quintic surface of general type; then it must be normal, and it has only elliptic double or triple points as essential singularities. Then we classify all such surfaces in terms of the classification of the elliptic double and triple points. We give many examples in order to verify the existence of various types of quintic surfaces of general type. We also make a study of the double or triple covering of a quintic surface over ${{\mathbf{P}}^2}$ obtained by the projection from a triple or double point on the surface. This reduces the classification of the surfaces to the classification of branch loci satisfying certain conditions. Finally we derive some properties of the Hilbert schemes of some types of quintic surfaces.
Deformations of complete minimal surfaces
Harold
Rosenberg
475-489
Abstract: A notion of deformation is defined and studied for complete minimal surfaces in ${R^3}$ and ${R^3}/G,G$ a group of translations. The catenoid, Enneper's surface, and the surface of Meeks-Jorge, modelled on a $3$-punctured sphere, are shown to be isolated. Minimal surfaces of total curvature $4\pi$ in ${R^3}/Z$ and $ {R^3}/{Z^2}$ are studied. It is proved that the helicoid and Scherk's surface are isolated under periodic perturbations.
Some remarks on deformations of minimal surfaces
Harold
Rosenberg;
Éric
Toubiana
491-499
Abstract: We consider complete minimal surfaces (c.m.s.'s) in ${R^3}$ and their deformations. $ {M_1}$ is an $\varepsilon$-deformation of ${M_0}$ if ${M_1}$ is a graph over ${M_0}$ in an $ \varepsilon$ tubular neighborhood of ${M_0}$ and ${M_1}$ is $ \varepsilon \;{C^1}$-close to ${M_0}$. A minimal surface $M$ is isolated if all c.m.s.'s which are sufficiently small deformations of $ M$ are congruent to $ M$. In this paper we construct an example of a nonisolated c.m.s. It is modelled on a $4$-punctured sphere and is of finite total curvature. On the other hand, we prove that a c.m.s. discovered by Meeks and Jorge, modelled on the sphere punctured at the fourth roots of unity, is isolated.
Unicity of a holomorphic functional calculus in infinite dimensions
José E.
Galé
501-508
Abstract: L. Waelbroeck gives a holomorphic functional calculus for Banach algebras and analytic functions on Banach spaces. The properties of this calculus extend the well-known properties for the case of several complex variables. In this last situation, W. Zame has obtained a theorem of unicity where the famous condition of compatibility is dropped. We obtain a theorem analogous to Zame's for Waelbroeck's calculus restricted to a certain algebra of germs of functions. We consider Banach spaces whose topological duals have the bounded approximation property. Also, results of the same kind as above are given for bornological algebras.
Nonlinear oblique boundary value problems for nonlinear elliptic equations
Gary M.
Lieberman;
Neil S.
Trudinger
509-546
Abstract: We consider the nonlinear oblique derivative boundary value problem for quasilinear and fully nonlinear uniformly elliptic partial differential equations of second order. The elliptic operators satisfy natural structure conditions as introduced by Trudinger in the study of the Dirichlet problem while for the boundary operators we formulate general structure conditions which embrace previously considered special cases such as the capillarity condition. The resultant existence theorems include previous work such as that of Lieberman on quasilinear equations and Lions and Trudinger on Neumann boundary conditions.
Weakly definable types
L. A. S.
Kirby;
A.
Pillay
547-563
Abstract: We study some generalizations of the notion of a definable type, first in an abstract setting in terms of ultrafilters on certain Boolean algebras, and then as applied to model theory.
Definable sets in ordered structures. I
Anand
Pillay;
Charles
Steinhorn
565-592
Abstract: This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the $ \mathcal{O}$-minimal structures. The definition of this class and the corresponding class of theories, the strongly $\mathcal{O}$-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of $\mathcal{O}$-minimal ordered groups and rings. Several other simple results are collected in $ \S3$. The primary tool in the analysis of $ \mathcal{O}$-minimal structures is a strong analogue of "forking symmetry," given by Theorem 4.2. This result states that any (parametrically) definable unary function in an $\mathcal{O}$-minimal structure is piecewise either constant or an order-preserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all ${\aleph _0}$-categorical $ \mathcal{O}$-minimal structures (Theorem 6.1).
Definable sets in ordered structures. II
Julia F.
Knight;
Anand
Pillay;
Charles
Steinhorn
593-605
Abstract: It is proved that any 0-minimal structure $M$ (in which the underlying order is dense) is strongly 0-minimal (namely, every $ N$ elementarily equivalent to $M$ is 0-minimal). It is simultaneously proved that if $M$ is 0-minimal, then every definable set of $ n$-tuples of $ M$ has finitely many "definably connected components."
Congruences on regular semigroups
Francis
Pastijn;
Mario
Petrich
607-633
Abstract: Let $S$ be a regular semigroup and let $ \rho$ be a congruence relation on $S$. The kernel of $\rho$, in notation $\ker \rho$, is the union of the idempotent $ \rho$-classes. The trace of $\rho$, in notation $\operatorname{tr}\,\rho $, is the restriction of $ \rho$ to the set of idempotents of $S$. The pair $(\ker \rho ,\operatorname{tr}\,\rho )$ is said to be the congruence pair associated with $\rho$. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple $((\rho \vee \mathcal{L})/\mathcal{L},\ker \rho ,(\rho \vee \mathcal{R})/\mathcal{R})$ is said to be the congruence triple associated with $\rho$. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of $S$. For congruence relations $\rho$ and $\theta$, put $\rho {T_l}\theta \;[\rho {T_r}\theta ,\rho T\theta ]$ if and only if $\rho \vee \mathcal{L} = \theta \vee \mathcal{L}\;[\rho \vee \mathcal{R} = \theta \vee \mathcal{R},\operatorname{tr}\rho = \operatorname{tr}\theta ]$. Then $ {T_l},{T_r}$ and $ T$ are complete congruences on the congruence lattice of $S$ and $ T = {T_l} \cap {T_r}$.
Closed geodesics on a Riemann surface with application to the Markov spectrum
A. F.
Beardon;
J.
Lehner;
M.
Sheingorn
635-647
Abstract: This paper determines those Riemann surfaces on which each nonsimple closed geodesic has a parabolic intersection--that is, an intersection in the form of a loop enclosing a puncture or a deleted disk. An application is made characterizing the simple closed geodesic on $H/\Gamma (3)$ in terms of the Markov spectrum. The thrust of the situation is this: If we call loops about punctures or deleted disks boundary curves, then if the surface has "little" topology, each nonsimple closed geodesic must contain a boundary curve. But if there is "enough" topology, there are nonsimple closed geodesics not containing boundary curves.
Total stability of sets for nonautonomous differential systems
Zhivko S.
Athanassov
649-663
Abstract: The principal purpose of this paper is to present sufficient conditions for total stability, or stability under constantly acting perturbations, of sets of a sufficiently general kind for nonautonomous ordinary differential equations. To do this, two Liapunov-like functions with specific properties are used. The obtained results include and considerably improve the classical results on total stability of isolated equilibrium points. Applications are presented to study the stability of nonautonomous Lurie-type nonlinear equations.
The divergence theorem
W. F.
Pfeffer
665-685
Abstract: We define a well-behaved multidimensional Riemann type integral such that the divergence of any vector field continuous in a compact interval and differentiable in its interior is integrable, and the integral equals the flux of the vector field out of the interval.
Boundary behavior of positive solutions of the heat equation on a semi-infinite slab
B. A.
Mair
687-697
Abstract: In this paper, the abstract Fatou-Naim-Doob theorem is used to investigate the boundary behavior of positive solutions of the heat equation on the semi-infinite slab $X = {{\mathbf{R}}^{n - 1}} \times {{\mathbf{R}}_ + } \times (0,T)$. The concept of semifine limit is introduced, and relationships are obtained between fine, semifine, parabolic, one-sided parabolic and two-sided parabolic limits at points on the parabolic boundary of $ X$. A Carleson-Calderón-type local Fatou theorem is also obtained for solutions on a union of two-sided parabolic regions.
An extremal problem for analytic functions with prescribed zeros and $r$th derivative in $H\sp \infty$
A.
Horwitz;
D. J.
Newman
699-713
Abstract: Let $({\alpha _1}, \ldots ,{\alpha _n})$ be $ n$ points in the unit disc $ U$. Suppose $g$ is analytic in $U$, $g({\alpha _1}) = \cdots = g({\alpha _n}) = 0$ (multiplicities included), and $\Vert g\prime\Vert _{\infty } \leq 1$. Then we prove that $\vert g(z)\vert \leq \vert\phi (z)\vert$ for all $z \in U$, where $\phi ({\alpha _1}) = \cdots = \phi ({\alpha _n}) = 0$ and $ \phi\prime(z)$ is a Blaschke product of order $n - 1$. We extend this result in a natural way to convex domains $D$ with analytic boundary. For $D$ not convex we show that there is no extremal function $\phi$.
Axiom $3$ modules
Paul
Hill;
Charles
Megibben
715-734
Abstract: By introducing the concept of a knice submodule, a refinement of the notion of nice subgroup, we are able to formulate a version of the third axiom of countability appropriate to the study of $p$-local mixed groups in the spirit of the well-known characterization of totally projective $p$-groups. Our Axiom $3$ modules, in fact, form a class of $ {{\mathbf{Z}}_{\mathbf{p}}}$-modules, encompassing the totally projectives in the torsion case, for which we prove a uniqueness theorem and establish closure under direct summands. Indeed Axiom $3$ modules turn out to be precisely the previously classified Warfield modules. But with the added power of the third axiom of countability characterization, we derive numerous new results, including the resolution of a long-standing problem of Warfield and theorems in the vein of familiar criteria due to Kulikov and Pontryagin.
Torsion free groups
Paul
Hill;
Charles
Megibben
735-751
Abstract: In this paper we introduce the class of torsion free $k$-groups and the notion of a knice subgroup. Torsion free $k$-groups form a class of groups more extensive than the separable groups of Baer, but they enjoy many of the same closure properties. We establish a role for knice subgroups of torsion free groups analogous to that played by nice subgroups in the study of torsion groups. For example, among the torsion free groups, the balanced projectives are characterized by the fact that they satisfy the third axiom of countability with respect to knice subgroups. Separable groups are characterized as those torsion free $k$-groups with the property that all finite rank, pure knice subgroups are direct summands. The introduction of these new classes of groups and subgroups is based on a preliminary study of the interplay between primitive elements and $\ast$-valuated coproducts. As a by-product of our investigation, new proofs are obtained for many classical results on separable groups. Our techniques lead naturally to the discovery that a balanced subgroup of a completely decomposable group is itself completely decomposable provided the corresponding quotient is a separable group of cardinality not exceeding ${\aleph _1}$; that is, separable groups of cardinality ${\aleph _1}$ have balanced projective dimension $\leq 1$.
Algebraic relations among solutions of linear differential equations
Michael F.
Singer
753-763
Abstract: Using power series methods, Harris and Sibuya [3,4] recently showed that if $k$ is an ordinary differential field of characteristic zero and $y \ne 0$ is an element of a differential extension of $ k$ such that $ y$ and $1/y$ satisfy linear differential equations with coefficients in $k$, then $y\prime/y$ is algebraic over $k$. Using differential galois theory, we generalize this and characterize those polynomial relations among solutions of linear differential equations that force these solutions to have algebraic logarithmic derivatives. We also show that if $ f$ is an algebraic function of genus $\geq 1$ and if $y$ and $f(y)$ or $y$ and $ {e^{\int y}}$ satisfy linear differential equations, then $y$ is an algebraic function.
$L\sp p$ inequalities for stopping times of diffusions
R. Dante
DeBlassie
765-782
Abstract: Let ${X_t}$ be a solution to a stochastic differential equation. Easily verified conditions on the coefficients of the equation give ${L^p}$ inequalities for stopping times of ${X_t}$ and the maximal function. An application to Brownian motion with radial drift is also discussed.
On algebras with convolution structures for Laguerre polynomials
Yūichi
Kanjin
783-794
Abstract: In this paper we treat the convolution algebra connected with Laguerre polynomials which was constructed by Askey and Gasper [1]. For this algebra, we study the maximal ideal space, Wiener's general Tauberian theorem, spectral synthesis and Helson sets. We also study Sidon sets and idempotent measures for the algebras with dual convolution structures.
Generic dynamics and monotone complete $C\sp \ast$-algebras
Dennis
Sullivan;
B.
Weiss;
J. D. Maitland
Wright
795-809
Abstract: Let $R$ be any ergodic, countable generic equivalence relation on a perfect Polish space $ X$. It follows from the main theorem of $\S1$ that, modulo a meagre subset of $ X,R$ may be identified with the relation of orbit equivalence ensuing from a canonical action of $ {\mathbf{Z}}$. Answering a longstanding problem of Kaplansky, Takenouchi and Dyer independently gave cross-product constructions of Type III $A{W^\ast}$-factors which were not von Neumann algebras. As a specialization of a much more general result, obtained in $\S3$, we show that the Dyer factor is isomorphic to the Takenouchi factor.
Quantization and Hamiltonian $G$-foliations
L.
Pukanszky
811-847
Abstract: As it was recognized twenty five years ago by A. A. Kirillov, in the unitary representation theory of nilpotent Lie groups a crucial role is played by orbits of the coadjoint representation. B. Kostant noted that, for any connected Lie group, these orbits admit a symplectic structure and lend themselves to an intrinsic characterization. The present author later observed, that already for the purposes of unitary representation theory of solvable Lie groups, this concept has to be enlarged and replaced by that of a generalized orbit. One objective of this paper is their intrinsic characterization. Other results prepare the way for the geometric construction of the corresponding unitary representations, to be developed later.
Connected locally connected toposes are path-connected
I.
Moerdijk;
G. C.
Wraith
849-859
Abstract: A conjecture of A. Joyal is proved, which states that, in contrast to topological spaces, toposes which are connected and locally connected are also path-connected. The reason for this phenomenon is the triviality of cardinality considerations in the topos-theoretic setting; any inhabited object pulls back to an enumerable object under some open surjective geometric morphism. This result points towards a homotopy theory for toposes.