Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions
C. Ward
Henson;
Lee A.
Rubel
1-32
Abstract: In this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition $x+y$, multiplication $x \cdot y$, and two-place exponentiation $ x^y$. For a restricted class of such functions, we show that every true identity follows from the natural set of eleven axioms. The rates of growth of such functions, in the case of a single independent variable $x$, as $x \to \infty $, are also studied, and we give an algorithm for the Hardy relation of eventual domination, again for a restricted class of functions. Value distribution of analytic functions of one and of several complex variables, especially the Nevanlinna characteristic, plays a major role in our proofs.
Decompositions of the maximal ideal space of $L\sp{\infty }$
Pamela
Gorkin
33-44
Abstract: In this paper we show the existence of one point maximal antisymmetric sets for $ {H^\infty } + C$.
The generalized Zahorski class structure of symmetric derivatives
Lee
Larson
45-58
Abstract: A generalized Zahorski class structure is demonstrated for symmetric derivatives. A monotonicity theorem is proved and a condition sufficient to ensure that a symmetric derivative has the Darboux property is presented.
Estimates for eigenfunctions and eigenvalues of nonlinear elliptic problems
Chris
Cosner
59-75
Abstract: We consider solutions to the nonlinear eigenvalue problem $\displaystyle (*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\tex... ...,\quad \vec u{\text{ = }}0,\quad {\text{on}}\partial \Omega ,\quad \vec{u} = 0,$ where (*) is a quasilinear strongly coupled second order elliptic system of partial differential equations and $ \Omega \subseteq \mathbf{R}^{n}$ is a smooth bounded domain. We obtain lower bounds for $\lambda$ in the case where $f(x,\vec u)$ has linear growth, and relations between $\lambda ,\Omega $, and ess sup$ \vert\vec u\vert$ when $f(x,\vec u)$ has sub- or superlinear growth. The estimates are based on integration by parts and application of certain Sobolev inequalities. We briefly discuss extensions to higher order systems.
Inducible periodic homeomorphisms of tree-like continua
Juan A.
Toledo
77-108
Abstract: In this paper we prove that every periodic homeomorphism on a tree-like continuum can be strongly induced on an inverse sequence composed of a certain kind of graph that we call ``bellows''. We introduce the concepts of ``#-graph'' of a periodic homeomorphism and of ``perfect'' homeomorphism. A theorem concerning the parallel inducing of two periodic homeomorphisms having orbit spaces with the same multiplicity structure is also proved. The results are related to conjugacy and to the pseudo-arc.
Noncommutative topological dynamics. I
Daniel
Avitzour
109-119
Abstract: Ergodicity and minimality are defined for $C^{\ast}$-flows $(G,A)$ where $G$ is a group acting on a $C^{\ast}$-algebra $A$ by *-automorphisms. Elementary properties are proved and several examples are given. In particular, an example shows that there are arbitrarily large $ C^{\ast}$-algebras admitting a minimal action of the integers.
Noncommutative topological dynamics. II
Daniel
Avitzour
121-135
Abstract: This part deals with almost periodic and weakly mixing ${C^ \ast }$-flows, and with disjointness and weak disjointness of $ {C^ \ast }$-flows (flows on ${C^ \ast }$-algebras). The main result is a generalization to ${C^ \ast }$-flows of Keynes and Robertson's characterization of minimal weakly mixing flows. Examples are discussed exhibiting anomalous behaviour of disjointness in the $ {C^ \ast }$-flow case.
Brauer's height conjecture for $p$-solvable groups
David
Gluck;
Thomas R.
Wolf
137-152
Abstract: We complete the proof of the height conjecture for $p$-solvable groups, using the classification of finite simple groups.
The free boundary of a semilinear elliptic equation
Avner
Friedman;
Daniel
Phillips
153-182
Abstract: The Dirichlet problem $ \Delta u = \lambda \,f(u)$ in a domain $ \Omega ,\,u = 1$ on $\partial\Omega$ is considered with $ f(t) = 0$ if $t \leq 0,\,f(t) > 0$ if $t > 0,\,f(t) \sim {t^p}$ if $t \downarrow 0,0 < p < 1;f(t)$ is not monotone in general. The set $\{ u = 0\}$ and the ``free boundary'' $\partial \{ u = 0\}$ are studied. Sharp asymptotic estimates are established as $\lambda \to \infty$. For suitable $f$, under the assumption that $ \Omega$ is a two-dimensional convex domain, it is shown that $\{ u = 0\}$ is a convex set. Analogous results are established also in the case where $\partial u/\partial v + \mu (u - 1) = 0$ on $\partial \Omega $.
The ill-posed Hele-Shaw model and the Stefan problem for supercooled water
Emmanuele
DiBenedetto;
Avner
Friedman
183-204
Abstract: The Hele-Shaw flow of a slow viscous fluid between slightly separated plates is analyzed in the ill-posed case when the fluid recedes due to absorption through a core $ G$. Necessary and sufficient conditions are given on the initial domain occupied by the fluid to ensure the existence of a solution. Regularity of the free boundary is established in certain rather general cases. Similar results are obtained for the analogous parabolic version, which models the one-phase Stefan problem for supercooled water.
Classifying torsion-free subgroups of the Picard group
Andrew M.
Brunner;
Michael L.
Frame;
Youn W.
Lee;
Norbert J.
Wielenberg
205-235
Abstract: Torsion-free subgroups of finite index in the Picard group are the fundamental groups of hyperbolic $3$-manifolds. The Picard group is a polygonal product of finite groups. Recent work by Karrass, Pietrowski and Solitar on the subgroups of a polygonal product make it feasible to calculate all the torsion-free subgroups of any finite index. This computation is carried out here for index 12 and 24, where there are, respectively, 2 and 17 nonisomorphic subgroups. The manifolds are identified by using surgery.
Group-graded rings, smash products, and group actions
M.
Cohen;
S.
Montgomery
237-258
Abstract: Let $A$ be a $k$-algebra graded by a finite group $G$, with ${A_1}$ the component for the identity element of $ G$. We consider such a grading as a ``coaction'' by $G$, in that $A$ is a $k{[G]^ \ast }$-module algebra. We then study the smash product $A\char93 k{[G]^ \ast }$; it plays a role similar to that played by the skew group ring $R\, \ast \,G$ in the case of group actions, and enables us to obtain results relating the modules over $ A,\,{A_1}$, and $A\char93 k{[G]^ \ast }$. After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of $A$ and ${A_1}$. In particular we generalize Lorenz and Passman's theorem on incomparability of primes in crossed products. We also answer a question of Bergman on graded Jacobson radicals.
The number of factorizations of numbers less than $x$ into divisors greater than $y$
Douglas
Hensley
259-274
Abstract: Let $A(x,\,y)$ be the number in the title. There is a function $h:[0,\,\infty ) \to [0,\,2]$, decreasing and convex, with $h(0) = 2$ and $\operatorname{lim}_{r \to \infty }h(r) = 0$, such that if $r = \operatorname{log} y/\sqrt {\operatorname{log} \,x}$ then as $x \to \infty$ with $ r$ fixed, $\displaystyle A(x,y) = \frac{{C(r)x\,\operatorname{exp} \left( {h(r)\sqrt {\ope... ...{log} \,x)}^{3/4}}}}\left( {1 + O{{(\operatorname{log} \,x)}^{ - 1/4}}} \right)$ . The estimate is uniform on intervals $0 < r \leq {R_0}$. As corollaries we have for $\operatorname{log} \,y = \theta {(\operatorname{log} \,x)^{1/4}}$, $\displaystyle \lim \limits_{x \to \infty } \,\frac{{A(x,\,y)}} {{A(x,\,1)/y}} = {e^{{\theta ^2}/2}}$ ,and if $ \operatorname{log} \,y = o {(\operatorname{log} \,x)^{1/4}}$ then $ A(x,\,y) \approx A(x,\,1)/y$.
The radius ratio and convexity properties in normed linear spaces
D.
Amir;
C.
Franchetti
275-291
Abstract: The supremum of the ratios of the self-radius ${r_A}(A)$ of a convex bounded set in a normed linear space $X$ to its absolute radius ${r_X}(A)$ is related to the supremum of the relative projection constants of the maximal subspaces of $ X$. Necessary conditions and sufficient conditions for these suprema to be smaller than 2 are given. These conditions are selfadjoint superproperties similar to $B$-convexity, superreflexivity and $ P$-convexity.
Free products of inverse semigroups
Peter R.
Jones
293-317
Abstract: A structure theorem is provided for the free product $ S\,{\operatorname{inv}}\,T$ of inverse semigroups $S$ and $T$. Each element of $ S\,{\operatorname{inv}}\,T$ is uniquely expressible in the form $\varepsilon (A)a$, where $A$ is a certain finite set of ``left reduced'' words and either $a = 1$ or $ a = {a_1} \cdots {a_m}$ is a ``reduced'' word with $aa_m^{ - 1} \in A$. (The word ${a_1} \cdots {a_m}$ in $ S\,{\operatorname{sgp}}\,T$ is called reduced if no letter is idempotent, and left reduced if exactly ${a_m}$ is idempotent; the notation $\varepsilon (A)$ stands for $\Pi \{ a{a^{ - 1}}:\,a \in A\}$.) Under a product remarkably similar to Scheiblich's product for free inverse semigroups, the corresponding pairs $ (A,\,a)$ form an inverse semigroup isomorphic with $ S\,{\operatorname{inv}}\,T$. This description enables various properties of $ S\,{\operatorname{inv}}\,T$ to be determined. For example $ (S\:{\operatorname{inv}}\:T)\backslash (S \cup T)$ is always completely semisimple and each of its subgroups is isomorphic with a finite subgroup of $S$ or $T$. If neither $S$ nor $T$ has a zero then $ (S\:{\operatorname{inv}}\:T)$ is fundamental, but in general fundamentality itself is not preserved.
Equivalence problems in projective differential geometry
Kichoon
Yang
319-334
Abstract: Equivalence problems for abstract, and induced, projective structures are investigated. (i) The notion of induced projective structures on submanifolds of a projective space is rigorously defined. (ii) Equivalence problems for such structures are discussed; in particular, it is shown that nonplanar surfaces in $ \mathbf{R}{P^3}$ are all projectively equivalent to each other. (iii) The imbedding problem of abstract projective structures is studied; in particular, we show that every abstract projective structure on a $2$-manifold arises as an induced structure on an arbitrary nonplanar surface in $\mathbf{R}{P^3}$; this result should be contrasted to that of Chern (see [6]).
Positive solutions of nonlinear elliptic equations---existence and nonexistence of solutions with radial symmetry in $L\sb{p}({\bf R}\sp{N})$
J. F.
Toland
335-354
Abstract: It is shown that when $r$ is nonincreasing, radially symmetric, continuous and bounded below by a positive constant, the solution set of the nonlinear elliptic eigenvalue problem $\displaystyle - \Delta u = \lambda u + r{u^{1 + \sigma }},\qquad u > 0\qquad {\... ...athbf{R}^N},\qquad u \to 0\qquad {\text{as}}\,{\text{\vert x\vert}} \to \infty$ , contains a continuum $ \mathcal{C}$ of nontrivial solutions which is unbounded in $\mathbf{R}\, \times \,{L_p}({\mathbf{R}^N})$ for all $p \geq 1$. Various estimates of the $ {L_p}$ norm of $ u$ are obtained which depend on the relative values of $\sigma$ and $p$, and the Pohozaev and Sobolev embedding constants.
Ford and Dirichlet regions for discrete groups of hyperbolic motions
P. J.
Nicholls
355-365
Abstract: It is shown that for a discrete group of hyperbolic motions of the unit ball of $ {\mathbf{R}^n}$, there is a single construction of fundamental regions which gives the Ford and Dirichlet regions as special cases and which also yields fundamental regions based at limit points. It is shown how the region varies continuously with the construction. The construction is connected with a class of limit points called Garnett points. The size of the set of such points is investigated.
Approximation of infinite-dimensional Teichm\"uller spaces
Frederick P.
Gardiner
367-383
Abstract: By means of an exhaustion process it is shown that Teichmüller's metric and Kobayashi's metric are equal for infinite dimensional Teichmüller spaces. By the same approximation method important estimates coming from the Reich-Strebel inequality are extended to the infinite dimensional cases. These estimates are used to show that Teichmüller's metric is the integral of its infinitesimal form. They are also used to give a sufficient condition for a sequence to be an absolute maximal sequence for the Hamilton functional. Finally, they are used to give a new sufficient condition for a sequence of Beltrami coefficients to converge in the Teichmüller metric.
Knots prime on many strings
Steven A.
Bleiler
385-401
Abstract: A study is made of the factorization of prime knots into tangles. Several infinite families of knots which do not factor into prime tangles are examined, and a new characterization of knot primality is developed.
Discontinuous translation invariant functionals
Sadahiro
Saeki
403-414
Abstract: Let $G$ be an infinite $ \sigma$-compact locally compact group. We shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant Fréchet spaces of Radon measures on $G$. These spaces include the convolution measure algebra $M(G)$, the Lebesgue spaces ${L^p}(G)$, where $1 \leq p \leq \infty$, and certain combinations thereof. Among other things, it will be shown that $ C(G)$ has many discontinuous translation invariant functionals, provided that $ G$ is amenable. This solves a problem of G. H. Meisters.
Linearization and mappings onto pseudocircle domains
Andrew
Haas
415-429
Abstract: We demonstrate the existence of linearizations for groups of conformal and anticonformal homeomorphisms of Riemann surfaces. The finitely generated groups acting on plane domains are classified in terms of specific linearizations. This extends Maskit's work in the directly conformal case. As an application we prove that there exist conformal representations of finite genus open Riemann surfaces for which accessible boundary points are either isolated or lie on circular arcs of pseudocircular boundary components. In many cases these are actually circle domains. Along the way we extend the applicability of Carathéodory's boundary correspondence theorem for prime ends.