Extremal length and harmonic functions on Riemann surfaces
Carl David
Minda
1-22
Abstract: Expressions for several conformally invariant pseudometrics on a Riemann surface $R$ are given in terms of three new forms of reduced extremal distance. The pseudometrics are defined by means of various subclasses of the set of all harmonic functions on $R$ having finite Dirichlet integral. The reduced extremal distance between two points is defined on $R$, on the Alexandroff one-point compactification of $R$ and on the Kerékjártó-Stoïlow compactification of $R$. These reduced extremal distances are computed in terms of harmonic functions having specified singularities and boundary behavior. The key to establishing this connection with harmonic functions is a general theorem dealing with extremal length on a compact bordered Riemann surface and its extensions to noncompact bordered surfaces. These results are used to obtain new tests for degeneracy in the classification theory of Riemann surfaces. Finally, some of the results are illustrated for a hyperbolic simply connected Riemann surface.
Applications of bundle map theory
Daniel Henry
Gottlieb
23-50
Abstract: This paper observes that the space of principal bundle maps into the universal bundle is contractible. This fact is added to I. M. James' Bundle map theory which is slightly generalized here. Then applications yield new results about actions on manifolds, the evaluation map, evaluation subgroups of classifying spaces of topological groups, vector bundle injections, the Wang exact sequence, and $H$-spaces.
The Hochschild homology of complete intersections
Klaus
Wolffhardt
51-66
Abstract: Let $\tilde{R}$ be the algebra of all convergent (or of all strictly convergent) power series in $ N$ variables over a commutative field $K$ of characteristic 0 with a valuation, e.g. $ \tilde{R} = K[{X_1}, \cdots ,{X_N}]$. With each $K$-algebra $R \cong \tilde{R}/\mathfrak{a}$ we associate a bigraded $R$-algebra $E$. By the powers of $ \mathfrak{a}$ a filtration of the Poincaré complex of $\tilde{R}$ is induced, and $E$ is the first term of the corresponding spectral sequence. If $ \mathfrak{a}$ is generated by a prime sequence in $ \tilde{R},R$ is called a complete intersection, and $E$--with an appropriate simple grading--is isomorphic to the Hochschild homology of $R$. The result is applied to hypersurfaces.
Differentiable structures on function spaces
Nishan
Krikorian
67-82
Abstract: A ${C^s}$ differentiable manifold structure is constructed for spaces of maps from a compact ${C^r}$ manifold $M$ to a ${C^{r + s}}$ manifold $N$. The method (1) is inspired by Douady; (2) does not require any additional structure on $ N$ (such as sprays); (3) includes the case when $N$ is an analytic manifold and concludes that the mapping space is also an analytic manifold; (4) can be used to treat all the classical mapping spaces ($ {C^r}$ functions, $ {C^r}$ functions with Hölder conditions, and Sobolev functions). Several interesting aspects of these manifolds are investigated such as their tangent spaces, their behavior with respect to functions, and realizations of Lie group structures on them. Differentiable structures are also exhibited for spaces of compact maps with noncompact domain.
The tangential Cauchy-Riemann complex on spheres
G. B.
Folland
83-133
Abstract: This paper investigates the ${\overline \partial _b}$ complex of Kohn and Rossi on the unit sphere in complex $n$-space (considered as the boundary of the unit ball). The methods are Fourier-analytic, exploiting the fact that the unitary group $U(n)$ acts homogeneously on the complex. We decompose the spaces of sections into irreducible components under the action of $U(n)$ and compute the action of ${\overline \partial _b}$ on each irreducible piece. We then display the connection between the ${\overline \partial _b}$ complex and the Dolbeault complexes of certain line bundles on complex projective space. Precise global regularity theorems for ${\overline \partial _b}$ are proved, including a Sobolev-type estimate for norms related to ${\overline \partial _b}$. Finally, we solve the $\overline \partial$-Neumann problem on the unit ball and obtain a proof by explicit calculations of the noncoercive nature of this problem.
Banach space bifurcation theory
David
Westreich
135-156
Abstract: We consider the bifurcation problem for the nonlinear operator equation $x = \lambda Lx + T(\lambda ,x,y)$ in a real Banach space $X$. Here $ {\lambda _0}$ is an eigenvalue of the bounded linear operator $L,X = N(I - {\lambda _0}L) \oplus R(I - {\lambda _0}L),T \in {C^1}$ and $T$ is of higher order in $x$. New techniques are developed to simplify the solution of the bifurcation problem. When ${\lambda _0}$ is a simple eigenvalue, ${\lambda _0}$ is shown to be a bifurcation point of the homogeneous equation (i.e. $ y \equiv 0$) with respect to 0. All solutions near $({\lambda _0},0)$ are shown to be of the form $(\lambda (\epsilon),x(\epsilon)),0 \leqslant \vert\epsilon\vert < {\epsilon_0},\lambda (\epsilon)$ and $ x(\epsilon)$ are continuous and $ \lambda (\epsilon)$ and $x(\epsilon)$ are in ${C^n}$ or real analytic as $T$ is in $ {C^{n + 1}}$ or is real analytic. When $T$ is real analytic and $\lambda (\epsilon){\lambda _0}$ then there are at most two solution branches, and each branch is an analytic function of $\lambda$ for $\lambda \ne {\lambda _0}$. If $T$ is odd and analytic, for each $\lambda \in ({\lambda _0} - \delta ,{\lambda _0})$ (or $\lambda \in ({\lambda _0},{\lambda _0} + \delta )$) there exist two nontrivial solutions near 0 and there are no solutions near 0 for $ \lambda \in ({\lambda _0},{\lambda _0} + \delta )$ (or $\lambda \in ({\lambda _0} - \delta ,{\lambda _0})$). We then demonstrate that in each sufficiently small neighborhood of a solution of the homogeneous bifurcation problem there are solutions of the nonhomogeneous equation (i.e. $y \not\equiv 0$) depending continuously on a real parameter and on $y$. If $ {\lambda _0}$ is an eigenvalue of odd multiplicity we prove it is a point of bifurcation of the homogeneous equation. With a strong restriction on the projection of $T$ onto the null space of $I - {\lambda _0}L$ we show ${\lambda _0}$ is a bifurcation point of the homogeneous equation when $ {\lambda _0}$ is a double eigenvalue. Counterexamples to some of our results are given when the hypotheses are weakened.
Topological types of polynomial differential equations
L.
Markus
157-178
Abstract: Consider a first order system of real ordinary differential equations, with polynomial coefficients, having no critical points in the number space ${R^n}$. Two such differential systems are called topologically equivalent in case there exists a homeomorphism of ${R^n}$ onto itself carrying the sensed (not parametrized) solutions of the first system onto the solution family of the second system. Let ${B^n}(m)$ be the cardinal number of topological equivalence classes for systems in $ {R^n}$ with polynomial coefficients of degree at most $m$. The author proves that ${B^2}(m)$ is finite and obtains explicit upper and lower bounds in terms of $m$. Also examples are given to show that ${B^n}(m)$ is noncountable for $n \geqslant 3$ and $ m \geqslant 6$.
Symmetrization of distributions and its application. II. Liouville type problem in convolution equations
Kuang-ho
Chen
179-194
Abstract: The symmetrization of distributions corresponding to a bounded $n - 1$ dimensional ${C^\infty }$-submanifold of a ${C^\infty }$-manifold is constructed. This device reduces the consideration of distributions in $ {R^n}$ to the one of distributions in ${R^1}$, i.e. the symmetrized distributions. Using the relation between the inverse Fourier transform of a symmetrized distribution and the one of the original (nonsymmetrized) distribution, we determine the rate of decay at infinity of solutions to a general convolution equation necessary to assure uniqueness. Using a result in the division problem for distributions, we achieve the following result: If $u \in C({R^n})$ is a solution of the convolution equation $S \ast u = f,f \in \mathcal{D}({R^n})$, with some suitable $ u \in \mathcal{D}({R^n})$, provided $u$ decays sufficiently fast at infinity.
Inductive limits of finite dimensional $C\sp{\ast} $-algebras
Ola
Bratteli
195-234
Abstract: Inductive limits of ascending sequences of finite dimensional ${C^ \ast }$-algebras are studied. The ideals of such algebras are classified, and a necessary and sufficient condition for isomorphism of two such algebras is obtained. The results of Powers concerning factor states and representations of UHF-algebras are generalized to this case. A study of the current algebra of the canonical anticommutation relations is then being made.
A decomposition for combinatorial geometries
Thomas H.
Brylawski
235-282
Abstract: A construction based on work by Tutte and Grothendieck is applied to a decomposition on combinatorial pregeometries in order to study an important class of invariants. The properties of this Tutte decomposition of a pregeometry into a subgeometry $G\backslash e$ and contraction $G/e$ is explored in a categorically integrated view using factored strong maps. After showing that direct sum decomposition distributes over the Tutte decomposition we construct a universal pair $(R,t)$ where $R$ is a free commutative ring with two generators corresponding to a loop and an isthmus; and $t$, the Tutte polynomial assigns a ring element to each pregeometry. Evaluations of $ t(G)$ give the Möbius function, characteristic polynomial, Crapo invariant, and numbers of subsets, bases, spanning and independent sets of $G$ and its Whitney dual. For geometries a similar decomposition gives the same information as the chromatic polynomial throwing new light on the critical problem. A basis is found for all linear identities involving Tutte polynomial coefficients. In certain cases including Hartmanis partitions one can recover all the Whitney numbers of the associated geometric lattice $L(G)$ from $t(G)$ and conversely. Examples and counterexamples show that duals, minors, connected pregeometries, series-parallel networks, free geometries (on which many invariants achieve their upper bounds), and lower distributive pregeometries are all characterized by their polynomials. However, inequivalence, Whitney numbers, and representability are not always invariant. Applying the decomposition to chain groups we generalize the classical two-color theorem for graphs to show when a geometry can be imbedded in binary affine space. The decomposition proves useful also for graphical pregeometries and for unimodular (orientable) pregeometries in the counting of cycles and co-boundaries.
Radial limit sets on the torus
Laurence D.
Hoffmann
283-290
Abstract: Let ${U^N}$ denote the unit polydisc and $ {T^N}$ the unit torus in the space of $N$ complex variables. A subset $A$ of ${T^N}$ is called an (RL)-set (radial limit set) if to each positive continuous function $ \rho$ on ${T^N}$, there corresponds a function $ f$ in ${H^\infty }({U^N})$ such that the radial limit $\vert f{\vert^ \ast }$ of the absolute value of $ f$ equals $\rho$, a.e. on ${T^N}$ and everywhere on $A$. If $N > 1$, the question of characterizing (RL)-sets is open, but two positive results are obtained. In particular, it is shown that ${T^N}$ contains an (RL)-set which is homeomorphic to a cartesian product $K \times {T^{N - 1}}$, where $K$ is a Cantor set. Also, certain countable unions of ``parallel'' copies of ${T^{N - 1}}$ are shown to be (RL)-sets in $ {T^N}$. In one variable, every subset of $T$ is an (RL)-set; in fact, there is always a zero-free function $f$ in $ {H^\infty }(U)$ with the required properties. It is shown, however, that there exist a circle $ A \subset {T^2}$ and a positive continuous function $\rho$ on ${T^2}$ to which correspond no zero-free $ f$ in ${H^\infty }({U^2})$ with $\vert f{\vert^ \ast } = \rho$ a.e. on $ {T^2}$ and everywhere on $ A$.
Regular overrings of regular local rings
Judith
Sally
291-300
Abstract: The local factorization theorem of Zariski and Abhyankar characterizes all $ 2$-dimensional regular local rings which lie between a given $2$-dimensional regular local ring $ R$ and its quotient field as finite quadratic transforms of $R$. This paper shows that every regular local ring $R$ of dimension $n > 2$ has infinitely many minimal regular local overrings which cannot be obtained by a monoidal transform of $R$. These overrings are localizations of rings generated over $R$ by certain quotients of elements of an $ R$-sequence. Necessary and sufficient conditions are given for this type of extension of $R$ to be regular.
Irreducible representations of the $C\sp{\ast} $-algebra generated by an $n$-normal operator
John W.
Bunce;
James A.
Deddens
301-307
Abstract: For $A$ an $n$-normal operator on Hilbert space, we determine the irreducible representations of ${C^ \ast }(A)$, the $ {C^ \ast }$-algebra generated by $A$ and the identity. For $A$ a binormal operator, we determine an explicit description of the topology on the space of unitary equivalence classes of irreducible representations of ${C^ \ast }(A)$.
Automorphisms of a free associative algebra of rank $2$. II
Anastasia J.
Czerniakiewicz
309-315
Abstract: Let $R$ be a commutative domain with 1. $ R\langle x,y\rangle$ stands for the free associative algebra of rank 2 over $R;R[\tilde x,\tilde y]$ is the polynomial algebra over $R$ in the commuting indeterminates $ \tilde x$ and $ \tilde y$. We prove that the map Ab$: \operatorname{Aut} (R\langle x,y\rangle ) \to \operatorname{Aut} (R[\tilde x,\tilde y])$ induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [5], Nagata [7] and van der Kulk [8]* that describes the automorphisms of $ F[\tilde x,\tilde y]$ ($ F$ a field) we are able to conclude that every automorphism of $F\langle x,y\rangle$ is tame (i.e. a product of elementary automorphisms).
On the evaluation of Brewer's character sums
Reinaldo E.
Giudici;
Joseph B.
Muskat;
Stanley F.
Robinson
317-347
Abstract: A decade ago in this journal B. W. Brewer defined a sequence of polynomials ${V_n}(x,1)$ and for $n = 4$ and 5 evaluated $\displaystyle \sum\limits_{x = 1}^p {{}_\chi ({V_n}(x,1))},$ $\chi$ the nonprincipal quadratic character of the prime $p$, in closed form. A. L Whiteman derived these results by means of cyclotomy. Brewer subsequently defined ${V_n}(x,Q)$. This paper applies cyclotomy to the more general polynomials and provides evaluations for several more values of $n$. Relevant quadratic decompositions of primes are studied.
Some asymptotic fixed point theorems
Roger D.
Nussbaum
349-375
Abstract: By an asymptotic fixed point theorem we mean a theorem in functional analysis in which the existence of fixed points of a map $ f$ is established with the aid of assumptions on the iterates ${f^n}$ of $f$. We prove below some new theorems of this type, and we obtain as corollaries results of F. E. Browder, G. Darbo, R. L. Frum-Ketkov, W. A. Horn and others. We also state a number of conjectures about fixed point theorems at the end of the paper.
The finiteness of $I$ when ${\it R}[{\it X}]/{\it I}$ is flat
Jack
Ohm;
David E.
Rush
377-408
Abstract: Let $R$ be a commutative ring with identity, let $X$ be an indeterminate, and let $I$ be an ideal of the polynomial ring $ R[X]$. Let $\min I$ denote the set of elements of $I$ of minimal degree and assume henceforth that $ \min I$ contains a regular element. Then $R[X]/I$ is a flat $R$-module implies $I$ is a finitely generated ideal. Under the additional hypothesis that $R$ is quasi-local integrally closed, the stronger conclusion that $I$ is principal holds. (An example shows that the first statement is no longer valid when $\min I$ does not contain a regular element.) Let $c(I)$ denote the content ideal of $I$, i.e. $c(I)$ is the ideal of $R$ generated by the coefficients of the elements of $I$. A corollary to the above theorem asserts that $ R[X]/I$ is a flat $ R$-module if and only if $ I$ is an invertible ideal of $R[X]$ and $c(I) = R$. Moreover, if $R$ is quasi-local integrally closed, then the following are equivalent: (i) $R[X]/I$ is a flat $R$-module; (ii) $R[X]/I$ is a torsion free $R$-module and $c(I) = R$; (iii) $I$ is principal and $c(I) = R$. Let $\xi$ denote the equivalence class of $ X$ in $R[X]/I$, and let $\langle 1,\xi , \cdots ,{\xi ^t}\rangle$ denote the $R$-module generated by $1,\xi , \cdots ,{\xi ^t}$. The following statements are also equivalent: (i) $\langle 1,\xi , \cdots ,{\xi ^t}\rangle$ is flat for all $ t \geqslant 0$; (ii) $ \langle 1,\xi , \cdots ,{\xi ^t}\rangle$ is flat for some $t \geqslant 0$ for which $1,\xi , \cdots ,{\xi ^t}$ are linearly dependent over $R$; (iii) $I = ({f_1}, \cdots ,{f_n}),{f_i} \in \min I$, and $c(I) = R$; (iv) $ c(\min I) = R$. Moreover, if $R$ is integrally closed, these are equivalent to $ R[X]/I$ being a flat $ R$-module. A certain symmetry enters in when $\xi$ is regular in $R[\xi ]$, and in this case (i)-(iv) are also equivalent to the assertion that $R[\xi ]$ and $R[1/\xi ]$ are flat $R$-modules.
Analytic capacity and approximation problems
A. M.
Davie
409-444
Abstract: We consider some problems concerning analytic capacity as a set function, which are relevant to approximation problems for analytic functions on plane sets. In particular we consider the question of semiadditivity of capacity. We obtain positive results in some special cases and give applications to approximation theory. In general we establish some equivalences among various versions of the semiadditivity question and certain questions in approximation theory.
Approximation on disks
Kenneth John
Preskenis
445-467
Abstract: Let $D$ be a closed disk in the complex plane, $ f$ a complex valued continuous function on $D$ and $ {R_f}(D) =$ the uniform closure on $D$ of rational functions in $z$ and $f$ which are finite. Among other results we obtain the following. Theorem. If $f$ is of class ${C^1}$ in a neighborhood of $D$ and $\vert{f_{\bar z}}\vert > \vert{f_z}\vert$ everywhere (i.e., $f$ is an orientation reversing immersion of $ D$ in the plane), then ${R_f}(D) = C(D)$. Theorem. Let $ f$ be a polynomial in $ z$ and $\bar z$. If for each a in $D,f - \Sigma {(j!)^{ - 1}}{D^j}f(a){(z - a)^j} = {(\bar z - \bar a)^k}g$ with $\vert{g_{\bar z}}\vert > \vert{g_z}\vert$ at he zeros of $g$ in $D$ where $Df = {f_z}$, then ${R_f}(D) = C(D)$. Corollary. Let $ f$ be a polynomial in $ z$ and $\bar z$ and let $ \vert{f_{z\bar z}}(0)\vert < \vert{f_{\bar z\,\bar z}}(0)\vert/2$. Then there exists an $r > 0$ such that, for $D = (\vert z\vert \leqslant r),{R_f}(D) = C(D)$. The proofs of the theorems use measures and the conditions involved in the theorems are independent of each other. Concerning the corollary, results of E. Bishop and G. Stolzenberg show that ${f_{\bar z}}(0) = 0$ and $\vert{f_{\bar z\,\bar z}}(0)\vert < \vert{f_{z\bar z}}(0)\vert$, then there exists no $ r$ such that ${R_f}(D) = C(D)$ where $D = (\vert z\vert \leqslant r)$. Let $F = ({f_1}, \cdots ,{f_n})$ be a map on $B$ = unit polydisk in ${{\mathbf{C}}^n}$ with values in ${{\mathbf{C}}^n},{P_F}$ = uniform closure on $ B$ of polynomials in ${z_1}, \cdots {z_n},{f_1}, \cdots ,{f_n}$. Theorem. If $F$ is of class ${C^1}$ in a neighborhood of $B,{F_{\bar z}}$ is invertible and if for each $a = ({a_1}, \cdots ,{a_n})$ in $ B$, there exist complex constants $\{ {c_j}\} ,\{ {d_{ij}}\} ,i,j = 1, \cdots ,n$, such that $\Sigma {c_j}({z_j} - {a_j})({f_j}(z) - {f_j}(a)) + \Sigma {d_{ij}}({z_i} - {a_i})({z_j} - {a_j})$ has positive real part for all $z \ne a$, then $\{ (\zeta ,F(\zeta )):\zeta \in B\}$ is a polynomially convex set. Corollary. If $ F = (f,g)$ where $ f(z,w) = \bar z + cz\bar z + d{\bar z^2} + q\bar zw,g(z,w) = \bar w + sw\bar w + t{\bar w^2} + p\bar wz$ and the coefficients satisfy $ \vert\bar c + d\vert + \vert d\vert + \vert q\vert < 1$ and $\vert\bar s + t\vert + \vert t\vert + \vert p\vert < 1$, then $ {P_F} = C(B)$. Corollary. If $ F(z) = \bar z + R(z)$ where $ R = ({R_1}, \cdots ,{R_n})$ is of class ${C^2}$ and satisfies the Lipschitz condition $\vert R(\zeta ) - R(\eta )\vert \leqslant k\vert\zeta - \eta\vert$ with $k < 1$, then $ {P_F} = C(B)$. This last corollary is a result of Hörmander and Wermer. The proof of the theorem uses methods from several complex variables.
Analytic continuation of Eisenstein series
Joseph
Lewittes
469-490
Abstract: The classical Eisenstein series are essentially of the form $ \operatorname{Im} z > 0,{r_1},{r_2}$ rational and $s$ an integer $> 2$. In this paper we show that if $s$ is taken to be complex the series, with ${r_1},{r_2}$ any real numbers, defines an analytic function of $(z,s)$ for $\operatorname{Im} z > 0,\operatorname{Re} s > 2$. Furthermore this function has an analytic continuation over the entire $s$ plane, exhibted explicitly by a convergent Fourier expansion. A formula for the transformation of the function when $z$ is subjected to a modular transformation is obtained and the special case of $s$ an integer is studied in detail.
Boundary links and an unlinking theorem
M. A.
Gutiérrez
491-499
Abstract: This paper gives a homotopic theoretic criterion for a higher dimensional link to be trivial.
Infinite games with imperfect information
Michael
Orkin
501-507
Abstract: We consider an infinite, two person zero sum game played as follows: On the $n$th move, players $A,B$ select privately from fixed finite sets, ${A_n},{B_n}$, the result of their selections being made known before the next selection is made. After an infinite number of selections, a point in the associated sequence space, $\Omega$, is produced upon which $B$ pays $A$ an amount determined by a payoff function defined on $\Omega$. In this paper we extend a result of Blackwell and show that if the payoff function is the indicator function of a set in the Boolean algebra generated by the $ {G_\delta }$'s (with respect to a natural topology on $\Omega$) then the game in question has a value.