The Gauss map for surfaces. I. The affine case
Joel L.
Weiner
431-446
Abstract: Let $M$ be a connected oriented surface and let $G_2^c$ be the Grassmannian of oriented $ 2$-planes in Euclidean $ (2 + c)$-space, ${{\mathbf{E}}^{2 + c}}$. Smooth maps $t:M \to G_2^c$ are studied to determine whether or not they are Gauss maps. Both local and global results are obtained. If $t$ is a Gauss map of an immersion $X:\;M \to {{\mathbf{E}}^{2 + c}}$, we study the extent to which $t$ uniquely determines $X$ under certain circumstances.
The Gauss map for surfaces. II. The Euclidean case
Joel L.
Weiner
447-466
Abstract: We study smooth maps $t:\;M \to G_2^c$ of a Riemann surface $M$ into the Grassmannian $G_2^c$ of oriented $2$-planes in $ {{\mathbf{E}}^{2 + c}}$ and determine necessary and sufficient conditons on $ t$ in order that it be the Gauss map of a conformal immersion $X:\;M \to {{\mathbf{E}}^{2 + c}}$. We sometimes view $t$ as an oriented riemannian vector bundle; it is a subbundle of $ {\mathbf{E}}_M^{2 + c}$, the trivial bundle over $M$ with fibre $ {{\mathbf{E}}^{2 + c}}$. The necessary and sufficient conditions obtained for simply connected $M$ involve the curvatures of $t$ and $ {t^ \bot }$, the orthogonal complement of $t$ in $ {\mathbf{E}}_M^{2 + c}$, as well as certain components of the tension of $ t$ viewed as a map $t:\;M \to {S^C}(1)$, where ${S^C}(1)$ is a unit sphere of dimension $ C$ that contains $ G_2^c$ as a submanifold in a natural fashion. If $t$ satisfies a particular necessary condition, then the results take two different forms depending on whether or not $t$ is the Gauss map of a conformal minimal immersion. The case $ t:\;M \to G_2^2$ is also studied in some additional detail.
The least $r$-free number in an arithmetic progression
Kevin S.
McCurley
467-475
Abstract: Let ${n_r}(a,q)$ be the least $r$-free number in the arithmetic progession $ a$ modulo $q$. Several results are proved that give lower bounds for $ {n_r}(a,q)$, improving on previous results due to Erdös and Warlimont. In addition, a heuristic argument is given, leading to two conjectures that would imply that the results of the paper are close to best possible.
Extensions of a theorem of Wintner on systems with asymptotically constant solutions
William F.
Trench
477-483
Abstract: A theorem of Wintner concerning sufficient conditions for a system
A simple theory of differential calculus in locally convex spaces
Richard A.
Graff
485-509
Abstract: A theory of differential calculus for nonlinear maps between general locally convex spaces is developed. All convergence notions are topological, and only familiarity with basic results from point set topology, differential calculus in Banach spaces, and locally convex space theory is assumed. The chain rule for continuous $k$th order differentiability, smoothness of inverse functions, and function space continuity properties of higher order derivatives are examined. It is shown that this theory extends the classical Fréchet theory of differential calculus for maps between Banach spaces.
Rational singularities and almost split sequences
Maurice
Auslander
511-531
Abstract: The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimensional representations of a finite subgroup $G$ of $\operatorname{GL} (2,{\mathbf{C}})$, where ${\mathbf{C}}$ is the complex numbers, is isomorphic to the $AR$ quiver of the reflexive modules of the quotient singularity associated with $G$.
Best rational approximations of entire functions whose Maclaurin series coefficients decrease rapidly and smoothly
A. L.
Levin;
D. S.
Lubinsky
533-545
Abstract: Let $f = \Sigma _{j = 0}^\infty {a_j}{z^j}$ be an entire function which satisfies $\displaystyle \vert{a_{j - 1}}a{ _{j + 1}}/a_j^2\vert \leqslant {\rho ^2},\qquad j = 1,2,3, \ldots ,$ where $0 < \rho < {\rho _0}$ and $ {\rho _0} = 0.4559 \ldots$ is the positive root of the equation $ 2\Sigma _{j = 1}^\infty {\rho ^{{j^2}}} = 1$. Let $r > 0$ be fixed. Let ${W_{LM}}$ denote the rational function of type $ (L,M)$ of best approximation to $f$ in the uniform norm on $\vert z\vert \leqslant r$. We show that for any sequence of nonnegative integers $\{ {M_L}\} _{L = 1}^\infty $ that satisfies $ {M_L} \leqslant 10L,\,L = 1,2,3, \ldots$, the rational approximations ${W_{L{M_L}}}$ converge to $f$ throughout $ {\mathbf{C}}$ as $L \to \infty$. In particular, convergence takes place for the diagonal sequence and for the row sequences of the Walsh array for $f$.
Martingale transforms and related singular integrals
Rodrigo
Bañuelos
547-563
Abstract: The operators obtained by taking conditional expectation of continuous time martingale transforms are studied, both on the circle $ T$ and on ${{\mathbf{R}}^n}$. Using a Burkholder-Gundy inequality for vector-valued martingales, it is shown that the vector formed by any number of these operators is bounded on ${L^p}({{\mathbf{R}}^n}),\,1 < p < \infty$, with constants that depend only on $p$ and the norms of the matrices involved. As a corollary we obtain a recent result of Stein on the boundedness of the Riesz transforms on $ {L^p}({{\mathbf{R}}^n}),\,1 < p < \infty$, with constants independent of $ n$.
Baer's lemma and Fuchs's problem 84a
Ulrich
Albrecht
565-582
Abstract: An indecomposable, torsion-free, reduced abelian group $A$ has the properties that (i) each subgroup $B$ of an $A$-projective group with ${S_A}(B) = B$ is $A$-projective and (ii) each subgroup $B$ of a group $G$ with $ {S_A}(G) + B = G$ and $ G/B$ $A$-projective is a direct summand if and only if $A$ is self-small and flat as a left $ E(A)$-module, and $ E(A)$ is right hereditary. Furthermore, a group-theoretic characterization is given for torsion-free, reduced abelian groups with a right and left Noetherian, hereditary endomorphism ring. This is applied to Fuchs' Problem 84a. Finally, various applications of the results of this paper are given.
Solvability of differential equations with linear coefficients of real type
Rainer
Felix
583-591
Abstract: Let $L$ be the infinitesimal generator associated with a flow on a manifold $M$. Regarding $L$ as an operator on a space of testfunctions we deal with the question if $L$ has closed range. (Questions of this kind are investigated in [4, 1, 2].) We provide conditions under which $L + \mu 1:\mathcal{S}(M) \to \mathcal{S}(M)$, $ \mu \in {\mathbf{C}}$, has closed range, where $M = {{\mathbf{R}}^n} \times K$, $K$ being a compact manifold; here $\mathcal{S}(M)$ is the Schwartz space of rapidly decreasing smooth functions. As a consequence we show that the differential operator $ {\Sigma _{i,j}}{a_{ij}}{x_j}(\partial /\partial {x_i}) + b$ defines a surjective mapping of the space $ \mathcal{S}({{\mathbf{R}}^n})$ of tempered distributions onto itself provided that all eigenvalues of the matrix $({a_{ij}})$ are real. (In the case of imaginary eigenvalues this is not true in general [3].)
Conditions suffisantes de r\'esolubilit\'e locale pour des op\'erateurs invariants \`a gauche sur des groupes nilpotents. II
Pierre
Lévy-Bruhl
593-603
Abstract: On donne des conditions suffisantes de résolubilité locale pour des opérateurs différentiels invariants à gauche sur certains groupes de Lie nilpotents gradués. Ces conditions portent sur l'image de l'opérateur par certaines représentations unitaires irréductibles du groupe.
A solution of Warner's 3rd problem for representations of holomorphic type
Floyd L.
Williams
605-612
Abstract: In response to one of ten problems posed by G. Warner, we assign (to the extent that it is possible) a geometric or cohomological interpretation-- in the sense of Langlands--to the multiplicty in ${L^2}(\Gamma \backslash G)$ of an irreducible unitary representation $\pi$ of a semisimple Lie group $G$, where $\Gamma$ is a discrete subgroup of $G$, in the case when $\pi$ has a highest weight.
Topological properties of Souslin subsets
R. W.
Hansell
613-622
Abstract: Let $X$ be a subparacompact regular space such that the projection map $p:X \times {\mathbf{P}} \to X$, where ${\mathbf{P}}$ is the space of irrational numbers, preserves collections of sets having a $\sigma $-locally finite refinement. It is shown that $p$ then preserves generalized ${F_\sigma }$-sets. It follows that, if $ X$ has any tpological property which is hereditary with respect to generalized ${F_\sigma }$-sets, then every Souslin subset of $ X$ will also have this property in the relative topology. Such topological properties include nearly all covering properties (paracompactness, metacompactness, etc.), as well as normality, collectionwise normality, and the Lindelöf property. We show that the above mapping property will hold whenever $X$ is a $ P(\omega )$-space, thus, in particular, when $X$ is any Souslin (hence any Baire) subset of a compact space crossed with a metrizable space. Additional topological properties of Souslin subsets, such as topological completeness, realcompactness, $ z$-embeddedness, and the properties of being a $P$-space or $\Sigma$-space, are also considered.
A Martin boundary in the plane
Thomas S.
Salisbury
623-642
Abstract: Let $E$ be an open connected subset of Euclidean space, with a Green function, and let $ \lambda$ be harmonic measure on the Martin boundary $\Delta$ of $E$. We will show that, except for a $\lambda \otimes \lambda $-null set of $(x,y) \in {\Delta ^2}$, $x$ is an entrance point for Brownian motion conditioned to leave $E$ at $y$. R. S. Martin gave examples in dimension $ 3$ or higher, for which there exist minimal accessible Martin boundary points $ x \ne y$ for which this condition fails. We will give a similar example in dimension $2$.
Entire functions which are infinitely integer-valued at a finite number of points
P.
Lockhart;
E. G.
Straus
643-654
Abstract: This paper determines arithmetic limits for the growth rates of entire functions which are infinitely integer valued on a finite set $S$. The characterization of such functions with growth rate less than the arithmetic limit is complete if there exist exponential polynomials which are infinitely integer valued on $S$.
Simple homotopy type of finite $2$-complexes with finite abelian fundamental group
M. Paul
Latiolais
655-662
Abstract: Theorem 1. Let $K$ be a finite $2$-dimensional $CW$-complex with $ {\pi _1}(K)$ finite and abelian. Then every element of the Whitehead group of $K$ is realizable as the torsion of a self-homotopy equivalence on $K$. Theorem 2. Homotopy equivalence and simple homotopy equivalence are the same for finite $ 2$-dimensional $ CW$-complexes with finite abelian fundamental groups.
Almost Mizohata operators
Nicholas
Hanges
663-675
Abstract: The range of every almost Mizohata operator is characterized via an integral formula.
The Stefan problem with heating: appearance and disappearance of a mushy region
M.
Bertsch;
P.
de Mottoni;
L. A.
Peletier
677-691
Abstract: We consider a material which is initially in the solid state and then, due to heating, starts to melt. We describe the appearance of a so-called mushy region, i.e., a region in which the material is in neither a solid nor a liquid state. The main result is that after a finite time the mushy region has disappeared and only the solid and the liquid phases have remained.
A new construction of noncrossed product algebras
Bill
Jacob;
Adrian R.
Wadsworth
693-721
Abstract: New examples of noncrossed product division algebras are obtained, using methods different from all previous noncrossed product constructions. The examples are division algebras over intersections of $p$-Henselian valued fields, and they have Schur index ${p^m}$ and exponent ${p^n}$ for any prime number $p$ and any integers $m \geqslant n \geqslant 2\;(n \geqslant 3\;{\text{if}}\;p = 2)$. The basic tools used in the construction are valuation theory and Galois cohomology; no generic methods are applied and there is no p.i. theory. Along the way, local-global principles are proved for central simple algebras over intersections of $ p$-Henselian valued fields.
Potentials producing maximally sharp resonances
Evans M.
Harrell;
Roman
Svirsky
723-736
Abstract: We consider quantum-mechanical potentials consisting of a fixed background plus an additional piece constrained only by having finite height and being supported in a given finite region in dimension $d \leqslant 3$. We characterize the potentials in this class that produce the sharpest resonances. In the one-dimensional or spherically symmetric specialization, a quite detailed description is possible. The maximally sharp resonances that we find are, roughly speaking, caused by barrier confinement of a metastable state, although in some situations they call for interactions in the interior of the confining barrier as well.
Equilibrium points of nonatomic games over a Banach space
M. Ali
Khan
737-749
Abstract: Schmeidler's results on the equilibrium points of nonatomic games with strategy sets in Euclidean $n$-space are generalized to nonatomic games with strategy sets in a Banach space. Our results also extend previous work of the author which assumed the Banach space to be separable and its dual to possess the Radon-Nikodým property. Our proofs use recent results in functional analysis.
Principal lines on surfaces immersed with constant mean curvature
C.
Gutiérrez;
J.
Sotomayor
751-766
Abstract: Configurations of lines of principal curvature on constant mean curvature immersed surfaces are studied. Analytical models for these configurations near general isolated umbilical points and particular types of ends are found. From the existence of transversal invariant measures for the foliations by principal lines, established here, follows that the union of recurrent lines of principal curvature is an open set. Examples illustrating all possible cases are given.
Weak limits of stopped diffusions
J. R.
Baxter;
R. V.
Chacon;
N. C.
Jain
767-792
Abstract: We consider the following homogenization problem: Let Brownian motion in $ {{\mathbf{R}}^d}$, $d \geqslant 3$, be killed on the surface of many small absorbing bodies (standard diffusion equation with Dirichlet boundary conditions). We investigate the limit as the number of bodies approaches infinity and the size of the bodies approaches 0. By taking a weak limit of stopping times we replace a convergence problem on the state space by an identification of the limit on the sample space. This technique then gives results without smoothness assumptions which were previously necessary.
On the dichotomy problem for tensor algebras
J.
Bourgain
793-798
Abstract: Let $I$, $J$ be discrete spaces and $E \subset I \times J$. Then either $E$ is a $V$-Sidon set (in the sense of $[{\mathbf{2}},\S11]$), or the restriction algebra $ A(E)$ is analytic. The proof is based on probabilistic methods, involving Slépian's lemma.
Maximal functions associated with curves and the Calder\'on-Zygmund method of rotations
Shuichi
Sato
799-806
Abstract: Let ${\delta _t}(t > 0)$ be a dilation in $ {{\mathbf{R}}^n}(n \geqslant 2)$ defined by $\displaystyle {\delta _t}x = ({t^{{\alpha _1}}}{x_1},{t^{{\alpha _2}}}{x_2}, \ldots ,{t^{{\alpha _n}}}{x_n})\qquad (x = ({x_1},{x_2}, \ldots ,{x_n})),$ where $ {\alpha _i} > 0(i = 1,2, \ldots ,n)$ and ${\alpha _i} \ne {\alpha _j}$ if $i \ne j$. For $\nu \in {{\mathbf{R}}^n}$ with $\vert\nu \vert = 1$, let $ {\Gamma _\nu }:(0,\infty ) \to {{\mathbf{R}}^n}$ be a curve defined by ${\Gamma _\nu }(t) = {\delta _t}\nu \,(0 < t < \infty )$. Using maximal functions associated with the curves $ {\Gamma _\nu }$, we define an operator $M$ which is a nonisotropic analogue of the one studied in R. Fefferman [2]. It is proved that $M$ is a bounded operator on ${L^p}({{\mathbf{R}}^n})$ for some $p$ with $1 < p < 2$. As its application we prove the $ {L^p}$ boundedness of operators of the form ${T^{\ast}}(f)(x) = {\sup _{\varepsilon > 0}}\vert{T_\varepsilon }(f)(x)\vert$, where ${T_\varepsilon }$ is an integral operator associated with a variable kernel with mixed homogeneity.
On multiplication algebras
David R.
Finston
807-818
Abstract: The basic properties of multiplication algebras of nonassociative algebras over rings are introduced, including a discussion of multiplication algebras of tensor products of algebras. A characterization of semisimple artinian multiplication algebras is given along with a discussion of the nature of the simple factors of a multiplication algebra modulo its Jacobson radical. A criterion distinguishing the multiplication algebras of certain associative algebras is proved. Examples are included to illustrate certain proved results.
The distribution of solutions to equations over finite fields
Todd
Cochrane
819-826
Abstract: Let ${\mathbb{F}_q}$ be the finite field in $q = {p^f}$ elements, $\underline F (\underline x )$ be a $ k$-tuple of polynomials in ${\mathbb{F}_q}[{x_1}, \ldots ,{x_n}]$, $ V$ be the set of points in $ \mathbb{F}_q^n$ satisfying $\underline F (\underline x ) = \underline 0$ and $ S$, $T$ be any subsets of $\mathbb{F}_q^n$. Set $\phi (V,\underline 0 ) = \vert V\vert - {q^{n - k}}$, $\displaystyle \phi (V,\underline y ) = \sum\limits_{\underline x \in V} {e\left... ...ot \underline y )} \right)\quad {\text{for}}\;\underline y \ne \underline 0 ,}$ and $\Phi (V) = {\max _{\underline y }}\vert\phi (V,\underline y )\vert$. We use finite Fourier series to show that $ (S + T) \cap V$ is nonempty if $ \vert S\vert\vert T\vert > {\Phi ^2}(V){q^{2k}}$. In case $q = p$ we deduce from this, for example, that if $C$ is a convex subset of ${\mathbb{R}^n}$ symmetric about a point in $ {\mathbb{Z}^n}$, of diameter $< 2p$ (with respect to the sup norm), and $\operatorname{Vol} (C) > {2^{2n}}\Phi (V){p^k}$, then $C$ contains a solution of $ \underline F (\underline x ) \equiv \underline 0 (\bmod p)$. We also show that if $B$ is a box of points in $\mathbb{F}_q^n$ not contained in any $ (n - 1)$-dimensional subspace and $\vert B\vert > 4 \cdot {2^{nf}}\Phi (V){q^k}$, then $B \cap V$ contains $n$ linearly independent points.
Some results concerning the boundary zero sets of general analytic functions
Robert D.
Berman
827-836
Abstract: Two results concerning the boundary zero sets of analytic functions on the unit disk $\Delta$ are proved. First we consider nonconstant analytic functions $f$ on $\Delta$ for which the radial limit function ${f^{\ast}}$ is defined at each point of the unit circumference $C$. We show that a subset $E$ of $C$ is the zero set of $ {f^{\ast}}$ for some such function $f$ if and only if it is a ${\mathcal{G}_\delta }$ that is not metrically dense in any open arc of $C$. We then give a precise version of an asymptotic radial uniqueness theorem and its converse. The constructions given in the proofs of each of these theorems employ an approximation theorem of Arakeljan.
Growth rates and critical exponents of classes of binary combinatorial geometries
Joseph P. S.
Kung
837-859
Abstract: We prove that a binary geometry of rank $n\;(n \geqslant 2)$ not containing $M({K_5})$ and ${F_7}$ (respectively, $M({K_5})$ and ${C_{10}}$) as a minor has at most $3n - 3$ (respectively, $ 4n - 5$) points. Here, $ M({K_5})$ is the cycle geometry of the complete graph on five vertices, $ {F_7}$ the Fano plane, and $ {C_{10}}$ a certain rank $ 4$ ten-point geometry containing the dual Fano plane $ F_7^{\ast}$ as a minor. Our technique is elementary and uses the notion of a bond graph. From these results, we deduce upper bounds on the critical exponents of these geometries.