Reflection, removable singularities, and approximation for partial differential equations. II
Leon
Ehrenpreis
1-45
Abstract: Let ${\Omega ^j}$ be domains in ${R^n}$. For each $j$ we are given a system ${{\mathbf{D}}^j}$ of linear constant coefficient operators and a function ${f^j}$ on $ {\Omega ^j}$ satisfying ${{\mathbf{D}}^j}{f^j} = 0$. When the $ {f^j}$ satisfy certain compatibility conditions on the intersections ${{\mathbf{D}}^j}$ on larger domains. As a consequence of our methods we are able to sharpen Hartogs' theorems to allow for continuation of solutions of overdetermined systems over noncompact sets.
On the invariance of $q$-convexity and hyperconvexity under finite holomorphic surjections
Nguyẽn Văn
Khuê;
Lê Văn
Thành
47-54
Abstract: In this note we have proved that 0-convexity and hyperconvexity are invariant under finite holomorphic surjections. Invariance of cohomological $q$-convexity for the case of finite dimension also has been established.
The $L\sp 2$-boundedness of pseudodifferential operators
I. L.
Hwang
55-76
Abstract: We give a new proof of the Calderon-Vaillancourt theorem. We also obtain the ${L^2}$-continuity of $a(x,D)$ if its symbol $a(x,\xi )$ satisfies some suitable conditions.
Variations on Lusin's theorem
Jack B.
Brown;
Karel
Prikry
77-86
Abstract: We prove a theorem about continuous restrictions of Marczewski measurable functions to large sets. This theorem is closely related to the theorem of Lusin about continuous restrictions of Lebesgue measurable functions to sets of positive measure and the theorem of Nikodym and Kuratowski about continuous restrictions of functions with the Baire property (in the wide sense) to residual sets. This theorem is used to establish Lusin-type theorems for universally measurable functions and functions which have the Baire property in the restricted sense. The theorems are shown (under assumption of the Continuum Hypothesis) to be "best possible" within a certain context.
Almost split sequences for rational double points
Maurice
Auslander;
Idun
Reiten
87-97
Abstract: Let $R$ be a complete local two-dimensional integrally closed noetherian nonregular Gorenstein domain with maximal ideal $m$ over an algebraically closed field $ k$, such that $R/m \simeq k$. Then the category of finitely generated reflexive modules is known to have almost split sequences, and our main result is that the $ AR$-quiver is an extended Dynkin quiver $ {\tilde A_n}$, ${\tilde D_n}$, $ {\tilde E_6}$, ${\tilde E_7}$, or $ {\tilde E_8}$.
Half-canonical series on algebraic curves
Montserrat
Teixidor i Bigas
99-115
Abstract: Denote by $\mathcal{M}_g^r$ the locus in the moduli space of curves of genus $g$ of those curves which have a theta-characteristic of (projective) dimension at least $r$. We give an upper bound for the dimension of $ \mathcal{M}_g^r$ and we determine this dimension completely for $r \leqslant 4$. For $r \leqslant 4$, we prove also that a generic point in every component of $\mathcal{M}_g^r$ has a single theta-characteristic of this dimension.
On certain $3$-generator Artin groups
Craig C.
Squier
117-124
Abstract: We describe the three $3$-generator Artin groups that correspond to the three sets $\{ p,q,r\}$ of positive integer solutions of ${p^{ - 1}} + {q^{ - 1}} + {r^{ - 1}} = 1$. In each case, we show that the Artin group is a free product with amalgamation or HNN extension involving finitely generated free groups and subgroups of finite index.
Periodic points and automorphisms of the shift
Mike
Boyle;
Wolfgang
Krieger
125-149
Abstract: The automorphism group of a topological Markov shift is studied by way of periodic points and unstable sets. A new invariant for automorphisms of dynamical systems, the gyration function, is used to characterize those automorphisms of finite subsystems of the full shift on $n$ symbols which can be extended to a composition of involutions of the shift. It is found that for any automorphism $U$ of a subshift of finite type $S$, for all large integers $ M$ the map $U{S^M}$ is a topological Markov shift whose unstable sets equal those of $S$. This fact yields, by way of canonical measures and dimension groups, information about dynamical properties of $U{S^k}$ such as the zeta function and entropy.
Complete groups with nonabelian composition factors
Jay
Zimmerman
151-159
Abstract: A finite group is said to be complete if it has trivial center and if every automorphism is an inner automorphism. A finite group with nonabelian composition factors has a unique completely reducible radical (CR radical). We consider finite groups with nonabelian composition factors whose CR radical consists of complete simple groups and we give necessary and sufficient conditions for such a group to be complete. This involves finding group theoretic conditions which are necessary and sufficient for a finite centerless group to occur as a self-normalizing subgroup of a direct product of symmetric groups.
Interpolating sequences in the polydisc
Bo
Berndtsson;
Sun-Yung A.
Chang;
Kai-Ching
Lin
161-169
Abstract: Let ${H^\infty }({D^n})$ denote the set of all bounded analytic functions defined on the polydisc $ {D^n}$ of ${{\mathbf{C}}^n}$. In this note, we give a sufficient condition for sequences of points in ${D^n}$ to be interpolating sequences for ${H^\infty }({D^n})$. We also discuss some conditions for interpolation of general domains.
Special points on first order partial differential equations and the deformations of solutions
Marek
Kossowski
171-184
Abstract: The object of this paper is to identify four cases of special behavior in a class of first order PDE for a real valued function. (The class of PDE may be thought of as perturbation of PDE with singular solutions.) In each case we show how invariants of the PDE determine properties of solutions. The properties of solutions examined here are the structure of critical points and singularities induced by cotangent projection. These properties are described in the sense of constructing local models for solutions and characterizing their behavior under small deformations. We will find two cases where deformations exhibit bifurcation phenomena, and describe generic deformations.
Martin boundaries of random walks: ends of trees and groups
Massimo A.
Picardello;
Wolfgang
Woess
185-205
Abstract: Consider a transient random walk ${X_n}$ on an infinite tree $T$ whose nonzero transition probabilities are bounded below. Suppose that ${X_n}$ is uniformly irreducible and has bounded step-length. (Alternatively, $ {X_n}$ can be regarded as a random walk on a graph whose metric is equivalent to the metric of $T$.) The Martin boundary of ${X_n}$ is shown to coincide with the space $ \Omega$ of all ends of $ T$ (or, equivalently, of the graph). This yields a boundary representation theorem on $\Omega$ for all positive eigenfunctions of the transition operator, and a nontangential Fatou theorem which describes their boundary behavior. These results apply, in particular, to finitely supported random walks on groups whose Cayley graphs admit a uniformly spanning tree. A class of groups of this type is constructed.
The set of continuous functions with everywhere convergent Fourier series
M.
Ajtai;
A. S.
Kechris
207-221
Abstract: This paper deals with the descriptive set theoretic properties of the class $ \operatorname{EC}$ of continuous functions with everywhere convergent Fourier series. It is shown that this set is a complete coanalytic set in $C(T)$. A natural coanalytic rank function on $ \operatorname{EC}$ is studied that assigns to each $f \in \operatorname{EC}$ a countable ordinal number, which measures the "complexity" of the convergence of the Fourier series of $f$. It is shown that there exist functions in $ \operatorname{EC}$ (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on $ \operatorname{EC}$ with ${\omega _1}$ distinct levels.
Totally ramified valuations on finite-dimensional division algebras
J.-P.
Tignol;
A. R.
Wadsworth
223-250
Abstract: Division algebras $ D$ with valuation $ v$ are studied, where $ D$ is finite-dimensional and totally ramified over its center $F$ (i.e., the ramification index of $ v$ over $v{\vert _F}$ equals $[D:F]$). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When $v{\vert _F}$ is Henselian a full classification of the $F$-subalgebras of $D$ is given. When $F$ has a Henselian valuation $v$ with separably closed residue field and $A$ is any tame central simple $F$-algebra, an algorithm is given for computing the underlying division algebra of $ A$ from a suitable subgroup of $ {A^{\ast}}/{F^{\ast}}$. Some examples are constructed using this valuation theory, including the first example of finite-dimensional $ F$-central division algebras ${D_1}$ and ${D_2}$ with ${D_1}{ \otimes _F}{D_2}$ not a division ring, but $ {D_1}$ and ${D_2}$ having no common subfield $K \supsetneqq F$.
On a problem concerning permutation polynomials
Gerhard
Turnwald
251-267
Abstract: Let $S(f)$ denote the set of integral ideals $ I$ such that $ f$ is a permutation polynomial modulo $I$, where $f$ is a polynomial over the ring of integers of an algebraic number field. We obtain a classification for the sets $S$ which may be written in the form $S(f)$.
On the distribution of the number of prime factors of sums $a+b$
P.
Erdős;
H.
Maier;
A.
Sárközy
269-280
Abstract: We continue a series of investigations by A. Balog and two of the authors (P. Erdös and A. Sárközy) on the arithmetic properties of the elements $a + b$, where $a \in {\mathbf{A}}$, $b \in {\mathbf{B}}$, $ {\mathbf{A}}$ and ${\mathbf{B}}$ "dense sequences." The present paper transfers the famous Erdös-Kac theorem on the normal distribution of the number of distinct prime factors of integers to such "sum sequences."
Small zeros of quadratic forms over number fields
Jeffrey D.
Vaaler
281-296
Abstract: Let $F$ be a nontrivial quadratic form in $ N$ variables with coefficients in a number field $k$ and let $A$ be a $K \times N$ matrix over $k$. We show that if the simultaneous equations $ F({\mathbf{x}}) = 0$ and $A{\mathbf{x}} = 0$ hold on a subspace $\mathfrak{X}$ of dimension $L$ and $L$ is maximal, then such a subspace $\mathfrak{X}$ can be found with the height of $\mathfrak{X}$ relatively small. In particular, the height of $ \mathfrak{X}$ can be explicitly bounded by an expression depending on the height of $F$ and the height of $A$. We use methods from geometry of numbers over adèle spaces and local to global techniques which generalize recent work of H. P. Schlickewei.
Approximation by equivariant homeomorphisms. I
Mark
Steinberger;
James
West
297-317
Abstract: Locally linear (= locally smoothable) actions of finite groups on finite dimensional manifolds are considered in which two incident components of fixed point sets of subgroups either coincide or one has codimension at least three in the other. For these actions, an equivariant $ \alpha$-approximation theorem is proved using engulfing techniques. As corollaries are obtained equivariant "fibrations are bundles" and "controlled $h$-cobordism" theorems, as well as an equivariant version of Edwards' cell-like mapping theorem and the vanishing of the set of transfer-invariant $ G$-homotopy topological structures, rel boundary, on ${T^n} \times {D_\rho }$ (when ${T^n}$ is the $n$-torus with trivial $G$ action and ${D_\rho }$ is a representation disc).
The Conner-Floyd map for formal $A$-modules
Keith
Johnson
319-332
Abstract: A generalization of the Conner-Floyd map from complex cobordism to complex $ K$-theory is constructed for formal $A$-modules when $A$ is the ring of algebraic integers in a number field or its $p$-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal $ A$-modules and to confirm a conjecture of D. Ravenel.
A residual property of certain linear groups
Peter F.
Stebe
333-340
Abstract: An extension of residual finiteness, residual finiteness with respect to nests, is demonstrated for certain subgroups of $ GL(n,Z)$, the polycyclic by finite groups. It is also shown that groups containing a free subgroup of rank greater than $1$ cannot have the property. It is not settled whether or not there are other solvable by finite groups, subgroups allowed by Tits' theorem, that are residually finite with respect to nests.
Splitting of closed ideals in $({\rm DFN})$-algebras of entire functions and the property $({\rm DN})$
Reinhold
Meise;
B. Alan
Taylor
341-370
Abstract: For a plurisubharmonic weight function $p$ on $ {{\mathbf{C}}^n}$ let ${A_p}({{\mathbf{C}}^n})$ denote the (DFN)-algebra of all entire functions on $ {{\mathbf{C}}^n}$ which do not grow faster than a power of $\exp (p)$. We prove that the splitting of many finitely generated closed ideals of a certain type in $ {A_p}({{\mathbf{C}}^n})$, the splitting of a weighted $\overline \partial $-complex related with $ p$, and the linear topological invariant (DN) of the strong dual of ${A_p}({{\mathbf{C}}^n})$ are equivalent. Moreover, we show that these equivalences can be characterized by convexity properties of $p$, phrased in terms of greatest plurisubharmonic minorants. For radial weight functions $ p$, this characterization reduces to a covexity property of the inverse of $ p$. Using these criteria, we present a wide range of examples of weights $ p$ for which the equivalences stated above hold and also where they fail.
The proximal normal formula in Banach space
J. M.
Borwein;
J. R.
Giles
371-381
Abstract: Approximation by proximal normals to the Clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis. We give simple general versions of such formulae in infinite dimensional Banach spaces which satisfy different geometrical conditions. Our first class, of spaces with uniformly Gâteaux differentiable norm includes the Hilbert space case and the formulae is attained through dense subsets. Our second class, of reflexive Kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space. Our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of Ekeland and Lebourg. Resume. L'approximation par les normales proximales au sous-différentiel généralisé de Clarke pour une fonction de distance produit d'un ensemble non-vide fermé et le cône normal à l'ensemble produit des normales proximales sont objets d'importance pour l'analyse non-régulière. Nous donnons deux versions simples et générales de telles formules dans les espaces de Banach de dimensions infinies. Premièrement, nous examinons la classe des espaces avec norme uniformément Gâteaux-dérivable qui comprend les espaces de Hilbert. Deuxièmement, nous examinons la classe espaces réflexifs Kadec et lisses. Enfin, notre méthode produit l'existence des solutions pour une classe de problèmes à l'optimisation.