A composite coincidence degree with applications to boundary value problems of neutral equations
L. H.
Erbe;
W.
Krawcewicz;
J. H.
Wu
459-478
Abstract: We present a topological degree theory for the nonlinear problem $L(I - B)(x) = G(x)$ with applications to a class of boundary value problems of neutral equations, where $ L$ is an unbounded Fredholm operator of index zero, $B$ is condensing and $G$ is $L$-compact.
$7$-dimensional nilpotent Lie algebras
Craig
Seeley
479-496
Abstract: All $ 7$-dimensional nilpotent Lie algebras over $ \mathbb{C}$ are determined by elementary methods. A multiplication table is given for each isomorphism class. Distinguishing features are given, proving that the algebras are pairwise nonisomorphic. Moduli are given for the infinite families which are indexed by the value of a complex parameter.
Bass series of local ring homomorphisms of finite flat dimension
Luchezar L.
Avramov;
Hans-Bjørn
Foxby;
Jack
Lescot
497-523
Abstract: Nontrivial relations between Bass numbers of local commutative rings are established in case there exists a local homomorphism $\phi :R \to S$ which makes $S$ into an $R$-module of finite flat dimension. In particular, it is shown that an inequality $ \mu _R^{i + {\text{depth}}\;R} \leq \mu _s^{i + {\text{depth}}\;S}$ holds for all $i \in \mathbb{Z}$. This is a consequence of an equality involving the Bass series $I_R^M(t) = \sum\nolimits_{i \in \mathbb{Z}} {\mu _R^i(M){t^i}}$ of a complex $M$ of $R$-modules which has bounded above and finite type homology and the Bass series of the complex of $ S$-modules $ M{\underline{\underline \otimes } _R}S$, where $ \underline{\underline{\otimes}}$ denotes the derived tensor product. It is proved that there is an equality of formal Laurent series $ I_s^{M{{\underline{\underline \otimes } }_R}S}(t) = I_R^M(t){I_{F(\phi )}}(t)$, where $F(\phi)$ is the fiber of $\phi$ considered as a homomorphism of commutative differential graded rings. Coefficientwise inequalities are deduced for $ I_S^{M{{\underline{\underline \otimes } }_R}S}(t)$, and Golod homomorphisms are characterized by one of them becoming an equality.
Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings
Matej
Brešar
525-546
Abstract: Biadditive mappings $B: R \times R \to R$ where $ R$ is a prime ring with certain additional properties, satisfying $B(x,x)x = xB(x,x)$ for all $x \in R$, are characterized. As an application we determine the structures of commutativity-preserving mappings, Lie isomorphisms, and Lie derivations of certain prime rings.
The Martin kernel and infima of positive harmonic functions
Zoran
Vondraček
547-557
Abstract: Let $D$ be a bounded Lipschitz domain in $ {{\mathbf{R}}^n}$ and let $ K(x,z)$, $x \in D$, $z \in \partial D$, be the Martin kernel based at $ {x_0} \in D$. For $ x,y \in D$, let $ k(x,y) = \inf \{ h(x):h\;$positive$\;$harmonic$\;$in$\; D, h(y) = 1\}$. We show that the function $ k$ completely determines the family of positive harmonic functions on $ D$. Precisely, for every $z \in \partial D$, ${\lim _{y \to z}}k(x,y)/k({x_0},y) = K(x,z)$. The same result is true for second-order uniformly elliptic operators and Schrödinger operators.
On the Brauer group of toric varieties
Frank R.
DeMeyer;
Timothy J.
Ford
559-577
Abstract: We compute the cohomological Brauer group of a normal toric variety whose singular locus has codimension less than or equal to $2$ everywhere.
Rosenlicht fields
John
Shackell
579-595
Abstract: Let $\phi$ satisfy an algebraic differential equation over $ {\mathbf{R}}$. We show that if $\phi$ also belongs to a Hardy field, it possesses an asymptotic form which must be one of a restricted number of types. The types depend only on the order of the differential equation. For a particular equation the types are still more restricted. In some cases one can conclude that no solution of the given equation lies in a Hardy field, and in others that a particular asymptotic form is the only possibility for such solutions. This therefore gives a new method for obtaining asymptotic solutions of nonlinear differential equations. The techniques used are in part derived from the work of Rosenlicht in Hardy fields.
Excluding infinite trees
P. D.
Seymour;
Robin
Thomas
597-630
Abstract: For each infinite cardinal $\kappa$ we give several necessary and sufficient conditions for a graph not to contain a minor isomorphic to the infinite $\kappa$-branching tree in terms of a certain kind of a "tree-decomposition," in terms of a "path-decomposition," and also in terms of a "cops-and-robber game." We also give necessary and sufficient conditions for a graph not to contain a subgraph isomorphic to a subdivision of the same tree.
Spectral symmetry of the Dirac operator in the presence of a group action
H. D.
Fegan;
B.
Steer
631-647
Abstract: Let $G$ be a compact Lie group of rank two or greater which acts on a spin manifold $M$ of dimension $4k + 3$ through isometries with finite isotropy subgroups at each point. Define the Dirac operator, $P$, on $M$ relative to the split connection. Then we show that $P$ has spectral $G$-symmetry. This is first established in a number of special cases which are both of interest in their own right and necessary to establish the more general case. Finally we consider changing the connection and show that for the Levi-Civita connection the equivariant eta function evaluated at zero is constant on $G$.
Fractal dimensions and singularities of the Weierstrass type functions
Tian You
Hu;
Ka-Sing
Lau
649-665
Abstract: A new type of fractal measures $ {\mathcal{K}^s}$, $1 \leq s \leq 2$, defined on the subsets of the graph of a continuous function is introduced. The $\mathcal{K}$-dimension defined by this measure is 'closer' to the Hausdorff dimension than the other fractal dimensions in recent literatures. For the Weierstrass type functions defined by $W(x) = \sum\nolimits_0^\infty {{\lambda ^{ - \alpha i}}g({\lambda ^i}x)}$, where $\lambda > 1$, $0 < \alpha < 1$, and $g$ is an almost periodic Lipschitz function of order greater than $\alpha$, it is shown that the $\mathcal{K}$-dimension of the graph of $ W$ equals to $2 - \alpha$, this conclusion is also equivalent to certain rate of the local oscillation of the function. Some problems on the 'knot' points and the nondifferentiability of $W$ are also discussed.
Estimates for some Kakeya-type maximal operators
Jose
Barrionuevo
667-682
Abstract: We use an abstract version of a theorem of Kolmogorov-Seliverstov-Paley to obtain sharp ${L^2}$ estimates for maximal operators of the form: $\displaystyle {\mathcal{M}_\mathcal{B}}f(x) = \mathop {\sup }\limits_{x \in S \in \mathcal{B}} \frac{1}{{\vert S\vert}}\int_S {\vert f(x - y)\vert dy}$ . We consider the cases where $ \mathcal{B}$ is the class of all rectangles in $ {{\mathbf{R}}^n}$ congruent to some dilate of ${[0,1]^{n - 1}} \times [0,{N^{ - 1}}]$; the class congruent to dilates of ${[0,{N^{ - 1}}]^{n - 1}} \times [0,1]$ ; and, in $ {{\mathbf{R}}^2}$ , the class of all rectangles with longest side parallel to a particular countable set of directions that include the lacunary and the uniformly distributed cases.
The Mandelbrot set and $\sigma$-automorphisms of quotients of the shift
Pau
Atela
683-703
Abstract: In this paper we study how certain loops in the parameter space of quadratic complex polynomials give rise to shift-automorphisms of quotients of the set $ {\Sigma _2}$ of sequences on two symbols. The Mandelbrot set ${\mathbf{M}}$ is the set of parameter values for which the Julia set of the corresponding polynomial is connected. Blanchard, Devaney, and Keen have shown that closed loops in the complement of the Mandelbrot set give rise to shift-automorphisms of ${\Sigma _2}$ , i.e., homeomorphisms of ${\Sigma _2}$ that commute with the shift map. We study what happens when the loops are not entirely in the complement of the Mandelbrot set. We consider closed loops that cross the Mandelbrot set at a single main bifurcation point, surrounding a component of ${\mathbf{M}}$ attached to the main cardioid. If $ n$ is the period of this component, we identify a period- $n$ orbit of $ {\Sigma _2}$ to a single point. The loop determines a shift-automorphism of this quotient space of $ {\Sigma _2}$ . We give these maps explicitly.
Piecewise ${\rm SL}\sb 2{\bf Z}$ geometry
Peter
Greenberg
705-720
Abstract: Piecewise $ {\text{SL}}_2{\mathbf{Z}}$ geometry studies properties of the plane invariant under $ {\text{pl}}$-homeomorphisms which, locally, have the form $x \mapsto Ax + b$ , with $A \in {\text{SL}}_2{\mathbf{Z}}$, $b \in {{\mathbf{Q}}^2}$ , and whose singular lines are rational. In this paper, invariants of polygons are obtained, relations with Pick's theorem are described, and a conjecture is posed.
Harmonic calculus on p.c.f. self-similar sets
Jun
Kigami
721-755
Abstract: The main object of this paper is the Laplace operator on a class of fractals. First, we establish the concept of the renormalization of difference operators on post critically finite (p.c.f. for short) self-similar sets, which are large enough to include finitely ramified self-similar sets, and extend the results for Sierpinski gasket given in [10] to this class. Under each invariant operator for renormalization, the Laplace operator, Green function, Dirichlet form, and Neumann derivatives are explicitly constructed as the natural limits of those on finite pre-self-similar sets which approximate the p.c.f. self-similar sets. Also harmonic functions are shown to be finite dimensional, and they are characterized by the solution of an infinite system of finite difference equations.
Modular forms of weight $\frac12$ defined on products of $p$-adic upper half-planes
Anne
Schwartz
757-773
Abstract: We continue Stark's study of modular forms defined on products of $ p$-adic upper half-planes. Specifically, we restrict to the case of the number field $\mathbb{Q}$ and one finite prime. In this setting we develop a multiplier system for modular forms of weight $ \frac{1} {2}$, and provide an example of such a form.
Pseudocompact and countably compact abelian groups: Cartesian products and minimality
Dikran N.
Dikranjan;
Dmitrii B.
Shakhmatov
775-790
Abstract: Denote by $\mathcal{G}$ the class of all Abelian Hausdorff topological groups. A group $G \in \mathcal{G}$ is minimal (totally minimal) if every continuous group isomorphism (homomorphism) $i:G \to H$ of $G$ onto $H \in \mathcal{G}$ is open. For $G \in \mathcal{G}$ let $\kappa (G)$ be the smallest cardinal $\tau \geq 1$ such that the minimality of $ {G^\tau }$ implies the minimality of all powers of $G$. For $\mathcal{Q} \subset \mathcal{G}$, $\mathcal{Q} \ne \emptyset$, we set $\kappa (\mathcal{Q}) = \sup \{ \kappa (G):G \in \mathcal{G}\}$ and denote by $\alpha (\mathcal{Q})$ the smallest cardinal $\tau \geq 1$ having the following property: If $ \{ {G_i}:i \in I\} \subset \mathcal{Q}$, $ I \ne \emptyset$, and each subproduct $\prod {\{ {G_i}:i \in J\} }$, with $J \subset I$, $ J \ne \emptyset$, and $\vert J\vert \leq \tau $, is minimal, then the whole product $ \prod {\{ {G_i}:i \in I\} }$ is minimal. These definitions are correct, and $ \kappa (G) \leq {2^\omega }$ and $\kappa (\mathcal{Q}) \leq \alpha (\mathcal{Q}) \leq {2^\omega }$ for all $G \in \mathcal{G}$ and any $\mathcal{Q} \subset \mathcal{G}$, $\mathcal{Q} \ne \emptyset$, while it can happen that $\kappa (\mathcal{Q}) < \alpha (\mathcal{Q})$ for some $ \mathcal{Q} \subset \mathcal{G}$. Let $\mathcal{C} = \{ G \in \mathcal{G}:G\;{\text{is}}\;{\text{countably}}\;{\text{compact}}\} $ and $\mathcal{P} = \{ G \in \mathcal{G}:G\;{\text{is}}\;{\text{pseudocompact}}\} $. If $G \in \mathcal{C}$ is minimal, then $G \times H$ is minimal for each minimal (not necessarily Abelian) group $H$; in particular, ${G^n}$ is minimal for every natural number $ n$. We show that $\alpha (\mathcal{C}) = \omega$, and so either $\kappa (\mathcal{C}) = 1$ or $\kappa (\mathcal{C}) = \omega$. Under Lusin's Hypothesis ${2^{{\omega _1}}} = {2^\omega }$ we construct $\{ {G_n}:n \in \mathbb{N}\} \subset \mathcal{P}$ and $ H \in \mathcal{P}$ such that: (i) whenever $ n \in \mathbb{N}$, $ G_n^n$ is totally minimal, but $G_n^{n + 1}$ is not even minimal, so $\kappa ({G_n}) = n + 1$; and (ii) $ {H^n}$ is totally minimal for each natural number $n$, but $ {H^\omega}$ is not even minimal, so $ \kappa (H) = \omega$. Under $ {\text{MA}} + \neg {\text{CH}}$, conjunction of Martin's Axiom with the negation of the Continuum Hypothesis, we construct $G \in \mathcal{P}$ such that $ {G^\tau }$ is totally minimal for each $ \tau < {2^\omega }$, while ${G^{{2^\omega }}}$ is not minimal, so $ \kappa (G) = {2^\omega }$. This yields $\alpha (\mathcal{P}) = \kappa (\mathcal{P}) = {2^\omega }$ under ${\text{MA}} + \neg {\text{CH}}$. We also present an example of a noncompact minimal group $G \in \mathcal{C}$, which should be compared with the following result obtained by the authors quite recently: Totally minimal groups $G \in \mathcal{C}$ are compact.
The Lagrangian Gauss image of a surface in Euclidean $3$-space
Marek
Kossowski
791-803
Abstract: We describe a correspondence between special nonimmersed surfaces in Euclidean $3$-space and exact Lagrangian immersions in the cotangent bundle of the unit sphere. This provides several invariants for such nonimmersive maps: the degree of the Gauss map, the Gauss-Maslov class, and the polarization index. The objectives of this paper are to compare these invariants in the cases where the underlying map immerses or fails to immerse and to describe the extend to which these invariants can be prescribed.
Weak topologies for the closed subsets of a metrizable space
Gerald
Beer;
Roberto
Lucchetti
805-822
Abstract: The purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space. It can be shown that all of the standard hyperspace topologies--including the Hausdorff metric topology, the Vietoris topology, the Attouch-Wets topology, the Fell topology, the locally finite topology, and the topology of Mosco convergence--arise as weak topologies generated by families of geometric functionals defined on closed sets. A key ingredient is the simple yet beautiful interplay between topologies determined by families of gap functionals and those determined by families of Hausdorff excess functionals.
The geometric structure of skew lattices
Jonathan
Leech
823-842
Abstract: A skew lattice is a noncommutative associative analogue of a lattice and as such may be viewed both as an algebraic object and as a geometric object. Whereas recent papers on skew lattices primarily treated algebraic aspects of skew lattices, this article investigates their intrinsic geometry. This geometry is obtained by considering how the coset geometries of the maximal primitive subalgebras combine to form a global geometry on the skew lattice. While this geometry is derived from the algebraic operations, it can be given a description that is independent of these operations, but which in turn induces them. Various aspects of this geometry are investigated including: its general properties; algebraic and numerical consequences of these properties; connectedness; the geometry of skew lattices in rings; connections between primitive skew lattices and completely simple semigroups; and finally, this geometry is used to help classify symmetric skew lattices on two generators.
Besov spaces on domains in ${\bf R}\sp d$
Ronald A.
DeVore;
Robert C.
Sharpley
843-864
Abstract: We study Besov spaces $ B_q^\alpha ({L_p}(\Omega ))$, $0 < p,q,\alpha < \infty$, on domains $\Omega$ in $ {\mathbb{R}^d}$ . We show that there is an extension operator $\mathcal{E}$ which is a bounded mapping from $B_q^\alpha ({L_p}(\Omega ))$ onto $B_q^\alpha ({L_p}({\mathbb{R}^d}))$. This is then used to derive various properties of the Besov spaces such as interpolation theorems for a pair of $B_q^\alpha ({L_p}(\Omega ))$, atomic decompositions for the elements of $B_q^\alpha ({L_p}(\Omega ))$, and a description of the Besov spaces by means of spline approximation.
The limit sets of some infinitely generated Schottky groups
Richard
Schwartz
865-875
Abstract: Let $P$ be a packing of balls in Euclidean space ${E^n}$ having the property that the radius of every ball of $P$ lies in the interval $[1/k,k]$. If $G$ is a Schottky group associated to $P$, then the Hausdorff dimension of the topological limit set of $G$ is less than a uniform constant $C(k,n) < n$. In particular, this limit set has zero volume.
Support theorems for Radon transforms on real analytic line complexes in three-space
Jan
Boman;
Eric Todd
Quinto
877-890
Abstract: In this article we prove support theorems for Radon transforms with arbitrary nonzero real analytic measures on line complexes (three-dimensional sets of lines) in ${\mathbb{R}^3}$. Let $f$ be a distribution of compact support on ${\mathbb{R}^3}$. Assume $Y$ is a real analytic admissible line complex and ${Y_0}$ is an open connected subset of $ Y$ with one line in $ {Y_0}$ disjoint from supp$\;f$. Under weak geometric assumptions, if the Radon transform of $f$ is zero for all lines in ${Y_0}$, then supp$ \;f$ intersects no line in $ {Y_0}$. These theorems are more general than previous results, even for the classical transform. We also prove a support theorem for the Radon transform on a nonadmissible line complex. Our proofs use analytic microlocal analysis and information about the analytic wave front set of a distribution at the boundary of its support.
The Weil-Petersson symplectic structure at Thurston's boundary
A.
Papadopoulos;
R. C.
Penner
891-904
Abstract: The Weil-Petersson Kähler structure on the Teichmüller space $\mathcal{T}$ of a punctured surface is shown to extend, in an appropriate sense, to Thurston's symplectic structure on the space $ \mathcal{M}{\mathcal{F}_0}$ of measured foliations of compact support on the surface. We introduce a space $ {\widetilde{\mathcal{M}\mathcal{F}}_0}$ of decorated measured foliations whose relationship to $ \mathcal{M}{\mathcal{F}_0}$ is analogous to the relationship between the decorated Teichmüller space $\tilde{\mathcal{T}}$ and $ \mathcal{T}$. $ \widetilde{\mathcal{M}{\mathcal{F}_0}}$ is parametrized by a vector space, and there is a natural piecewise-linear embedding of $ \mathcal{M}{\mathcal{F}_0}$ in $ \widetilde{\mathcal{M}{\mathcal{F}_0}}$ which pulls back a global differential form to Thurston's symplectic form. We exhibit a homeomorphism between $ \tilde{\mathcal{T}}$ and $ {\widetilde{\mathcal{M}\mathcal{F}}_0}$ which preserves the natural two-forms on these spaces. Following Thurston, we finally consider the space $ \mathcal{Y}$ of all suitable classes of metrics of constant Gaussian curvature on the surface, form a natural completion $ \overline{\mathcal{Y}}$ of $\mathcal{Y}$, and identify $ \overline{\mathcal{Y}} - \mathcal{Y}$ with $ \mathcal{M}{\mathcal{F}_0}$. An extension of the Weil-Petersson Kähler form to $ \mathcal{Y}$ is found to extend continuously by Thurston's symplectic pairing on $ \mathcal{M}{\mathcal{F}_0}$ to a two-form on $ \overline{\mathcal{Y}}$ itself.