Above and below subgroups of a lattice-ordered group
Richard N.
Ball;
Paul
Conrad;
Michael
Darnel
1-40
Abstract: In an $ l$-group $G$, this paper defines an $l$-subgroup $A$ to be above an $l$-subgroup $B$ (or $B$ to be below $A$) if for every integer $n$, $a \in A$, and $b \in B$, $n(\vert a\vert \wedge \vert b\vert) \leqslant \vert a\vert$. It is shown that for every $ l$-subgroup $A$, there exists an $l$-subgroup $B$ maximal below $A$ which is closed, convex, and, if the $ l$-group $G$ is normal-valued, unique, and that for every $l$-subgroup $B$ there exists an $l$-subgroup $A$ maximal above $B$ which is saturated: if $0 = x \wedge y$ and $ x + y \in A$, then $ x \in A$. Given $ l$-groups $A$ and $B$, it is shown that every lattice ordering of the splitting extension $ G = A \times B$, which extends the lattice orders of $A$ and $B$ and makes $A$ lie above $B$, is uniquely determined by a lattice homomorphism $\pi$ from the lattice of principal convex $ l$-subgroups of $ A$ into the cardinal summands of $B$. This extension is sufficiently general to encompass the cardinal sum of two $l$-groups, the lex extension of an $ l$-group by an $ o$-group, and the permutation wreath product of two $l$-groups. Finally, a characterization is given of those abelian $l$-groups $G$ that split off below: whenever $ G$ is a convex $ l$-subgroup of an $ l$-group $H$, $H$ is then a splitting extension of $G$ by $A$ for any $l$-subgroup $A$ maximal above $G$ in $H$.
Equivariant Morse theory for flows and an application to the $N$-body problem
Filomena
Pacella
41-52
Abstract: In this paper, using Conley's index and equivariant cohomology, some Morse type inequalities are deduced for a flow equivariant with respect to the action of a compact topological group. In the case of a gradient flow induced by a nondegenerate smooth function these inequalities coincide with those described by R. Bott. The theory is applied to the study of the central configurations of $N$-bodies.
Weighted inequalities for the one-sided Hardy-Littlewood maximal functions
E.
Sawyer
53-61
Abstract: Let ${M^ + }f(x) = {\sup _{h > 0}}(1/h)\int_x^{x + h} {\vert f(t)\vert\,dt}$ denote the one-sided maximal function of Hardy and Littlewood. For $w(x) \geqslant 0$ on $R$ and $1 < p < \infty$, we show that $ {M^ + }$ is bounded on $ {L^p}(w)$ if and only if $ w$ satisfies the one-sided $ {A_p}$ condition: $\displaystyle \left( {A_p^ + } \right)\qquad \left[ {\frac{1} {h}\int_{a - h}^a... ...1} {h}\int_a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right]^{p - 1}} \leqslant C$ for all real $a$ and positive $h$. If in addition $v(x) \geqslant 0$ and $\sigma = {v^{ - 1/(p - 1)}}$,then ${M^ + }$ is bounded from ${L^p}(v)$ to ${L^p}(w)$ if and only if $\displaystyle \int_I {{{[{M^ + }({\chi _I}\sigma )]}^p}w \leqslant C\int_I {\sigma < \infty } }$ for all intervals $I = (a,b)$ such that $\int_{ - \infty }^a {w > 0}$. The corresponding weak type inequality is also characterized. Further properties of $A_p^ +$ weights, such as $A_p^ + \Rightarrow A_{p - \varepsilon }^ +$ and $A_p^ + = (A_1^ + ){(A_1^ - )^{1 - p}}$, are established.
\`A propos de ``wedges'' et d'``edges'', et de prolongements holomorphes
Jean-Pierre
Rosay
63-72
Abstract: Holomorphic extensions in wedges of continuous functions defined on edges, which are extensions in the distributional sense, are shown to be genuine continuous extensions, and a ${\mathcal{C}^1}$ version of the edge of the wedge theorem is proved.
Weighted nonlinear potential theory
David R.
Adams
73-94
Abstract: The potential theoretic idea of the "thinness of a set at a given point" is extended to the weighted nonlinear potential theoretic setting--the weights representing in general singularities/degeneracies--and conditions on these weights are given that guarantee when two such notions are equivalent at the given point. When applied to questions of boundary regularity for solutions to (degenerate) elliptic second-order partial differential equations in bounded domains, this result relates the boundary Wiener criterion for one operator to that of another, and in the linear case gives conditions for boundary regular points to be the same for various operators. The methods also yield two weight norm inequalities for Riesz potentials $\displaystyle {\left( {\int {{{({I_\alpha }{\ast}f)}^q}v\,dx} } \right)^{1/q}} \leqslant {\left( {\int {{f^p}w\,dx} } \right)^{1/p}},$ $1 < p \leqslant q < \infty$, which at least in the first-order case $(\alpha = 1)$ have found some use in a number of places in analysis.
Isometries on $L\sb {p,1}$
N. L.
Carothers;
B.
Turett
95-103
Abstract: The extreme points of the sphere of the Lorentz function space ${L_{p,1}}[0,1]$ are computed. As an application, the linear isometries from ${L_{p,1}}$ into itself are completely described.
Equivariant minimal immersions of $S\sp 2$ into $S\sp {2m}(1)$
Norio
Ejiri
105-124
Abstract: We classify the directrix curves associated with equivariant minimal immersions of ${S^2}$ into $ {S^{2m}}(1)$ and obtain some applications.
Unknotting information from $4$-manifolds
T. D.
Cochran;
W. B. R.
Lickorish
125-142
Abstract: Results of S. K. Donaldson, and others, concerning the intersection forms of smooth $4$-manifolds are used to give new information on the unknotting numbers of certain classical knots. This information is particularly sensitive to the signs of the knot crossings changed in an unknotting process.
Attracting orbits in Newton's method
Mike
Hurley
143-158
Abstract: It is well known that the dynamical system generated by Newton's Method applied to a real polynomial with all of its roots real has no periodic attractors other than the fixed points at the roots of the polynomial. This paper studies the effect on Newton's Method of roots of a polynomial "going complex". More generally, we consider Newton's Method for smooth real-valued functions of the form ${f_\mu }(x) = g(x) + \mu $, $\mu$ a parameter. If ${\mu _0}$ is a point of discontinuity of the map $\mu \to$ (the number of roots of ${f_\mu }$), then, in the presence of certain nondegeneracy conditions, we show that there are values of $\mu$ near ${\mu _0}$ for which the Newton function of $ {f_\mu }$ has nontrivial periodic attractors.
The axioms of supermanifolds and a new structure arising from them
Mitchell J.
Rothstein
159-180
Abstract: An analysis of supermanifolds over an arbitrary graded-commmutative algebra is given, proceeding from a set of axioms the first of which is that the derivations of the structure sheaf of a supermanifold are locally free. These axioms are satisfied not by the sheaf of $ {G^\infty }$ functions, as has been asserted elsewhere, but by an extension of this sheaf. A given $ {G^\infty }$ manifold may admit many supermanifold extensions, and it is unknown at present whether there are ${G^\infty }$ manifolds that admit no such extension. When the underlying graded-commutative algebra is commutative, the axioms reduce to the Berezin-Kostant supermanifold theory.
Absolute subretracts and weak injectives in congruence modular varieties
Brian A.
Davey;
L. G.
Kovács
181-196
Abstract: Absolute subretracts and weak injectives in congruence modular varieties of universal algebras are investigated by focusing attention on the directly indecomposibles. The proofs rely on a congruence modular version of generalized direct products (direct products with amalgamation) and on the generalized Jónsson Lemma for congruence modular varieties. The results have immediate application to varieties of groups or rings.
Eigenvalues below the essential spectra of singular elliptic operators
W. D.
Evans;
Roger T.
Lewis
197-222
Abstract: A new technique is developed for determining if the number of eigenvalues below the essential spectrum of a singular elliptic differential operator is finite. A method is given for establishing lower bounds for the least point of the essential spectrum in terms of the behavior of the coefficients and weight near the singularities. Higher-order operators are included in these results as well as second-order Schrödinger operators.
The blow-up boundary for nonlinear wave equations
Luis A.
Caffarelli;
Avner
Friedman
223-241
Abstract: Consider the Cauchy problem for a nonlinear wave equation $\square u = F(u)$ in $N$ space dimensions, $N \leqslant 3$, with $ F$ superlinear and nonnegative. It is well known that, in general, the solution blows up in finite time. In this paper it is shown, under some assumptions on the Cauchy data, that the blow-up set is a space-like surface $t = \phi (x)$ with $\phi (x)$ continuously differentiable.
A regularity theorem for minimizing hypersurfaces modulo $\nu$
Frank
Morgan
243-253
Abstract: It is proved that an $(n - 1)$-dimensional, area-minimizing flat chain modulo $\nu$ in $ {{\mathbf{R}}^n}$, with smooth extremal boundary of at most $\nu /2$ components, has an interior singular set of Hausdorff dimension at most $ n - 8$. Similar results hold for more general integrands.
Countable-dimensional universal sets
Roman
Pol
255-268
Abstract: The main results of this paper are a construction of a countable union of zero dimensional sets in the Hilbert cube whose complement does not contain any subset of finite dimension $n \geqslant 1$ (Theorem 2.1, Corollary 2.3) and a construction of universal sets for the transfinite extension of the Menger-Urysohn inductive dimension (Theorem 2.2, Corollary 2.4).
Chaotic functions with zero topological entropy
J.
Smítal
269-282
Abstract: Recently Li and Yorke introduced the notion of chaos for mappings from the class ${C^0}(I,I)$, where $I$ is a compact real interval. In the present paper we give a characterization of the class $M \subset {C^0}(I,I)$ of mappings chaotic in this sense. As is well known, $ M$ contains the mappings of positive topological entropy. We show that $ M$ contains also certain (but not all) mappings that have both zero topological entropy and infinite attractors. Moreover, we show that the complement of $M$ consists of maps that have only trajectories approximate by cycles. Finally, it turns out that the original Li and Yorke notion of chaos can be replaced by (an equivalent notion of) $\delta$-chaos, distinguishable on a certain level $\delta > 0$.
BMO rational approximation and one-dimensional Hausdorff content
Joan
Verdera
283-304
Abstract: Let $X \subset {\mathbf{C}}$ be compact and let $f \in \operatorname{VMO} ({\mathbf{C}})$. We give necessary and sufficient conditions on $f$ and $X$ for ${f_{\vert X}}$ to be the limit of a sequence of rational functions without poles on $X$ in the norm of $\operatorname{BMO} (X)$, the space of functions of bounded mean oscillation on $X$. We also characterize those compact $X \subset {\mathbf{C}}$ with the property that the restriction to $X$ of each function in $\operatorname{VMO} ({\mathbf{C}})$, which is holomorphic on $\mathop X\limits^ \circ $, is the limit of a sequence of rational functions with poles off $ X$. Our conditions involve the notion of one-dimensional Hausdorff content. As an application, a result related to the inner boundary conjecture is proven.
The Radon transform on ${\rm SL}(2,{\bf R})/{\rm SO}(2,{\bf R})$
D. I.
Wallace;
Ryuji
Yamaguchi
305-318
Abstract: Let $G$ be $SL(2,{\mathbf{R}})$. $G$ acts on the upper half-plane $\mathcal{H}$ by the Möbius transformation, providing $ \mathcal{H}$ with the Riemannian metric structure along with the Laplacian, $ \Delta$. We study the integral transform along each geodesic. $G$ acts on $ \mathcal{P}$, the space of all geodesics, in a natural way, providing $\mathcal{P}$ with its invariant measure and its own Laplacian. ( $ \mathcal{P}$ actually is a coset space of $G$.) Therefore the above transform can be viewed as a map from a suitable function space on $\mathcal{H}$ to a suitable function space on $\mathcal{P}$. We prove a number of properties of this transform, including the intertwining properties with its Laplacians and its relation to the Fourier transforms.
Analytic perturbation of the Taylor spectrum
Zbigniew
Slodkowski
319-336
Abstract: Let ${T_1}(z), \ldots ,{T_m}(z)$, $z \in G \subset {{\mathbf{C}}^k}$, be analytic families of bounded operators in a complex Banach space $X$, such that for each $z \in G$ the operators ${T_i}(z)$ and ${T_j}(z)$, $ i,j = 1, \ldots ,n$, commute. Main result: If $K(z)$ denotes the Taylor spectrum of the tuple $ ({T_1}(z), \ldots ,{T_m}(z))$, then the set-valued function $K:G \to {2^{{\mathbf{C}}m}}$ is analytic. Analyticity of such set-valued functions is defined here by a simultaneous local maximum property of $ k$-tuples of complex polynomials on the graph of $K$.
Regularity results for an elliptic-parabolic free boundary problem
M.
Bertsch;
J.
Hulshof
337-350
Abstract: We study an elliptic-parabolic free boundary problem in one space dimension. We give several regularity results for both the weak solution and the free boundary. In particular conditions are given which ensure that the free boundary is a ${C^1}$-curve.
On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type
Wei-Ming
Ni;
Izumi
Takagi
351-368
Abstract: We derive a priori estimates for positive solutions of the Neumann problem for some semilinear elliptic systems (i.e., activator-inhibitor systems in biological pattern formation theory), as well as for semilinear single equations related to such systems. By making use of these a priori estimates, we show that under certain assumptions, there is no positive nonconstant solutions for single equations or for activator-inhibitor systems when the diffusion coefficient (of the activator, in the case of systems) is sufficiently large; we also study the existence of nonconstant solutions for specific domains.
Vector fields in the vicinity of a circle of critical points
J.-F.
Mattei;
M. A.
Teixeira
369-381
Abstract: In this paper the $ {C^S}$-conjugacy between vector fields on $ {{\mathbf{R}}^2}$ having a circle of critical points is studied.