Fredholm, Hodge and Liouville theorems on noncompact manifolds
Robert
Lockhart
1-35
Abstract: Fredholm, Liouville, Hodge, and ${L^2}$-cohomology theorems are proved for Laplacians associated with a class of metrics defined on manifolds that have finitely many ends. The metrics are conformal to ones that are asymptotically translation invariant. They are not necessarily complete. The Fredholm results are, of necessity, with respect to weighted Sobolev spaces. Embedding and compact embedding theorems are also proved for these spaces.
Uniform distribution of two-term recurrence sequences
William Yslas
Vélez
37-45
Abstract: Let ${u_0},\,{u_1},\,A,\,B$ be rational integers and for $n \geqslant 2$ define ${u_n} = A{u_{n - 1}} + B{u_{n - 2}}$. The sequence $({u_n})$ is clearly periodic modulo $ m$ and we say that $ ({u_n})$ is uniformly distributed modulo $m$ if for every $s$, every residue modulo $m$ occurs the same number of times in the sequence of residues ${u_s},\,{u_{s + 1}},\, \ldots ,\,{u_{s + N - 1}}$, where $N$ is the period of $({u_n})$ modulo $m$. If $({u_n})$ is uniformly distributed modulo $ m$ then $m$ divides $N$, so we write $N = mf$. Several authors have characterized those $ m$ for which $ ({u_n})$ is uniformly distributed modulo $m$. In fact in this paper we will show that a much stronger property holds when $m = {p^k},\,p$, a prime. Namely, if $({u_n})$ is uniformly distributed modulo $ {p^k}$ with period $ {p^k}f$, then every residue modulo ${p^k}$ appears exactly once in the sequence ${u_s},\,{u_{s + f}},\, \ldots ,\,{u_{s + ({p^k} - 1)f}}$, for every $s$. We also characterize those composite $ m$ for which this more stringent property holds.
$F$-purity and rational singularity in graded complete intersection rings
Richard
Fedder
47-62
Abstract: A simple criterion is given for determining ``almost completely'' whether the positively graded complete intersection ring $R = K[{X_1},\, \ldots ,\,{X_{n + t}}]/({G_i},\, \ldots ,\,{G_t})$, of dimension $n$, has an $F$-pure type singularity at $m = ({X_1},\, \ldots ,\,{X_{n + t}})$. Specifically, if $\operatorname{deg} ({X_i}) = {\alpha _i} > 0$ for $1 \leq i \leq n + t$ and $ \operatorname{deg} ({G_i}) = {d_i} > 0$ for $ i \leq i \leq t$, then there exists an integer $\delta$ determined by the singular locus of $ R$ such that: (1) $ R$ has $F$-pure type if $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} < \delta$. (2) $R$ does not have $F$-pure type if $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > 0$. The characterization given by this theorem is particularly effective if the singularity of $R$ at $m$ is isolated. In that case, $\delta = 0$ so that only the condition $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0$ is not solved by the above result. In particular, it follows from work of Kei-ichi Watanabe that if $ R$ has an isolated rational singularity, then $R$ has $F$-pure type. The converse is also ``almost true'' with the only exception being the case where $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0$. In proving this criterion, a weak but more stable form of $F$-purity, called $F$-contractedness, is defined and explored. $ R$ is $F$-contracted (in characteristic $ p > 0$) if every system of parameters for $m$ is contracted with respect to the Frobenius map $F:\,R \to R$. Just as for $F$-purity, the notion of $F$-contracted type is defined in characteristic 0 by reduction to characteristic $ p$. The two notions of $ F$-pure (type) and $ F$-contracted (type) coincide when $R$ is Gorenstein; whence, in particular, when $ R$ is a complete intersection ring.
The binary matroids with no $4$-wheel minor
James G.
Oxley
63-75
Abstract: The cycle matroids of wheels are the fundamental building blocks for the class of binary matroids. Brylawski has shown that a binary matroid has no minor isomorphic to the rank-3 wheel $ M({\mathcal{W}_3})$ if and only if it is a series-parallel network. In this paper we characterize the binary matroids with no minor isomorphic to $M({\mathcal{W}_4})$. This characterization is used to solve the critical problem for this class of matroids and to extend results of Kung and Walton and Welsh for related classes of binary matroids.
Hypergeometric functions over finite fields
John
Greene
77-101
Abstract: In this paper the analogy between the character sum expansion of a complex-valued function over $ {\text{GF}}(p)$ and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields. It is shown that such functions satisfy many summation and transformation formulas analogous to their classical counterparts.
T-degrees, jump classes, and strong reducibilities
R. G.
Downey;
C. G.
Jockusch
103-136
Abstract: It is shown that there exist r.e. degrees other than 0 and $\mathbf{0}^{\prime}$ which have a greatest r.e. $ 1$-degree. This solves an old question of Rogers and Jockusch. We call such degrees $1$-topped. We show that there exist incomplete $1$-topped degrees above any low r.e. degree, but also show that no nonzero low degree is $1$-topped. It then follows by known results that all incomplete $1$-topped degrees are low$_{2}$ but not low. We also construct cappable nonzero $1$-topped r.e. degrees and examine the relationships between $1$-topped r.e. degrees and high r.e. degrees. Finally, we give an analysis of the ``local'' relationships of r.e. sets under various strong reducibilities. In particular, we analyze the structure of r.e. ${\text{wtt-}}$ and $ {\text{tt}}$-degrees within a single r.e. $ {\text{T}}$-degree. We show, for instance, that there is an r.e. degree which contains a greatest r.e. $ {\text{wtt-}}$-degree and a least r.e. $ {\text{tt}}$-degree yet does not consist of a single r.e. ${\text{wtt}}$-degree. This depends on a new construction of a nonzero r.e. $ {\text{T}}$-degree with a least $ {\text{tt}}$-degree, which proves to have several further applications.
A regularity result for viscosity solutions of Hamilton-Jacobi equations in one space dimension
R.
Jensen;
P. E.
Souganidis
137-147
Abstract: Viscosity solutions of Hamilton-Jacobi equations need only to be continuous. Here we prove that, in the special case of a one-dimensional stationary problem, under quite general assumptions, Lipschitz continuous viscosity solutions have right and left derivatives at every point. Moreover, these derivatives have some kind of continuity properties.
Cohomology theories on spaces
E.
Spanier
149-161
Abstract: In this paper a previously proven uniqueness theorem for nonnegative cohomology theories on the same space is extended to cohomology theories on the same finite-dimensional space. In this form it is applicable to extraordinary cohomology theories. An example is given to show that the theorem does not hold without finite dimensionality.
The \'etale cohomology of $p$-torsion sheaves. I
William Anthony
Hawkins
163-188
Abstract: This paper generalizes a formula of Grothendieck, Ogg, and Shafarevich that expresses the Euler-Poincaré characteristic of a constructible sheaf of ${F_l}$-modules on a smooth, proper curve, over an algebraically closed field $k$ of characteristic $p > 0$, as a sum of local and global terms, where $l \ne p$. The primary focus is on removing the restriction on $l$. We begin with calculations for $ p$-torsion sheaves trivialized by $p$-extensions, but using etale cohomology to give a unified proof for all primes $l$. In the remainder of this work, only $ p$-torsion sheaves are considered. We show the existence on ${X_{{\text{et}}}}$, $X$ a scheme of characteristic $ p$, of a short exact sequence of sheaves, involving the tangent space at the identity of a finite, flat, height 1, commutative group scheme, and the subsheaf fixed by the $p$th power endomorphism; the latter turns out to be an etale group scheme. A corollary gives complete results on the Euler-Poincaré characteristic of a constructible sheaf of $ {F_p}$-modules on a smooth, proper curve, over an algebraically closed field $ k$ of characteristic $ p > 0$, when the generic stalk has rank $p$. Explicit computations are given for the Euler characteristics of such $p$-torsion sheaves on ${P^1}$ and a result on elliptic surfaces is included. A study is made of the comparison of the $ p$-ranks of abelian extensions of curves. Several examples of $p$-ranks for nonhyperelliptic curves are discussed. The paper concludes with a brief sketch of results on certain constructible sheaves of $ {F_q}$-modules, $q={p^r},\,r \ge 1$.
The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data
Stephanos
Venakides
189-226
Abstract: We study the initial value problem for the Korteweg-de Vries equation $\displaystyle ({\text{i}})\quad {u_t} - 6u{u_x} + {\varepsilon ^2}{u_{xxx}} = 0$ in the limit of small dispersion, i.e., $\varepsilon \to 0$. When the unperturbed equation $\displaystyle ({\text{ii}})\quad {u_t} - 6u{u_x} = 0$ develops a shock, rapid oscillations arise in the solution of the perturbed equation (i) In our study: a. We compute the weak limit of the solution of (i) for periodic initial data as $ \varepsilon \to 0$. b. We show that in the neighborhood of a point $ (x,\,t)$ the solution $ u(x,\,t,\,\varepsilon)$ can be approximated either by a constant or by a periodic or by a quasiperiodic solution of equation (i). In the latter case the associated wavenumbers and frequencies are of order $ O(1/\varepsilon )$. c. We compute the number of phases and the wave parameters associated with each phase of the approximating solution as functions of $x$ and $t$. d. We explain the mechanism of the generation of oscillatory phases. Our computations in a and c are subject to the solution of the Lax-Levermore evolution equations (7.7). Our results in b-d rest on a plausible averaging assumption.
On root invariants of periodic classes in ${\rm Ext}\sb A({\bf Z}/2,{\bf Z}/2)$
Paul
Shick
227-237
Abstract: We prove that if a class in the cohomology of the mod 2 Steenrod algebra is $ \operatorname{mod}\,2$-periodic in the sense of [10], then its root invariant must be $ {\upsilon _{n + 1}}$-periodic, where $ {\upsilon _{n}}$ denotes the $n$th generator of ${\pi _ \ast }({\text{BP}})$.
Sets of uniqueness in compact, $0$-dimensional metric groups
D. J.
Grubb
239-249
Abstract: An investigation is made of sets of uniqueness in a compact 0-dimensional space. Such sets are defined by pointwise convergence of sequences of functions that generalize partial sums of trigonometric series on Vilenkin groups. Several analogs of classical uniqueness theorems are proved, including a version of N. Bary's theorem on countable unions of closed sets of uniqueness.
A cohomological pairing of half-forms
P. L.
Robinson
251-261
Abstract: Blattner and Rawnsley have constructed half-forms for regular polarizations of arbitrary index. We show how to pair these half-forms into a line bundle fashioned purely from the symplectic data, with no assumption on the intersection of the polarizations. Our pairing agrees with the regular BKS pairing when the polarizations are positive.
The structure of $\sigma$-ideals of compact sets
A. S.
Kechris;
A.
Louveau;
W. H.
Woodin
263-288
Abstract: Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where $\sigma$-ideals of compact sets are encountered very often as notions of small or exceptional sets, we undertake in this paper a descriptive set theoretic study of $\sigma$-ideals of compact sets in compact metrizable spaces. In the first part we study the complexity of such ideals, showing that the structural condition of being a $\sigma$-ideal imposes severe definability restrictions. A typical instance is the dichotomy theorem, which states that $\sigma$-ideals which are analytic or coanalytic must be actually either complete coanalytic or else ${G_\delta}$. In the second part we discuss (generators or as we call them here) bases for $ \sigma$-ideals and in particular the problem of existence of Borel bases for coanalytic non-Borel $\sigma$-ideals. We derive here a criterion for the nonexistence of such bases which has several applications. Finally in the third part we develop the connections of the definability properties of $ \sigma$-ideals with other structural properties, like the countable chain condition, etc.
On scrambled sets for chaotic functions
A. M.
Bruckner;
Thakyin
Hu
289-297
Abstract: Some recent research has raised questions concerning the possible sizes of scrambled sets for chaotic functions. We answer these questions by showing that a scrambled set can have full measure, but cannot be residual although a scrambled set can be second category in every interval. We also indicate relationships that exist between chaotic functions and transitive functions.
The moduli of compact continuations of an open Riemann surface of genus one
M.
Shiba
299-311
Abstract: Let $(R,\,\{ A,\,B\} )$ be a marked open Riemann surface of genus one. Denote by $(T,\,\{ {A_T},\,{B_T}\} ,i)$ a pair of a marked torus $ (T,\,\{ {A_T},\,{B_T}\} )$ and a conformal embedding $i$ of $R$ into $T$ with $i(A)$ and $i(B)$ homotopic respectively to $ {A_T}$ and ${B_T}$. We say that $(T,\,\{ {A_T},\,{B_T}\} ,i)$ and $(T^{\prime},\,\{ {A_T^{\prime}},\,{B_T^{\prime}}\} ,i^{\prime})$ are equivalent if $i^{\prime} \circ {i^{ - 1}}$ extends to a conformal mapping of $T$ onto $ {T^\prime}$. The equivalence classes are called compact continuations of $(R,\,\{ A,\,B\} )$ and the set of moduli of compact continuations of $(R,\,\{ A,\,B\} )$ is denoted by $M = M(R,\,\{ A,\,B\} )$. Then $M$ is a closed disk in the upper half plane. The radius of $M$ represents the size of the ideal boundary of $ R$ and gives a generalization of Schiffer's span for planar domains; in particular, it vanishes if and only if $R$ belongs to the class ${O_{AD}}$. On the other hand, any holomorphic differential on $R$ with distinguished imaginary part produces in a canonical manner a compact continuation of $(R,\,\{ A,\,B\} )$. Such a compact continuation is referred to as a hydrodynamic continuation of $(R,\,\{ A,\,B\} )$. The boundary of $M$ parametrizes in a natural way the space of hydrodynamic continuations; i.e., the hydrodynamic continuations have extremal properties.
Isotype submodules of $p$-local balanced projective groups
Mark
Lane
313-325
Abstract: By giving necessary and sufficient conditions for two isotype submodules of a $p$-local balanced projective group to be equivalent, we are able to introduce a general theory of isotype submodules of $p$-local balanced projective groups (or $ IB$ modules). Numerous applications of the above result are available particularly for the special class of $IB$ modules introduced by Wick (known as SKT modules). We first show that the class of SKT modules is closed under direct summands, and then we are able to show that if $H$ appears as an isotype submodule of the $ p$-local balanced projective group $G$ such that $G/H$ is the coproduct of countably generated torsion groups, then $H$ is an SKT module. Finally we show that $ IB$ modules satisfy general structural properties such as transitivity, full transitivity, and the equivalence of ${p^\alpha }$-high submodules.
Positive solutions of systems of semilinear elliptic equations: the pendulum method
Joseph
Glover
327-342
Abstract: Conditions are formulated which guarantee the existence of positive solutions for systems of the form \begin{displaymath}\begin{gathered}- \Delta {u_1} + {f_1}({u_1}, \ldots ,\,{u_n}... ...}({u_1}, \ldots ,\,{u_n}) = {\mu _n}, \end{gathered} \end{displaymath} , where $\Delta$ is the Laplacian (with Dirichlet boundary conditions) on an open domain in ${\mathbf{R}^d}$, and where each ${\mu_i}$ is a positive measure. The main tools used are probabilistic potential theory, Markov processes, and an iterative scheme which is not a generalization of the one used for quasimonotone systems. Quasimonotonicity is not assumed and new results are obtained even for the case where $\partial {f_k}/\partial {x_j} > 0$ for every $ k$ and $j$.
Global solvability on compact nilmanifolds of three or more steps
Jacek M.
Cygan;
Leonard F.
Richardson
343-373
Abstract: We apply the methods of representation theory of nilpotent Lie groups to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations $Df = g$ in $ {C^\infty}$ of a compact nilmanifold of three or more steps. We investigate which algebraically well-defined conditions on $D$ in the complexified Lie algebra imply that smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution. This extends and improves the results of an earlier two step paper.
Convergence of series of scalar- and vector-valued random variables and a subsequence principle in $L\sb 2$
S. J.
Dilworth
375-384
Abstract: Let $({d_n})_{n = 1}^\infty$ be a martingale difference sequence in ${L_0}(X)$, where $X$ is a uniformly convex Banach space. We investigate a necessary condition for convergence of the series $ \sum {_{n = 1}^\infty {a_n}{d_n}}$. We also prove a related subsequence principle for the convergence of a series of square-integrable scalar random variables.
Hamiltonian analysis of the generalized problem of Bolza
F. H.
Clarke
385-400
Abstract: On étudie le problème généralisé de Bolza en calcul des variations. Presented at the International Conference on the Calculus of Variations held to honour the memory of Leonida Tonelli, Scuola Normale Superiore, Pisa, March 1986. On obtient des conditions nécessaires en forme hamiltonienne, sous des hypothèses moins exigeantes qu'antérieurement, en particulier sans qualification sur les contraintes. Le lien avec les problèmes de contrôle optimal est développé, ainsi que l'apport de ces conditions à la théorie de la régularité de la solution. We obtain necessary conditions in Hamiltonian form for the generalized problem of Bolza in the calculus of variations. These are proven in part by an extension to Hamiltonians of Tonelli's method of auxiliary Lagrangians. One version of the conditions is of a new character since it is obtained in the absence of any constraint qualification on the data. A new regularity theorem is shown to be a consequence of the necessary conditions.
New results on automorphic integrals and their period functions
Richard A.
Cavaliere
401-412
Abstract: Automorphic integrals, being generalizations of automorphic forms on discrete subgroups of $ SL(2,\,\mathbf{R})$, share properties similar to those of forms. In this article I obtain a natural boundary result for integrals which is similar to that which holds for forms. If an automorphic integral on a given group behaves like a form on a subgroup of finite index (i.e., the period functions are identically zero), then in fact the integral must be a form on the whole group. Specializing to modular integrals with integer dimension I obtain a lower bound on the number of poles of the period functions which, of necessity, lie in quadratic extensions of the rationals.
The isometry groups of manifolds and the automorphism groups of domains
Rita
Saerens;
William R.
Zame
413-429
Abstract: We prove that every compact Lie group can be realized as the (full) automorphism group of a strictly pseudoconvex domain and as the (full) isometry group of a compact, connected, smooth Riemannian manifold.