Symmetries of planar growth functions. II
William J.
Floyd
447-502
Abstract: Let $G$ be a finitely generated group, and let $\Sigma$ be a finite generating set of $ G$. The growth function of $(G,\Sigma )$ is the generating function $ f(z) = \sum\nolimits_{n = 0}^\infty {{a_n}{z^n}}$, where ${a_n}$ is the number of elements of $ G$ with word length $ n$ in $\Sigma$. Suppose that $G$ is a cocompact group of isometries of Euclidean space $ {\mathbb{E}^2}$ or hyperbolic space $ {\mathbb{H}^2}$, and that $ D$ is a fundamental polygon for the action of $G$. The full geometric generating set for $ (G,D)$ is $\{ g \in G:g \ne 1$ and $gD \cap D \ne \emptyset \} $. In this paper the recursive structure for the growth function of $(G,\Sigma )$ is computed, and it is proved that the growth function $f$ is reciprocal $ (f(z) = f(1/z))$ except for some exceptional cases when $D$ has three, four, or five sides.
Homogeneous chaos, $p$-forms, scaling and the Feynman integral
G. W.
Johnson;
G.
Kallianpur
503-548
Abstract: In a largely heuristic but fascinating recent paper, Hu and Meyer have given a "formula" for the Feynman integral of a random variable $f$ on Wiener space in terms of the expansion of $ f$ in Wiener chaos. The surprising properties of scaling in Wiener space make the problem of rigorously connecting this formula with the usual definition of the analytic Feynman integral a subtle one. One of the main tools in carrying this out is our definition of the 'natural extension' of $ p$th homogeneous chaos in terms of the 'scale-invariant lifting' of $ p$-forms on the white noise space ${L^2}({\mathbb{R}_ + })$ connected with Wiener space. The key result in our development says that if $ {f_p}$ is a symmetric function in ${L^2}(\mathbb{R}_ + ^p)$ and ${\psi _p}({f_p})$ is the associated $p$-form on ${L^2}({\mathbb{R}_ + })$, then ${\psi _p}({f_p})$ has a scaled $ {L^2}$-lifting if and only if the '$k$th limiting trace' of ${f_p}$ exists for $k = 0,1, \ldots ,[p/2]$. This necessary and sufficient condition for the lifting of a $p$-form on white noise space to a random variable on Wiener space is a worthwhile contribution to white noise theory apart from any connection with the Feynman integral since $p$-forms play a role in white noise calculus analogous to the role played by $p$th homogeneous chaos in Wiener calculus. Various $k$-traces arise naturally in this subject; we study some of their properties and relationships. The limiting $k$-trace plays the most essential role for us.
On a theorem of Muckenhoupt and Wheeden and a weighted inequality related to Schr\"odinger operators
C.
Pérez
549-562
Abstract: We extend in several directions a theorem of B. Muckenhoupt and R. Wheeden relating the ${L^p}$-norms of Riesz potentials and fractional maximal operators. We apply these results to give a simple proof and sharpen a weighted inequality for Schrödinger operators of Chang, Wilson and Wolff.
Normality in $X\sp 2$ for compact $X$
G.
Gruenhage;
P. J.
Nyikos
563-586
Abstract: In 1977, the second author announced the following consistent negative answer to a question of Katětov: Assuming $ {\text{MA}} + \neg {\text{CH}}$, there is a compact nonmetric space $ X$ such that $ {X^2}$ is hereditarily normal. We give the details of this example, and construct another example assuming $ {\text{CH}}$. We show that both examples can be constructed so that ${X^2}\backslash \Delta$ is perfectly normal. We also construct in $ {\text{ZFC}}$ a compact nonperfectly normal $X$ such that $ {X^2}\backslash \Delta$ is normal.
A qualitative uncertainty principle for unimodular groups of type ${\rm I}$
Jeffrey A.
Hogan
587-594
Abstract: It has long been known that if $f \in {L^2}({{\mathbf{R}}^n})$ and the supports of and its Fourier transform $ \hat f$ are bounded then $f = 0$ almost everywhere. More recently it has been shown that the same conclusion can be reached under the weaker condition that the supports of $f$ and $\hat f$ have finite measure. These results may be thought of as qualitative uncertainty principles since they limit the "concentration" of the Fourier transform pair $ (f,\hat f)$. Little is known, however, of analogous results for functions on locally compact groups. A qualitative uncertainty principle is proved here for unimodular groups of type I.
$EHP$ spectra and periodicity. I. Geometric constructions
Brayton
Gray
595-616
Abstract: The techniques used in $EHP$ calculation are studied, and lead to the notion of an $EHP$ spectrum. A simple inductive procedure suggests the existence of higher order $EHP$ spectra in which the first differential corresponds to ${v_n}$ multiplication. The next case $(n = 1)$ is constructed using the work of Cohen, Moore, and Neisendorfer. Some of the expected universal properties are proven.
$EHP$ spectra and periodicity. II. $\Lambda$-algebra models
Brayton
Gray
617-640
Abstract: The results of part I suggest that for small $m$, the Smith-Toda spectrum $V(m)$ can be approximated by spaces having universal mapping properties and interlocking fibrations. For each $m$, a $\Lambda$-algebra model representing the Bousfield-Kan $E\prime$ term for these spaces is constructed, and all of the ideal results are proven on the chain level.
Solutions to the nonautonomous bistable equation with specified Morse index. I. Existence
Nicholas D.
Alikakos;
Peter W.
Bates;
Giorgio
Fusco
641-654
Abstract: We investigate the existence of unstable solutions of specified Morse index for the equation ${\varepsilon ^2}{u_{xx}} - f(x,u) = 0$ on a finite interval and Neumann boundary conditions.
A Frobenius characterization of rational singularity in $2$-dimensional graded rings
Richard
Fedder
655-668
Abstract: A ring $ R$ is said to be $ F$-rational if, for every prime $P$ in $R$, the local ring ${R_P}$ has the property that every system of parameters ideal is tightly closed (as defined by Hochster-Huneke). It is proved that if $R$ is a $2$-dimensional graded ring with an isolated singularity at the irrelevant maximal ideal $m$, then the following are equivalent: (1) $R$ has a rational singularity at $ m$. (2) $R$ is $F$-rational. (3) $a(R) < 0$. Here $a(R)$ (as defined by Goto-Watanabe) denotes the least nonvanishing graded piece of the local cohomology module ${H_m}(R)$. The proof of this result relies heavily on the properties of derivations of $ R$, and suggests further questions in that direction; paradigmatically, if one knows that $D(a)$ satisfies a certain property for every derivation $D$, what can one conclude about the original ring element $a$?
Cohomology classes associated to anomalies
Gregory Lambros
Peterson
669-704
Abstract: One of the proposed settings for the description of anomalies in the setting of gauge field theories is a local bicomplex associated to a principal fiber bundle $G \to P \to M$. It is a bigraded algebra with two differentials which is invariantly defined, i.e. independent of local coordinates of $ M$. I denote it by $S_M^{ \bullet , \bullet }$. Briefly, $ S_M^{p,q}$ consists of local $p$-multilinear operators from the gauge algebra into $ q$-forms on $M$ which depend on a connection $ A$ in a local manner; local means that the operators decrease supports. The gauge algebra is the Lie algebra of the gauge group, which consists of diffeomorphisms of $ P$ that respect the action of $G$ and cover the identity diffeomorphism of $ M$. In this setting, the anomalies are described as integrals over $ M$ whose integrands can be shown to be representatives of total cohomology classes in $ {H^1}(S_M^{ \bullet , \bullet })$. The main reason for restricting to a local bicomplex is due to Peetre's theorem. It states that local operators are differential operators over open sets $U \subset M$. This property is both mathematically natural and required by physical considerations. This paper explores the computation of the total cohomology of the local bicomplex by beginning with the coordinate description of the differential operators and then determining which of these differential operators can be used to construct invariantly defined objects. What is accomplished is the description of the differential operators which are invariant under the action of the local diffeomorphisms of ${\mathbb{R}^n}$ and the computation of their total cohomology over open sets $ U \subset M$. The main result is that $\displaystyle H_d^ \bullet ({(S_U^{ \bullet , \bullet })^{{\operatorname{Diff}_... ...}({\mathbb{R}^n})}}) \simeq H_d^ \bullet (W{(\mathfrak{g})_{[\tfrac{n} {2}]}}),$ where ${(S_U^{ \bullet , \bullet })^{{{\operatorname{Diff}}_{{\text{loc}}}}({\mathbb{R}^n})}}$ denotes the invariant differential operators over the open set $U$ and $ W{(\mathfrak{g})_{[\tfrac{n} {2}]}}$ is the Weil algebra of $\mathfrak{g}$, the Lie algebra of $G$ truncated at $[\tfrac{n} {2}]$, the greatest integer less than or equal to half the dimension of $M$. This shows that the cohomology groups over open sets are nonzero only in the range $n \leq q \leq n + r$ where $r$ is the dimension of the Lie algebra $\mathfrak{g}$ , and in this range they are all finite dimensional. This result is globalized in the special case that the associated fiber bundle $ \operatorname{ad}^\ast\;P$ is trivializable.
Vojta's refinement of the subspace theorem
Wolfgang M.
Schmidt
705-731
Abstract: Vojta's refinement of the Subspace Theorem says that given linearly independent linear forms $ {L_1}, \ldots , {L_n}$ in $ n$ variables with algebraic coefficients, there is a finite union $U$ of proper subspaces of ${\mathbb{Q}^n}$, such that for any $\varepsilon > 0$ the points $ \underline{\underline x} \in {\mathbb{Z}^n}\backslash \{ \underline{\underline 0} \}$ with (1) $ \vert{L_1}(\underline{\underline x} ) \cdots {L_n}(\underline{\underline x} )\vert\; < \;\vert\underline{\underline x} {\vert^{ - \varepsilon }}$ lie in $U$, with finitely many exceptions which will depend on $ \varepsilon$ . Put differently, if $ X(\varepsilon )$ is the set of solutions of (1), if $\bar X(\varepsilon )$ is its closure in the subspace topology (whose closed sets are finite unions of subspaces) and if $ \bar X\prime(\varepsilon )$ consists of components of dimension $> 1$ , then $\bar X\prime(\varepsilon ) \subset U$ . In the present paper it is shown that $ \bar X\prime(\varepsilon )$ is in fact constant when $ \varepsilon$ lies outside a simply described finite set of rational numbers. More generally, let $k$ be an algebraic number field and $ S$ finite set of absolute values of $k$ containing the archimedean ones. For $\upsilon \in S$ let $L_1^\upsilon, \ldots ,L_m^\upsilon$ be linear forms with coefficients in $k$, and for $\underline{\underline x} \in {K^n}\backslash \{ \underline{\underline 0} \}$ with height $ {H_k}(\underline{\underline x} ) > 1$ define ${a_{\upsilon i}}(\underline{\underline x} )$ by $\vert L_i^\upsilon(\underline{\underline x} )\vert _\upsilon/\vert\underline{\... ...{\underline x} )^{ - {a_{\upsilon i}}(\underline{\underline x} )/{d_\upsilon}}}$ where the ${d_\upsilon}$ are the local degrees. The approximation set $A$ consists of tuples $\underline{\underline a} = \{ {a_{\upsilon i}}\} \;(\upsilon \in S,1 \leqq i \leqq m)$ such that for every neighborhood $O$ of $\underline{\underline a} $ the points $\underline{\underline x} $ with $\{ {a_{{v_i}}}\{ \underline{\underline x} )\} \in O$ are dense in the subspace topology. Then $ A$ is a polyhedron whose vertices are rational points.
Harmonic analysis and ultracontractivity
Michael
Cowling;
Stefano
Meda
733-752
Abstract: Let ${({T_t})_{t > 0}}$ be a symmetric contraction semigroup on the spaces ${L^p}(M)\;(1 \leq p \leq \infty )$, and let the functions $\phi$ and $\psi$ be "regularly related". We show that ${({T_t})_{t > 0}}$ is $\phi$-ultracontractive, i.e., that ${({T_t})_{t > 0}}$ satisfies the condition $ {\left\Vert {{T_t}f} \right\Vert _\infty } \leq C\phi {(t)^{ - 1}}{\left\Vert f \right\Vert _1}$ for all $f$ in ${L^1}(M)$ and all $t$ in $ {{\mathbf{R}}^ + }$, if and only if the infinitesimal generator $\mathcal{G}$ has Sobolev embedding properties, namely, ${\left\Vert {\psi {{(\mathcal{G})}^{ - \alpha }}f} \right\Vert _q} \leq C{\left\Vert f \right\Vert _p}$ for all $f$ in ${L^p}(M)$, whenever $1 < p < q < \infty$ and $\alpha = 1/p - 1/q$ . We establish some new spectral multiplier theorems and maximal function estimates. In particular, we give sufficient conditions on $ m$ for $m(\mathcal{G})$ to map ${L^p}(M)$ to ${L^q}(M)$, and for the example where there exists $ \mu$ in ${{\mathbf{R}}^ + }$ such that $\phi (t) = {t^\mu }$ for all $t$ in ${{\mathbf{R}}^ + }$ , we give conditions which ensure that the maximal function ${\sup _{t > 0}}\vert{t^\alpha }{T_t}f( \bullet )\vert$ is bounded.
On the discrete series of generalized Stiefel manifolds
Jian-Shu
Li
753-766
Abstract: A study of the discrete series of generalized Stiefel manifolds is made using the oscillator representation. New infinite families of such discrete series are constructed.
Lyapunov graphs and flows on surfaces
K. A.
de Rezende;
R. D.
Franzosa
767-784
Abstract: In this paper, a characterization of Lyapunov graphs associated to smooth flows on surfaces is presented. We first obtain necessary and sufficient conditions for a Lyapunov graph to be associated to Morse-Smale flows and then generalize them to smooth flows. The methods employed in the proofs are of interest in their own right for they introduce the use of the Conley index in this context. Moreover, an algorithmic geometric construction of flows on surfaces is described.
Prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature
Zhi Ren
Jin
785-810
Abstract: Let $({M^n},g)$ be a complete noncompact Riemannian manifold with the curvature bounded between two negative constants. Given a function $K$ on ${M^n}$, in terms of the behaviors of $K$ at infinite, we give a fairly complete answer to when the $K$ can be the scalar curvature function of a complete metric ${g_1}$ which is conformal to $g$.
Weighted norm inequalities for general operators on monotone functions
Shanzhong
Lai
811-836
Abstract: In this paper we characterize the weights $w,v$ for which $ {\left\Vert {{S_\phi }f} \right\Vert _{p,w}} \leq C{\left\Vert f \right\Vert _{q,v}}$, for $f$ nonincreasing, where $ {S_\phi }f = \smallint _0^\infty {\phi (x,y)f(y)dy} $.
Threshold growth dynamics
Janko
Gravner;
David
Griffeath
837-870
Abstract: We study the asymptotic shape of the occupied region for monotone deterministic dynamics in $d$-dimensional Euclidean space parametrized by a threshold $ \theta > 0$, and a Borel set $ \mathcal{N} \subset {\mathbb{R}^d}$ with positive and finite Lebesgue measure. If $ {A_n}$ denotes the oocupied set of the dynamics at integer time $n$, then $ {A_{n + 1}}$ is obtained by adjoining any point $x$ for which the volume of overlap between $x + \mathcal{N}$ and ${A_n}$ exceeds $\theta$. Except in some degenerate cases, we prove that ${n^{ - 1}}{A_n}$ converges to a unique limiting "shape" $L$ starting from any bounded initial region $ {A_0}$ that is suitably large. Moreover, $L$ is computed as the polar transform for $ 1/w$, where $w$ is an explicit width function that depends on $ \mathcal{N}$ and $ \theta$. It is further shown that $L$ describes the limiting shape of wave fronts for certain cellular automaton growth rules related to lattice models of excitable media, as the threshold and range of interaction increase suitably. In the case of box $({l^\infty })$ neighborhoods on ${\mathbb{Z}^2}$, these limiting shapes are calculated and the dependence of their anisotropy on $ \theta$ is examined. Other specific two- and three-dimensional examples are also discussed in some detail.
Function spaces of completely metrizable spaces
Jan
Baars;
Joost
de Groot;
Jan
Pelant
871-883
Abstract: Let $X$ and $Y$ be metric spaces and let $\phi :{C_p}(X) \to {C_p}(Y)$ (resp. $ \phi :C_p^\ast(X) \to C_p^\ast(Y)$) be a continuous linear surjection. We prove that $Y$ is completely metrizable whenever $ X$ is. As a corollary we obtain that complete metrizability is preserved by ${l_p}$ (resp. $l_p^\ast$-equivalence) in the class of all metric spaces. This solves Problem 35 in [2] (raised by Arhangel'skiĭ).