The role of countable dimensionality in the theory of cell-like relations
Fredric D.
Ancel
1-40
Abstract: Consider only metrizable spaces. The notion of a slice-trivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact $ U{V^\infty }$ point images to be slice-trivial. Theorem 4.5 posits a number of necessary and sufficient conditions for a map to be a hereditary shape equivalence. Several applications of these two theorems are made, including the following. Theorem 5.1. A cell-like map $ f:X \to Y$ is a hereditary shape equivalence if there is a sequence $\{ {K_n}\}$ of closed subsets of $ Y$ such that (1) $Y - \bigcup\nolimits_{n = 1}^\infty {{K_n}}$ is countable dimensional, and (2) $ f\vert{f^{ - 1}}({K_n}):{f^{ - 1}}({K_n}) \to {K_n}$ is a hereditary shape equivalence for each $n \geq 1$. Theorem 5.9. If $ f:X \to Y$ is a proper onto map whose point inverses are $U{V^\infty }$ sets, then $Y$ is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if $Y$ is countable dimensional, then $ Y$ is an absolute neighborhood retract. Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.
Bowen-Ruelle measures for certain piecewise hyperbolic maps
Lai-Sang
Young
41-48
Abstract: We consider a class of piecewise ${C^2}$ Lozi-like maps and prove the existence of invariant measures with absolutely continuous conditional measures on unstable manifolds
Functions of $\Phi$-bounded variation and Riemann-Stieltjes integration
Michael
Schramm
49-63
Abstract: A notion of generalized bounded variation is introduced which simultaneously generalizes many of those previously examined. It is shown that the class of functions arising from this definition is a Banach space with a suitable norm. Appropriate variation functions are defined and examined, and an analogue of Helly's theorem is estabished. The significance of this class to convergence of Fourier series is briefly discussed. A result concerning Riemann-Stieltjes integrals of functions of this class is proved.
Strong laws of large numbers for products of random matrices
Steve
Pincus
65-89
Abstract: This work, on products of random matrices, is inspired by papers of Furstenberg and Kesten (Ann. Math. Statist. 31 (1960), 457-469) and Furstenberg (Trans. Amer. Math. Soc. 108 (1963), 377-428). In particular, a formula was known for almost sure limits for normalized products of random matrices in terms of a stationary measure. However, no explicit computational techniques were known for these limits, and little was known about the stationary measures. We prove two main theorems. The first assumes that the random matrices are upper triangular and computes the almost sure limits in question. For the second, we assume the random matrices are $2 \times 2$ and Bernoulli, i.e., random matrices whose support is two points. Then the second theorem gives an asymptotic result for the almost sure limits, with rates of convergence in some cases.
Mixed projection inequalities
Erwin
Lutwak
91-105
Abstract: A number of sharp geometric inequalities for polars of mixed projection bodies (zonoids) are obtained. Among the inequalities derived is a polar projection inequality that has the projection inequality of Petty as a special case. Other special cases of this polar projection inequality are inequalities (between the volume of a convex body and that of the polar of its $i$th projection body) that are strengthened forms of the classical inequalities between the volume of a convex body and its projection measures (Quermassintegrale). The relation between the Busemann-Petty centroid inequality and the Petty projection inequality is shown to be similar to the relation that exists between the Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry. Some mixed integral inequalities are derived similar in spirit to inequalities obtained by Chakerian and others.
${\rm BMO}(\rho)$ and Carleson measures
Wayne Stewart
Smith
107-126
Abstract: This paper concerns certain generalizations of $ {\text{BMO}}$, the space of functions of bounded mean oscillation. Let $ \rho$ be a positive nondecreasing function on $ (0,\infty )$ with $\rho (0 + ) = 0$. A locally integrable function on $ {{\mathbf{R}}^m}$ is said to belong to $ {\text{BMO}}(\rho)$ if its mean oscillation over any cube $Q$ is $ O(\rho (l(Q)))$, where $ l(Q)$ is the edge length of $Q$. Carleson measures are known to be closely related to $ {\text{BMO}}$. Generalizations of these measures are shown to be similarly related to the spaces $ {\text{BMO}}(\rho)$. For a cube $Q$ in $ {{\mathbf{R}}^m},\;\vert Q\vert$ denotes its volume and $R(Q)$ is the set $ \{ (x,y) \in {\mathbf{R}}_ + ^{m + 1}:x \in Q,\;0 < y < l(Q)\}$. A measure $ \mu$ on ${\mathbf{R}}_ + ^{m + 1}$ is called a $ \rho$-Carleson measure if $ \vert\mu \vert(R(Q)) = O(\rho (l(Q))\vert Q\vert)$, for all cubes $Q$. L. Carleson proved that a compactly supported function in $ {\text{BMO}}$ can be represented as the sum of a bounded function and the balyage, or sweep, of some Carleson measure. A generalization of this theorem involving ${\text{BMO}}(\rho )$ and $\rho$-Carleson measures is proved for a broad class of growth functions, and this is used to represent $ {\text{BMO}}(\rho )$ as a dual space. The proof of the theorem is based on a proof of J. Garnett and P. Jones of Carleson's theorem. Another characterization of ${\text{BMO}}(\rho )$ using $\rho $-Carleson measures is a corollary. This result generalizes a characterization of $ {\text{BMO}}$ due to C. Fefferman. Finally, an atomic decomposition of the predual of $ {\text{BMO}}(\rho )$ is given.
Finite time analyticity for the two- and three-dimensional Rayleigh-Taylor instability
C.
Sulem;
P.-L.
Sulem
127-160
Abstract: The Rayleigh-Taylor instability refers to the dynamics of the interface between two ideal irrotational fluids of different densities superposed one over the other and in relative motion. The well-posedness of this problem is considered for two- and three-dimensional flows in the entire space and in the presence of a horizontal bottom. In the entire space, finite time analyticity of the interface is proven when the initial interface has sufficiently small gradients and is flat at infinity. In the presence of a horizontal bottom, the initial interface corrugations has also to be small initially but it is not required to vanish at infinity.
Projections on tensor product spaces
E. J.
Halton;
W. A.
Light
161-165
Abstract: $(S,\Sigma ,\mu ),(T,\Theta ,\upsilon )$ are finite, nonatomic measure spaces. $G$ and $H$ are finite-dimensional subspaces of ${L_1}(S)$ and ${L_1}(T)$ respectively. Both $G$ and $H$ contain the constant functions. It is shown that the relative projection constant of $ {L_1}(S) \otimes H + G \otimes {L_1}(T)$ in $ {L_1}(S \times T)$ is at least $3$.
On a.e. convergence of solutions of hyperbolic equations to $L\sp p$-initial data
Alberto
Ruiz
167-188
Abstract: We consider the Cauchy data problem $ u(x,0) = 0$, $\partial u(x,0)/\partial t = f(x)$, for a strongly hyperbolic second order equation in $ n$th spatial dimension, $n \geq 3$, with ${C^\infty }$ coefficients. Almost everywhere convergence of the solution of this problem to initial data, in the appropriate sense is proved for $f$ in ${L^p}$, $2n/(n + 1) < p < 2(n - 2)/(n - 3)$. The basic techniques are ${L^p}$-estimates for some maximal operators associated to the problem (see [4]), and the asymptotic expansion of the Riemann function given by D. Ludwig (see [9]).
On invariant finitely additive measures for automorphism groups acting on tori
S. G.
Dani
189-199
Abstract: Consider the natural action of a subgroup $H$ of $ {\text{GL}}(n,{\mathbf{Z}})$ on $ {{\mathbf{T}}^n}$. We relate the $H$-invariant finitely additive measures on $ ({{\mathbf{T}}^n},\mathcal{L})$ where $ \mathcal{L}$ is the class of all Lebesgue measurable sets, to invariant subtori $ C$ such that the $ H$-action on either $ C$ or ${{\mathbf{T}}^n}/C$ factors to an action of an amenable group. In particular, we conclude that if $H$ is a nonamenable group acting irreducibly on $ {{\mathbf{T}}^n}$ then the normalised Haar measure is the only $H$-invariant finitely additive probability measure on $ ({{\mathbf{T}}^n},\mathcal{L})$ such that $ \mu (R) = 0$, where $ R$ is the (countable) subgroup consisting of all elements of finite order; this answers a question raised by J. Rosenblatt. Along the way we analyse $H$-invariant finitely additive measures defined for all subsets of $ {{\mathbf{T}}^n}$ and deduce, in particular, that the Haar measure extends to an $ H$-invariant finitely additive measure defined on all sets if and only if $ H$ is amenable.
Minimal leaves in foliations
Daniel M.
Cass
201-213
Abstract: The paper defines a property of open Riemannian manifolds, called quasi-homogeneity. This property is quasi-isometry invariant and is shown to hold for any manifold which appears as a minimal leaf in a foliation. Examples are given of surfaces which are not quasi-homogeneous. One such is the well-known noncompact leaf of Reeb's foliation of ${S^3}$. These surfaces have bounded geometry.
The duration of transients
S.
Pelikan
215-221
Abstract: A transformation $ T$ defined on $X \subset {{\mathbf{R}}^n}$ for which $T(X) \supset X$ is considered. A transient in $X$ is a trajectory $x,Tx, \ldots ,{T^m}x \subset X$ so that ${T^{m + 1}}x \notin X$. In this case, $ m$ is the duration of the transient. A method for estimating the average duration of transients is given, and an example of a transformation with exceedingly long transients is described.
On the restriction of the Fourier transform to curves: endpoint results and the degenerate case
Michael
Christ
223-238
Abstract: For smooth curves $ \Gamma$ in ${{\mathbf{R}}^n}$ with certain curvature properties it is shown that the composition of the Fourier transform in $ {{\mathbf{R}}^n}$ followed by restriction to $\Gamma$ defines a bounded operator from ${L^p}({{\mathbf{R}}^n})$ to ${L^q}(\Gamma )$ for certain $p,q$. The curvature hypotheses are the weakest under which this could hold, and $ p$ is optimal for a range of $q$. In the proofs the problem is reduced to the estimation of certain multilinear operators generalizing fractional integrals, and they are treated by means of rearrangement inequalities and interpolation between simple endpoint estimates.
Periodic solutions of Hamilton's equations and local minima of the dual action
Frank H.
Clarke
239-251
Abstract: The dual action is a functional whose extremals lead to solutions of Hamilton's equations. Up to now, extremals of the dual action have been obtained either through its global minimization or through application of critical point theory. A new methodology is introduced in which local minima of the dual action are found to exist. Applications are then made to the existence of Hamiltonian trajectories having prescribed period.
The algebra of the finite Fourier transform and coding theory
R.
Tolimieri
253-273
Abstract: The role of the finite Fourier transform in the theory of error correcting codes has been explored in a recent text by Richard Blahut. In this work we study how the finite Fourier transform relates to certain polynomial identities involving weight enumerator polynomials of linear codes. These include the generalized MacWilliams identities and theorems originally due to ${\text{R}}$. Gleason concerning polynomial algebras containing weight enumerator polynomials. The Heisenberg group model of the finite Fourier transform provides certain algebras of classical theta functions which will be applied to reprove Gleason's results.
Period doubling and the Lefschetz formula
John
Franks
275-283
Abstract: This article gives an application of the Lefschetz fixed point theorem to prove, under certain hypotheses, the existence of a family of periodic orbits for a smooth map. The family has points of periods ${2^k}p$ for some $p$ and all $k \geq 0$. There is a version of the result for a parametrized family $f_t$ which shows that these orbits are "connected" in parametrized space under appropriate hypotheses.
A method for investigating geometric properties of support points and applications
Johnny E.
Brown
285-291
Abstract: A normalized univalent function $f$ is a support point of $S$ if there exists a continuous linear functional $ L$ (which is nonconstant on $S$) for which $f$ maximizes $\operatorname{Re} L(g),g \in S$. For such functions it is known that $\Gamma = {\text{C}} - f(U)$ is a single analytic arc that is part of a trajectory of a certain quadratic differential $ Q(w)\;d{w^2}$. A method is developed which is used to study geometric properties of support points. This method depends on consideration of $\operatorname{Im} \{ {w^2}Q(w)\}$ rather than the usual $\operatorname{Re} \{ {w^2}Q(w)\}$. Qualitative, as well as quantitative, applications are obtained. Results related to the Bieberbach conjecture when the extremal functions have initial real coefficients are also obtained.
Fractional integrals on weighted $H\sp p$ and $L\sp p$ spaces
Jan-Olov
Strömberg;
Richard L.
Wheeden
293-321
Abstract: We study the two weight function problem $ \parallel {I_\alpha }f{\parallel _{H_u^q}} \leqslant c\parallel f{\parallel _{H_v^p}},0 < p \leqslant q < \infty$ , for fractional integrals on Hardy spaces. If $u$ and $v$ satisfy the doubling condition and $0 < p \leqslant 1$, we obtain a necessary and sufficient condition for the norm inequality to hold. If $1 < p < \infty$ we obtain a necessary condition and a sufficient condition, and show these are the same under various additional conditions on $ u$ and $v$. We also consider the corresponding problem for $L_u^q$ and $L_v^p$, and obtain a necessary and sufficient condition in some cases.
Of planar Eulerian graphs and permutations
Gadi
Moran
323-341
Abstract: Infinite planar Eulerian graphs are used to show that for $v > 0$ the covering number of the infinite simple group $ {H_v} = S/{S^v}$ is two. Here $S$ denotes the group of all permutations of a set of cardinality $ {\aleph _v},{S^v}$ denotes its subgroup consisting of the permutations moving less than ${\aleph _v}$ elements, and the covering number of a (simple) group $G$ is the smallest positive integer $ n$ satisfying $ {C^n} = G$ for every nonunit conjugacy class $C$ in $G$.
Local operators and derivations on $C\sp \ast$-algebras
C. J. K.
Batty
343-352
Abstract: The variations on a theme of locality for a pair of operators $ (H,K)$ on a $ {C^\ast}$-algebra $\mathfrak{A}$ are expressed algebraically. If $ K$ is a $ \ast$-derivation generating an action of $ \mathbb{R}$ on $\mathfrak{A}$, and $H$ is $\ast$-linear and $K$-local, then, under certain restrictions, $ H$ is shown to be very closely related to $K$.
Fuchsian groups and algebraic number fields
P. L.
Waterman;
C.
Maclachlan
353-364
Abstract: Given the signature of a finitely-generated Fuchsian group, we find the minimal extension of the rationals for which there is a Fuchsian group having the required signature, whose matrix entries lie in this field.
Conway's field of surreal numbers
Norman L.
Alling
365-386
Abstract: Conway introduced the Field $ {\mathbf{No}}$ of numbers, which Knuth has called the surreal numbers. ${\mathbf{No}}$ is a proper class and a real-closed field, with a very high level of density, which can be described by extending Hausdorff's ${\eta _\xi }$ condition. In this paper the author applies a century of research on ordered sets, groups, and fields to the study of $ {\mathbf{No}}$. In the process, a tower of subfields, $\xi {\mathbf{No}}$, is defined, each of which is a real-closed subfield of $ {\mathbf{No}}$ that is an ${\eta _\xi }$-set. These fields all have Conway partitions. This structure allows the author to prove that every pseudo-convergent sequence in ${\mathbf{No}}$ has a unique limit in ${\mathbf{No}}$.
An idempotent completion functor in homotopy theory
Harold M.
Hastings
387-402
Abstract: We observe that Artin-Mazur style $R$-completions ($R$ is a commutative ring with identity) induce analogous idempotent completions on the weak prohomotopy category pro-Ho(Top). Because Ho(Top) is a subcategory of pro-Ho(Top) and pro-Ho(Top) is closely related to the topologized homotopy category of J. F. Adams and D. Sullivan, our construction represents the Sullivan completions as homotopy limits of idempotent functors. In addition, we show that the Sullivan completion is idempotent on those spaces (in analogy with the Bousfield and Kan ${R_\infty }$-completion on $ R$-good spaces) for which its cohomology with coefficients in $ R$ agrees with that of our Artin-Mazur style completion. Finally, we rigidify the Artin-Mazur completion to obtain an idempotent Artin-Mazur completion on a category of generalized prospaces which preserves fibration and suitably defined cofibration sequences. (Our previous results on idempotency and factorization lift to the rigid completion.) Our results answer questions of Adams, Sullivan, and, later, A. Deleanu.
On the model equations which describe nonlinear wave motions in a rotating fluid
Jong Uhn
Kim
403-417
Abstract: This paper concerns mathematical aspects of the two model equations describing nonlinear wave motions in a rotating fluid. We establish local existence of solutions and show that singularities occur in a finite time under certain hypotheses. We also show that these equations admit nonconstant travelling wave solutions.
Compact group actions and maps into $K(\pi,1)$-spaces
Daniel H.
Gottlieb;
Kyung B.
Lee;
Murad
Özaydin
419-429
Abstract: Let $G$ act on an aspherical manifold $ M$. If $G$ is a compact Lie group acting effectively and homotopically trivially then $ G$ must be abelian. We prove a much more general form of this result, thus extending results of Donnelly and Schultz. Our method gives us a splitting result for torus actions complementing a result of Conner and Raymond. We also generalize a theorem of Schoen and Yau on homotopy equivariance.