The strong maximal function on a nilpotent group
Michael
Christ
1-13
Abstract: An analogue of the strong maximal function of Jessen, Marcinkiewicz, and Zygmund is shown to be bounded on ${L^p}$, for all $p > 1$, on a nilpotent Lie group.
The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points
Bo
Deng
15-53
Abstract: The complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained. It is topologically conjugate to a non-Bernoulli shift called $\{ {\sum,\sigma } \}$. We also obtain a more or less complete picture, referred to as the net weaving bifurcation, when the fixed point of such a system is undergoing the generic saddle-node bifurcation. The idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork, transcritical, periodic doubling, and Hopf bifurcations are all homotopically conjugate to our shift dynamics $ \{ {\sum,\sigma } \}$ in a neighborhood of a transverse homoclinic orbit. These bifurcations are also examined in the context of the spectral decomposition with respect to the maximal indecomposable nonwandering sets.
Tensor products and Grothendieck type inequalities of operators in $L\sb p$-spaces
Bernd
Carl;
Andreas
Defant
55-76
Abstract: Several results in the theory of $(p,q)$-summing operators are improved by a unified but elementary tensor product concept.
Stable splittings of the dual spectrum of the classifying space of a compact Lie group
Chun-Nip
Lee
77-111
Abstract: For a compact Lie group $G$, there is a map from the $G$-equivariant fixed point spectrum of the zero sphere to the dual spectrum of the classifying space of $G, DB{G_ + }$. When $G$ is finite, the affirmative solution to Segal's conjecture states that this map is an equivalence upon appropriate completion of the source. In the case of a compact Lie group, we obtain splitting results of $DB{G_ + }$ via this map upon taking $p$-adic completions.
Nests of subspaces in Banach space and their order types
Alvaro
Arias;
Jeff
Farmer
113-130
Abstract: This paper addresses some questions which arise naturally in the theory of nests of subspaces in Banach space. The order topology on the index set of a nest is discussed, as well as the method of spatial indexing by a vector; sufficient geometric conditions for the existence of such a vector are found. It is then shown that a continuous nest exists in any Banach space. Applications and examples follow; in particular, an extension of the Volterra nest in $ {L^\infty }[ {0,1} ]$ to a continuous one, a continuous nest in a Banach space having no two elements isomorphic to one another, and a characterization of separable ${\mathcal{L}_p}$-spaces in terms of nests.
The helical transform as a connection between ergodic theory and harmonic analysis
Idris
Assani;
Karl
Petersen
131-142
Abstract: Direct proofs are given for the formal equivalence of the $ {L^2}$ boundedness of the maximal operators corresponding to the partial sums of Fourier series, the range of a discrete helical walk, partial Fourier coefficients, and the discrete helical transform. Strong $(2, 2)$ for the double maximal (ergodic) helical transform is extended to actions of ${\mathbb{R}^d}$ and $ {\mathbb{Z}^d}$. It is also noted that the spectral measure of a measure-preserving flow has a continuity property at $\infty$, the Local Ergodic Theorem satisfies a Wiener-Wintner property, and the maximal helical transform is not weak $(1, 1)$.
A simple proof of the fundamental theorem of Kirby calculus on links
Ning
Lu
143-156
Abstract: In this paper, we relate surgeries on links and Heegaard decompositions, relate framed links and surface mapping classes, and give a simple proof of the fundamental theorem of Kirby calculus on links by the presentation of the surface mapping class groups.
A Chern character in cyclic homology
Luca Quardo
Zamboni
157-163
Abstract: We show that inner derivations act trivially on the cyclic cohomology of the normalized cyclic complex $ \mathcal{C}(\Omega)/\mathcal{D}(\Omega)$ where $\Omega$ is a differential graded algebra. This is then used to establish the fact that the map introduced in $ [$ GJ$ ]$ defines a Chern character in $K$ theory.
The structure of rings in some varieties with definable principal congruences
G. E.
Simons
165-179
Abstract: We study varieties of rings with identity that satisfy an identity of the form $xy = yp(x,y)$, where every term of the polynomial $p$ has degree greater than one. These varieties are interesting because they have definable principal congruences and are residually small. Let $\mathcal{V}$ be such a variety. The subdirectly irreducible rings in $ \mathcal{V}$ are shown to be finite local rings and are completely described. This results in structure theorems for the rings in $\mathcal{V}$ and new examples of noncommutative rings in varieties with definable principal congruences. A standard form for the defining identity is given and is used to show that $ \mathcal{V}$ also satisfies an identity of the form $xy = q(x,y)x$. Analogous results are shown to hold for varieties satisfying $ xy = q(x,y)x$.
On the resolution of a curve lying on a smooth cubic surface in ${\bf P}\sp 3$
Salvatore
Giuffrida;
Renato
Maggioni
181-201
Abstract: Let $C$ be any reduced and irreducible curve lying on a smooth cubic surface in ${\mathbb{P}^3}$. In this paper we determine the graded Betti numbers of the ideal sheaf ${\mathcal{J}_C}$.
On the genus of smooth $4$-manifolds
Alberto
Cavicchioli
203-214
Abstract: The projective complex plane and the "twisted" ${S^3}$ bundle over ${S^1}$ are proved to be the unique closed prime connected (smooth or PL) $4$-manifolds of genus two. Then the classification of the nonorientable $4$-manifolds of genus $4$ is given. Finally the genus of a manifold $ M$ is shown to be related with the $2$nd Betti number of $M$ and some applications are proved in the general (resp. simply-connected) case.
A quasiregular analogue of a theorem of Hardy and Littlewood
Craig A.
Nolder
215-226
Abstract: Suppose that $ f$ is analytic in the unit disk. A theorem of Hardy and Littlewood relates the Hölder continuity of $f$ over the unit disk to the growth of the derivative. We prove here a quasiregular analogue of this result in certain domains in $n$-dimensional space. We replace values of the derivative with a local integral average. In the process we generalize a result on the continuity of quasiconformal mappings due to Nakki and Palka. We also present another proof of the relationship between the growth of the derivative and quasiregular mappings in BMO.
An open collar theorem for $4$-manifolds
Craig R.
Guilbault
227-245
Abstract: Let ${M^4}$ be an open $4$-manifold with boundary. Conditions are given under which ${M^4}$ is homeomorphic to $\partial M \times [0,1)$. Applications include a $4$-dimensional weak $h$-cobordism theorem and a classification of weakly flat embeddings of $2$-spheres in ${S^4}$. Specific examples of $(n - 2)$-spheres embedded in $ {S^n}$ (including $ n = 4$) are also discussed.
L'espace des pseudo-arcs d'une surface
Robert
Cauty
247-263
Abstract: We prove that, for any surface $M$, the space of pseudo-arcs contained in $ M$ is homeomorphic to $M \times {l^2}$.
Frames associated with an abelian $l$-group
James J.
Madden
265-279
Abstract: Every archimedean $ l$-group (lattice-ordered group) with weak unit is shown to be isomorphic to a sub-$l$-group of the $l$-group of continuous realvalued functions on a Tychonoff locale canonically associated with the $ l$-group. This strengthens the classical Yosida representation theorem in a useful way. The proof uses methods from universal algebra and is constructive.
Two-dimensional Cremona groups acting on simplicial complexes
David
Wright
281-300
Abstract: We show that the $ 2$-dimensional Cremona group $\displaystyle \operatorname{Cr}_2 = \operatorname{Aut}_k\;k(X,Y)$ acts on a $ 2$-dimensional simplicial complex $C$, which has as vertices certain models in the function field $k(X,Y)$. The fundamental domain consists of one face $ F$. This yields a structural description of $ \operatorname{Cr}_2$ as an amalgamation of three subgroups along pairwise intersections. The subgroup ${\text{GA}}_2 = \operatorname{Aut}_k\;k[X,Y]$ (integral Cremona group) acts on $C$ by restriction. The face $ F$ has an edge $ E$ such that the ${\text{GA}}_2$ translates of $E$ form a tree $T$. The action of ${\text{GA}}_2$ on $T$ yields the well-known structure theory for ${\text{GA}}_2$ as an amalgamated free product, using Serre's theory of groups acting on trees.
A Haar-type theory of best $L\sb 1$-approximation with constraints
András
Kroó;
Darrell
Schmidt
301-319
Abstract: A general setting for constrained ${L^1}$-approximation is presented. Let $ {U_n}$ be a finite dimensional subspace of $C[a,b]$ and $L$ be a linear operator from ${U_n}$ to ${C^r}(K)\;(r = 0,1)$ where $K$ is a finite union of disjoint, closed, bounded intervals. For $\upsilon,u \in {C^r}(K)$ with $\upsilon < u$, the approximating set is $ {\tilde U_n}(\upsilon,u) = \{ p \in {U_n}:\upsilon \leq Lp \leq u\;{\text{on}}\;K\}$ and the norm is $\Vert f\Vert _w = \int_a^b {\vert f\vert w\,dx}$ where $w$ a positive continuous function on $[a,b]$. We obtain necessary and sufficient conditions for ${\tilde U_n}(\upsilon,u)$ to admit unique best $ \Vert\;\cdot\;\Vert _w$-approximations to all $ f \in C[a,b]$ for all positive continuous $w$ and all $\upsilon,u \in {C^r}(K)\;(r = 0,1)$ satisfying a nonempty interior condition. These results are applied to several ${L^1}$-approximation problems including polynomial and spline approximation with restricted derivatives, lacunary polynomial approximation with restricted derivatives, and others.
Gauge invariant quantization on Riemannian manifolds
Zhang Ju
Liu;
Min
Qian
321-333
Abstract: For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold $M$, a family of differential operators is given, which acts on the space of smooth sections of a vector bundle on $M$. Such a correspondence may be considered as a rule to quantize classical systems moving in a Riemannian manifold or in a gauge field. Some applications of our construction are also given in this paper.
On the $p$-adic completions of nonnilpotent spaces
A. K.
Bousfield
335-359
Abstract: This paper deals with the $p$-adic completion $ {F_{p\infty }}X$ developed by Bousfield-Kan for a space $X$ and prime $p$. A space $X$ is called ${F_p}$-good when the map $X \to {F_{p\infty }}X$ is a $\bmod$-$p$ homology equivalence, and called ${F_p}$-bad otherwise. General examples of ${F_p}$-good spaces are established beyond the usual nilpotent or virtually nilpotent ones. These include the polycyclic-by-finite spaces. However, the wedge of a circle with a sphere of positive dimension is shown to be ${F_p}$-bad. This provides the first example of an ${F_p}$-bad space of finite type and implies that the $p$-profinite completion of a free group on two generators must have nontrivial higher $\bmod$-$p$ homology as a discrete group. A major part of the paper is devoted to showing that the desirable properties of nilpotent spaces under the $ p$-adic completion can be extended to the wider class of $p$-seminilpotent spaces.
Normal form and linearization for quasiperiodic systems
Shui-Nee
Chow;
Kening
Lu;
Yun Qiu
Shen
361-376
Abstract: In this paper, we consider the following system of differential equations: $\displaystyle \dot \theta = \omega + \Theta (\theta,z), \quad \dot z = Az + f(\theta,z),$ where $ \theta \in {C^m}$, $\omega = ({\omega _1}, \ldots,{\omega _m}) \in {R^m}$, $z \in {C^n}$, $A$ is a diagonalizable matrix, $f$ and $\Theta$ are analytic functions in both variables and $2\pi$-periodic in each component of the vector $\theta,\Theta = O(\vert z\vert)$ and $f = O(\vert z{\vert^2})$ as $z \to 0$. We study the normal form of this system of the equations and prove that this system can be transformed to a system of linear equations $\displaystyle \dot \theta = \omega, \quad \dot z = Az$ by an analytic transformation provided that the eigenvalues of $A$ and the frequency $\omega$ satisfy certain small-divisor conditions.
Multipliers of families of Cauchy-Stieltjes transforms
R. A.
Hibschweiler;
T. H.
MacGregor
377-394
Abstract: For $\alpha > 0$ let ${\mathcal{F}_\alpha }$ denote the class of functions defined for $\vert z\vert < 1$ by integrating $1/{(1 - xz)^\alpha }$ against a complex measure on $ \vert x\vert= 1$. A function $g$ holomorphic in $\vert z\vert < 1$ is a multiplier of ${\mathcal{F}_\alpha }$ if $f \in {\mathcal{F}_\alpha }$ implies $gf \in {\mathcal{F}_\alpha }$. The class of all such multipliers is denoted by $ {\mathcal{M}_\alpha }$. Various properties of ${\mathcal{M}_\alpha }$ are studied in this paper. For example, it is proven that $\alpha < \beta$ implies $ {\mathcal{M}_\alpha } \subset {\mathcal{M}_\beta }$, and also that $ {\mathcal{M}_\alpha } \subset {H^\infty }$. Examples are given of bounded functions which are not multipliers. A new proof is given of a theorem of Vinogradov which asserts that if $f^{\prime}$ is in the Hardy class $ {H^1}$, then $f \in {\mathcal{M}_1}$. Also the theorem is improved to $ f^{\prime} \in {H^1}$ implies $ f \in {\mathcal{M}_\alpha }$, for all $ \alpha > 0$. Finally, let $\alpha > 0$ and let $f$ be holomorphic in $ \vert z\vert < 1$. It is known that $f$ is bounded if and only if its Cesàro sums are uniformly bounded in $ \vert z\vert \leq 1$. This result is generalized using suitable polynomials defined for $\alpha > 0$.
An asymptotic estimate for heights of algebraic subspaces
Jeffrey Lin
Thunder
395-424
Abstract: We count the number of subspaces of affine space with a given dimension defined over an algebraic number field with height less than or equal to $B$. We give an explicit asymptotic estimate for the number of such subspaces as $B$ goes to infinity, where the constants involved depend on the classical invariants of the number field (degree, discriminant, class number, etc.). The problem is reformulated as an estimate for the number of lattice points in a certain bounded domain.
A simplified trace formula for Hecke operators for $\Gamma\sb 0(N)$
Shepley L.
Ross
425-447
Abstract: Let $N$ and $n$ be relatively prime positive integers, let $ \chi$ be a Dirichlet character modulo $N$, and let $k$ be a positive integer. Denote by ${S_k}(N,\chi)$ the space of cusp forms on ${\Gamma _0}(N)$ of weight $k$ and character $\chi$, a space denoted simply $ {S_k}(N)$ when $ \chi$ is the trivial character. Beginning with Hijikata's formula for the trace of ${T_n}$ acting on $ {S_k}(N,\chi)$, we develop a formula which essentially reduces the computation of this trace to looking up values in a table. From this formula we develop very simple formulas for (1) the dimension of $ {S_k}(N,\chi)$ and (2) the trace of ${T_n}$ acting on ${S_k}(N)$.
Noetherian ring extensions with trace conditions
Robert B.
Warfield
449-463
Abstract: Finite ring extensions of Noetherian rings with certain restrictions on the corresponding trace ideals are studied. This setting includes finite free extensions and extensions arising from actions of finite groups when the order of the group is invertible. In this setting we establish the following results which were previously obtained (for finite extensions without trace conditions) only under strong restrictions on the rings involved. Let $R \subset S$ be an extension of Noetherian rings such that $S$ is finitely generated as a left $R$-module and such that the left trace ideal of $S$ in $R$ is equal to $R$. If $S$ is right fully bounded, or is a Jacobson ring, then $R$ has the same property; furthermore, $R$ and $S$ have the same classical Krull dimension. If $S$ is finitely generated as both a right and a left $ R$-module, if both trace ideals of $S$ in $R$ are equal to $R$, and if $S$ satisfies the strong second layer condition, then this condition also holds in $R$. Finally, we compare the link graphs of $ R$ and $S$