Holomorphic extension and decomposition from a totally real manifold
Zai Fei
Ye
1-33
Abstract: This paper is to develop an elementary cohomological approach for decomposing a function into boundary values of holomorphic functions and for discussing the corresponding microlocal analysis and hyperfunction theory.
Irredundant sets in Boolean algebras
Stevo
Todorčević
35-44
Abstract: It is shown that every uncountable Boolean algebra $A$ contains an uncountable subset $ I$ such that no $ a$ of $I$ is in the subalgebra generated by $I\backslash \{ a\}$ using an additional axiom of set theory. It is also shown that a use of some such axiom is necessary.
$\Delta$-sets
R. W.
Knight
45-60
Abstract: A model of ${\text{ZFC}}$ is constructed in which there exists a subset of the Moore plane that is countably paracompact but not normal. The method used in the construction is forcing using uncountable sets of finite partial functions, $ {\omega _1}$ and ${\omega _2}$ are shown to be preserved using a fusion lemma.
Classification of the Tor-algebras of codimension four almost complete intersections
Andrew R.
Kustin
61-85
Abstract: Let $(R,m,k)$ be a local ring in which $ 2$ is a unit. Assume that every element of $k$ has a square root in $k$. We classify the algebras $ \operatorname{Tor}_ \bullet ^R(R/J,k)$ as $J$ varies over all grade four almost complete intersection ideals in $R$ . The analogous classification has already been found when $J$ varies over all grade four Gorenstein ideals [21], and when $J$ varies over all ideals of grade at most three [5, 30]. The present paper makes use of the classification, in [21], of the Tor-algebras of codimension four Gorenstein rings, as well as the (usually nonminimal) $ {\text{DG}}$-algebra resolution of a codimension four almost complete intersection which is produced in [25 and 26].
Complex multiplication cycles and a conjecture of Be\u\i linson and Bloch
Chad
Schoen
87-115
Abstract: A generalization of the conjecture of Birch and Swinnerton-Dyer is investigated using complex multiplication cycles on a particular Kuga fiber variety. A weak finiteness result consistent with the conjecture is proved. The image of complex multiplication cycles under the Abel-Jacobi map is computed explicitly. The results provide numerical evidence supporting the conjecture. They also give evidence for a relationship between complex multiplication cycles and a modular form of weight $5/2$ and raise questions for further investigation.
Localizing with respect to self-maps of the circle
Carles
Casacuberta;
Georg
Peschke
117-140
Abstract: We describe a general procedure to construct idempotent functors on the pointed homotopy category of connected ${\text{CW}}$-complexes, some of which extend $ P$-localization of nilpotent spaces, at a set of primes $P$. We focus our attention on one such functor, whose local objects are ${\text{CW}}$-complexes $X$ for which the $p$th power map on the loop space $\Omega X$ is a self-homotopy equivalence if $p \notin P$. We study its algebraic properties, its behaviour on certain spaces, and its relation with other functors such as Bousfield's homology localization, Bousfield-Kan completion, and Quillen's plus-construction.
Periodic seeded arrays and automorphisms of the shift
Ezra
Brown
141-161
Abstract: The automorphism group $\operatorname{Aut}({\Sigma _2})$ of the full $ 2$-shift is conjectured to be generated by the shift and involutions. We approach this problem by studying a certain family of automorphisms whose order was unknown, but which we show to be finite and for which we find factorizations as products of involutions. The result of this investigation is the explicit construction of a subgroup $\mathcal{H}$ of $\operatorname{Aut}({\Sigma _2})$ ; $\mathcal{H}$ is generated by certain involutions $ {g_n}$, and turns out to have a number of curious properties. For example, $ {g_n}$ and ${g_k}$ commute unless $n$ and $k$ are consecutive integers, the order of $ {g_{n + k}} \circ \cdots \circ {g_k}$ is independent of $k$, and $ \mathcal{H}$ contains elements of all orders. The investigation is aided by the development of results about certain new types of arrays of 0's and $1$'s called periodic seeded arrays, as well as the use of Boyle and Krieger's work on return numbers and periodic points.
Cayley-Bacharach schemes and their canonical modules
Anthony V.
Geramita;
Martin
Kreuzer;
Lorenzo
Robbiano
163-189
Abstract: A set of $ s$ points in ${\mathbb{P}^d}$ is called a Cayley-Bacharach scheme ( ${\text{CB}}$-scheme), if every subset of $s - 1$ points has the same Hilbert function. We investigate the consequences of this "weak uniformity." The main result characterizes $ {\text{CB}}$-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a $ {\text{CB}}$-scheme $ X$ has to satisfy growth conditions which are only slightly weaker than the ones given by Harris and Eisenbud for points with the uniform position property. We also characterize ${\text{CB}}$-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points. Applications include efficient algorithms for checking whether a given set of points is a $ {\text{CB}}$-scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hilbert functions of Cohen-Macaulay domains.
$3$-manifolds which admit finite group actions
Shi Cheng
Wang
191-203
Abstract: We prove several results which support the following conjectures: $ (1)$ Any smooth action of a finite group on a geometric $3$-manifold can be conjugated to preserve the geometric structure. $(2)$ Every irreducible closed $3$-manifold $M$ with infinite $ {\pi _1}(M)$ is finitely covered by a Haken $3$-manifold.
A metric deformation and the first eigenvalue of Laplacian on $1$-forms
Takashi
Otofuji
205-220
Abstract: We search for a higher-dimensional analogue of Calabi's example of a metric deformation, quoted by Cheeger, which inspired him to prove an inequality between the first eigenvalue of the Laplacian on functions and an isoperimetric constant. We construct an example of a metric deformation on ${S^n}$, ${n} \geq 5$, where the first eigenvalue of the Laplacian on functions remains bounded above from zero, and the first eigenvalue of the Laplacian on $1$-forms tends to zero. This metric deformation makes the sphere in the limit into a manifold with a cone singularity, which is an intermediate point on a path of deformation from an (${S^n}$, some metric) to an ( ${S^{n - 1}} \times {S^1}$, some metric).
Analyse quasi-sure et l'estimation du noyau de la chaleur pour temps petit
Shizan
Fang
221-241
Abstract: The Ito functional can be redefined out of a slim set by the natural way. Quasi-sure analysis is used to deal with the heat kernel asymptotic problems.
$\omega$-chaos and topological entropy
Shi Hai
Li
243-249
Abstract: We present a new concept of chaos, $\omega$-chaos, and prove some properties of $ \omega$-chaos. Then we prove that $\omega$-chaos is equivalent to positive entropy on the interval. We also prove that $\omega$-chaos is equivalent to the definition of chaos given by Devaney on the interval.
$2$-weights for unitary groups
Jian Bei
An
251-278
Abstract: This paper gives a description of the local structures of $2$-radical subgroups in a finite unitary group and proves Alperin's weight conjecture for finite unitary groups when the characteristic of modular representation is even.
Approximation of approximate fibrations by bundle maps
Y. H.
Im
279-295
Abstract: In this paper, we give some conditions under which approximate fibrations can be approximated by locally trivial bundle maps.
Reflecting Brownian motion in a cusp
R. Dante
DeBlassie;
Ellen H.
Toby
297-321
Abstract: Let $C$ be the cusp $\{ (x,y):x \geq 0$, $- {x^\beta } \leq y \leq {x^\beta }\}$ where $ \beta > 1$. Set $\partial {C_1} = \{ (x,y):x \geq 0, y = - {x^\beta }\}$ and $\partial {C_2} = \{ (x,y):x \geq 0$, $y = {x^\beta }\}$. We study the existence and uniqueness in law of reflecting Brownian motion in $ C$. The angle of reflection at $ \partial {C_j}\backslash \{ 0\}$ (relative to the inward unit normal) is a constant ${\theta _j} \in \left( { - \frac{\pi } {2},\frac{\pi } {2}} \right)$, and is positive iff the direction of reflection has a negative first component in all sufficiently small neighborhoods of 0. When $ {\theta _1} + {\theta _2} \leq 0$, existence and uniqueness in law hold. When $ {\theta _1} + {\theta _2} > 0$, existence fails. We also obtain results for a large class of asymmetric cusps. We make essential use of results of Warschawski on the differentiability at the boundary of conformal maps.
Some complete $\Sigma\sp 1\sb 2$ sets in harmonic analysis
Howard
Becker;
Sylvain
Kahane;
Alain
Louveau
323-336
Abstract: We prove that several specific pointsets are complete $\Sigma _2^1$ (complete PCA). For example, the class of ${N_0}$-sets, which is a hereditary class of thin sets that occurs in harmonic analysis, is a pointset in the space of compact subsets of the unit circle; we prove that this pointset is complete $\Sigma _2^1$. We also consider some other aspects of descriptive set theory, such as the nonexistence of Borel (and consistently with $ {\text{ZFC}}$, the nonexistence of universally measurable) uniformizing functions for several specific relations. For example, there is no Borel way (and consistently, no measurable way) to choose for each ${N_0}$-set, a trigonometric series witnessing that it is an ${N_0}$-set.
The Koebe semigroup and a class of averaging operators on $H\sp p({\bf D})$
Aristomenis G.
Siskakis
337-350
Abstract: We study on the Hardy space ${H^p}$ the operators ${T_F}$ given by $\displaystyle {T_F}(f)(z) = \frac{1} {z}\int_0^z {f(\zeta )\frac{1} {{F(\zeta )}}\;d\zeta }$ where $ F(z)$ is analytic on the unit disc $ \mathbb{D}$ and has $ \operatorname{Re} F(z) \geq 0$. Each such operator is closely related to a strongly continuous semigroup of weighted composition operators. By studying first an extremal such semigroup (the Koebe semigroup) we are able to obtain the upper bound ${\left\Vert {{T_F}} \right\Vert _p} \leq 2p\operatorname{Re} (1/F(0)) + \vert\operatorname{Im} (1/F(0))\vert$ for the norm. We also show that $ {T_F}$ is compact on $ {H^p}$ if and only if the measure $\mu$ in the Herglotz representation of $ 1/F$ is continuous.
Probing L-S category with maps
Barry
Jessup
351-360
Abstract: For any map $X\xrightarrow{f}Y$, we introduce two new homotopy invariants, $ {\text{dcat}}\;f$ and ${\text{rcat}}\;f$. The classical category ${\text{cat}}\;f$ is a lower bound for both, while $ {\text{dcat}}\;f \leq {\text{cat}}\;X$ and ${\text{rcat}}\;f \leq {\text{cat}}\;Y$. When $ Y$ is an Eilenberg-Mac Lane space, $f$ represents a cohomology class and ${\text{dcat}}\;f$ often gives a good estimate for $ {\text{cat}}\;X$. We prove that if $\Omega \in {H^n}(M;\mathbb{Z})$ is the fundamental class of a compact, simply connected $ n$-manifold, then $ {\text{dcat}}\;\Omega = {\text{cat}}\;M$. Similarly, when $X$ is sphere, then $f$ is a homotopy class and while ${\text{cat}}\;f = 1$, ${\text{rcat}}\;f$ can be a good approximation to ${\text{cat}}\;Y$. We show that if $ \alpha \in {\pi _2}(\mathbb{C}{P^n})$ is nonzero, then ${\text{rcat}}\;\alpha = n$. Rational analogues are introduced and we prove that for $u \in {H^\ast}(X;\mathbb{Q})$, ${\text{dcat}_0}\;u = 1 \Leftrightarrow {u^2} = 0$ and $u$ is spherical.
The limiting behavior of the Kobayashi-Royden pseudometric
Shulim
Kaliman
361-371
Abstract: We study the limit of the sequence of Kobayashi metrics of Riemann surfaces (when these Riemann surfaces form an analytic fibration in such a way that the total space of fibration becomes a complex surface), as the fibers approach the center fiber which is not in general smooth. We prove that if the total space is a Stein surface and the smooth part of the center fiber contains a component biholomorphic to a quotient of the disk by a Fuchsian group of first kind, then the Kobayashi metrics of the near-by fibers converge to the Kobayashi metric of this component as fibers tend to the center fiber.
Rees algebras of ideals having small analytic deviation
Sam
Huckaba;
Craig
Huneke
373-402
Abstract: In this article we identify two large families of ideals of a Cohen-Macaulay (sometimes Gorenstein) local ring whose Rees algebras are Cohen-Macaulay. Our main results imply, for example, that if $(R,M)$ is a regular local ring and $P$ is a prime ideal of $R$ such that ${P^n}$ is unmixed for all $n \geq 1$, then the Rees algebra $ R[Pt]$ is Cohen-Macaulay if either $ \dim (R/P) = 2$, or $\dim (R/P) = 3,R/P$ is Cohen-Macaulay, and $ R/P$ is integrally closed.
The constrained least gradient problem in ${\bf R}\sp n$
Peter
Sternberg;
Graham
Williams;
William P.
Ziemer
403-432
Abstract: We consider the constrained least gradient problem $\displaystyle \inf \left\{ {\int_\Omega {\vert\nabla u\vert dx:u \in {C^{0,1}}(... ...a u\vert \leq 1\;{\text{a.e.}},u = g\;{\text{on}}\;\partial \Omega } } \right\}$ which arises as the relaxation of a nonconvex problem in optimal design. We establish the existence of a solution by an explicit construction in which each level set is required to solve an obstacle problem. We also establish the uniqueness of solutions and discuss their structure.
Calcul du spectre d'une nilvari\'et\'e de rang deux et applications
Hubert
Pesce
433-461
Abstract: Résumé. On calcule, en utilisant la théorie des orbites de Kirillov, le spectre d'une nilvariété compacte de rang deux. Puis on utilise ce calcul pour étudier et caractériser les déformations isospectrales de ces variétés.