Sums of linked ideals
Bernd
Ulrich
1-42
Abstract: It is shown that the sum of two geometrically linked ideals in the linkage class of a complete intersection is again an ideal in the linkage class of a complete intersection. Conversely, every Gorenstein ideal (of height at least two) in the linkage class of a complete intersection can be obtained as a "generalized localization" of a sum of two geometrically linked ideals in the linkage class of a complete intersection. We also investigate sums of doubly linked Gorenstein ideals. As an application, we construct a perfect prime ideal which is strongly nonobstructed, but not strongly Cohen-Macaulay, and a perfect prime ideal which is not strongly nonobstructed, but whose entire linkage class is strongly Cohen-Macaulay.
Unknotted homology classes on unknotted surfaces in $S\sp 3$
Bruce
Trace
43-56
Abstract: Suppose $ F$ is a closed, genus $ g$ surface which is standardly embedded in ${S^3}$. Let $\gamma$ denote a primitive element in $ {H_1}(F)$ which satisfies ${\theta _F}(\gamma ,\gamma ) = 0$ where ${\theta _F}$ is the Seifert pairing on $ F$. We obtain a number theoretic condition which is equivalent to $\gamma$ being realizable by a curve (in $F$) which is unknotted in ${S^3}$. Various related observations are included.
The minimal normal extension for $M\sb z$ on the Hardy space of a planar region
John
Spraker
57-67
Abstract: Multiplication by the independent variable on ${H^2}(R)$ for $R$ a bounded open region in the complex plane $\mathbb{C}$ is a subnormal operator. This paper characterizes its minimal normal extension $ N$. Any normal operator is determined by a scalar-valued spectral measure and a multiplicity function. It is a consequence of some standard operator theory that a scalar-valued spectral measure for $N$ is harmonic measure for $R$, $\omega$. This paper investigates the multiplicity function $m$ for $N$. It is shown that $m$ is bounded above by two $\omega $-a.e., and necessary and sufficient conditions are given for $m$ to attain this upper bound on a set of positive harmonic measure. Examples are given which indicate the relationship between $N$ and the boundary of $R$.
Some weighted inequalities on product domains
Henry
Lin
69-85
Abstract: We extend the results of R. Fefferman [3] on the bidisc to higher product domains via induction. As an application, we extend the weighted inequality for Calderon-Zygmund operators on the bidisc to higher product domains, and we also extend the result of the Littlewood-Paley operator corresponding to the arbitrary disjoint rectangles to the weighted case.
Abelian and nondiscrete convergence groups on the circle
A.
Hinkkanen
87-121
Abstract: A group $ G$ of homeomorphisms of the unit circle onto itself is a convergence group if every sequence of elements of $G$ contains a subsequence, say ${{\text{g}}_n}$, such that either (i) $ {{\text{g}}_n} \to {\text{g}}$ and ${\text{g}}_n^{ - 1} \to {{\text{g}}^{ - 1}}$ uniformly on the circle where $ {\text{g}}$ is a homeomorphism, or (ii) ${{\text{g}}_n} \to {{\text{x}}_0}$ and $ {\text{g}}_n^{ - 1} \to {{\text{y}}_0}$ uniformly on compact subsets of the complements of $ \{ {{\text{y}}_0}\}$ and $ \{ {{\text{x}}_0}\}$, respectively, for some points ${{\text{x}}_0}$ and $ {{\text{y}}_0}$ of the circle (possibly ${{\text{x}}_0}{\text{ = }}{{\text{y}}_0}$). For example, a group of $K$-quasisymmetric maps, for a fixed $K$, is a convergence group. We show that if $G$ is an abelian or nondiscrete convergence group, then there is a homeomorphism $f$ such that $f \circ G \circ {f^{ - 1}}$ is a group of Màbius transformations.
Outlet points and homogeneous continua
Paweł
Krupski;
Janusz R.
Prajs
123-141
Abstract: (1) A proof is presented for Bing's conjecture that homogeneous, treelike continua are hereditarily indecomposable. As a consequence, each homogeneous curve admits the continuous decomposition into the maximal terminal, homeomorphic, homogeneous, hereditarily indecomposable, treelike subcontinua. (2) A homogeneous, hereditarily unicoherent continuum contains either an arc or arbitrarily small, nondegenerate, indecomposable subcontinua. (3) A treelike continuum with property $K$ which is homogeneous with respect to confluent light mappings contains no two nondegenerate subcontinua with the one-point intersection.
Homogeneous continua in Euclidean $(n+1)$-space which contain an $n$-cube are $n$-manifolds
Janusz R.
Prajs
143-148
Abstract: Let $X$ be a homogeneous continuum and let ${E^n}$ be Euclidean $n$-space. We prove that if $X$ is properly contained in a connected $ (n + 1)$-manifold, then $ X$ contains no $ n$-dimensional umbrella (i.e. a set homeomorphic to the set $\{ ({x_1}, \ldots ,{x_{n + 1}}) \in {E^{n + 1}}:x_1^2 + \cdots + x_{n + 1}^2 \leq 1$ and ${x_{n + 1}} \leq 0$ and either ${x_1} = \cdots = {x_n} = 0$ or ${x_{n + 1}} = 0\}$). Combining this fact with an earlier result of the author we conclude that if $ X$ lies in ${E^{n + 1}}$ and topologically contains $ {E^n}$, then $ X$ is an $ n$-manifold.
Leray functor and cohomological Conley index for discrete dynamical systems
Marian
Mrozek
149-178
Abstract: We introduce the Leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological Conley index of an isolated invariant set of a homeomorphism on a locally compact metric space. We prove the homotopy and additivity properties for this index and compute the index in some examples. As one of applications we prove the existence of nonconstant, bounded solutions of the Euler approximation of a certain system of ordinary differential equations.
Existence of weak solutions for the Navier-Stokes equations with initial data in $L\sp p$
Calixto P.
Calderón
179-200
Abstract: The existence of weak solutions for the Navier-Stokes equations for the infinite cylinder with initial data in $ {L^p}$ is considered in this paper. We study the case of initial data in ${L^p}({R^n})$, $2 < p < n$, and $n = 3,4$. An existence theorem is proved covering these important cases and therefore, the "gap" between the Hopf-Leray theory $(p = 2)$ and that of Fabes-Jones-Riviere $ (p > n)$ is bridged. The existence theorem gives a new method of constructing global solutions. The cases $p = n$ are treated at the end of the paper.
Addendum to the paper: ``Existence of weak solutions for the Navier-Stokes equations with initial data in $L\sp p$'' [Trans. Amer. Math. Soc. {\bf 318} (1990), no. 1, 179--200; MR0968416 (90k:35199)]
Calixto P.
Calderón
201-207
Abstract: This paper considers the existence of global weak solutions for the Navier-Stokes equations in the infinite cylinder $ {{\mathbf{R}}^n} \times {{\mathbf{R}}_ + }$ with initial data in $ {L^r}$, $n \geq 3$, $1 < r < \infty$. An imbedding theorem as well as related initial value problems are also studied, thus completing results in [2].
A classification of Baire class $1$ functions
A. S.
Kechris;
A.
Louveau
209-236
Abstract: We study in this paper various ordinal ranks of (bounded) Baire class $ 1$ functions and we show their essential equivalence. This leads to a natural classification of the class of bounded Baire class $ 1$ functions ${\mathcal{B}_1}$ in a transfinite hierarchy $ \mathcal{B}_1^\xi (\xi < {\omega _1})$ of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of $ \Delta _2^0$ sets via transfinite differences of closed sets can be carried out. The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class $ 1$ function $f$ are introduced and used in this study.
Inequalities for eigenvalues of selfadjoint operators
Stephen M.
Hook
237-259
Abstract: We establish several inequalities for eigenvalues of selfadjoint operators in Hilbert space. The results are quite general. In particular, let $\Omega$ be a region in ${{\mathbf{R}}^n},\partial \Omega$ its boundary and $ \Delta$ the Laplace operator in $ {{\mathbf{R}}^n}$. Let $ p(x)$ be a polynomial of degree $m$ having nonnegative real coefficients. We show that if the problems (1) $- \Delta u = \lambda u$ in $\Omega ;u = 0$ on $ \partial \Omega$; (2) $p( - \Delta )\upsilon = \mu \upsilon$ in $\Omega ;\upsilon$ and its first $m - 1$ derivatives$ =0$ on$ \partial \Omega$; and (3) $ {( - \Delta )^m}w = vw$ in $ \Omega ;w$ and its first $m - 1$ derivatives$ =0$ on$ \partial \Omega$ are selfadjoint with discrete spectra of finite multiplicity ${\lambda _1} \leq {\lambda _2} \leq \cdots$ etc. then (4) $p(\Gamma _i^{1/m}) \geq {\mu _i} \geq p({\lambda _i})$ for each index $i$. The set of problems (1), (2), (3) and the result (4) is only one example of our more general result. The above problems (1), (2), and (3) can be thought of as related through the single operator given by the Laplacian. We also establish results for eigenvalues for unrelated operators. Let $ A$, $B$ and $A + B$ be selfadjoint on domains ${D_A},{D_B}$, and $ {D_{A + B}}$ with $ {D_{A + B}} \subseteq {D_A} \cap {D_B}$. If $A$, $B$, and $A + B$ have discrete spectra $\{ {\lambda _i}\} _{i = 1}^\infty ,\{ {\mu _i}\} _{i = 1}^\infty$ and $\{ {\Gamma _i}\} _{i = 1}^\infty$ arranged in ascending order, as above, then inequality (5) $\sum\nolimits_{i = 1}^n {{\Gamma _i}} \geq \sum\nolimits_{i = 1}^n {({\lambda _i} + {v_i})}$ is established for each positive integer $n$.
Algebraically invariant extensions of $\sigma$-finite measures on Euclidean space
Krzysztof
Ciesielski
261-273
Abstract: Let $G$ be a group of algebraic transformations of $ {{\mathbf{R}}^n}$, i,e., the group of functions generated by bijections of $ {{\mathbf{R}}^n}$ of the form $ ({f_1}, \ldots ,{f_n})$ where each ${f_i}$ is a rational function with coefficients in ${\mathbf{R}}$ in $n$-variables. For a function $\gamma :G \to (0,\infty )$ we say that a measure $ \mu$ on ${{\mathbf{R}}^n}$ is $\gamma$-invariant when $\mu (g[A]) = \gamma (g)\cdot\mu (A)$ for every $ g \in G$ and every $ \mu$-measurable set $ A$. We will examine the question: "Does there exist a proper $\gamma $-invariant extension of $ \mu ?$ We prove that if $ \mu$ is $\sigma $-finite then such an extension exists whenever $G$ contains an uncountable subset of rational functions $H \subset {({\mathbf{R}}({X_1}, \ldots ,{X_n}))^n}$ such that $\mu (\{ x:{h_1}(x) = {h_2}(x)\} ) = 0$ for all $ {h_1},{h_2} \in H,{h_1} \ne {h_2}$. In particular if $G$ is any uncountable subgroup of affine transformations of ${{\bf {R}}^n},\gamma (g{\text{)}}$ is the absolute value of the Jacobian of $g \in G$ and $\mu$ is a $\gamma$-invariant extension of the $ n$-dimensional Lebesgue measure then $\mu$ has a proper $\gamma$-invariant extension. The conclusion remains true for any $\sigma$-finite measure if $G$ is a transitive group of isometries of ${{\mathbf{R}}^n}$. An easy strengthening of this last corollary gives also an answer to a problem of Harazisvili.
Unique continuation for $\Delta+v$ and the C. Fefferman-Phong class
Sagun
Chanillo;
Eric
Sawyer
275-300
Abstract: We show that the strong unique continuation property holds for the inequality $\left\vert {\Delta u} \right\vert \leq \left\vert \upsilon \right\vert\left\vert u \right\vert$, where the potential $\upsilon (x)$ satisfies the C. Fefferman-Phong condition in a certain range of $p$ values. We also deal with the situation of $ u(x)$ vanishing at infinity. These are all consequences of appropriate Carleman inequalities.
Massey products in the cohomology of groups with applications to link theory
David
Stein
301-325
Abstract: Invariants of links in ${S^3}$ are developed using a modification of the Massey product of one-dimensional classes in the cohomology of certain groups. The theory yields two types of invariants, invariants which depend upon a collection of meridians, or basing, of a link, and invariants which do not. The invariants, which are independent of the basing, are compared with John Milnor's $\overline \mu $-invariants. For two component links, a collection of ostensibly based invariants is shown to be independent of the basing. If the linking number of the components of such a link is zero, the resulting invariants may be equivalent to the Sato-Levine-Cochran invariants.
On the nonimmersion of products of real projective spaces
Hyun-Jong
Song;
W. Stephen
Wilson
327-334
Abstract: In this paper we utilize $B{P^*}(\;)$, a generalized cohomology theory associated with the Brown-Peterson spectrum to prove a nonimmersion theorem for products of real projective spaces.
A Hurewicz spectral sequence for homology
David A.
Blanc
335-354
Abstract: For any connected space $ {\mathbf{X}}$ and ring $ R$, we describe a first-quadrant spectral sequence converging to ${\tilde H_*}({\bf {X}};R)$, whose $ {E^2}$-term depends only on the homotopy groups of $ {\mathbf{X}}$ and the action of the primary homotopy operations on them. We show that (for simply connected $ {\mathbf{X}}$) the $ {E^2}$-term vanishes below a line of slope $1/2$; computing part of the ${E^2}$-term just above this line, we find a certain periodicity, which shows, in particular, that this vanishing line is best possible. We also show how the differentials in this spectral sequence can be used to compute certain Toda brackets.
Weighted norm estimates for the Fourier transform with a pair of weights
Jan-Olov
Strömberg;
Richard L.
Wheeden
355-372
Abstract: We prove weighted norm inequalities of the form $\displaystyle {\left\Vert {\hat f} \right\Vert _{L_u^q}} \leq C{\left\Vert f \right\Vert _{H_\upsilon ^p}},\quad 0 < p \leq q < \infty ,$ for the Fourier transform on ${{\mathbf{R}}^n}$. For some weight functions $ \upsilon$, the Hardy space $H_\upsilon ^p$ on the right can be replaced by $L_\upsilon ^p$. The proof depends on making an atomic decomposition of $ f$ and using cancellation properties of the atoms.
Cells and the reflection representation of Weyl groups and Hecke algebras
J. Matthew
Douglass
373-399
Abstract: Let $\mathcal{H}$ be the generic algebra of the finite crystallographic Coxeter group $W$, defined over the ring $ \mathbb{Q}[{u^{1/2}},{u^{ - 1/2}}]$. First, the two-sided cell corresponding to the reflection representation of $\mathcal{H}$ is shown to consist of the nonidentity elements of $W$ having a unique reduced expression. Next, the matrix entries of this representation are computed in terms of certain Kazhdan-Lusztig polynomials. Finally, the Kazhdan-Lusztig polynomials just mentioned are described in case $W$ is of type ${{\text{A}}_{l - 1}}$ or ${{\text{B}}_l}$.
A one-phase hyperbolic Stefan problem in multi-dimensional space
De Ning
Li
401-415
Abstract: The hyperbolic heat transfer model is obtained by replacing the classical Fourier's law with the relaxation relation $ \tau \vec qt + \vec q = - k\nabla T$. The sufficient and necessary conditions are derived for the local existence and uniqueness of classical solutions for multi- ${\text{D}}$ Stefan problem of hyperbolic heat transfer model where phase change is accompanied with delay of latent heat storage.