Generalized iteration of forcing
M.
Groszek;
T.
Jech
1-26
Abstract: Generalized iteration extends the usual notion of iterated forcing from iterating along an ordinal to iterating along any partially ordered set. We consider a class of forcings called perfect tree forcing. The class includes Axiom A forcings with a finite splitting property, such as Cohen, Laver, Mathias, Miller, Prikry-Silver, and Sacks forcings. If $ \mathcal{P}$ is a perfect tree forcing, there is a decomposition $\mathcal{Q} * \mathcal{R}$ such that $\mathcal{Q}$ is countably closed, $\mathcal{R}$ has the countable chain condition, and $ \mathcal{Q} * \mathcal{R}$ adds a $ \mathcal{P}$-generic set. Theorem. The mixed-support generalized iteration of perfect tree forcing decompositions along any well-founded partial order preserves ${\omega _1}$. Theorem. If ${\text{ZFC}}$ is consistent, so is $ {\text{ZFC + }}{{\text{2}}^\omega }$ is arbitrarily large + whenever $\mathcal{P}$ is a perfect tree forcing and $\mathcal{D}$ is a collection of ${\omega _1}$ dense subsets of $\mathcal{P}$, there is a $\mathcal{D}$-generic filter on $\mathcal{P}$.
Zeros of solutions and of Wronskians for the differential equation $L\sb ny+p(x)y=0$
Uri
Elias
27-40
Abstract: The equation which is studied here is ${L_n}y + p(x)y = 0,a \leq x \leq b$, where $ {L_n}$ is a disconjugate differential operator and $p(x)$ is of a fixed sign. We prove that certain solutions of the equation and corresponding odd-order minors of the Wronskian have an equal number of zeros, and we apply this property to oscillation problems.
Relative cohomology and projective twistor diagrams
S. A.
Huggett;
M. A.
Singer
41-57
Abstract: The use of relative cohomology in the investigation of functionals on tensor products of twistor cohomology groups is considered and yields a significant reduction in the problem of looking for contours for the evaluation of (projective) twistor diagrams. The method is applied to some simple twistor diagrams and is used to show that the standard twistor kernel for the first order massless scalar ${\phi^4}$ vertex admits a (cohomological) contour for only one of the physical channels. A new kernel is constructed for the ${\phi^4}$ vertex which admits contours for all channels.
Box-spaces and random partial orders
Béla
Bollobás;
Graham
Brightwell
59-72
Abstract: Winkler [2] studied random partially ordered sets, defined by taking $ n$ points at random in $ {[0,1]^d}$, with the order on these points given by the restriction of the order on ${[0,1]^d}$. Bollobás and Winkler [1] gave several results on the height of such a random partial order. In this paper, we extend these results to a more general setting. We define a box-space to be, roughly speaking, a partially ordered measure space such that every two intervals of nonzero measure are isomorphic up to a scale factor. We give some examples of box-spaces, including (i) ${[0,1]^d}$ with the usual measure and order, and (ii) Lorentzian space-time with the order given by causality. We show that, for every box-space, there is a constant $d$ which behaves like the dimension of the space. In the second half of the paper, we study random partial orders defined by taking a Poisson distribution on a box-space. (This is of course essentially the same as taking $n$ random points in a box-space.) We extend the results of Bollobás and Winkler to these random posets. In particular we show that, for a box-space $ X$ of dimension $ d$, there is a constant $ {m_X}$ such that the length of a longest chain tends to ${m_X}{n^{1/d}}$ in probability.
Strong Bertini theorems
Steven
Diaz;
David
Harbater
73-86
Abstract: We show that the singular locus of the general member of a linear system has dimension less than that predicted by Bertini's theorem, provided that the base locus is scheme-theoretically smooth. As corollaries, we obtain a result about complete intersection varieties containing a given subvariety and a result concerning liaison.
Nonsingular algebraic curves in ${\bf R}{\rm P}\sp 1\times{\bf R}{\rm P}\sp 1$
Sachiko
Matsuoka
87-107
Abstract: We give some restrictions for the mutual position of the connected components of a nonsingular algebraic curve in the product space ${\mathbf{R}}{P^1} \times {\mathbf{R}}{P^1}$ of two real projective lines. We obtain our main theorem by calculating the Brown invariant of a certain quadratic form determined by the algebraic curve. Moreover, we consider a double covering of $ {\mathbf{C}}{P^1} \times {\mathbf{C}}{P^1}$ branched along the complexification of our curve and antiholomorphic involutions that are the lifts of the complex conjugation.
Boundary value problems for degenerate elliptic-parabolic equations of the fourth order
Robert G.
Root
109-134
Abstract: We consider boundary value problems for the fourth-order linear equation $\displaystyle {A^{ijrs}}{u_{ijrs}} + {A^{ijr}}{u_{ijr}} + {A^{ij}}{u_{ij}} - \gamma {({a^{ij}}{u_i})_j} + {A^i}{u_i} + Fu = f\quad {\text{in}}\overline \Omega$ with smooth coefficients. The fourth-order part may degenerate on arbitrary subsets of $ \overline \Omega$ i.e., $ {A^{ijrs}}(x){m_{ij}}{m_{rs}} \geq 0$ for all $n \times n$ matrices $ M$, with no restriction on where equality occurs. We assume the ${a^{ij}}$ part of the operator is uniformly elliptic (of second order) on $\Omega$ while $\gamma$ is a parameter allowing us to increase modulus of ellipticity as much as needed. As in Fichera's second-order elliptic-parabolic equations [see, for example, Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. (8) 5 (1956), 1-30], because of the degeneracy, there may be characteristic portions of the boundary; however, we restrict our attention to the noncharacteristic case. We define a weak solution to the Dirichlet problem and obtain existence and uniqueness results. The question of regularity is addressed; elliptic regularization is used to obtain a Sobolev-type global regularity result. The equation models an anisotropic, inhomogeneous plate under tension that can lose stiffness at any point and in any direction. The regularity result has the satisfying physical interpretation that sufficient tension results in a smooth solution.
On vector bundles on $3$-folds with sectional genus $1$
Edoardo
Ballico
135-147
Abstract: Here we give a classification (in characteristic zero) of pairs $(V,E)$ with $V$ being a smooth, connected, complete $ 3$-fold and $E$ a rank-$2$ spanned ample vector bundle on $ V$ with sectional genus $ 1$. The proof uses the partial classification of Fano $3$-folds and Mori theory.
Determinant expression of Selberg zeta functions. I
Shin-ya
Koyama
149-168
Abstract: We show that for $ {\text{PSL}}(2,{\mathbf{R}})$ and its congruence subgroup, the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacian. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.
On the interior of subsemigroups of Lie groups
K. H.
Hofmann;
W. A. F.
Ruppert
169-179
Abstract: Let $G$ denote a Lie group with Lie algebra $\mathfrak{g}$ and with a subsemigroup $S$ whose infinitesimal generators generate $ \mathfrak{g}$. We construct real analytic curves $\gamma :{{\mathbf{R}}^ + } \to S$ such that $\dot \gamma (0)$ is a preassigned tangent vector of $S$ at the origin and that $\gamma(t)$ is in the interior of $S$ for all positive $t$. Among the consequences, we find that the boundary of $S$ has to be reasonably well behaved. Our procedure involves the construction of certain linear generating sets from a given Lie algebra generating set, and this may be of independent interest.
On the characteristic classes of actions of lattices in higher rank Lie groups
Garrett
Stuck
181-200
Abstract: We show that under certain assumptions, the measurable cohomology class of the linear holonomy cocycle of a foliation yields information about the characteristic classes of the foliation. Combined with the results of a previous paper, this yields vanishing theorems for characteristic classes of certain actions of lattices in higher rank semisimple Lie groups.
Similarity orbits and the range of the generalized derivation $X\to MX-XN$
Allen
Schweinsberg
201-211
Abstract: If $M$ and $N$ are bounded operators on infinite dimensional complex Hilbert spaces $ \mathcal{H}$ and $\mathcal{K}$, let $\tau (X) = MX - XN$ for $X$ in $ \mathcal{L}(\mathcal{K},\mathcal{H})$. The closure of the range of $\tau$ is characterized when $ M$ and $N$ are normal. There is a close connection between the range of $\tau$ and operators $C$ for which $ [\begin{array}{*{20}{c}} M \& C 0 \& N \end{array} ]$ is in the closure of the similarity orbit of $ [\begin{array}{*{20}{c}} M \& 0 0 ^ N \end{array} ]$. This latter set is characterized and compared with the closure of the range of $\tau$.
Optimal H\"older and $L\sp p$ estimates for $\overline\partial\sb b$ on the boundaries of real ellipsoids in ${\bf C}\sp n$
Mei-Chi
Shaw
213-234
Abstract: Let $D$ be a real ellipsoid in ${{\mathbf{C}}^n},n \geq 3$, with defining function $\rho (z) = \sum\nolimits_{k = 1}^n {(x_k^{2{n_k}} + y_k^{2{m_k}})} - 1$, ${z_k} = {x_k} + i{y_k}$, where ${n_k},{m_k} \in N$. In this paper we study the sharp Hàlder and ${L^p}$ estimates for the solutions of the tangential Cauchy-Riemann equations ${\overline \partial _b}$ on the boundary $\partial D$ of $D$ using the integral kernel method. In particular, we proved that if $\alpha \in L_{(0,1)}^\infty (\partial D)$ such that ${\overline \partial _b}\alpha = 0$ on $\partial D$ in the distribution sense, then there exists a $u \in {\Lambda _{1/2m}}(\partial D)$ satisfying $ {\overline \partial _b}u = \alpha$ and ${\left\Vert u \right\Vert _{{\Lambda _{1/2m}}(\partial D)}} \leq c{\left\Vert \alpha \right\Vert _{{L^\infty }(\partial D)}}$ for some constant $ c > 0$ independent of $ \alpha$, where ${\Lambda _{1/2m}}(\partial D)$ is the Lipschitz space with exponent $\frac{1} {{2m}}$ and $ 2m = {\max _{1 \leq k \leq n}}\min (2{n_k},2{m_k})$ is the type of the domain $ D$.
Hausdorff dimension of divergent Teichm\"uller geodesics
Howard
Masur
235-254
Abstract: Let $g > 1$ be given and let $k = ({k_1}, \ldots ,{k_n})$ be an $ n$-tuple of positive integers whose sum is $4g - 4$. Denote by ${Q_k}$ the set of all holomorphic quadratic differentials on compact Riemann surfaces of genus $ g$ whose zeros have orders ${k_1}, \ldots, {k_n}$. $Q_k$ is called a stratum inside the cotangent space of all holomorphic quadratic differentials over the Teichmüller space of genus $g$. Let ${Q_k}/\operatorname{Mod} (g)$ be the moduli space where $ \operatorname{Mod} (g)$ is the mapping class group. Each $q \in {Q_k}$ defines a Teichmüller geodesic. Theorem. There exists $\delta > 0$ so that for almost all $q \in {Q_k}$, the set of $\theta$, such that the geodesic defined by $ {e^{i\theta }}q$ eventually leaves every compact set in ${Q_k}/\operatorname{Mod} (g)$, has Hausdorff dimension $\theta$.
The decompositions of Schur complexes
Hyoung J.
Ko
255-270
Abstract: This paper presents a method for finding the characteristic-free Pieri type decompositions of Schur modules, Weyl modules, and Schur complexes. We also introduce several new combinatorial rules for computing the Littlewood-Richardson coefficients.
A convergent framework for the multicomponent KP-hierarchy
G. F.
Helminck;
G. F.
Post
271-292
Abstract: In this paper we describe how to construct convergent solutions of the multicomponent KP-hierarchy, starting from a certain open subset of the Grassmann manifold of a special kind of Banach space, and derive an expression of its solutions in terms of Fredholm determinants. Further we show that the simplest nonscalar reduction of the present hierarchy leads to the AKNS-hierarchy.
Extending discrete-valued functions
John
Kulesza;
Ronnie
Levy;
Peter
Nyikos
293-302
Abstract: In this paper, we show that for a separable metric space $X$, every continuous function from a subset $S$ of $X$ into a finite discrete space extends to a continuous function on $X$ if and only if every continuous function from $ S$ into any discrete space extends to a continuous function on $X$. We also show that if there is no inner model having a measurable cardinal, then there is a metric space $X$ with a subspace $S$ such that every $2$-valued continuous function from $S$ extends to a continuous function on all of $X$, but not every discrete-valued continuous function on $S$ extends to such a map on $X$. Furthermore, if Martin's Axiom is assumed, such a space can be constructed so that not even $ \omega$-valued functions on $S$ need extend. This last result uses a version of the Isbell-Mrowka space $\Psi$ having a ${C^ * }$-embedded infinite discrete subset. On the other hand, assuming the Product Measure Extension Axiom, no such $\Psi$ exists.
Periodicity and decomposability of basin boundaries with irrational maps on prime ends
Russell B.
Walker
303-317
Abstract: Planar basin boundaries of iterated homeomorphisms induce homeomorphisms on prime ends. When the basin is connected, simply connected, and has a compact connected boundary, the space of prime ends is a topological circle. If the induced homeomorphism on prime ends has rational rotation number, the basin boundary contains periodic orbits. Several questions as to basin boundary periodics, decomposability, and minimality, when the induced map on prime ends has irrational rotation number, are answered by construction of both homeomorphisms and diffeomorphisms. Examples in the literature of basin boundaries with interesting prime end dynamics have been sparse. Prime end dynamics has drawn recent interest as a natural tool for the study of strange attractors.
On the distance of subspaces of $l\sp n\sb p$ to $l\sp k\sb p$
William B.
Johnson;
Gideon
Schechtman
319-329
Abstract: It is proved that if $l_p^n$ is well-isomorphic to $X \oplus Y$ and $X$ either has small dimension or is a Euclidean space, then $Y$ is well-isomorphic to $l_p^k$, $ k = \operatorname{dim} Y$. The proofs use new forms of the finite dimensional decomposition method. It is shown that the constant of equivalence between a normalized $K$-unconditional basic sequence in $l_p^n$ and a subsequence of the unit vector basis of $l_p^n$ is greatest, up to a constant depending on $ K$, when the sequence spans a $2$-Euclidean space.
Stable patterns in a viscous diffusion equation
A.
Novick-Cohen;
R. L.
Pego
331-351
Abstract: We consider a pseudoparabolic regularization of a forward-backward nonlinear diffusion equation ${u_t} = \Delta (f(u) + \nu {u_t})$, motivated by the problem of phase separation in a viscous binary mixture. The function $f$ is non-monotone, so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases. We find that any bounded measurable steady state solution $u(x)$ satisfying $f(u) = {\text{constant}}$,
Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture
Alberto
Albano;
Sheldon
Katz
353-368
Abstract: We study the deformation theory of lines on the Fermat quintic threefold. We formulate an infinitesimal version of the generalized Hodge conjecture, and use our analysis of lines to prove it in a special case.
Noether-Lefschetz locus for surfaces
Sung-Ock
Kim
369-384
Abstract: We generalize M. Green's Explicit Noether-Lefschetz Theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces. Moreover, we give a new proof of the Density Theorem due to C. Ciliberto, J. Harris, and R. Miranda [5].
Markov partitions for expanding maps of the circle
Matthew
Stafford
385-403
Abstract: We study Markov partitions for orientation-preserving expanding maps of the circle whose rectangles are connected. Up to a reordering of basis elements, the class of induced matrices arising for such partitions is characterized. Then the study focuses on the subclass of partitions for which each boundary set is a periodic orbit. We show that, if the boundary orbit of a partition is well-distributed, the partition and its symmetries can be constructed. An accompanying result is concerned with double covers of the circle only. It says that, for a given period, all partitions bounded by ill-distributed orbits have the same induced matrix.
Maximal representations of surface groups in bounded symmetric domains
Luis
Hernández
405-420
Abstract: Let $\Gamma$ be the fundamental group of a hyperbolic surface of genus $g$; for $1 \le p \le q,PSU(p,q)$ is the group of isometries of a certain Hermitian symmetric space ${D_{p,q}}$ of rank $p$. There exists a characteristic number $c:\operatorname{Hom} (\Gamma ,PSU(p,q)) \to \mathbb{R}$, which is constant on each connected component and such that $\vert c(\rho )\vert \leq 4p\pi (g - 1)$ for every representation $\rho$. We show that representations with maximal characteristic number (plus some nondegeneracy condition if $p > 2$ leave invariant a totally geodesic subspace of ${D_{p,q}}$ isometric to ${D_{p,p}}$.
Infinitesimal rigidity for the action of ${\rm SL}(n,{\bf Z})$ on ${\bf T}\sp n$
James W.
Lewis
421-445
Abstract: Let $\Gamma = {\mathbf{SL}}(n,\mathbb{Z})$ or any subgroup of finite index. Then the action of $ \Gamma$ on ${\mathbb{T}^n}$ by automorphisms is infinitesimally rigid for $n \ge 7$, i.e., the cohomology ${\text{H}^1}(\Gamma ,\operatorname{Vec} ({\mathbb{T}^n})) = 0$, where $\operatorname{Vec} ({\mathbb{T}^n})$ denotes the module of $ {C^\infty }$ vector fields on $ {\mathbb{T}^n}$.
Symplectic double groupoids over Poisson $(ax+b)$-groups
Kentaro
Mikami
447-463
Abstract: First, we classify all the multiplicative Poisson structures on the $ (ax + b)$-group and determine their dual Poisson Lie groups. Next, we show the existence of symplectic groupoid over the Poisson $ (ax + b)$-group. Finally, by the Hamilton-Jacobi method we construct nontrivial symplectic double groupoids and conclude that for each pair of nondegenerate multiplicative Poisson structures of the $(ax + b)$-group there exists a symplectic double groupoid.