Errata to: ``Analysis and applications of holomorphic functions in higher dimensions''
R. Z.
Yeh
Ribbons and their canonical embeddings
Dave
Bayer;
David
Eisenbud
719-756
Abstract: We study double structures on the projective line and on certain other varieties, with a view to having a nice family of degenerations of curves and K3 surfaces of given genus and Clifford index. Our main interest is in the canonical embeddings of these objects, with a view toward Green's Conjecture on the free resolutions of canonical curves. We give the canonical embeddings explicitly, and exhibit an approach to determining a minimal free resolution.
Clifford indices of ribbons
David
Eisenbud;
Mark
Green
757-765
Abstract: We present a theory of "limit linear series" for rational ribbons-- that is, for schemes that are double structures on $ {P^1}$. This allows us to define a "linear series Clifford index" for ribbons. Our main theorem shows that this is the same as the Clifford index of ribbons studied by Eisenbud-Bayer in this same volume. This allows us to prove that the Clifford index is semicontinuous in degenerations from a smooth curve to a ribbon. A result of Fong [1993] then shows that ribbons may be deformed to smooth curves of the same Clifford index. Thus the Canonical Curve Conjecture of Green [1984] would follow, at least for a general smooth curve of each Clifford index, from the corresponding statement for ribbons.
Geometry of dots and ropes
Karen A.
Chandler
767-784
Abstract: An $\alpha $-dot is the first infinitesimal neighbourhood of a point with respect to an $(\alpha - 1)$-dimensional affine space. We define a notion of uniform position for a collection of dots in projective space, which in particular holds for a collection of dots arising as a general plane section of a higher-dimensional scheme. We estimate the Hilbert function of such a collection of dots, with the result that Theorem 1. Let $\Gamma$ be a collection of $ d$ $\alpha $-dots in uniform position in ${\mathbb{P}^n},\alpha \geqslant 2$. Then the Hilbert function $ {h_\Gamma }$ of $ \Gamma$ satisfies $\displaystyle {h_\Gamma }(r) \geqslant \min (rn + 1,2d) + (\alpha - 2)\min ((r - 1)n - 1,\,d)$ for $r \geqslant 3$. Equality occurs for some $r$ with $ rn + 2 \leqslant 2d$ if and only if ${\Gamma _{{\text{red}}}}$ is contained in a rational normal curve $C$, and the tangent directions to this curve at these points are all contained in $ \Gamma$. Equality occurs for some $r$ with $ (r - 1)n \leqslant d$ if and only if $\Gamma$ is contained in the first infinitesimal neighbourhood of $C$ with respect to a subbundle, of rank $\alpha - 1$ and of maximal degree, of the normal bundle of $C$ in $ {\mathbb{P}^n}$. This implies an upper bound on the degree of a subbundle of rank $\alpha - 1$ of the normal bundle of an irreducible nondegenerate smooth curve of degree $ d$ in ${\mathbb{P}^n}$, by a Castelnuovo argument.
On the Ramsey property of families of graphs
N.
Sauer
785-833
Abstract: For graphs $ A$ and $B$ the relation $A \to (B)_r^1$ means that for every $r$-coloring of the vertices of $ A$ there is a monochromatic copy of $B$ in $A$. $\operatorname{Forb} ({G_1},{G_2}, \ldots ,{G_n})$ is the family of graphs which do not embed any one of the graphs ${G_1},{G_2}, \ldots ,{G_n}$, a family $\mathcal{F}$ of graphs has the Ramsey property if for every graph $ B \in \mathcal{F}$ there is a graph $ A \in \mathcal{F}$ such that $A \to (B)_r^1$. Nešetřil and Rödl (1976) have proven that if either both graphs $G$ and $K$ are two-connected or the complements of both graphs $G$ and $K$ are two-connected then $\operatorname{Forb} (G,K)$ has the Ramsey property. We prove that if $ \overline G$ is disconnected and $K$ is disconnected then $\operatorname{Forb} (G,K)$ does not have the Ramsey property, except for four pairs of graphs $ (G,K)$. A family $\mathcal{F}$ of finite graphs is an age if there is a countable graph $G$ whose set of finite induced subgraphs is $\mathcal{F}$. We characterize those pairs of graphs $(G,H)$ for which $\operatorname{Forb} (G,H)$ is not an age but has the Ramsey property.
On the dimension and the index of the solution set of nonlinear equations
P. S.
Milojević
835-856
Abstract: We study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving Fredholm maps of positive index. The classes of maps under consideration are (pseudo) $A$-proper and either approximation-essential or equivariant approximation-essential. Applications are given to semilinear elliptic BVP's.
On the general notion of fully nonlinear second-order elliptic equations
N. V.
Krylov
857-895
Abstract: The general notion of fully nonlinear second-order elliptic equation is given. Its relation to so-called Bellman equations is investigated. A general existence theorem for the equations like ${P_m}({u_{{x^i}{x^j}}}) = \sum\nolimits_{k = 0}^{m - 1} {{c_k}(x){P_k}({u_{{x^i}{x^j}}})}$ is obtained as an example of an application of the general notion of fully nonlinear elliptic equations.
Some new observations on the G\"ollnitz-Gordon and Rogers-Ramanujan identities
Krishnaswami
Alladi
897-914
Abstract: Two new, short and elementary proofs of the Göllnitz-Gordon identities are presented by considering the odd-even split of the Euler Pentagonal Series and the Triangular Series of Gauss. Using this approach the equality of certain shifted partition functions are established. Next, the odd and even parts of the famous Rogers-Ramanujan series are shown to have interesting product representations ($\bmod 80$). From this, new shifted partition identities are derived.
Infinitesimal bending and twisting in one-dimensional dynamics
Frederick P.
Gardiner
915-937
Abstract: An infinitesimal theory for bending and earthquaking in one-dimensional dynamics is developed. It is shown that any tangent vector to Teichmüller space is the initial data for a bending and for an earthquaking ordinary differential equation. The discussion involves an analysis of infinitesimal bendings and earthquakes, the Hilbert transform, natural bounded linear operators from a Banach space of measures on the Möbius strip to tangent vectors to Teichmüller space, and the construction of a nonlinear right inverse for these operators. The inverse is constructed by establishing an infinitesimal version of Thurston's earthquake theorem.
Invariants of locally conformally flat manifolds
Thomas
Branson;
Peter
Gilkey;
Juha
Pohjanpelto
939-953
Abstract: Let $M$ be a locally conformally flat manifold with metric $g$. Choose a local coordinate system on $ M$ so $g = {e^{2h}}x\,dx \circ dx$ where $dx \circ dx$ is the Euclidean standard metric. A polynomial $P$ in the derivatives of $h$ with coefficients depending smoothly on $ h$ is a local invariant for locally conformally flat structures if the expression $ P({h_X})$ is independent of the choice of $X$. Form valued local invariants are defined similarly. In this paper, we study the properties of the associated de Rham complex. We show that any invariant form can be obtained from the previously studied local invariants of Riemannian structures by restriction. We show the cohomology of the de Rham complex of local invariants is trivial. We also obtain the following characterization of the Euler class. Suppose that for an invariant polynomial $P$, the integral $\int_{{T^m}} {P\vert d{v_g}\vert}$ vanishes for any locally conformally flat metric $g$ on the torus ${T^m}$. Then up to the divergence of an invariantly defined one form, the polynomial $ P$ is a constant multiple of the Euler integrand.
On the morphology of $\gamma$-expansions with deleted digits
Mike
Keane;
Meir
Smorodinsky;
Boris
Solomyak
955-966
Abstract: We investigate the size of the set of reals which can be represented in base $\gamma$ using only the digits $0,1,3$. It is shown that this set has Lebesgue measure zero for $\gamma \leqslant 1/3$ and equals an interval for $\gamma \geqslant 2/5$. Our main goal is to prove that it has Lebesgue measure zero for a certain countable subset of $(1/3,2/5)$.
The Hausdorff dimension of $\lambda$-expansions with deleted digits
Mark
Pollicott;
Károly
Simon
967-983
Abstract: In this article we examine the continuity of the Hausdorff dimension of the one parameter family of Cantor sets $ \Lambda (\lambda ) = \{ \sum\nolimits_{k = 1}^\infty {{i_k}{\lambda ^k}:{i_k} \in S\} }$, where $S \subset \{ 0,1, \ldots ,(n - 1)\}$. In particular, we show that for almost all (Lebesgue) $ \lambda \in [\tfrac{1} {n},\tfrac{1} {l}]$ we have that ${\dim _H}(\Lambda (\lambda )) = \frac{{\log l}} {{ - \log \lambda }}$ where $l = \operatorname{Card} (S)$. In contrast, we show that under appropriate conditions on $S$ we have that for a dense set of $\lambda \in [\tfrac{1} {n},\tfrac{1} {l}]$ we have $ {\dim _H}(\Lambda (\lambda )) < \frac{{\log l}} {{ - \log \lambda }}$.
Crossed products of ${\rm II}\sb 1$-subfactors by strongly outer actions
Carl
Winsløw
985-991
Abstract: We study the crossed product $A \rtimes G \supseteq B \rtimes G$ of an inclusion $A \supseteq B$ of type $ {\text{I}}{{\text{I}}_1}$-factors by a discrete strongly outer action $ G$. In particular, we find conditions under which the strong amenability of $A \supseteq B$ implies that of $A \rtimes G \supseteq B \rtimes G$, and vice versa.
Hypercyclic weighted shifts
Héctor N.
Salas
993-1004
Abstract: An operator $ T$ acting on a Hilbert space is hypercyclic if, for some vector $ x$ in the space, the orbit $ \{ {T^n}x:n \geqslant 0\}$ is dense. In this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic. As a consequence, we show within the class of weighted shifts that multi-hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic. For general hypercyclic operators the corresponding questions were posed by D. A. Herrero, and they still remain open. Using a different technique we prove that $I + T$ is hypercyclic whenever $ T$ is a unilateral backward weighted shift, thus answering in more generality a question recently posed by K. C. Chan and J. H. Shapiro.
On Cappell-Shaneson's homology $L$-classes of singular algebraic varieties
Shoji
Yokura
1005-1012
Abstract: S. Cappell and J. Shaneson (Stratifiable maps and topological invariants, J. Amer. Math. Soc. 4 (1991), 521-551) have recently developed a theory of homology $ L$-classes, extending Goresky-MacPherson's homology $L$-classes. In this paper we show that Cappell-Shaneson's homology $L$-classes for compact complex, possibly singular, algebraic varieties can be interpreted as a unique natural transformation from a covariant cobordism function $ \Omega$ to the ${\mathbf{Q}}$-homology functor $ {H_{\ast}}(;{\mathbf{Q}})$ satisfying a certain normalization condition, just like MacPherson's Chern classes and Baum-Fulton-MacPherson's Todd classes.
Distinguished K\"ahler metrics on Hirzebruch surfaces
Andrew D.
Hwang;
Santiago R.
Simanca
1013-1021
Abstract: Let ${\mathcal{F}_n}$ be a Hirzebruch surface, $n \geqslant 1$. Using the family of extremal metrics on these surfaces constructed by Calabi [1], we study a closely related scale-invariant variational problem, and show that only ${\mathcal{F}_1}$ admits an extremal Kähler metric which is critical for this new functional. Applying a result of Derdzinski [3], we prove that this metric cannot be conformally equivalent to an Einstein metric on $ {\mathcal{F}_1}$. When $n = 2$, we show there is a critical orbifold metric on the space obtained from ${\mathcal{F}_2}$ by blowing down the negative section.
Formation of diffusion waves in a scalar conservation law with convection
Kevin R.
Zumbrun
1023-1032
Abstract: We study the scalar conservation law, $c( - \infty )$ and $c( + \infty )$, behavior that has been observed numerically in solutions of the full equations. The interesting aspect of the analysis is that the asymptotic state of the solution is not known a priori, in contrast to cases treated previously.
A note on norm inequalities for integral operators on cones
Ke Cheng
Zhou
1033-1041
Abstract: Norm inequalities for the Riemann-Liouville operator ${R_r}f(x) = \int_{\langle 0,x\rangle } {\Delta _V^{r - 1}(x - t)f(t)dt}$ and Weyl operator ${W_r}f(x) = \int_{\langle x,\infty \rangle } {\Delta _V^{r - 1}(t - x)f(t)dt}$ on cones in ${R^d}$ have been obtained in the case $r \geqslant 1$ [7]. In this note, these inequalities are further extended to the case $r < 1$. The question of whether the Hardy operator $Hf(x) = \int_{\langle 0,x\rangle } {f(t)dt}$ on cones is bounded from ${L^p}(\Delta _V^\alpha (X))$ to ${L^q}(\Delta _V^\beta (x))\;(q < p)$ is also solved.
Tensor product of difference posets
Anatolij
Dvurečenskij
1043-1057
Abstract: A tensor product of difference posets, which generalize orthoalgebras and orthomodular posets, is defined, and an equivalent condition is presented. In particular, we show that a tensor product for difference posets with a sufficient system of probability measures exists, as well as a tensor product of any difference poset and any Boolean algebra, which is isomorphic to a bounded Boolean power.
A note on the problem of prescribing Gaussian curvature on surfaces
Wei Yue
Ding;
Jia Quan
Liu
1059-1066
Abstract: The problem of existence of conformal metrics with Gaussian curvature equal to a given function $K$ on a compact Riemannian $2$-manifold $M$ of negative Euler characteristic is studied. Let ${K_0}$ be any nonconstant function on $M$ with $ \max {K_0} = 0$, and let $ {K_\lambda } = {K_0} + \lambda$. It is proved that there exists a ${\lambda ^{\ast}} > 0$ such that the problem has a solution for $ K = {K_\lambda }$ iff $\lambda \in ( - \infty ,{\lambda ^{\ast}}]$. Moreover, if $\lambda \in (0,{\lambda ^{\ast}})$, then the problem has at least $2$ solutions.
Functions with bounded spectrum
Ha Huy
Bang
1067-1080
Abstract: Let $0 < p \leqslant \infty ,\,f(x) \in {L_p}({\mathbb{R}^n})$, and $ \operatorname{supp} Ff$ be bounded, where $F$ is the Fourier transform. We will prove in this paper that the sequence $ \vert\vert{D^\alpha }f\vert\vert _p^{1/\vert\alpha \vert},\,\alpha \geqslant 0$, has the same behavior as the sequence $\mathop {\lim }\limits_{\xi \in \operatorname{supp} Ff} \vert{\xi ^\alpha }{\vert^{1/\vert\alpha \vert}}$, $ \alpha \geqslant 0$. In other words, if we know all "far points" of $\operatorname{supp} Ff$, we can wholly describe this behavior without any concrete calculation of $\vert\vert{D^\alpha }f\vert{\vert _p},\,\alpha \geqslant 0$. A Paley-Wiener-Schwartz theorem for a nonconvex case, which is a consequence of the result, is given.