On meromorphic functions with finite logarithmic order
Peter
Tien-Yu
Chern
473-489
Abstract: By using a slow growth scale, the logarithmic order, with which to measure the growth of functions, we obtain basic results on the value distribution of a class of meromorphic functions of zero order.
Constant mean curvature surfaces in $M^2\times \mathbf{R}$
David
Hoffman;
Jorge
H. S.
de Lira;
Harold
Rosenberg
491-507
Abstract: The subject of this paper is properly embedded $H-$surfaces in Riemannian three manifolds of the form $M^2\times \mathbf{R}$, where $M^2$ is a complete Riemannian surface. When $M^2={\mathbf R}^2$, we are in the classical domain of $H-$surfaces in ${\mathbf R}^3$. In general, we will make some assumptions about $M^2$ in order to prove stronger results, or to show the effects of curvature bounds in $M^2$ on the behavior of $H-$surfaces in $M^2\times \mathbf{R}$.
Steinberg symbols modulo the trace class, holonomy, and limit theorems for Toeplitz determinants
Richard
W.
Carey;
Joel
D.
Pincus
509-551
Abstract: Suppose that $\phi=\psi z^\gamma$ where $\gamma\in Z_+$ and $\psi \in \text{\rm Lip}_\beta,\,{1\over 2}<\beta<1$, and the Toeplitz operator $T_\psi$ is invertible. Let $D_n(T_\phi)$ be the determinant of the Toeplitz matrix $((\hat\phi _{i,j}))=((\hat\phi _{i-j})),\quad 0\leq i,j\leq n ,$ where $\hat \phi_k={1\over 2\pi}\int_0^{2\pi} \phi(\theta)e^{-ik\theta}\, d\theta$. Let $P_n$ be the orthogonal projection onto $\ker {S^*}^{n+1}=\bigvee\{1,e^{i\theta}, e^{2i\theta},\ldots, e^{in\theta}\},$where $S=T_z$; set $Q_n=1-P_n$, let $H_\omega$ denote the Hankel operator associated to $\omega$, and set $\tilde\omega(t)=\omega({1\over t})$ for $t\in \mathbb{T}$. For the Wiener-Hopf factorization $\psi=f\bar g$ where $f, g$ and ${1\over f },{1\over g}\in \text{\rm Lip}_\beta\cap H^\infty(\mathbb{T} ), {1\over 2}<\beta<1$, put $E(\psi)=\exp\sum_{k=1}^\infty k(\log f)_k(\log \bar g)_{-k}$, $G(\psi)=\exp(\log\psi)_0.$ Theorem A. $D_n(T_\phi)=(-1)^{(n+1)\gamma} G(\psi)^{n+1}E(\psi) G({\bar g\over f})^\gamma$ $\cdot \det\bigg((T_{{f\over \bar g}z^{n+1}}\cdot [1-H_{\bar g\over f} Q_{n-\gam... ...^{\alpha-1},z^{\tau-1})\bigg)_{\gamma \times \gamma} \cdot [1+O(n^{1-2\beta})].$ Let $H^2(\mathbb{T} )= {\mathcal X}\dotplus {\mathcal Y}$ be a decomposition into $T_\phi T_{\phi^{-1}}$invariant subspaces, ${\mathcal X}= \bigcap_{n=1}^\infty\operatorname{ran} (T_\phi T_{\phi^{-1}})^n$and ${\mathcal Y}=\bigcup _{n=1}^\infty\ker (T_\phi T_{\phi^{-1}})^n$, so that $T_\phi T_{\phi^{-1}}$ restricted to ${\mathcal X}$ is invertible, ${\mathcal Y}$ is finite dimensional, and $T_\phi T_{\phi^{-1}}$ restricted to ${\mathcal Y}$ is nilpotent. Let $\{w_\alpha\}_1^\gamma$ be the basis $\{T_f z^\alpha\}_0^{\gamma-1}$ for the null space of $T_\phi T_{\phi^{-1}}$, and let $u_\alpha$ be the top vector in a Jordan root vector chain of length $m_\alpha+1$ lying over $(-1)^{m_\alpha}w_\alpha$, i.e., $(T_\phi T_{\phi^{-1}})^{m_\alpha}u_\alpha =(-1)^{m_\alpha}w_\alpha$where $m_\alpha=\max\{m\in Z_+:\exists x\,\text{\rm so that} (T_\phi T_{\phi^{-1}})^mx=w_\alpha\}^{-1}$. Theorem B. $E( \psi) G({\bar g\over f})^\gamma=$ $ {\prod_{\lambda\in\sigma(T_{\phi} T_{\phi^{-1}})\setminus \{0\}}\,\lambda}\over \det( u_\alpha,T_{1\over g}z^{\tau-1})$ $=\left (\bar g\cup f\times {\bar g\over f}\cup z^\gamma\right )(\mathbb{T} )$, the holonomy of a Deligne bundle with connection defined by the factorization $\phi= f\bar gz^\gamma$. Note that the generalizations of the Szegö limit theorem for $D_n(T_\phi)$which have appeared in the literature with $1$ instead of $[1-H_{\bar g\over f} Q_{n-\gamma} H_{({f\over \bar g})^{\tilde{}}}]^{-1}$ have the defect that the limit of ${D_n(T_\phi)\over (-1)^{(n+1)\gamma} G(\psi)^{n+1} \det(T_{{f\over \bar g}z^{n+1}}z^{\alpha-1},z^{\tau-1})}$ does not exist in general. An example is given with $D_n(T_\phi)\neq 0$yet $D_{\gamma-1}(T_{{f\over \bar g}z^{n+1}})=0$ for infinitely many $n$.
Norms and essential norms of linear combinations of endomorphisms
Pamela
Gorkin;
Raymond
Mortini
553-571
Abstract: We compute norms and essential norms of linear combinations of endomorphisms on uniform algebras.
MacNeille completions and canonical extensions
Mai
Gehrke;
John
Harding;
Yde
Venema
573-590
Abstract: Let $V$ be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if $V$ is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety $V$ is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.
Multi-scale Young measures
Pablo
Pedregal
591-602
Abstract: We introduce multi-scale Young measures to deal with problems where multi-scale phenomena are relevant. We prove some interesting representation results that allow the use of these families of measures in practice, and illustrate its applicability by treating, from this perspective, multi-scale convergence and homogenization of multiple integrals.
An explicit characterization of Calogero--Moser systems
Fritz
Gesztesy;
Karl
Unterkofler;
Rudi
Weikard
603-656
Abstract: Combining theorems of Halphen, Floquet, and Picard and a Frobenius type analysis, we characterize rational, meromorphic simply periodic, and elliptic KdV potentials. In particular, we explicitly describe the proper extension of the Airault-McKean-Moser locus associated with these three classes of algebro-geometric solutions of the KdV hierarchy with special emphasis on the case of multiple collisions between the poles of solutions. This solves a problem left open since the mid-1970s.
Newton polygons and local integrability of negative powers of smooth functions in the plane
Michael
Greenblatt
657-670
Abstract: Let $f(x,y)$ be any smooth real-valued function with $f(0,0)=0$. For a sufficiently small neighborhood $U$ of the origin, we study the number \begin{displaymath}\sup\left\{\epsilon:\int_U \vert f(x,y)\vert^{-\epsilon}<\infty\right\}. \end{displaymath} It is known that sometimes this number can be expressed in a natural way using the Newton polygon of $f$. We provide necessary and sufficient conditions for this Newton polygon characterization to hold. The behavior of the integral at the supremal $\epsilon$ is also analyzed.
Some quotient Hopf algebras of the dual Steenrod algebra
J.
H.
Palmieri
671-685
Abstract: Fix a prime $p$, and let $A$ be the polynomial part of the dual Steenrod algebra. The Frobenius map on $A$ induces the Steenrod operation $\widetilde{\mathscr{P}}^{0}$on cohomology, and in this paper, we investigate this operation. We point out that if $p=2$, then for any element in the cohomology of $A$, if one applies $\widetilde{\mathscr{P}}^{0}$ enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that ``enough times'' should be ``once.'' The bulk of the paper is a study of some quotients of $A$ in which the Frobenius is an isomorphism of order $n$. We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about $\widetilde{\mathscr{P}}^{0}$. The dual complete Steenrod algebra makes an appearance.
Equivariant Gysin maps and pulling back fixed points
Bernhard
Hanke;
Volker
Puppe
687-702
Abstract: We develop a new approach to the pulling back fixed points theorem of W. Browder and use it in order to prove various generalizations of this result.
On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo $5$ and generalizations
Alexander
Berkovich;
Frank
G.
Garvan
703-726
Abstract: In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic ${\mathcal O}(\pi)$ denotes the number of odd parts of the partition $\pi$and $\pi'$ is the conjugate of $\pi$. In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod $5$: \begin{align*}p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod{5}, p(n) &= p_0(n) + p_2(n), \end{align*} where $p_i(n)$ ($i=0,2$) denotes the number of partitions of $n$ with $\mathrm{srank}\equiv i\pmod{4}$ and $p(n)$ is the number of unrestricted partitions of $n$. Andrews asked for a partition statistic that would divide the partitions enumerated by $p_i(5n+4)$ ($i=0,2$) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the $2$-quotient-rank and the $5$-core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the $2$-quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod $5$. This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo $5$. Finally, we discuss some new formulas for partitions that are $5$-cores and discuss an intriguing relation between $3$-cores and the Andrews-Garvan crank.
Cusp size bounds from singular surfaces in hyperbolic 3-manifolds
C.
Adams;
A.
Colestock;
J.
Fowler;
W.
Gillam;
E.
Katerman
727-741
Abstract: Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, $\ell$-curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9. We also find particular upper bounds on meridian length, $\ell$-curve length and maximal cusp volume for hyperbolic knots in $\mathbb{S} ^3$ depending on crossing number. Particular improved bounds are obtained for alternating knots.
Isovariant Borsuk-Ulam results for pseudofree circle actions and their converse
Ikumitsu
Nagasaki
743-757
Abstract: In this paper we shall study the existence of an $S^1$-isovariant map from a rational homology sphere $M$ with pseudofree action to a representation sphere $SW$. We first show some isovariant Borsuk-Ulam type results. Next we shall consider the converse of those results and show that there exists an $S^1$-isovariant map from $M$ to $SW$ under suitable conditions.
Surfaces of general type with $p_g=q=1, K^2=8$
and bicanonical map of degree $2$
Francesco
Polizzi
759-798
Abstract: We classify the minimal algebraic surfaces of general type with $p_g=q=1, \; K^2=8$ and bicanonical map of degree $2$. It will turn out that they are isogenous to a product of curves, i.e. if $S$ is such a surface, then there exist two smooth curves $C, \; F$ and a finite group $G$ acting freely on $C \times F$ such that $S = (C \times F)/G$. We describe the $C, \; F$ and $G$that occur. In particular the curve $C$ is a hyperelliptic-bielliptic curve of genus $3$, and the bicanonical map $\phi$ of $S$ is composed with the involution $\sigma$ induced on $S$ by $\tau \times id: C \times F \longrightarrow C \times F$, where $\tau$ is the hyperelliptic involution of $C$. In this way we obtain three families of surfaces with $p_g=q=1, \; K^2=8$which yield the first-known examples of surfaces with these invariants. We compute their dimension and we show that they are three generically smooth, irreducible components of the moduli space $\mathcal{M}$ of surfaces with $p_g=q=1, \; K^2=8$. Moreover, we give an alternative description of these surfaces as double covers of the plane, recovering a construction proposed by Du Val.
Lagrangian submanifolds and moment convexity
Bernhard
Krötz;
Michael
Otto
799-818
Abstract: We consider a Hamiltonian torus action $T\times M \rightarrow M$ on a compact connected symplectic manifold $M$ and its associated momentum map $\Phi$. For certain Lagrangian submanifolds $Q\subseteq M$ we show that $\Phi (Q)$ is convex. The submanifolds $Q$ arise as the fixed point set of an involutive diffeomorphism $\tau :M\rightarrow M$ which satisfies several compatibility conditions with the torus action, but which is in general not anti-symplectic. As an application we complete a symplectic proof of Kostant's non-linear convexity theorem.
The general hyperplane section of a curve
Elisa
Gorla
819-869
Abstract: In this paper, we discuss some necessary and sufficient conditions for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non-arithmetically Cohen-Macaulay curve of $\mathbf{P}^3$. We prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non-arithmetically Cohen-Macaulay curve of $\mathbf{P}^3$, arise also as degree matrices of some smooth, integral, non-arithmetically Cohen-Macaulay curve, and we characterize the exceptions. We give a necessary condition on the graded Betti numbers of the general plane section of an arithmetically Buchsbaum (non-arithmetically Cohen-Macaulay) curve in $\mathbf{P}^n$. For curves in $\mathbf{P}^3$, we show that any set of Betti numbers that satisfies that condition can be realized as the Betti numbers of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay curve. We also show that the matrices that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, integral, (smooth) non-arithmetically Cohen-Macaulay space curve are exactly those that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay space curve and have positive subdiagonal. We also prove some bounds on the dimension of the deficiency module of an arithmetically Buchsbaum space curve in terms of the degree matrix of the general plane section of the curve, and we prove that they are sharp.
Nondegenerate $q$-biresolving textile systems and expansive automorphisms of onesided full shifts
Masakazu
Nasu
871-891
Abstract: We study nondegenerate, $q$-biresolving textile systems and using properties of them, we prove a conjecture of Boyle and Maass on arithmetic constraints for expansive automorphisms of onesided full shifts and positively expansive endomorphisms of mixing topological Markov shifts. A similar result is also obtained for expansive leftmost-permutive endomorphisms of onesided full shifts.
Representation formulae and inequalities for solutions of a class of second order partial differential equations
Lorenzo
D'Ambrosio;
Enzo
Mitidieri;
Stanislav
I.
Pohozaev
893-910
Abstract: Let $L$ be a possibly degenerate second order differential operator and let $\Gamma_\eta=d^{2-Q}$ be its fundamental solution at $\eta$; here $d$ is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of $-Lu\ge f(\xi,u)\ge 0$ on ${\mathbb{R}}^N$ to satisfy the representation formula \begin{displaymath}(\mbox R)\qquad\qquad\qquad\qquad\qquad u(\eta)\ge\int_{\mat... ...amma_\eta f(\xi,u) \,d\xi.\qquad\qquad\qquad\qquad\qquad\qquad \end{displaymath} We prove that (R) holds provided $f(\xi,\cdot)$ is superlinear, without any assumption on the behavior of $u$ at infinity. On the other hand, if $u$ satisfies the condition \begin{displaymath}\liminf_{R\rightarrow\infty} {-\int}_{R\le d(\xi)\le 2R}\vert u(\xi)\vert d\xi =0,\end{displaymath} then (R) holds with no growth assumptions on $f(\xi,\cdot)$.
Uniform bounds under increment conditions
Michel
Weber
911-936
Abstract: We apply a majorizing measure theorem of Talagrand to obtain uniform bounds for sums of random variables satisfying increment conditions of the type considered in Gál-Koksma Theorems. We give some applications.