Kac-Moody Lie algebras, spectral sequences, and the Witt formula
Seok-Jin
Kang
463-493
Abstract: In this work, we develop a homological theory for the graded Lie algebras, which gives new information on the structure of the Lorentzian Kac-Moody Lie algebras. The technique of the Hochschild-Serre spectral sequences offers a uniform method of studying the higher level root multiplicities and the principally specialized affine characters of Lorentzian Kac-Moody Lie algebras.
Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in ${\bf R}\sp n$. II
Kevin
McLeod
495-505
Abstract: We prove a uniqueness result for the positive solution of $\Delta u + f(u) = 0$ in ${\mathbb{R}^n}$ which goes to 0 at $\infty$. The result applies to a wide class of nonlinear functions $f$, including the important model case $f(u) = - u + {u^p}$ , $1 < p < (n + 2)/(n - 2)$. The result is proved by reducing to an initial-boundary problem for the ${\text{ODE}}\;u'' + (n - 1)/r + f(u) = 0$ and using a shooting method.
Continuous dependence of nonmonotonic discontinuous differential equations
Daniel C.
Biles
507-524
Abstract: Continuous dependence of solutions for a class of nonmonotonic, discontinuous differential equations is studied. First, a local existence theorem due to $Z$. Wu is extended to a larger class. Then, a result concerning continuous dependence for this larger class is proven. This employs a type of convergence similar to Gihman's Convergence Criterion, which is defined to be $\displaystyle {\text{For all}}\;a,b\;{\text{and}}\;y\quad \mathop {\lim }\limits_{n \to \infty } \int_a^b {{f_n}(s,y)ds = } \int_a^b {{f_\infty }(s,y)\,ds}.$ The significance of Gihman's Convergence Criterion is that for certain classes of differential equations it has been found to be necessary and sufficient for continuous dependence. Finally, examples are presented to motivate and clarify this continuous dependence result.
Determinants of Laplacians on the space of conical metrics on the sphere
Hala Khuri
King
525-536
Abstract: On a compact surface with smooth boundary, the determinant of the Laplacian associated to a smooth metric on the surface (with Dirichlet boundary conditions if the boundary is nonempty) is a well-defined isospectral invariant. As a function on the moduli space of such surfaces, it is a smooth function whose boundary behavior in certain cases is well understood; see [OPS and K]. In this paper, we restrict ourselves to a certain class of singular metrics on closed surfaces called conical metrics. We show that the determinant of the associated Laplacian is still well defined and that it is a real analytic function on a suitably restricted subset of the space of conical metrics on the sphere.
An isoperimetric inequality for Artin groups of finite type
Kay
Tatsuoka
537-551
Abstract: We show that Artin groups of finite type satisfy a quadratic isoperimetric inequality. Moreover we describe an explicit algorithm to solve the word problem in quadratic time.
The cohomology algebra of a commutative group scheme
Robert
Fossum;
William
Haboush
553-565
Abstract: Let $k$ be a commutative ring with unit of characteristic $p > 0$ and let $G = \operatorname{Spec}(A)$ be an affine commutative group scheme over $k$. Let ${{\text{H}}^ \bullet }(G)$ be the graded Hochschild algebraic group cohomology algebra and, for $ M$ a rational $ G$-module, let ${{\text{H}}^ \bullet }(G,M)$ denote the graded Hochschild cohomology ${{\text{H}}^ \bullet }(G)$-module. We show that $ {{\text{H}}^ \bullet }(G)$ is, in general, a graded Hopf algebra. When $G = {{\mathbf{G}}_{a,k}}$, let ${\alpha _{{p^\nu }}}$ denote the subgroup of ${p^\nu }$-nilpotents and let ${{\text{F}}_\nu }$ denote the $\nu$th power of the Frobenius. We show that for any finite $M$ that there is a $\nu$ such that $\displaystyle {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}},M) \cong {{\text{H}}^... ...otimes _k}{\text{F}}_{\nu }^\ast ({{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}}))$ where ${\text{F}}_\nu ^\ast$ is the endomorphism of $ {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}})$ induced by ${F_v}$. As a consequence, we can show that ${{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}},M)$ is a finitely generated module over $ {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}})$ when $M$ is a finite dimensional vector space over $k$.
The Ehrenfeucht-Fra\"\i ss\'e-game of length $\omega\sb 1$
Alan
Mekler;
Saharon
Shelah;
Jouko
Väänänen
567-580
Abstract: Let $\mathfrak{A}$ and $ \mathfrak{B}$ be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraïssé-game of length $ {\omega _1}$ of $\mathfrak{A}$ and $ \mathfrak{B}$ which we denote by $ {\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$. This game is like the ordinary Ehrenfeucht-Fraïssé-game of ${L_{\omega \omega }}$ except that there are ${\omega _1}$ moves. It is clear that ${\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$ is determined if $ \mathfrak{A}$ and $\mathfrak{B}$ are of cardinality $\leq {\aleph _1}$. We prove the following results: Theorem 1. If $V = L$, then there are models $\mathfrak{A}$ and $\mathfrak{B}$ of cardinality ${\aleph _2}$ such that the game ${\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$ is nondetermined. Theorem 2. If it is consistent that there is a measurable cardinal, then it is consistent that ${\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$ is determined for all $\mathfrak{A}$ and $ \mathfrak{B}$ of cardinality $ \leq {\aleph _2}$. Theorem 3. For any $\kappa \geq {\aleph _3}$ there are $\mathfrak{A}$ and $ \mathfrak{B}$ of cardinality $\kappa$ such that the game ${\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$ is nondetermined.
Certain hypergeometric series related to the root system $BC$
R. J.
Beerends;
E. M.
Opdam
581-609
Abstract: We show that the generalized hypergeometric function $_2{F_1}$ of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type $ BC$. In addition we prove that the Jacobi polynomials of matrix argument correspond to the Jacobi polynomials associated with the root system of type $BC$. We also give a precise relation between Jack polynomials and the Jacobi polynomials associated with the root system of type $A$. As a side result one obtains generalized hook-length formulas which are related to Harish-Chandra's ${\mathbf{c}}$-function and one can prove a conjecture due to Macdonald relating two inner products on a space of symmetric functions.
On the evaluation map
Aniceto
Murillo
611-622
Abstract: The evaluation map of a differential graded algebra or of a space is described under two different approaches. This concept turns out to have geometric implications: (i) A $ 1$-connected topological space, with finite-dimensional rational homotopy, has finite-dimensional rational cohomology if and only if it has nontrivial evaluation map. (ii) Let $ E\xrightarrow{\rho }B$ be a fibration of simplyconnected spaces. If the rational cohomology of the fibre is finite dimensional and the evaluation map of the base is different from zero, then the evaluation map of the total space is nonzero. Also, if $\rho$ is surjective in rational homotopy and the evaluation map of $E$ is nontrivial, then the evaluation map of the fibre is different from zero.
Finitely decidable congruence modular varieties
Joohee
Jeong
623-642
Abstract: A class $\mathcal{V}$ of algebras of the same type is said to be finitely decidable iff the first order theory of the class of finite members of $\mathcal{V}$ is decidable. Let $\mathcal{V}$ be a congruence modular variety. In this paper we prove that if $\mathcal{V}$ is finitely decidable, then the following hold. (1) Each finitely generated subvariety of $\mathcal{V}$ has a finite bound on the cardinality of its subdirectly irreducible members. (2) Solvable congruences in any locally finite member of $\mathcal{V}$ are abelian. In addition we obtain various necessary conditions on the congruence lattices of finite subdirectly irreducible algebras in $\mathcal{V}$.
On property I for knots in $S\sp 3$
Xingru
Zhang
643-657
Abstract: This paper deals with the question of which knot surgeries on $ {S^3}$ can yield $ 3$-manifolds homeomorphic to, or with the same fundamental group as, the Poincaré homology $3$-sphere.
Homogeneity for open partitions of pairs of reals
Qi
Feng
659-684
Abstract: We prove a partition theorem for analytic sets of reals, namely, if $A \subseteq \mathbb{R}$ is analytic and $ {[A]^2} = {K_0} \cup {K_1}$ with ${K_0}$ relatively open, then either there is a perfect 0-homogeneous subset or $A$ is a countable union of $1$-homogeneous subsets. We also show that such a partition property for coanalytic sets is the same as that each uncountable coanalytic set contains a perfect subset. A two person game for this partition property is also studied. There are some applications of such partition properties.
Hypersurfaces with constant mean curvature in the complex hyperbolic space
Susana
Fornari;
Katia
Frensel;
Jaime
Ripoll
685-702
Abstract: A classical theorem of A. D. Alexandrov characterized round spheres is extended to the complex hyperbolic space $ {\mathbf{C}}{{\mathbf{H}}^2}$ of constant holomorphic sectional curvature. A detailed description of the horospheres and equidistant hypersurfaces in $ {\mathbf{C}}{{\mathbf{H}}^2}$ determining in particular their stability, is also given.
Geometric curvature bounds in Riemannian manifolds with boundary
Stephanie B.
Alexander;
I. David
Berg;
Richard L.
Bishop
703-716
Abstract: An Alexandrov upper bound on curvature for a Riemannian manifold with boundary is proved to be the same as an upper bound on sectional curvature of interior sections and of sections of the boundary which bend away from the interior. As corollaries those same sectional curvatures are related to estimates for convexity and conjugate radii; the Hadamard-Cartan theorem and Yau's isoperimetric inequality for spaces with negative curvature are generalized.
Noncommutative K\"othe duality
Peter G.
Dodds;
Theresa K.-Y.
Dodds;
Ben
de Pagter
717-750
Abstract: Using techniques drawn from the classical theory of rearrangement invariant Banach function spaces we develop a duality theory in the sense of Köthe for symmetric Banach spaces of measurable operators affiliated with a semifinite von Neumann algebra equipped with a distinguished trace. A principal result of the paper is the identification of the Köthe dual of a given Banach space of measurable operators in terms of normality.
Gauss map of minimal surfaces with ramification
Min
Ru
751-764
Abstract: We prove that for any complete minimal surface $M$ immersed in ${R^n}$, if in $ C{P^{n - 1}}$ there are $q > n(n + 1)/2$ hyperplanes ${H_j}$ in general position such that the Gauss map of $M$ is ramified over ${H_j}$ with multiplicity at least ${e_j}$ for each $j$ and $\displaystyle \sum\limits_{j = 1}^q {\left({1 - \frac{{(n - 1)}} {{{e_j}}}} \right) > n(n + 1)/2}$ , then $ M$ must be flat.
Bass numbers of local cohomology modules
Craig L.
Huneke;
Rodney Y.
Sharp
765-779
Abstract: Let $A$ be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of $A$ itself, but with respect to an arbitrary ideal of $A$. The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that $A$ is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true.
Brown-Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups
Akira
Kono;
Nobuaki
Yagita
781-798
Abstract: The Steenrod algebra structures of $ {H^\ast}(BG;Z/p)$ for compact Lie groups are studied. Using these, Brown-Peterson cohomology and Morava $K$-theory are computed for many concrete cases. All these cases have properties similar as torsion free Lie groups or finite groups, e.g., $B{P^{odd}}(BG) = 0$.
Zeros of the successive derivatives of Hadamard gap series
Robert M.
Gethner
799-807
Abstract: A complex number $ z$ is in the final set of an analytic function $f$, as defined by Pólya, if every neighborhood of $z$ contains zeros of infinitely many $ {f^{(n)}}$. If $ f$ is a Hadamard gap series, then the part of the final set in the open disk of convergence is the origin along with a union of concentric circles.
The number of irreducible factors of a polynomial. I
Christopher G.
Pinner;
Jeffrey D.
Vaaler
809-834
Abstract: Let $F(x)$ be a polynomial with coefficients in an algebraic number field $k$. We estimate the number of irreducible cyclotomic factors of $F$ in $k[x]$, the number of irreducible noncyclotomic factors of $F$, the number of $n$th roots of unity among the roots of $F$, and the number of primitive $ n$th roots of unity among the roots of $F$. All of these quantities are counted with multiplicity and estimated by expressions which depend explicitly on $k$, on the degree of $F$ and height of $F$, and (when appropriate) on $ n$. We show by constructing examples that some of our results are essentially sharp.
Uniform algebras generated by holomorphic and pluriharmonic functions
Alexander J.
Izzo
835-847
Abstract: It is shown that if ${f_1}, \ldots ,{f_n}$ are pluriharmonic on ${B_n}$ (the open unit ball in ${\mathbb{C}^n})$ and ${C^1}$ on $ {\bar B_n}$, and the $n \times n$ matrix $(\partial {f_j}/\partial {\bar z_k})$ is invertible at every point of ${B_n}$, then the norm-closed algebra generated by the ball algebra $ A({\bar B_n})$ and ${f_1}, \ldots ,{f_n}$ is equal to $C({\bar B_n})$. Extensions of this result to more general strictly pseudoconvex domains are also presented.
On the Toda and Kac-van Moerbeke systems
F.
Gesztesy;
H.
Holden;
B.
Simon;
Z.
Zhao
849-868
Abstract: Given a solution of the Toda lattice we explicitly construct a solution of the Kac-van Moerbeke system related to each other by a Miura-type transformation. As an illustration of our method we derive the $N$-soliton solutions of the Kac-van Moerbeke lattice.
When Cantor sets intersect thickly
Brian R.
Hunt;
Ittai
Kan;
James A.
Yorke
869-888
Abstract: The thickness of a Cantor set on the real line is a measurement of its "size". Thickness conditions have been used to guarantee that the intersection of two Cantor sets is nonempty. We present sharp conditions on the thicknesses of two Cantor sets which imply that their intersection contains a Cantor set of positive thickness.
The trace of the heat kernel in Lipschitz domains
Russell M.
Brown
889-900
Abstract: We establish the existence of an asymptotic expansion as $t \to {0^ + }$ for the trace of the heat kernel for the Neumann Laplacian in a bounded Lipschitz domain. The proof of an asymptotic expansion for the heat kernel for the Dirichlet Laplacian is also sketched. The treatment of the Dirichlet Laplacian extends work of Brossard and Carmona who obtained the same result in ${C^1}$-domains.
Inequalities for mixed projection bodies
Erwin
Lutwak
901-916
Abstract: Mixed projection bodies are related to ordinary projection bodies (zonoids) in the same way that mixed volumes are related to ordinary volume. Analogs of the classical inequalities from the Brunn-Minkowski Theory (such as the Minkowski, Brunn-Minkowski, and Aleksandrov-Fenchel inequalities) are developed for projection and mixed projection bodies.