Modular representation theory of finite groups with T.I. Sylow $p$-subgroups
H. I.
Blau;
G. O.
Michler
417-468
Abstract: Let $p$ be a fixed prime, and let $ G$ be a finite group with a T.I. Sylow $p$-subgroup $P$. Let $ N = {N_G}(P)$ and let $ k(G)$ be the number of conjugacy classes of $G$. If $z(G)$ denotes the number of $p$-blocks of defect zero, then we show in this article that $ z(G) = k(G) - k(N)$. This result confirms a conjecture of J. L. Alperin. Its proof depends on the classification of the finite simple groups. Brauer's height zero conjecture and the Alperin-McKay conjecture are also verified for finite groups with a T.I. Sylow $p$-subgroup.
Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities
John R.
Stembridge
469-498
Abstract: We apply the theory of Hall-Littlewood functions to prove several multiple basic hypergeometric series identities, including some previously known generalizations of the Rogers-Ramanujan identities due to G. E. Andrews and D. M. Bressoud. The techniques involve the adaptation of a method due to I. G. Macdonald for calculating partial fraction expansions of certain types of symmetric formal power series. Macdonald originally used this method to prove a pair of generating function identities for plane partitions conjectured by MacMahon and Bender-Knuth. We show that this method can also be used to prove another pair of plane partition identities recently obtained by R. A. Proctor.
Defect relations for degenerate meromorphic maps
Wan Xi
Chen
499-515
Abstract: Using a concept called subgeneral position and adapting a weight function created by E. I. Nochka, this work proves the Cartan's conjecture on defect relations for a degenerate meromorphic map from a parabolic manifold into a projective space.
Weighted inequalities for one-sided maximal functions
F. J.
Martín-Reyes;
P.
Ortega Salvador;
A.
de la Torre
517-534
Abstract: Let $M_g^ +$ be the maximal operator defined by $\displaystyle M_g^ + f(x) = \mathop {\sup }\limits_{h > 0} \left( {\int_x^{x + ... ... f(t)\vert g(t)dt} } \right){\left( {\int_x^{x + h} {g(t)dt} } \right)^{ - 1}},$ where $g$ is a positive locally integrable function on ${\mathbf{R}}$. We characterize the pairs of nonnegative functions $(u,v)$ for which $M_g^ +$ applies ${L^p}(v)$ in ${L^p}(u)$ or in weak- ${L^p}(u)$. Our results generalize Sawyer's (case $ g = 1$) but our proofs are different and we do not use Hardy's inequalities, which makes the proofs of the inequalities self-contained.
The $v\sb 1$-periodic homotopy groups of an unstable sphere at odd primes
Robert D.
Thompson
535-559
Abstract: The $\bmod \;p$ ${v_1}$-periodic homotopy groups of a space $ X$ are defined by considering the homotopy classes of maps of a Moore space into $ X$ and then inverting the Adams self map. In this paper we compute the $ p$ $ {v_1}$-periodic homotopy groups of an odd dimensional sphere, localized at an odd prime. This is done by showing that these groups are isomorphic to the stable $\bmod \;p$ ${v_1}$-periodic homotopy groups of $B\Sigma _p^{2(p - 1)n}$, the $2(p - 1)n$ skeleton of the classifying space for the symmetric group ${\Sigma _p}$. There is a map ${\Omega ^{2n + 1}}{S^{2n + 1}} \to {\Omega ^\infty }(J \wedge B\Sigma _p^{2(p - 1)n})$, where $ J$ is a spectrum constructed from connective $K$-theory, and the image in homotopy is studied.
Algebraic distance graphs and rigidity
M.
Homma;
H.
Maehara
561-572
Abstract: An algebraic distance graph is defined to be a graph with vertices in $ {E^n}$ in which two vertices are adjacent if and only if the distance between them is an algebraic number. It is proved that an algebraic distance graph with finite vertex set is complete if and only if the graph is "rigid". Applying this result, we prove that (1) if all the sides of a convex polygon $\Gamma$ which is inscribed in a circle are algebraic numbers, then the circumradius and all diagonals of $\Gamma$ are also algebraic numbers, (2) the chromatic number of the algebraic distance graph on a circle of radius $r$ is $\infty$ or $2$ accordingly as $r$ is algebraic or not. We also prove that for any $ n > 0$, there exists a graph $G$ which cannot be represented as an algebraic distance graph in ${E^n}$.
Jumps of orderings
C. J.
Ash;
C. G.
Jockusch;
J. F.
Knight
573-599
Abstract: Here it is shown that for each recursive ordinal $\alpha \geqslant 2$ and each Turing degree ${\mathbf{d}} > {{\mathbf{0}}^{(\alpha )}}$, there is a linear ordering ${\mathbf{A}}$ such that $ {\mathbf{d}}$ is least among the $\alpha$th jumps of degrees of (open diagrams of) isomorphic copies of $ {\mathbf{A}}$ and for $\beta < \alpha$, the set of $\beta$th jumps of degrees of copies of ${\mathbf{A}}$ has no least element.
Topological entropy of fixed-point free flows
Romeo F.
Thomas
601-618
Abstract: Topological entropy was introduced as an invariant of topological conjugacy and also as an analogue of measure theoretic entropy. Topological entropy for one parameter flows on a compact metric spaces is defined by Bowen. General statements are proved about this entropy, but it is not easy to calculate the topological entropy, and to show it is invariant under conjugacy. For all this I would like to try to pose a new direction and study a definition for the topological entropy that involves handling the technical difficulties that arise from allowing reparametrizations of orbits. Some well-known results are proved as well using this definition. These results enable us to prove some results which seem difficult to prove using Bowen's definition. Also we show here that this definition is equivalent to Bowen's definition for any flow without fixed points on a compact metric space. Finally, it is shown that the topological entropy of an expansive flow can be defined globally on a local cross sections.
The heat equation for Riemannian foliations
Seiki
Nishikawa;
Mohan
Ramachandran;
Philippe
Tondeur
619-630
Abstract: Let $\mathcal{F}$ be a Riemannian foliation on a closed oriented manifold $M$, with the transversal Laplacian ${\Delta _B}$ acting on the basic forms $\Omega _B^r(\mathcal{F})$ of degree $r \geqslant 0$. We construct the fundamental solution $ e_B^r(x,y,t)$ for the basic heat operator $\partial /\partial t + {\Delta _B}$, and prove existence and uniqueness for the solution of the heat equation on $ \Omega _B^r(\mathcal{F})$. As an application we give a new proof for the deRham-Hodge decomposition theorem for $ {\Delta _B}$ in $\Omega _B^r(\mathcal{F})$, generalizing the approach to the classical deRham-Hodge theorem pioneered by Milgram and Rosenbloom.
Dirac manifolds
Theodore James
Courant
631-661
Abstract: A Dirac structure on a vector space $V$ is a subspace of $V$ with a skew form on it. It is shown that these structures correspond to subspaces of $V \oplus {V^{\ast}}$ satisfying a maximality condition, and having the property that a certain symmetric form on $V \oplus {V^{\ast}}$ vanishes when restricted to them. Dirac structures on a vector space are analyzed in terms of bases, and a generalized Cayley transformation is defined which takes a Dirac structure to an element of $O(V)$. Finally a method is given for passing a Dirac structure on a vector space to a Dirac structure on any subspace. Dirac structures on vector spaces are generalized to smooth Dirac structures on a manifold $P$, which are defined to be smooth subbundles of the bundle $ TP \oplus {T^{\ast}}P$ satisfying pointwise the properties of the linear case. If a bundle $L \subset TP \oplus {T^{\ast}}P$ defines a Dirac structure on $P$, then we call $L$ a Dirac bundle over $P$. A $3$-tensor is defined on Dirac bundles whose vanishing is the integrability condition of the Dirac structure. The basic examples of integrable Dirac structures are Poisson and presymplectic manifolds; in these cases the Dirac bundle is the graph of a bundle map, and the integrability tensors are $[B,B]$ and $d\Omega$ respectively. A function $f$ on a Dirac manifold is called admissible if there is a vector field $X$ such that the pair $(X,df)$ is a section of the Dirac bundle $ L$; the pair $ (X,df)$ is called an admissible section. The set of admissible functions is shown to be a Poisson algebra. A process is given for passing Dirac structures to a submanifold $Q$ of a Dirac manifold $P$. The induced bracket on admissible functions on $Q$ is in fact the Dirac bracket as defined by Dirac for constrained submanifolds.
Link homotopy with one codimension two component
Paul A.
Kirk
663-688
Abstract: Link maps with one codimension two component are studied and an invariant of link maps modulo link homotopy is constructed using ideas from knot theory and immersion theory. This invariant is used to give examples of nontrivial link homotopy classes and to show that there are infinitely many distinct link homotopy classes in many dimensions. A link map with the codimension two component embedded is shown to be nullhomotopic. These ideas are applied to the special case of $2$-spheres in ${S^4}$ to give simple examples of the failure of the Whitney trick in dimension $4$.
Lie algebra representations of dimension $<p\sp 2$
Helmut
Strade
689-709
Abstract: Various methods of representation theory of modular Lie algebras are improved. As an application the structure of the Lie algebras having a faithful irreducible module of dimension $< {p^2}$ is determined. Applications to the classification theory of modular simple Lie algebras are given.
On the Dirichlet space for finitely connected regions
Kit Chak
Chan
711-728
Abstract: This paper is devoted to the study of the Dirichlet space $\operatorname{Dir} (G)$ for finitely connected regions $G$; we are particularly interested in the algebra of bounded multiplication operators on this space. Results in different directions are obtained. One direction deals with the structure of closed subspaces invariant under all bounded multiplication operators. In particular, we show that each such subspace contains a bounded function. For regions with circular boundaries we prove that a finite codimensional closed subspace invariant under multiplication by $ z$ must be invariant under all bounded multiplication operators, and furthermore it is of the form $p\operatorname{Dir} (G)$, where $p$ is a polynomial with all its roots lying in $G$. Another direction is to study cyclic and noncyclic vectors for the algebra of all bounded multiplication operators. Typical results are: if $f \in \operatorname{Dir} (G)$ and $f$ is bounded away from zero then $ f$ is cyclic; on the other hand, if the zero set of the radial limit function of $ f$ on the boundary has positive logarithmic capacity, then $f$ is not cyclic. Also, some other sufficient conditions for a function to be cyclic are given. Lastly, we study transitive operator algebras containing all bounded multiplication operators; we prove that they are dense in the algebra of all bounded operators in the strong operator topology.
The method of negative curvature: the Kobayashi metric on ${\bf P}\sb 2$ minus $4$ lines
Michael J.
Cowen
729-745
Abstract: Bloch, and later H. Cartan, showed that if ${H_1}, \ldots ,{H_{n + 2}}$ are $n + 2$ hyperplanes in general position in complex projective space ${{\mathbf{P}}_n}$, then $ {{\mathbf{P}}_n} - {H_1} \cup \cdots \cup {H_{n + 2}}$ is (in current terminology) hyperbolic modulo $\Delta$, where $\Delta$ is the union of the hyperplanes $({H_{^1}} \cap \cdots \cap {H_k}) \oplus ({H_{k + 1}} \cap \cdots \cap {H_{n + 2}})$ for $2 \leqslant k \leqslant n$ and all permutations of the ${H_i}$. Their results were purely qualitative. For $ n = 1$, the thrice-punctured sphere, it is possible to estimate the Kobayashi metric, but no estimates were known for $n \geqslant 2$. Using the method of negative curvature, we give an explicit model for the Kobayashi metric when $n = 2$.
Local behavior of solutions of quasilinear elliptic equations with general structure
J.-M.
Rakotoson;
William P.
Ziemer
747-764
Abstract: This paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form $ \operatorname{div} A(x,u,\nabla u) + B(x,u,\nabla u)$, where $A$ and $B$ are Borel measurable, are solutions to the equation $ \operatorname{div} A(x,u,\nabla u) + B(x,u,\nabla u) = \mu$ for some nonnegative Radon measure $\mu$. Among other things, it is shown that if $ u$ is a Hölder continuous solution to this equation, then the measure $ \mu$ satisfies the growth property $\mu [B(x,r)] \leqslant M{r^{n - p + \varepsilon }}$ for all balls $B(x,r)$ in $ {{\mathbf{R}}^n}$. Here $\varepsilon$ depends on the Hölder exponent of $ u$ while $p > 1$ is given by the structure of the differential operator. Conversely, if $ \mu$ is assumed to satisfy this growth condition, then it is shown that $ u$ satisfies a Harnack-type inequality, thus proving that $u$ is locally bounded. Under the additional assumption that $A$ is strongly monotonic, it is shown that $ u$ is Hölder continuous.
Amenability of weighted convolution algebras on locally compact groups
Niels
Grønbæk
765-775
Abstract: We give a direct transition from the existence of a bounded right approximate identity in the diagonal ideal for a weighted convolution algebra on a locally compact group to the existence of translation invariant means on an associated weighted ${L^\infty }$-space, thus giving a characterization of amenability for such an algebra.
$\Lambda(q)$ processes
Ron C.
Blei
777-786
Abstract: Motivated by some classical notions in harmonic analysis, $\Lambda (q)$ processes are introduced in the context of a study of stochastic interdependencies. An extension of a classical theorem of Salem and Zygmund regarding random Fourier series is obtained. The Littlewood exponent of $\Lambda (q)$ processes is estimated and, in some archetypical cases, computed.
Holomorphic maps which preserve intrinsic metrics or measures
Ian
Graham
787-803
Abstract: Suppose that $ M$ is a domain in a taut complex manifold $M'$, and that $\Omega$ is a strictly convex bounded domain in $ {{\mathbf{C}}^n}$. We consider the following question: given a holomorphic map $F:M \to \Omega$ which is an isometry for the infinitesimal Kobayashi metric at one point, must $ F$ be biholomorphic? With an additional technical assumption on the behavior of the Kobayashi distance near points of $\partial M$, we show that $F$ gives a biholomorphism of $ M$ with an open dense subset of $\Omega$. Moreover, $F$ extends as a homeomorphism from a larger domain $\tilde M$ to $\Omega$. We also give some related results--refinements of theorems of Bland and Graham and Fornaess and Sibony, and the answer to a question of Graham and Wu.
Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions
Zhou Ping
Xin
805-820
Abstract: This paper concerns the large time behavior toward planar rarefaction waves of the solutions for scalar viscous conservation laws in several dimensions. It is shown that a planar rarefaction wave is nonlinearly stable in the sense that it is an asymptotic attractor for the viscous conservation law. This is proved by using a stability result of rarefaction wave for scalar viscous conservation laws in one dimension and an elementary $ {L^2}$-energy method.
Remarks and corrections for: ``Groups acting on affine algebras'' [Trans. Amer. Math. Soc. {\bf 310} (1988), no. 2, 485--497; MR0940913 (89i:16029)]
Daniel R.
Farkas
821-823