Potent axioms
Matthew
Foreman
1-28
Abstract: This paper suggests alternatives to the ordinary large cardinal axioms of set theory. These axioms can be viewed as generalizations of large cardinals and exhibit many of the same phenomena. They are shown to imply the G.C.H., every set of reals in $L({\mathbf{R}})$ is Lebesgue measurable, and various results in combinatorics, algebra and model theory.
Extensions of Verma modules
Kevin J.
Carlin
29-43
Abstract: A spectral sequence is introduced which computes extensions in category $\mathcal{O}$ in terms of derived functors associated to coherent translation functors. This is applied to the problem of computing extensions of one Verma module by another when the highest weights are integral and regular. Some results are obtained which are consistent with the Gabber-Joseph conjecture. The main result is that the highest-degree nonzero extension is one-dimensional. The spectral sequence is also applied to the Kazhdan-Lusztig conjecture and related to the work of Vogan in this area.
Bounds on the dimension of variations of Hodge structure
James A.
Carlson
45-64
Abstract: We derive upper bounds on the dimension of a variation of Hodge structure of weight two and show that these bounds are sharp. Using them we exhibit maximal geometric variations of Hodge structure. Analogous results for higher weight are obtained in the presence of a nondegeneracy hypothesis, and variations coming from hypersurfaces are shown to be nondegenerate. Maximal geometric variations of higher weight are also constructed.
Clarke's gradients and epsilon-subgradients in Banach spaces
Jay S.
Treiman
65-78
Abstract: A new characterization of Clarke's normal cone to a closed set in a Banach space is given. The normal cone is characterized in terms of weak-star limits of epsilon normals. A similar characterization of Clarke's generalized gradients is also presented. Restrictions must be placed on the Banach spaces to make the formulas valid.
Hyperreflexivity and a dual product construction
David R.
Larson
79-88
Abstract: We show that an example of a nonhyperreflexive CSL algebra recently constructed by Davidson and Power is a special case of a general and natural reflexive subspace construction. Completely different techniques of proof are needed because of absence of symmetry. It is proven that if $\mathcal{S}$ and $ \mathcal{I}$ are reflexive proper linear subspaces of operators acting on a separable Hilbert space, then the hyperreflexivity constant of ${({\mathcal{S}_ \bot } \otimes {\mathcal{I}_ \bot })^ \bot }$ is at least as great as the product of the constants of $ \mathcal{S}$ and $\mathcal{I}$.
On sieved orthogonal polynomials. III. Orthogonality on several intervals
Mourad E. H.
Ismail
89-111
Abstract: We introduce two generalizations of Chebyshev polynomials. The continuous spectrum of either is $\{ x: - 2\sqrt c /(1 + c) \leqslant {T_k}(x) \leqslant 2\sqrt c /(1 + c)\}$, where $c$ is a positive parameter. The weight function of the polynomials of the second kind is ${\{ 1 - ({(1 + c)^2}/4\operatorname{c} )T_k^2(x)\} ^{1/2}}/\vert{U_{k - 1}}(x)\vert$ when $c \geqslant 1$. When $c < 1$ we pick up discrete masses located at the zeros of ${U_{k - 1}}(x)$. The weight function of the polynomials of the first kind is also included. Sieved generalizations of the symmetric Pollaczek polynomials and their $q$-analogues are also treated. Their continuous spectra are also the above mentioned set. The $ q$-analogues include a sieved version of the Rogers $q$-ultraspherical polynomials and another set of $q$-ultraspherical polynomials discovered by Askey and Ismail. Generating functions and explicit formulas are also derived.
Some generalized Brown-Gitler spectra
Paul G.
Goerss;
John D. S.
Jones;
Mark E.
Mahowald
113-132
Abstract: Brown-Gitler spectra for the homology theories associated with the spectra $K{{\mathbf{Z}}_p}^ \wedge $ , $bo$, and $bu$ are constructed. Complexes adapted to the new Brown-Gitler spectra are produced and a spectral sequence converging to stable maps into these spectra is constructed and examined.
$H\sp p$-classes on rank one symmetric spaces of noncompact type. I. Nontangential and probabilistic maximal functions
Patricio
Cifuentes
133-149
Abstract: Two kinds of $ {H^p}$-classes of harmonic functions are defined on a general rank one symmetric space of noncompact type. The first one is introduced by using a nontangential maximal function. The second is related to the diffusion generated by the Laplace-Beltrami operator. The equivalence of the two classes is proven for $0 < p < \infty$.
Classification of metabelian $p$-groups
Wu Nan
Chou
151-176
Abstract: Let $G$ be a two-generator metabelian group of exponent $p$ with class of nilpotence $c$, where $c \leqslant p - 1$ and $ p$ is an odd prime. In this paper, we shall consider the classification problem when $ \vert{G_2}/{G_3}\vert = p$, $ \vert{G_3}/{G_4}\vert = {p^2}$and $\vert{G_4}/{G_5}\vert \leqslant {p^2}$.
On linking double lines
Juan
Migliore
177-185
Abstract: A double line is a nonreduced locally Cohen-Macaulay scheme of degree two supported on a line in projective three-space. The heart of this work is to compute the associated Hartshorne-Rao module for such a curve. We can then say exactly when two such curves are in the same liaison class and in fact when they are directly linked. In particular, we find that $C$ is only self-linked in characteristic two.
Joint spectra and analytic set-valued functions
M.
Klimek
187-196
Abstract: We investigate analyticity of joint spectra of ${A^m}$-valued holomorphic mappings, where $ A$ denotes a complex Banach algebra. We show also that if $K$ is an analytic set-valued function whose values are compact subsets of ${{\mathbf{C}}^n}$ and $d$ is the transfinite diameter in ${{\mathbf{C}}^n}$, then the upper-semicontinuous regularization of $\log d(K)$ is plurisubharmonic. Moreover, we give higher dimensional extensions of Aupetit's Scarcity Theorem.
Anosov diffeomorphisms and expanding immersions. II
Lowell
Jones
197-216
Abstract: This paper continues the study of hyperbolic attractors, expanding immersions, and quotient solenoids which was begun in a previous paper of the same title. The main result states that certain hyperbolic attractors are topologically conjugate to an Anosov diffeomorphism.
Existence and nonoscillation theorems for an Emden-Fowler equation with deviating argument
William F.
Trench
217-231
Abstract: Sufficient conditions are given for a generalized Emden-Fowler equation with deviating argument to have nonoscillatory solutions with prescribed asymptotic behavior as $t \to \infty $. The integrability condition on the nonlinear term requires only conditional convergence, supplemented by a condition on the order of convergence, which is automatically satisfied in some important special cases. The exponent in the nonlinear term may be any real number. The deviating argument is not assumed to be purely advanced or retarded, and, in some cases, need not tend to infinity. Some of the results are global, in that the desired solution is shown to exist on a given interval, rather than only for sufficiently large $t$.
A bilaterally deterministic $\rho$-mixing stationary random sequence
Richard C.
Bradley
233-241
Abstract: A (nondegenerate) strictly stationary sequence $({X_k},\;k \in {\mathbf{Z}})$ of random variables is constructed such that the $\rho$-mixing (maximal correlation mixing) condition is satisfied and each ${X_k}$ is measurable with respect to the double tail $\sigma$-field.
Properties of relatively free inverse semigroups
N. R.
Reilly;
P. G.
Trotter
243-262
Abstract: The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if $ S$ is combinatorial (i.e., $\mathcal{H}$ is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, $S$ does not contain a copy of the bicyclic semigroup) or $E$-unitary (i.e., $E(S)$ is the kernel of the minimum group congruence) then the relatively free inverse semigroup $F{\mathcal{V}_X}$ on the set $X$ in the variety $\mathcal{V}$ generated by $S$ is also combinatorial, completely semisimple or $E$-unitary, respectively. If $ S$ is a fundamental (i.e., the only congruence contained in $\mathcal{H}$ is the identity congruence) and $ \vert X\vert \geqslant {\aleph _0}$, then $ F{\mathcal{V}_X}$ is also fundamental. $ F{\mathcal{V}_X}$ may not be fundamental if $ \vert X\vert < {\aleph _0}$. It is also shown that for any variety of groups $\mathcal{U}$ and for $\vert X\vert \geqslant {\aleph _0}$, there exists a variety of inverse semigroups $\mathcal{V}$ which is minimal with respect to the properties (i) $ F{\mathcal{V}_X}$ is fundamental and (ii) $\mathcal{V} \cap \mathcal{G} = \mathcal{U}$, where $ \mathcal{G}$ is the variety of groups. In the main result of the paper it is shown that there exists a variety $\mathcal{V}$ for which $F{\mathcal{V}_X}$ is not completely semisimple, thereby refuting a long standing conjecture.
On coupled multiparameter nonlinear elliptic systems
Robert Stephen
Cantrell
263-285
Abstract: This paper considers the system of nonlinear Dirichlet boundary value problems $\displaystyle \left\{ \begin{gathered}Lu(x) = \lambda f(u(x),v(x)) Lv(x) = \mu g(u(x),v(x)) \end{gathered} \right\},\qquad x \in \Omega ,$ a bounded domain in $ {{\mathbf{R}}^n}$. Here $ L$ is a strongly, uniformly elliptic linear partial differential operator, $ \lambda$, $\mu$ are real parameters, and $ f$, $g:{{\mathbf{R}}^2} \to R$ are smooth with $\displaystyle f(0,0) = 0 = g(0,0).$ A detailed analysis of the solution set to the system is given from the point of view of several parameter bifurcation theory.
Dominated permutations of subsequences of random variables
Aaron
Meyerowitz;
Mark
Schwartz
287-294
Abstract: The generalized strong law of large numbers of Komlós may be extended to include certain dominated permutations of the random variables. A further extension to larger classes of permutations is obtained through decompositions of sequences of positive integers.
On type of metric spaces
J.
Bourgain;
V.
Milman;
H.
Wolfson
295-317
Abstract: Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of ${l_p}$-cubes into the ${l_1}$-cube (Hamming cube) are discussed.
Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space
Ralph
Howard;
S. Walter
Wei
319-331
Abstract: Call a compact Riemannian manifold $M$ a strongly unstable manifold if it is not the range or domain of a nonconstant stable harmonic map and also the homotopy class of any map to or from $ M$ contains elements of arbitrarily small energy. If $M$ is isometrically immersed in Euclidean space, then a condition on the second fundamental form of $ M$ is given which implies $ M$ is strongly unstable. As compact isotropy irreducible homogeneous spaces have "standard" immersions into Euclidean space this allows a complete list of the strongly unstable compact irreducible symmetric spaces to be made.
Boundary uniqueness theorems in ${\bf C}\sp n$
Joseph A.
Cima;
Emil J.
Straube
333-339
Abstract: Let $n$-dimensional manifolds ${\Gamma _k},\,k = 1,2, \ldots$, be given in a smoothly bounded domain $\Omega \subset {{\mathbf{C}}^n}$. Assume that the ${\Gamma _k}$ "converge" to an $ n$-dimensional, totally real manifold $\Gamma \subseteq \partial \Omega$ and that a function $f$ analytic in $\Omega$ has the property that its traces $ {f_k}$ on ${\Gamma _k}$ have distributional limit zero as $k \to \infty $ (or assume that ${f_k} \to 0$ pointwise). Then under the assumption that $f$ is polynomially bounded near $P \in \Gamma$ by $ {(\operatorname{dist} (z,\partial \Omega ))^{ - 1}}$ we conclude that $ f$ is identically zero.
Lower bounds of the gap between the first and second eigenvalues of the Schr\"odinger operator
Qi Huang
Yu;
Jia Qing
Zhong
341-349
Abstract: In this paper the authors prove the following theorem: Let $ \Omega$ be a smooth strictly convex bounded domain in ${R^n}$ and $V:\Omega \to R$ a nonnegative convex function. Suppose $ {\lambda _1}$ and ${\lambda _2}$ are the first and second nonzero eigenvalues of the equation $\displaystyle - \Delta f + Vf = \lambda f,\qquad f{\vert _{\partial \Omega }} \equiv 0.$ Then ${\lambda _2} - {\lambda _1} \geqslant {\pi ^2}/{d^2}$, where $d$ is the diameter of $\Omega$.
The law of the iterated logarithm in uniformly convex Banach spaces
Michel
Ledoux
351-365
Abstract: We give necessary and sufficient conditions for a random variable $ X$ with values in a uniformly convex Banach space $B$ to satisfy the law of the iterated logarithm. Precisely, we show that a $B$-valued random variable $X$ satisfies the (compact) law of the iterated logarithm if and only if $E\{ \vert\vert X\vert{\vert^2}/{L_2}\vert\vert X\vert\vert\} < \infty$, the family $\{ \vert{x^{\ast}}(X){\vert^2};\,{x^{\ast}} \in {B^{\ast}},\,\vert\vert{x^{\ast}}\vert\vert = 1\}$ is uniformly integrable and ${S_n}/{a_n} \to 0$ in probability.
An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra
Zbigniew
Slodkowski
367-377
Abstract: It is shown that for every annulus $P = \{ z \in {{\mathbf{C}}^n}:\delta < \vert z\vert < r\}$, $ \delta > 0$, there exists a set-valued correspondence $z \to K(z):P \to {2^{{{\mathbf{C}}^n}}}$, whose graph is a bounded relatively closed subset of the manifold $\{ (z,w) \in P \times {{\mathbf{C}}^n}:{z_1}{w_1} + \cdots + {z_n}{w_n} = 1\} $ which can be covered by $ n$-dimensional analytic manifolds. The analytic set-valued selection $ K$ obtained thereby is then applied to several problems in complex analysis and spectral theory which involve solving the equation ${a_1}{x_1} + \cdots + {a_n}{x_n} = y$. For example, an elementary proof is given of the following special case of a theorem due to Oka: every bounded pseudoconvex domain in $ {{\mathbf{C}}^2}$ is a domain of holomorphy.
Corrigendum to: ``On isometric embeddings of graphs'' [Trans. Amer. Math. Soc. {\bf 288} (1985), no. 2, 527--536; MR0776391 (86f:05055b)]
R. L.
Graham;
P. M.
Winkler
379
Correction to: ``Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions'' [Trans. Amer. Math. Soc. {\bf 282} (1984), no. 1, 1--32; MR0728700 (85h:03015)]
C. Ward
Henson;
Lee A.
Rubel
381