Abstract functional-differential equations and reaction-diffusion systems
R. H.
Martin;
H. L.
Smith
1-44
Abstract: Several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a Banach space setting. The ideas are applied to reaction-diffusion systems that have time delays in the nonlinear reaction terms. The techniques presented here include differential inequalities, invariant sets, and Lyapunov functions, and therefore they provide for a wide range of applicability. The results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays.
The $27$-dimensional module for $E\sb 6$. III
Michael
Aschbacher
45-84
Abstract: This is the third in a series of five papers investigating the subgroup structure of the universal Chevalley group $G = {E_6}(F)$ of type ${E_6}$ over a field $F$ and the geometry induced on the $ 27$-dimensional $ FG$-module $V$ by the symmetric trilinear form $ f$ preserved by $ G$. The series uses the geometry on $V$ to describe and enumerate (up to a small list of ambiguities) all closed maximal subgroups of $ G$ when $F$ is finite or algebraically closed.
On Gel\cprime fand pairs associated with solvable Lie groups
Chal
Benson;
Joe
Jenkins;
Gail
Ratcliff
85-116
Abstract: Let $G$ be a locally compact group, and let $K$ be a compact subgroup of ${\operatorname{Aut}}(G)$, the group of automorphisms of $G$. There is a natural action of $K$ on the convolution algebra $ {L^1}(G)$, and we denote by $L_K^1(G)$ the subalgebra of those elements in $ {L^1}(G)$ that are invariant under this action. The pair $(K,G)$ is called a Gelfand pair if $ L_K^1(G)$ is commutative. In this paper we consider the case where $ G$ is a connected, simply connected solvable Lie group and $K \subseteq {\operatorname{Aut}}(G)$ is a compact, connected group. We characterize such Gelfand pairs $(K,G)$, and determine a moduli space for the associated $K$-spherical functions.
On the existence of central sequences in subfactors
Dietmar H.
Bisch
117-128
Abstract: We prove a relative version of [Co1, Theorem 2.1] for a pair of type $ {\text{I}}{{\text{I}}_1}$-factors $N \subset M$. This gives a list of necessary and sufficient conditions for the existence of nontrivial central sequences of $M$ contained in the subfactor $N$. As an immediate application we obtain a result by Bédos [Be, Theorem A], showing that if $N$ has property $\Gamma$ and $G$ is an amenable group acting freely on $ N$ via some action $ \sigma$, then the crossed product $ N{ \times _\sigma }G$ has property $\Gamma$. We also include a proof of a relative Mc Duff-type theorem (see [McD, Theorems $ 1$, $2$ and $3$]), which gives necessary and sufficient conditions implying that the pair $N \subset M$ is stable.
On the bihomogeneity problem of Knaster
Krystyna
Kuperberg
129-143
Abstract: The author constructs a locally connected, homogeneous, finitedimensional, compact, metric space which is not bihomogeneous, thus providing a compact counterexample to a problem posed by B. Knaster around 1921.
Iterated spinning and homology spheres
Alexander I.
Suciu
145-157
Abstract: Given a closed $ n$-manifold ${M^n}$ and a tuple of positive integers $ P$, let ${\sigma _P}M$ be the $P$-spin of $M$. If $ {M^n} \not\backsimeq{S^n}$ and $P \ne Q$ (as unordered tuples), it is shown that ${\sigma _P}M\not\backsimeq{\sigma _Q}M$ if either (1) $ {H_*}({M^n})\not\cong{H_*}({S^n})$, (2)${\pi _1}M$ finite, (3) $M$ aspherical, or (4) $n = 3$. Applications to the homotopy classification of homology spheres and knot exteriors are given.
Identities on quadratic Gauss sums
Paul
Gérardin;
Wen-Ch’ing Winnie
Li
159-182
Abstract: Given a local field $ F$, each multiplicative character $\theta$ of the split algebra $F \times F$ or of a separable quadratic extension of $F$ has an associated generalized Gauss sum $\gamma _\theta ^F$. It is a complex valued function on the character group of ${F^ \times } \times F$, meromorphic in the first variable. We define a pairing between such Gauss sums and study its properties when $F$ is a nonarchimedean local field. This has important applications to the representation theory of $ GL(2,F)$ and correspondences $ [{\text{GL}}3]$.
The Jacobian module of a Lie algebra
J. P.
Brennan;
M. V.
Pinto;
W. V.
Vasconcelos
183-196
Abstract: There is a natural way to associate to the commuting variety $ C(A)$ of an algebra $ A$ a module over a polynomial ring. It serves as a vehicle to study the arithmetical properties of $C(A)$, particularly Cohen-Macaulayness. The focus here is on Lie algebras and some of their representations.
Dichromatic link invariants
Jim
Hoste;
Mark E.
Kidwell
197-229
Abstract: We investigate the skein theory of oriented dichromatic links in $ {S^3}$. We define a new chromatic skein invariant for a special class of dichromatic links. This invariant generalizes both the two-variable Alexander polynomial and the twisted Alexander polynomial. Alternatively, one may view this new invariant as an invariant of oriented monochromatic links in ${S^1} \times {D^2}$, and as such it is the exact analog of the twisted Alexander polynomial. We discuss basic properties of this new invariant and applications to link interchangeability. For the full class of dichromatic links we show that there does not exist a chromatic skein invariant which is a mutual extension of both the two-variable Alexander polynomial and the twisted Alexander polynomial.
Composite ribbon number one knots have two-bridge summands
Steven A.
Bleiler;
Mario
Eudave Muñoz
231-243
Abstract: A composite ribbon knot which can be sliced with a single band move has a two-bridge summand.
Gel\cprime fer functions, integral means, bounded mean oscillation, and univalency
Shinji
Yamashita
245-259
Abstract: A Gelfer function $ f$ is a holomorphic function in $ D = \{ \left\vert z \right\vert < 1\}$ such that $f(0) = 1$ and $ f(z) \ne - f(w)$ for all $ z$, $w$ in $D$. The family $G$ of Gelfer functions contains the family $ P$ of holomorphic functions $f$ in $D$ with $f(0) = 1$ and Re $f > 0$ in $D$. If $f$ is holomorphic in $D$ and if the ${L^2}$ mean of $f'$ on the circle $\{ \left\vert z \right\vert = r\}$ is dominated by that of a function of $G$ as $r \to 1 - 0$, then $f \in BMOA$. This has two recent and seemingly different results as corollaries. A core of the proof is the fact that ${\operatorname{log}}f \in BMOA$ if $f \in G$. Besides the properties obtained concerning $f \in G$ itself, we shall investigate some families of functions where the roles played by $ P$ in Univalent Function Theory are replaced by those of $G$. Some exact estimates are obtained.
Nonmonomial characters and Artin's conjecture
Richard
Foote
261-272
Abstract: If $E/F$ is a Galois extension of number fields with solvable Galois group $G$, the main result of this paper proves that if the Dedekind zeta-function of $ E$ has a zero of order less than $ {\mathcal{M}_G}$ at the complex point ${s_0} \ne 1$, then all Artin $ L$-series for $ G$ are holomorphic at $ {s_0}$ -- here ${\mathcal{M}_G}$ is the smallest degree of a nonmonomial character of any subgroup of $G$. The proof relies only on certain properties of $L$-functions which are axiomatized to give a purely character-theoretic statement of this result.
Isometric isomorphisms between Banach algebras related to locally compact groups
F.
Ghahramani;
A. T.
Lau;
V.
Losert
273-283
Abstract: Let ${G_1}$, ${G_2}$ be locally compact groups. We prove in this paper that if $T$ is an isometric isomorphism from the Banach algebra ${\text{LUC}}{({G_1})^*}$ (the continuous dual of the Banach space of left uniformly continuous functions on ${G_1}$, equipped with Arens multiplication) onto $ {\text{LUC}}{({G_2})^*}$, then $T$ maps $M({G_1})$ onto $M({G_2})$ and $ {L^1}({G_1})$ onto ${L^1}({G_2})$. We also prove that any isometric isomorphism from $ {L^1}{({G_1})^{**}}$ (second conjugate algebra of $ {L^1}({G_1})$) onto $ {L^1}{({G_2})^{**}}$ maps ${L^1}({G_1})$ onto $ {L^1}({G_2})$.
Maximal polynomials and the Ilieff-Sendov conjecture
Michael J.
Miller
285-303
Abstract: In this paper, we consider those complex polynomials which have all their roots in the unit disk, one fixed root, and all the roots of their first derivatives as far as possible from a fixed point. We conjecture that any such polynomial has all the roots of its derivative on a circle centered at the fixed point, and as many of its own roots as possible on the unit circle. We prove a part of this conjecture, and use it to define an algorithm for constructing some of these polynomials. With this algorithm, we investigate the 1962 conjecture of Sendov and the 1969 conjecture of Goodman, Rahman and Ratti and (independently) Schmeisser, obtaining counterexamples of degrees $6$, $8$, $10$, and $12$ for the latter.
Complex interpolation for normed and quasi-normed spaces in several dimensions. III. Regularity results for harmonic interpolation
Zbigniew
Slodkowski
305-332
Abstract: The paper continues the study of one of the complex interpolation methods for families of finite-dimensional normed spaces ${\{ {{\mathbf{C}}^n},\vert\vert \cdot \vert{\vert _z}\} _{z \in G}}$, where $ G$ is open and bounded in $ {{\mathbf{C}}^k}$. The main result asserts that (under a mild assumption on the datum) the norm function $(z,w) \to \vert\vert w\vert\vert _z^2$ belongs to some anisotropic Sobolew class and is characterized by a nonlinear PDE of second order. The proof uses the duality theorem for the harmonic interpolation method (obtained earlier by the author). A new, simpler proof of this duality relation is also presented in the paper.
Weakly almost periodic functions and thin sets in discrete groups
Ching
Chou
333-346
Abstract: A subset $ E$ of an infinite discrete group $G$ is called (i) an ${R_W}$-set if any bounded function on $G$ supported by $E$ is weakly almost periodic, (ii) a weak $ p$-Sidon set $(1 \leq p < 2)$ if on ${l^1}(E)$ the ${l^p}$-norm is bounded by a constant times the maximal ${C^*}$-norm of ${l^1}(G)$, (iii) a $T$-set if $xE \cap E$ and $Ex \cap E$ are finite whenever $x \ne e$, and (iv) an $FT$-set if it is a finite union of $ T$-sets. In this paper, we study relationships among these four classes of thin sets. We show, among other results, that (a) every infinite group $G$ contains an ${R_W}$-set which is not an $FT$-set; (b) countable weak $p$-Sidon sets, $1 \leq p < 4/3$ are $FT$-sets.
Continuous spatial semigroups of $*$-endomorphisms of ${\germ B}({\germ H})$
Robert T.
Powers;
Geoffrey
Price
347-361
Abstract: To each continuous semigroup of $*$-endomorphisms $\alpha$ of $\mathfrak{B}\left( \mathfrak{H} \right)$ with an intertwining semigroup of isometries there is associated a $*$-representation $\pi$ of the domain $\mathfrak{O}(\delta )$ of the generator of $ \alpha$. It is shown that the Arveson index $ {d_ * }(\alpha )$ is the number of times the representation $ \pi$ contains the identity representation of $\mathfrak{O}(\delta )$. This result is obtained from an analysis of the relation between two semigroups of isometries, $U$ and $S$, satisfying the condition $S{(t)^*}U(t) = {e^{ - \lambda t}}I$ for $ t \geq 0$ and $\lambda > 0$.
Jacobi polynomials as generalized Faber polynomials
Ahmed I.
Zayed
363-378
Abstract: Let ${\mathbf{B}}$ be an open bounded subset of the complex $z$-plane with closure $\overline {\mathbf{B}}$ whose complement ${\overline {\mathbf{B}} ^c}$ is a simply connected domain on the Riemann sphere. $z = \psi (w)$ map the domain $\left\vert w \right\vert > \rho \quad (\rho > 0)$ one-to-one conformally onto the domain $ {\overline {\mathbf{B}} ^c}$ such that $ \psi (\infty ) = \infty$. Let $R(w) = \sum\nolimits_{n = 0}^\infty {{c_n}{w^{ - n}}}$, ${c_0} \ne 0$ be analytic in the domain $\left\vert w \right\vert > \rho$ with $R(w) \ne 0$. Let $ F(z) = \sum\nolimits_{n = 0}^\infty {{b_n}} {z^n}$, $F*(z) = \sum\nolimits_{n = 0}^\infty {\frac{1} {{{b_n}}}} {z^n}$ be analytic in $\left\vert z \right\vert < 1$ and analytically continuable to any point outside $\left\vert z \right\vert < 1$ along any path not passing through the points $z = 0,1,\infty$. The generalized Faber polynomials $\{ {P_n}(z)\} _{n = 0}^\infty$ of ${\mathbf{B}}$ are defined by $ \{ P_n^{(\alpha ,\beta )}(z)\} _{n = 0}^\infty$ are generalized Faber polynomials of any region $ {\mathbf{B}}$, then it must be the elliptic region $ \{ z:\vert z + 1\vert + \vert z - 1\vert < \rho + \frac{1}{\rho },\rho > 1\} ;$ (2) the only Jacobi polynomials that can be classified as generalized Faber polynomials are the Tchebycheff polynomials of the first kind, some normalized Gegenbauer polynomials, some normalized Jacobi polynomials of type $\{ P_n^{(\alpha ,\alpha + 1)}(z)\} _{n = 0}^\infty$, $\{ P_n^{(\beta + 1,\beta )}(z)\} _{n = 0}^\infty$ and there are no others, no matter how one normalizes them; (3) the Hermite and Laguerre polynomials cannot be generalized Faber polynomials of any region.
The nonstandard treatment of Hilbert's fifth problem
Joram
Hirschfeld
379-400
Abstract: We give a nonstandard proof that every locally Euclidean group is a Lie group. The heart of the proof is a strong nonstandard variant of Gleason's lemma for a class of groups that includes all locally Euclidean groups.
Generalized local Fatou theorems and area integrals
B. A.
Mair;
Stan
Philipp;
David
Singman
401-413
Abstract: Let $X$ be a space of homogeneous type and $ W$ a subset of $X \times (0,\infty )$. Then, under minimal conditions on $W$, we obtain a relationship between two modes of convergence at the boundary $X$ for functions defined on $ W$. This result gives new local Fatou theorems of the Carleson-type for solutions of Laplace, parabolic and Laplace-Beltrami equations as immediate consequences of the classical results. Lusin area integral characterizations for the existence of limits within these more general approach regions are also obtained.