Representations of compact groups on Banach algebras
David
Gurarie
1-55
Abstract: Let a compact group $ U$ act by automorphisms of a commutative regular and Wiener Banach algebra $\mathcal{A}$. We study representations ${R^\omega }$ of $U$ on quotient spaces $\mathcal{A}/I(\omega )$, where $\omega$ is an orbit of $U$ in the Gelfand space $X$ of $ \mathcal{A}$ and $I(\omega )$ is the minimal closed ideal with hull $\omega \subset X$. The main result of the paper is: if $\mathcal{A} = \,{\mathcal{A}_\rho }(X)$ is a weighted Fourier algebra on a LCA group $X = \hat A$ with a subpolynomial weight $ \rho$ on $A$, and $U$ acts by affine transformations on $ X$, then for any orbit $\omega \subset X$ the representation ${R^\omega }$ has finite multiplicity. Precisely, the multiplicity of $ \pi \in \hat U$ in ${R^\omega }$ is estimated as $k(\pi ;{R^\omega }) \leq c \cdot \deg (\pi )\;\forall \pi \in \hat U$ with a constant $c$ depending on $A$ and $\rho$. Applications of this result are given to topologically irreducible representations of motion groups and primary ideals of invariant subalgebras.
Homomorphisms and endomorphisms in varieties of pseudocomplemented distributive lattices (with applications to Heyting algebras)
M. E.
Adams;
V.
Koubek;
J.
Sichler
57-79
Abstract: According to a result by K. B. Lee, the lattice of varieties of pseudocomplemented distributive lattices is the $\omega + 1$ chain ${B_{ - 1}} \subset {B_0} \subset {B_1} \subset \cdots \subset {B_n} \subset \cdots \subset {B_\omega }$ in which the first three varieties are formed by trivial, Boolean, and Stone algebras respectively. In the present paper it is shown that any Stone algebra is determined within ${B_1}$ by its endomorphism monoid, and that there are at most two nonisomorphic algebras in $ {B_2}$ with isomorphic monoids of endomorphisms; the pairs of such algebras are fully characterized both structurally and in terms of their common endomorphism monoid. All varieties containing ${B_3}$ are shown to be almost universal. In particular, for any infinite cardinal $\kappa$ there are ${2^\kappa }$ nonisomorphic algebras of cardinality $\kappa$ in ${B_3}$ with isomorphic endomorphism monoids. The variety of Heyting algebras is also almost universal, and the maximal possible number of nonisomorphic Heyting algebras of any infinite cardinality with isomorphic endomorphism monoids is obtained.
Necessary and sufficient conditions for oscillations of higher order delay differential equations
G.
Ladas;
Y. G.
Sficas;
I. P.
Stavroulakis
81-90
Abstract: Consider the $n{\text{th}}$ order delay differential equation (1) $\displaystyle {x^{(n)}}(t) + {( - 1)^{n + 1}}\sum\limits_{i = 0}^k {{p_i}x(t - {\tau_i}) = 0, \qquad t \geq {t_0}},$ where the coefficients and the delays are constants such that $0 = {\tau_0} < {\tau_{1}}\, < \cdots < {\tau_k};{p_0}\, \geq 0,{p_i} > 0,i = 1,2,\ldots,k;k \geq 1$ and $n \geq 1$. The characteristic equation of (1) is (2) $\displaystyle {\lambda ^n} + {( - 1)^{n + 1}}\;\sum\limits_{i = 0}^k {{p_i}{e^{ - \lambda {\tau_i}}} = 0}.$ We prove the following theorem. Theorem. (i) For $n$ odd every solution of (1) oscillates if and only if (2) has no real roots. (ii) For $n$ even every bounded solution of (1) oscillates if and only if (2) has no real roots in $( - \infty ,0]$. The above results have straightforward extensions for advanced differential equations.
Factorizing the polynomial of a code
G.
Hansel;
D.
Perrin;
C.
Reutenauer
91-105
Abstract: We give an extension and a simplified presentation of a theorem of Schützenberger. This theorem describes the factorization of the commutative polynomial associated with a finite maximal code. It is the deepest result known so far in the theory of (variable-length) codes.
Riesz decompositions in Markov process theory
R. K.
Getoor;
J.
Glover
107-132
Abstract: Riesz decompositions of excessive measures and excessive functions are obtained by probabilistic methods without regularity assumptions. The decomposition of excessive measures is given for Borel right processes. The results for excessive functions are formulated within the framework of weak duality. These results extend and generalize the pioneering work of Hunt in this area.
On bases in the disc algebra
J.
Bourgain
133-139
Abstract: It is shown that the disc algebra has no Besselian basis. In fact, concrete minorations on certain Lebesgue functions are obtained. A consequence is the nonisomorphism of the disc algebra and the space of uniformly convergent Fourier series on the circle.
Convergence of multivariate polynomials interpolating on a triangular array
T. N. T.
Goodman;
A.
Sharma
141-157
Abstract: Given a triangular array of complex numbers, it is well known that for any function $f$ smooth enough, there is a unique polynomial $ {G_n}f$ of degree $ \leq n$ such that on each of the first $n + 1$ rows of the array the divided difference of $ {G_n}f$ coincides with that of $f$. This result has recently been generalized to give a unique polynomial ${\mathcal{G}_n}f$ in $k$ variables $(k > 1)$ of total degree $\leq n$ which interpolates a given function $f$ on a triangular array in ${C^k}$. In this paper we extend some results of A. O. Gelfond and derive formulas for ${\mathcal{G}_n}f$ and $f - {\mathcal{G}_n}f$ to prove some results on convergence of $ {\mathcal{G}_n}f$ to $ f$ as $n \to \infty$ under various conditions on $ f$ and on the triangular array.
Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems
Michael
Beals;
Michael
Reed
159-184
Abstract: The authors develop a calculus of pseudodifferential operators with nonsmooth coefficients in order to study the regularity of solutions to linear equations $P\,(x,D)\,u = f$. The regularity theorems are similar to those of Bony, but the calculus and the methods of proof are quite different. We apply the linear results to study the regularity properties of solutions to quasilinear partial differential equations.
On some nonextendable derivations of the gauge-invariant CAR algebra
Geoffrey L.
Price
185-201
Abstract: We provide examples of some approximately inner, commutative $ \ast$-derivations which are generators on the gauge-invariant CAR algebra but which have no closed densely-defined extensions to the CAR. Necessary conditions are given for a class of generators on the GICAR algebra to extend to closed $\ast$-derivations on the CAR.
Extremal problems for polynomials with exponential weights
H. N.
Mhaskar;
E. B.
Saff
203-234
Abstract: For the extremal problem: $\displaystyle {E_{n,r}}(\alpha ): = \min \parallel \exp ( - \vert x{\vert^\alpha })\,({x^n} + \cdots ){\parallel_{{L^r}}}, \qquad \alpha > 0,$ where $ {L^r}\,(0 < r \leqslant \infty )$ denotes the usual integral norm over ${\mathbf{R}}$, and the minimum is taken over all monic polynomials of degree $n$, we describe the asymptotic form of the error ${E_{n,r}}(\alpha )\;({\text{as}}\;n \to \infty )$ as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The case $r = 2$ yields new information regarding the polynomials $\{ {p_n}(\alpha ;x) = {\gamma_n}(\alpha )\,{x^n} + \cdots \}$ which are orthonormal on ${\mathbf{R}}$ with respect to $\exp ( - 2\vert x{\vert^\alpha })$. In particular, it is shown that a conjecture of Freud concerning the leading coefficients ${\gamma_n}(\alpha )$ is true in a Cesàro sense. Furthermore a contracted zero distribution theorem is proved which, unlike a previous result of Ullman, does not require the truth of the Freud's conjecture. For $ r = \infty ,\alpha > 0$ we also prove that, if $\deg {P_n}(x) \leqslant n$, the norm $\parallel \exp ( - \vert x\vert^{\alpha })\,{P_n}(x)\parallel_{{L^\infty }}$ is attained on the finite interval $\displaystyle \left[ { - {{(n/{\lambda_\alpha })}^{1/\alpha }},{{(n/{\lambda_\a... ...ambda_\alpha } = \Gamma (\alpha )/{2^{\alpha - 2}}{\{ \Gamma (\alpha /2)\} ^2}.$ Extensions of Nikolskii-type inequalities are also given.
The strong conclusion of the F. and M. Riesz theorem on groups
I.
Glicksberg
235-240
Abstract: Let $S$ be a closed proper generating subsemigroup of the dual $\Gamma$ of a locally compact abelian group $ G$. Then there are Haar singular measures on $G$ orthogonal to $S$ unless $G = {\mathbf{R}} \times \Delta$ or ${\mathbf{T}} \times \Delta $ with $\Delta$ discrete, and then all $ \mu$ orthogonal to $ S$ are Haar absolutely continuous.
The theory of $G\sp{\infty }$-supermanifolds
Charles P.
Boyer;
Samuel
Gitler
241-267
Abstract: A theory of supermanifolds is developed in which a supermanifold is an ordinary manifold associated with a certain integrable second order $G$-structure. A structure theorem is proved showing that every $ {G^\infty }$-supermanifold has a complete distributive lattice of foliations with flat affine leaves. Furthermore, an existence and uniqueness theorem for local flows of ${G^\infty }$ vector fields is proved.
Cyclic vectors in the Dirichlet space
Leon
Brown;
Allen L.
Shields
269-303
Abstract: We study the Hilbert space of analytic functions with finite Dirichlet integral in the open unit disc. We try to identify the functions whose polynomial multiples are dense in this space. Theorems 1 and 2 confirm a special case of the following conjecture: if $\vert f(z)\vert \geqslant \vert g\,(z)\vert$ at all points and if $g$ is cyclic, then $f$ is cyclic. Theorems 3-5 give a sufficient condition ($f$ is an outer function with some smoothness and the boundary zero set is at most countable) and a necessary condition (the radial limit can vanish only for a set of logarithmic capacity zero) for a function $ f$ to be cyclic.
Involutions with isolated fixed points on orientable $3$-dimensional flat space forms
E.
Luft;
D.
Sjerve
305-336
Abstract: In this paper we completely classify (up to conjugacy) all involutions $\iota: M \to M$, where $M$ is an orientable connected flat $ 3$-dimensional space form, such that $\iota$ has fixed points but only finitely many. If $ M_1,\ldots,M_6$ are the $ 6$ space forms then only $M_1, M_2, M_6$ admit such involutions. Moreover, they are unique up to conjugacy. The main idea behind the proof is to find incompressible tori $T \subseteq M$ so that either $\iota(T) = T$ or $\iota(T) \cap T = \varnothing$ and then cut $ M$ into simpler pieces. These results lead to a complete classification of $ 3$-manifolds containing $\mathbf{Z} \oplus \mathbf{Z} \oplus \mathbf{Z}$ in their fundamental groups.
Compressed algebras: Artin algebras having given socle degrees and maximal length
Anthony
Iarrobino
337-378
Abstract: J. Emsalem and the author showed in [18] that a general polynomial $ f$ of degree $ j$ in the ring $ \mathcal{R} = k[ {{y_1},\ldots,{y_r}} ]$ has $\left( {\begin{array}{*{20}{c}} {j + r - 1} {r - 1} \end{array} } \right)$ linearly independent partial derivates of order $ i$, for $i = 0,1,\ldots,t = [ {j/2} ]$. Here we generalize the proof to show that the various partial derivates of $ s$ polynomials of specified degrees are as independent as possible, given the room available. Using this result, we construct and describe the varieties $G(E)$ and $Z(E)$ parametrizing the graded and nongraded compressed algebra quotients $A = R/I$ of the power series ring $R = k[[{x_1},\ldots,{x_r}]]$, having given socle type $E$. These algebras are Artin algebras having maximal length $\dim {_{k}}A$ possible, given the embedding degree $r$ and given the socle-type sequence $ E = ({e_1},\ldots,{e_s})$, where ${e_i}$ is the number of generators of the dual module $\hat A$ of $A$, having degree $i$. The variety $Z(E)$ is locally closed, irreducible, and is a bundle over $G(E)$, fibred by affine spaces ${{\mathbf{A}}^N}$ whose dimension is known. We consider the compressed algebras a new class of interesting algebras and a source of examples. Many of them are nonsmoothable--have no deformation to $(k + \cdots + k)$--for dimension reasons. For some choices of the sequence $ E,{\text{D}}$. Buchsbaum, ${\text{D}}$. Eisenbud and the author have shown that the graded compressed algebras of socle-type $ E$ have almost linear minimal resolutions over $R$, with ranks and degrees determined by $ E$. Other examples have given type $e = {\dim_k}\;({\text{socle}}\;A)$ and are defined by an ideal $I$ with certain given numbers of generators in $R = k[[{x_1},\ldots\;,{x_r}]]$. An analogous construction of thin algebras $ A = R/({f_1},\ldots,{f_s})$ of minimal length given the initial degrees of $ {f_1},\ldots,{f_s}$ is compared to the compressed algebras. When $ r = 2$, the thin algebras are characterized and parametrized, but in general when $r > 3$, even their length is unknown. Although $k = {\mathbf{C}}$ through most of the paper, the results extend to characteristic $ p$.
Closed timelike geodesics
Gregory J.
Galloway
379-388
Abstract: It is shown that every stable free $t$-homotopy class of closed timelike curves in a compact Lorentzian manifold contains a longest curve which must be a closed timelike geodesic. This result enables one to obtain a Lorentzian analogue of a classical theorem of Synge. A criterion for stability is presented, and a theorem of Tipler is derived as a special case of the result stated above.
Slice links in $S\sp{4}$
Tim
Cochran
389-401
Abstract: We produce necessary and sufficient conditions of a homotopy-theoretic nature for a link of $2$-spheres in ${S^4}$ to be slice (i.e., cobordant to the unlink). We give algebraic conditions on the link group sufficient to guarantee sliceness, generalizing the known results for boundary links. The notion of a "stable link" is introduced and shown to be useful in constructing cobordisms in dimension $ 4$.
Contributions from conjugacy classes of regular elliptic elements in ${\rm Sp}(n,\,{\bf Z})$ to the dimension formula
Min King
Eie
403-410
Abstract: The dimension of the space of cusp forms on the degree $n$ Siegel upper half-space can be obtained from the Selberg trace formula; in this paper we compute the contribution from the conjugacy classes of regular elliptic elements in $ \operatorname{Sp}(n,{\mathbf{Z}})$ using Weyl's character formula for representations of $ {\text{GL}}(n,{\mathbf{C}})$.
On certain Boolean algebras $\mathcal{P}(\omega)/I$
Winfried
Just;
Adam
Krawczyk
411-429
Abstract: We consider possible isomorphisms between algebras of the form $ \mathcal{P}(\omega)/I$, assuming ${\rm {CH}}$. In particular, the solution of a problem of Erdös and Ulam is given. We include some remarks on the completeness of such algebras.