Counting interval graphs
Phil
Hanlon
383-426
Abstract: In this paper we enumerate interval graphs (up to isomorphism) along with labelled interval graphs, identity interval graphs, transitive interval graphs and various sorts of unit interval graphs. The enumeration makes use of a structural decomposition of interval graphs which leads to a characterization of those interval graphs having a unique interval representation. Several tables are included.
Large cardinals and strong model theoretic transfer properties
Matthew
Foreman
427-463
Abstract: In this paper we prove the following theorem: $[{\rm {Con}}({\rm {ZFC}}\,{\rm { + }}\,there\,is\,a\,{\rm {2 - }}huge\,cardinal) \Rightarrow for\,all\,n$ $\displaystyle {\rm {Con}}({\rm {ZFC + }}({\aleph _{n + 3}},{\aleph _{n + 2}},{\... ...{n + 1}}) \twoheadrightarrow ({\aleph _{n + 2}},{\aleph _{n + 1}},{\aleph _n}))$ . We do this by using iterated forcing to collapse the $2$-huge cardinal to $ {\aleph _{n + 1}}$ and extending the elementary embedding generically.
The metabelian $p$-groups of maximal class. II
R. J.
Miech
465-474
Abstract: This paper gives a classification, up to isomorphism, of the metabelian $p$-groups of maximal class. A recent idea is used to produce a considerable simplification of an earlier classification scheme for these groups.
Left definite multiparameter eigenvalue problems
Paul
Binding
475-486
Abstract: We study the problem $\displaystyle (\ast)\qquad{T_m}{x_m} = \sum\limits_{n = 1}^k {{\lambda _n}{V_{mn}}{x_m},\qquad 0 \ne } {x_m} \in {H_m},\,m = 1, \ldots ,k,$ where $ {T_m}$ and ${V_{mn}}$ are selfadjoint linear operators on separable Hilbert spaces ${H_m}$, with ${T_m}$ positive, $ T_m^{ - 1}$ compact and $ {V_{mn}}$ bounded. We assume ``left definiteness'' which involves positivity of certain linear combinations of cofactors in the determinant with $(m,\,n)$th entry $({x_m},\,{V_{mn}}{x_m})$. We establish a spectral theory for $(\ast)$ that is in some way simpler and more complete than those hitherto available for this case. In particular, we make use of operators ${B_n} = \Delta _n^{ - 1}{\Delta _0}$, where the ${\Delta _n}$ are determinantal operators on $ \otimes _{m = 1}^k{H_m}$. This complements an established approach to the alternative ``right definite'' problem (where ${\Delta _0}$ is positive) via the operators ${\Gamma_n} = \Delta _0^{ - 1}{\Delta _n}$.
On weighted norm inequalities for the Hilbert transform of functions with moments zero
Ernst
Adams
487-500
Abstract: Let $\tilde f$ denote the Hilbert transform of $f$, i.e. $\displaystyle \tilde f(x) = {\rm {p}}{\rm {.v}}{\rm {.}}\int {\frac{{f(t)}}{{x - t}}dt}$ and let $1 < p < \infty$. A weight function $ w$ is shown to satisfy $\displaystyle \int {\vert\tilde f(x)} {\vert^p}w(x)dx \le C{\int {\vert f(x)\vert} ^p}w(x)dx$ for all $f$ with the first $N$ moments zero, if and only if it is of the form $ w(x) = \vert q(x){\vert^p}U(x)$, where $q$ is a polynomial of degree at most $ N$ and $U \in {A_p}$.
Some varieties containing relation algebras
Roger
Maddux
501-526
Abstract: Three varieties of algebras are introduced which extend the variety $ RA$ of relation algebras. They are obtained from $RA$ by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or $NA$, $WA$, and $SA$, respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that $WA$ is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that $ WA$ is the closure of the variety of representable $RA$'s under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples.
Cellular maps between polyhedra
James P.
Henderson
527-537
Abstract: A compact subset $ X$ of a polyhedron $ P$ is cellular in $ P$ if there is a pseudoisotopy of $P$ shrinking precisely $X$ to a point. A proper surjection $f:P\rightarrow Q$ is cellular if each point inverse of $f$ is cellular in $P$. We give certain conditions under which cellular maps between polyhedra are approximable by homeomorphisms. An example of a cellular map which is not approximable by homeomorphisms is also given.
Smoothness of the boundary values of functions bounded and holomorphic in the disk
Shinji
Yamashita
539-544
Abstract: The non-Euclidean counterparts of Hardy-Littlewood's theorems on Lipschitz and mean Lipschitz functions are considered. Let $ 1\le p < \infty$ and $0 < \alpha \le 1$. For $f$ holomorphic and bounded, $\vert f\vert< 1$, in $\vert z\vert< 1$, the condition that is necessary and sufficient for $f$ to be continuous on $\vert z\vert\le 1$ with the boundary function $f({e^{it}}) \in \sigma {\Lambda _\alpha }$, the hyperbolic Lipschitz class. Furthermore, the condition that the $p$th mean of $f^{\ast}$ on the circle $\vert z\vert=r < 1$ is $O({(1 - r)^{\alpha - 1}})$ is necessary and sufficient for $f$ to be of the hyperbolic Hardy class $H_\sigma^{p}$ and for the radial limits to be of the hyperbolic mean Lipschitz class $ \sigma\Lambda_\alpha^{p}$.
Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators
J. C.
Lagarias
545-554
Abstract: This paper defines the notion of a best simultaneous Diophantine approximation to a vector $\alpha$ in $R^n$ with respect to a norm $\left\Vert \,\cdot\, \right\Vert$ on $ R^n$. Suppose $ \alpha$ is not rational and order the best approximations to $ \alpha$ with respect to $\left\Vert\, \cdot\, \right\Vert$ by increasing denominators $1=q_1 < q_2 < \cdots$. It is shown that these denominators grow at least at the rate of a geometric series, in the sense that $\displaystyle g\left( {\alpha ,\,\left\Vert {\,\cdot\,} \right\Vert} \right) = ... ...liminf\limits_{k \to \infty }} {({q_k})^{1/k}} \geq 1 + \frac{1}{{{2^{n + 1}}}}$ . Let $g\left( {\left\Vert\, \cdot\, \right\Vert} \right)$ denote the infimum of $g\left( {\alpha ,\,\left\Vert {\,\cdot\,} \right\Vert} \right)$ over all $\alpha$ in $R^n$ with an irrational coordinate. For the sup norm $ \left\Vert\, \cdot \,\right\Vert _s$ on $R^2$ it is shown that $g\left( {\left\Vert \, \cdot \, \right\Vert}_s \right)\ge\theta=1.270^{+}$ where $ \theta^4=\theta^{2}+1$.
Coextensions of regular semigroups by rectangular bands. II
John
Meakin;
K. S. S.
Nambooripad
555-568
Abstract: A construction of all coextensions of a regular semigroup $S$ by rectangular bands is obtained. The construction is analogous to Hall's construction of orthodox semigroups as spined products of bands and inverse semigroups and reduces to that construction when $S$ is inverse. The results are specialized to provide a construction of the category of all normal coextensions of a regular semigroup.
Spaces which look like quaternionic projective $n$-space
C. A.
McGibbon
569-587
Abstract: The projective $ n$-spaces which correspond to the various multiplicative structures on the three sphere are studied. Necessary and sufficient conditions for a projective $n$-space to extend to a projective $ n+1$-space are described. At each odd prime, an infinite family of exotic projective spaces is constructed. These exotic spaces are not homotopy equivalent, at the prime in question, to the classical quaternionic projective $ n$-space. It is also shown that these exotic projective $n$-spaces do not occur as the finite skeleton of a classifying space for a group with the homotopy type of the three sphere.
On actions of regular type on complex Stiefel manifolds
McKenzie Y.
Wang
589-610
Abstract: The usual unitary representations of the special unitary, symplectic, or special orthogonal groups define a sequence of smooth actions on the complex Stiefel manifolds called the regular linear models. If one of the above groups acts smoothly on the complex Stiefel manifold of orthonormal $2$-frames in $ \mathbf C^n$ for odd $ n$, and if the identity component of the principal isotropy type is of regular type, then it is shown under mild dimension restrictions that the orbit structure and the cohomology structure of the fixed point varieties (over the $\mod 2$ Steenrod algebra) resemble those of the regular linear models. The resemblance is complete in the cases of the special unitary and symplectic groups. There is an obstruction to complete resemblance in the case of the special orthogonal groups. An application of the above regularity theorems is given.
On actions of adjoint type on complex Stiefel manifolds
McKenzie Y.
Wang
611-628
Abstract: Let $G(m)$ denote ${\rm {SU}}(m)$ or $ {\rm {Sp}}(m)$. It is shown that when $m \geq 5\,G(m)$ cannot act smoothly on $W_{n,2}$, the complex Stiefel manifold of orthonormal $2$-frames in $ \mathbf C^n$, for $ n$ odd, with connected principal isotropy type equal to the class of maximal tori in $G(m)$. This demonstrates an important difference between $W_{n,2}$, $n$ odd, and $S^{2n-3}\times S^{2n-1}$ in the behavior of differentiable transformation groups. Exactly the same holds for ${\rm {SO}}(m)$ or Spin$(m)$ if it is further assumed that a maximal $2$-torus of $ {\rm {SO}}(m)$ has fixed points.$^{2}$
On group $C\sp{\ast} $-algebras of bounded representation dimension
Iain
Raeburn
629-644
Abstract: We consider the structure of group $C^{\ast}$-algebras whose irreducible representations have bounded dimension. We give some general results, including a description of the topology on the spectrum, and then calculate explicitly the $ C^{\ast}$-algebras of two specific groups, one of them a nonsymmorphic space group.
Generalization of continuous posets
Dan
Novak
645-667
Abstract: In this paper we develop a general theory of continuity in partially ordered sets. Among the interesting special cases of this theory is the theory of continuous lattices developed by D. Scott, J. Lawson and others.
Determining Thurston classes using Nielsen types
Jane
Gilman
669-675
Abstract: In previous work [3] we showed how the Thurston or Bers classifications of diffeomorphisms of surfaces could be obtained using the Nielsen types of the lifts of the diffeomorphism to the unit disc. In this paper we find improved conditions on the Nielsen types for the Thurston and Bers classes. We use them to verify that an example studied by Nielsen is a pseudo-Anosov diffeomorphism with stretching factor of degree 4. This example is of interest in its own right, but it also serves to illustrate exactly how the Nielsen types are used for verifying examples. We discuss the general usefulness of this method.
Quaternionic Kaehler manifolds
Lee
Whitt
677-692
Abstract: The topological classification of $4$- and $8$- (real) dimensional compact quaternionic Kaehler manifolds is given. There is only the torus in dimension 4. In dimension 8, there are 12 homeomorphism classes; representatives are given explicitly.
Connected algebraic monoids
Mohan S.
Putcha
693-709
Abstract: Let $S$ be a connected algebraic monoid with group of units $G$ and lattice of regular $ \mathcal{J}$-classes $\mathcal{U}(S)$. The connection between the solvability of $G$ and the semilattice decomposition of $ S$ into archimedean semigroups is further elaborated. If $S$ has a zero and if $\mathcal{U}(S)\le 7$, then it is shown that $ G$ is solvable if and only if $ \mathcal{U}(S)$ is relatively complemented. If $J\in \mathcal{U}(S)$, then we introduce two basic numbers $\theta(J)$ and $\delta(J)$ and study their properties. Crucial to this process is the theorem that for any indempotent $ e$ of $S$, the centralizer of $ e$ in $G$ is connected. Connected monoids with central idempotents are also studied. A conjecture about their structure is forwarded. It is pointed out that the maximal connected submonoids of $S$ with central idempotents need not be conjugate. However special maximal connected submonoids with central idempotents are conjugate. If $S$ is regular, then $S$ is a Clifford semigroup if and only if for all $f\in E(S)$, the set $\{ e\vert e \in E(S),\,e \geq f\}$ is finite. Finally the maximal semilattice image of any connected monoid is determined.
On the Schwarz reflection principle
J. S.
Hwang
711-719
Abstract: Recently, we have solved a long outstanding problem of A. J. Lohwater (1953) by showing that if $f(z)$ is meromorphic in $\vert z\vert< 1$ whose radial limits have modulus 1 for almost all points on an arc $A$ of $\vert z\vert< 1$, and if $P$ is a singular point of $f(z)$ on $A$, then every value of modulus 1 which is not in the range of $f(z)$ at $P$ is an asymptotic value of $f(z)$ at some point of each subarc of $ A$ containing the point $ P$. Lohwater proved this theorem for functions of bounded characteristic and he made a comment that his method is not, in general, applicable to functions of unbounded characteristic. In this paper, we shall present an alternative proof of the above theorem based on the very method of Lohwater.
Invariant operators and univalent functions
Reuven
Harmelin
721-731
Abstract: Necessary and sufficient conditions for univalence of meromorphic functions in certain domains in the complex plane are established in terms of some differential operators of degrees $\ge 3$, possessing the same invariance property as the Schwarzian derivative. Those operators include the derivatives of the Schwarzian derivative and Aharonov's invariants. Conditions for the existence of quasiconformal extensions are also achieved.
Unipotent characters of the even orthogonal groups over a finite field
George
Lusztig
733-751
Abstract: The characters of unipotent representations of a simple algebraic group over $F_q$ of type $\ne D_n$ on any regular semisimple element are explicitly known for large $q$. This paper deals with the remaining case: type $ D_n$.
Multiple critical points of perturbed symmetric functionals
Paul H.
Rabinowitz
753-769
Abstract: Variational problems which are invariant under a group of symmetries often possess multiple solutions. This paper studies the effect of perturbations which are not small and which destroy the symmetry for two classes of such problems and shows how multiple solutions persist despite the perturbation.
A nonshrinkable decomposition of $S\sp{n}$ involving a null sequence of cellular arcs
R. J.
Daverman;
J. J.
Walsh
771-784
Abstract: This paper presents a decomposition $G$ of $ S^n(n\ge 3)$ into points and a null sequence of cellular arcs such that $ S^n/G$ is not a manifold; furthermore, the union of the nondegenerate elements from $G$ lies in a $2$-cell in $S^n$ and the image in $S^n/G$ of this union has 0-dimensional closure. Examples of nonshrinkable decompositions with a null sequence of cellular arcs have been constructed in the case $n=3$ by D. S. Gillman and J. M. Martin and by R. H. Bing and M. Starbird. We construct another example in this dimension, for which all the arcs lie in the boundary of a crumpled cube $C$, and then produce higher dimensional examples by spinning $C$.
Asymptotic expansions of some integral transforms by using generalized functions
Ahmed I.
Zayed
785-802
Abstract: The technique devised by Wong to derive the asymptotic expansions of multiple Fourier transforms by using the theory of Schwartz distributions is extended to a large class of integral transforms. The extension requires establishing a general procedure to extend these integral transforms to generalized functions. Wong's technique is then applied to some of these integral transforms to obtain their asymptotic expansions. This class of integral transforms encompasses, among others, the Laplace, the Airy, the $K$ and the Hankel transforms.
Correction to: ``Closed $3$-manifolds with no periodic maps'' [Trans. Amer. Math. Soc. {\bf 221} (1976), no. 2, 403--418; MR {\bf 54} \#3703]
Frank
Raymond;
Jeffrey L.
Tollefson
803-807
Erratum to: ``Liapounoff's theorem for nonatomic, finitely-additive, bounded, finite-dimensional, vector-valued measures'' [Trans. Amer. Math. Soc. {\bf 266} (1981), no. 2, 499--514; MR 82f:28008]
Thomas E.
Armstrong;
Karel
Prikry
809
Erratum to: ``Embedding processes in Brownian motion in ${\bf R}\sp{n}$''
Neil
Falkner
811