Noncollision singularities in the four-body problem
Robert Orrin
Shelton
225-259
Abstract: It is shown that if there is a singularity in a solution of the four-body problem which is not a collision then the motion of the bodies near the singularity is nearly one-dimensional. This is established by grouping the bodies into natural clusters and showing the angular momentum of each cluster with respect to its center of mass tends to zero near the singularity. This is related to Sperling's proof of von Zeipel's theorem.
The Bergman norm and the Szeg\H{o} norm
Saburou
Saitoh
261-279
Abstract: Let G denote an arbitrary bounded regular region in the plane and ${H_2}\left( G \right)$ the analytic Hardy class on G with index 2. We show that the generalized isoperimetric inequality \begin{multline}\frac{1}{\pi }\,\iint\limits_G {{{\left\vert {\varphi \left( z \... ...t}^{2}}\,\left\vert dz \right\vert}\,\,\,\,\,\,\,(z\,=\,x\,+\,iy) \end{multline} holds for any $\varphi$ and $\psi \, \in \,{H_2}\left( G \right)$. We also determine necessary and sufficient conditions for equality.
Fourier inversion of invariant integrals on semisimple real Lie groups
Rebecca A.
Herb
281-302
Abstract: Let G be a connected, semisimple real Lie group with finite center. Associated with every regular semisimple element g of G is a tempered invariant distribution ${ \Lambda _g}$ given by an orbital integral. This paper gives an inductive formula for computing the Fourier transform of ${ \Lambda _g}$ in terms of the space of tempered invariant eigendistributions of G.
Extensions, restrictions, and representations of states on $C\sp{\ast} $-algebras
Joel
Anderson
303-329
Abstract: In the first three sections the question of when a pure state g on a $ {C^{\ast}}$-subalgebra B of a $ {C^{\ast}}$-algebra A has a unique state extension is studied. It is shown that an extension f is unique if and only if inf $ \left\Vert {b\left( {a\, - \,f\left( a \right)1} \right)b} \right\Vert\, = \,0$ for each a in A, where the inf is taken over those b in B such that $ 0\, \leqslant \,b\, \leqslant \,1$ and $g(b) = 1$. The special cases where B is maximal abelian and/or $A\, = \,B\left( H \right)$ are treated in more detail. In the remaining sections states of the form $T \mapsto \mathop {\lim }\limits_{\mathcal{u}} \left( {T{x_\alpha },\,{x_\alpha }} \right)$, where $\left\{ {{x_\alpha }} \right\}{\,_{\alpha \, \in \,\kappa }}$ is a set of unit vectors in H and $ {\mathcal{u}}$ is an ultrafilter are studied.
The equivalence of $\times \sp{t}C\approx \times \sp{t}D$ and $J\times C\approx J\times D$
Ronald
Hirshon
331-340
Abstract: Let C satisfy the maximal condition for normal subgroups and let $\times {\,^t}C\, \approx \, \times {\,^t}D$ for some positive integer t. Then $C\, \times \,J\, \approx \,D\, \times \,J$ where J is the infinite cyclic group. If $\times {\,^s}C\, \approx \, \times {\,^t}D$ and $s \geqslant \,t$, there exists a finitely generated free abelian group S such that C is a direct factor of $ D\, \times \,S$.
Distribution of zeros of orthogonal polynomials
Paul G.
Nevai
341-361
Abstract: The purpose of the paper is to investigate distribution of zeros of orthogonal polynomials given by a three term recurrence relation.
Ramsey's theorem for spaces
Joel H.
Spencer
363-371
Abstract: A short proof is given of the following known result. For all k, r, t there exists n so that if the t-spaces of an n-space are r-colored there exists a k-space all of whose t-spaces are the same color. Here t-space refers initially to a t-dimensional affine space over a fixed finite field. The result is also shown for a more general notion of t-space.
Metacompactness, paracompactness, and interior-preserving open covers
Heikki J. K.
Junnila
373-385
Abstract: In this paper metacompactness and paracompactness are characterized in terms of the existence of closure-preserving closed refinements and interior-preserving open star-refinements of interior-preserving directed open covers of a topological space. Several earlier results on metacompact spaces and paracompact spaces are obtained as corollaries to these characterizations. For a Tychonoff-space X, metacompactness of X is characterized in terms of orthocompactness of $ X\, \times \,\beta X$.
Maximal subspaces of Besov spaces invariant under multiplication by characters
R.
Johnson
387-407
Abstract: Unlike the familiar $ {L^p}$ spaces, neither the homogeneous Besov spaces nor the ${H^p}$ spaces, $0\, < \,p\, < \,\,1$, are closed under multiplication by the functions $ x\, \to \,{e^{i\left\langle {x,h} \right\rangle }}$. We determine the maximal subspace of these spaces which are closed under multiplication by these functions, which are the characters of $ {R^n}$.
The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension
Barry F.
Knerr
409-424
Abstract: We consider the Cauchy problem in one space dimension for a nonlinear degenerate parabolic partial differential equation. The connectedness of the support of the solution and estimates of the growth of the support as $t \,\to \,\infty$ are established.
The module of indecomposables for mod $2$ finite $H$-spaces
Richard
Kane
425-433
Abstract: The module of indecomposables obtained from the mod 2 cohomology of a finite H-space is studied. It is shown that this module is trivial in dimensions $ \equiv \,0$ (mod 4).
Nash rings on planar domains
Gustave A.
Efroymson
435-445
Abstract: Let D be a semialgebraic domain in ${R^2}$. Let ${N_D}$ denote the Nash ring of algebraic analytic functions on D. Let ${A_D}$ denote the ring of analytic functions on D. The main theorem of this paper implies that if $\mathcal{B}$ is a prime ideal of ${N_D}$, then $\mathcal{B}{A_D}$ is also prime. This result is proved by considering $p\left( {x,\,y} \right)$ in $\textbf{R}[{x,\,y}]$ and showing that $p({x,\,y})$ can be put into a form so that its factorization in ${N_D}$ is given by looking at its local factorization as a polynomial in y with coefficients which are analytic functions of x. Then for more general domains, a construction using the ``complex square root'' enables one to reduce to the case already considered.
Erratum to: ``The ninety-one types of isogonal tilings in the plane'' (Trans. Amer. Math. Soc. {\bf 242} (1978), 335--353)
Branko
Grünbaum;
G. C.
Shephard
446-446