Countable homogeneous tournaments
A. H.
Lachlan
431-461
Abstract: A tournament $ T$ is called homogeneous just in case every isomorphism of subtournaments of smaller cardinality can be lifted to an automorphism of $T$. It is shown that there are precisely three homogeneous tournaments of power ${\aleph_0}$. Some analogous results for $ 2$-tournaments are obtained.
An interface tracking algorithm for the porous medium equation
E.
DiBenedetto;
David
Hoff
463-500
Abstract: We study the convergence of a finite difference scheme for the Cauchy problem for the porous medium equation ${u_t} = {({u^m})_{x\,x}},m > 1$. The scheme exhibits the following two features. The first is that it employs a discretization of the known interface condition for the propagation of the support of the solution. We thus generate approximate interfaces as well as an approximate solution. The second feature is that it contains a vanishing viscosity term. This term permits an estimate of the form $\parallel {({u^{m - 1}})_{x\,x}}\;\parallel _{1,{\mathbf{R}}} \leqslant c/t$. We prove that both the approximate solution and the approximate interfaces converge to the correct ones. Finally error bounds for both solution and free boundaries are proved in terms of the mesh parameters.
Deformation and linkage of Gorenstein algebras
Andrew R.
Kustin;
Matthew
Miller
501-534
Abstract: General double linkage of Gorenstein algebras is defined. Rigidity, genericity, and regularity up to codimension six all pass across general double linkage. Rigid strongly unobstructed codimension four Gorenstein algebras which lie in different Herzog classes are produced.
Nondegenerate symmetric bilinear forms on finite abelian $2$-groups
Rick
Miranda
535-542
Abstract: Let ${\mathcal{B}_2}$ be the semigroup of isomorphism classes of finite abelian $2$-groups with a nondegenerate symmetric bilinear form having values in $Q/{\mathbf{Z}}$. Generators for ${\mathcal{B}_2}$ were given by C. T. C. Wall and the known relations among these generators were proved to be complete by A. Kawauchi and S. Kojima. In this article we describe a normal form for such bilinear forms, expressed in terms of Wall's generators, and as a by-product we obtain a simpler proof of the completeness of the known relations.
$L\sp{2}$-cohomology of noncompact surfaces
David R.
DeBaun
543-565
Abstract: This paper is motivated by the question of whether nonzero $ {L^2}$-harmonic differentials exist on coverings of a Riemann surface of genus $\geqslant 2$. Our approach will be via an analogue of the de Rham theorem. Some results concerning the invariance of ${L^2}$-homology and the intersection number of $ {L^2}$-cycles are demonstrated. A growth estimate for triangulations of planar coverings of the two-holed torus is derived. Finally, the equivalence between the existence of $ {L^2}$-harmonic one-cycles and the transience of random walks on a planar surface is shown.
Rate of approach to minima and sinks---the Morse-Smale case
Helena S.
Wisniewski
567-581
Abstract: The dynamical systems herein are Morse-Smale diffeomorphisms and flows on ${C^\infty }$ compact manifolds. We show the asymptotic rate of approach of orbits to the sinks of the systems to be bounded by an expression of the form $K\;\exp ( - DN)$, where $D$ may be any number smaller than $C = {\min_p}\{ 1/m\;\log \;\operatorname{Jac}\;{D_P}\,{f^m}\vert{W^u}(P)\}$. Here the minimum is taken over all nonsink $P$ in the nonwandering set of $f$, and $m$ is the period of $P$. We extend our theorems to the entire manifold, so that there is no restriction on the location of the initial points of trajectories.
Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab
B. A.
Mair
583-599
Abstract: This paper investigates the boundary behaviour of positive solutions of the equation $Lu = 0$, where $L$ is a uniformly parabolic second-order differential operator in divergence form having Hölder-continuous coefficients on $X = {{\mathbf{R}}^n} \times (0,T)$, where $0 < T < \infty$. In particular, the notion of semithinness for the potential theory on $X$ associated with $L$ is introduced, and the relationships between fine, semifine and parabolic convergence at points of $ {{\mathbf{R}}^n} \times \{ 0 \}$ are studied. The abstract Fatou-Naim-Doob theorem is used to deduce that every positive solution of $ Lu = 0$ on $X$ has parabolic limits Lebesgue-almost-everywhere on ${{\mathbf{R}}^n} \times \{ 0 \}$. Furthermore, a Carleson-type result is obtained for solutions defined on a union of parabolic regions.
Twistor CR manifolds and three-dimensional conformal geometry
Claude R.
LeBrun
601-616
Abstract: A CR (i.e. partially complex) $5$-manifold is contructed as a sphere bundle over an arbitrary $3$-manifold with conformal metric. This so-called twistor $CR$ manifold is show to capture completely the original geometry, and necessary and sufficient conditions are given for an abstract CR manifold to arise via the construction. The above correspondence is then used to prove that a twistor CR manifold is locally imbeddable as a real hypersurface in ${{\mathbf{C}}^3}$ only if it is real-analytic with respect to a suitable atlas.
Diffusion approximation and computation of the critical size
C.
Bardos;
R.
Santos;
R.
Sentis
617-649
Abstract: This paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation. To obtain this result, we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size. The paper also contains some simple numerical examples and some new results for the Milne problem.
Characteristic, maximum modulus and value distribution
W. K.
Hayman;
J. F.
Rossi
651-664
Abstract: Let $f$ be an entire function such that $\log M(r,f)\sim T(r,f)$ on a set $ E$ of positive upper density. Then $f$ has no finite deficient values. In fact, if we assume that $E$ has density one and $f$ has nonzero order, then the roots of all equations $f(z) = a$ are equidistributed in angles. In view of a recent result of Murai [6] the conclusions hold in particular for entire functions with Fejér gaps.
Minimal cyclic-$4$-connected graphs
Neil
Robertson
665-687
Abstract: A theory of cyclic-connectivity is developed, matroid dual to the standard vertex-connectivity. The cyclic-$4$-connected graphs minimal under the elementary operations of single-edge deletion or contraction and removal of a trivalent vertex are classified. These turn out to belong to three simple infinite families of indecomposable graphs, or to be decomposable into constituent subgraphs which themselves belong to three simple infinite families. This is modeled after W. T. Tutte's theorem classifying the minimal $3$-connected graphs under single-edge deletion or contraction as forming the single infinite family of "wheels." Such theorems serve two main purposes: (1) illustrating the structure of graphs in the class by isolating a type of extremal graph, and (2) by providing a set-up so that induction on $\vert E(G)\vert$ can be carried out effectively within the class.
Variational invariants of Riemannian manifolds
Jerrold
Siegel;
Frank
Williams
689-705
Abstract: This paper treats higher-dimensional analogues to the minimum geodesic distance in a compact Riemannian manifold $M$ with finite fundamental group. These invariants are based on the concept of homotopy distance in $M$. This defines a parametrized variational problem which is approached by globalizing the Morse theory of the spaces of paths between two points of $ M$ to the space of all paths in $M$. We develop machinery that we apply to calculate the invariants for numerous examples. In particular, we shall observe that knowledge of the invariants for the standard spheres determines the question of the existence of elements of Hopf invariant one.
Infinite crossed products and group-graded rings
D. S.
Passman
707-727
Abstract: In this paper, we precisely determine when a crossed product $R\;\ast\;G$ is semiprime or prime. Indeed we show that these conditions ultimately depend upon the analogous conditions for the crossed products $R\;\ast\;N$ of the finite subgroups $N$ of $G$ and upon the interrelationship between the normalizers of these subgroups and the ideal structure of $R$. The proof offered here is combinatorial in nature, using the $\Delta$-methods, and is entirely self-contained. Furthermore, since the argument applies equally well to strongly $G$-graded rings, we have opted to work in this more general context.
On positive solutions of some pairs of differential equations
E. N.
Dancer
729-743
Abstract: In this paper, we discuss the existence of solutions, with both components positive, of a Dirichlet problem for a coupled pair of partial differential equations. The main result is proved by using degree theory in cones. We also discuss the asymptotic behaviour of solutions as a parameter tends to zero.
Function theoretic results for complex interpolation families of Banach spaces
Richard
Rochberg
745-758
Abstract: The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vector valued holomorphic functions. Versions of the Schwarz lemma, Liouville's theorem, the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory.
On the arithmetic of projective coordinate systems
Christian
Herrmann
759-785
Abstract: A complete list of subdirectly irreducible modular (Arguesian) lattices generated by a frame of order $n \geq 4\;(n \geq 3)$ is given. Also, it is shown that a modular lattice variety containing the rational projective geometries cannot be both finitely based and generated by its finite dimensional members.
Difference equations, isoperimetric inequality and transience of certain random walks
Jozef
Dodziuk
787-794
Abstract: The difference Laplacian on a square lattice in ${{\mathbf{R}}^n}$ has been studied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient.
Tensor products for the de Sitter group
Robert P.
Martin
795-814
Abstract: The decomposition of the tensor product of a principal series representation with any other irreducible unitary representation of $G$ is determined for the simply connected double covering, $G = \operatorname{Spin}\,(4,1)$, of the DeSitter group.
Ramsey games
A.
Hajnal;
Zs.
Nagy
815-827
Abstract: The paper deals with game-theoretic versions of the partition relations $ \alpha \to (\beta )_2^{ < \tau }$ and $\alpha \to (\beta )_2^\tau $ introduced in [2]. The main results are summarized in the Introduction.
Rank change on adjoining real powers to Hardy fields
Maxwell
Rosenlicht
829-836
Abstract: This paper concerns asymptotic approximations and expansions in cases where the usual Poincaré power series in $ 1/x$ do not suffice because there may be more than one comparability class of functions that are very large or very small. The attempt to find asymptotic approximations in terms of real powers of given representatives of the comparability classes fails in general, but the situation can be saved by the adjunction of suitable real power products of the original functions, at the possible cost of an increase in the number of comparability classes.
Some applications of the topological characterizations of the sigma-compact spaces $l\sp{2}\sb{f}$ and $\Sigma $
Doug
Curtis;
Tadeusz
Dobrowolski;
Jerzy
Mogilski
837-846
Abstract: We use a technique involving skeletoids in $\sigma$-compact metric ARs to obtain some new examples of spaces homeomorphic to the $\sigma$-compact linear spaces $l_f^2$ and $\Sigma$. For example, we show that (1) every ${\aleph_0}$-dimensional metric linear space is homeomorphic to $l_f^2$; (2) every $\sigma$-compact metric linear space which is an AR and which contains an infinite-dimensional compact convex subset is homeomorphic to $ \Sigma$; and (3) every weak product of a sequence of $\sigma$-compact metric ARs which contain Hilbert cubes is homeomorphic to $ \Sigma$.
On the spectrum of $C\sb{0}$-semigroups
Jan
Prüss
847-857
Abstract: In this paper we give characterizations of the spectrum of a $ {C_0}$-semigroup $ {e^{At}}$ in terms of certain solution properties of the differential equation $ (\ast)\;u^{\prime} = Au + f$ and, in case $X$ is a Hilbert space, also in terms of properties of $ {(\lambda - A)^{ - 1}}$. We give several applications of these results including a study of the existence of dichotomic projections for $ (\ast)$.