Haar measure for measure groupoids
Peter
Hahn
1-33
Abstract: It is proved that Mackey's measure groupoids possess an analogue of Haar measure for locally compact groups; and many properties of the group Haar measure generalize. Existence of Haar measure for groupoids permits solution of a question raised by Ramsay. Ergodic groupoids with finite Haar measure are characterized.
The regular representations of measure groupoids
Peter
Hahn
35-72
Abstract: Techniques are developed to study the regular representation and $ \sigma$-regular representations of measure groupoids. Convolution, involution, a modular Hilbert algebra, and local and global versions of the regular representation are defined. The associated von Neumann algebras, each uniquely determined by the groupoid and the cocycle $\sigma$, provide a generalization of the group-measure space construction. When the groupoid is principal and ergodic, these algebras are factors. Necessary and sufficient conditions for the $ \sigma$-regular representations of a principal ergodic groupoid to be of type I, II, or III are given, as well as a description of the flow of weights; these are independent of $ \sigma$. To treat nonergodic groupoids, an ergodic decomposition theorem is provided.
Processes with independent increments on a Lie group
Philip
Feinsilver
73-121
Abstract: The Lévy-Khinchin representation for processes with independent increments is extended to processes taking values in a Lie group. The basis of the proof is to approximate continuous time processes by Markov chains. The processes involved are handled by the technique, developed by Stroock and Varadhan, of characterizing Markov processes by associated martingales.
The asymptotic behaviour of certain integral functions
P. C.
Fenton
123-140
Abstract: Let$f(z)$ be an integral function satisfying $\displaystyle {\int_{}^\infty \{\log \,m(r,f)\, - \,\cos \,\pi \rho \,\log \,M(r,f)\} ^ + }\frac{{dr}}{{{r^{\rho + 1}}}}\, < \,\infty$ and $\displaystyle 0\, < \,\mathop {\lim }\limits_{\overline {r\, \to \infty } } \,\frac{{\log \,M(r,f)}}{{{r^\rho }}}\, < \,\infty$ for some $ \rho :\,0\, < \,\rho \, < \,1$. It is shown that such functions have regular asymptotic behaviour outside a set of circles with centres ${\zeta _i}$ and radii ${t_i}$ for which $\displaystyle \sum\limits_{i = 1}^\infty {\frac{{{t_i}}}{{\left\vert {{\zeta _i}} \right\vert}}} < \infty$ .
Alternators of a right alternative algebra
Irvin Roy
Hentzel
141-156
Abstract: We show that in any right alternative algebra, the additive span of the alternators is nearly an ideal. We give an easy test to use to determine if a given set of additional identities will imply that the span of the alternators is an ideal. We apply our technique to the class of right alternative algebras satisfying the condition $(a,a,b) = \lambda [a,[a,b]]$. We show that any semiprime algebra over a field of characteristic $ \ne 2$, $\ne 3$ which satisfies the right alternative law and the above identity with $\lambda \ne 0$ is a subdirect sum of (associative and commutative) integral domains.
The cohomology of the symmetric groups
Benjamin Michael
Mann
157-184
Abstract: Let ${{\mathcal{S}}_n}$ be the symmetric group on n letters and SG the limit of the sets of degree +1 homotopy equivalences of the $n - 1$ sphere. Let p be an odd prime. The main results of this paper are the calculations of $ {H^{\ast}}({\mathcal{S}_n},\,Z/p)$ and $ {H^{\ast}}(SG,Z/p)$ as algebras, determination of the action of the Steenrod algebra, $ \mathcal{a}(p)$, on $ {H^{\ast}}({\mathcal{S}_n},\,Z/p)$ and $ {H^{\ast}}(SG,Z/p)$ and integral analysis of $ {H^{\ast}}({\mathcal{S}_n},\,Z,\,p)$ and $ {H^{\ast}}(SG,\,Z,\,p)$.
The invariant $\Pi \sp{0}\sb{\alpha }$ separation principle
Douglas E.
Miller
185-204
Abstract: We ``invariantize'' the classical theory of alternated unions to obtain new separation results in both invariant descriptive set theory and in infinitary logic. Application is made to the theory of definitions of countable models.
The traces of holomorphic functions on real submanifolds
Gary Alvin
Harris
205-223
Abstract: Suppose M is a real-analytic submanifold of complex Euclidean n = space and consider the following question: Given a real-analytic function f defined on M, is f the restriction to M of an ambient holomorphic function? If M is a C.R. submanifold the question has been answered completely. Namely, f is the trace of a holomorphic function if and only if f is a C.R. function. The more general situation in which M need not be a C.R. submanifold is discussed in this paper. A complete answer is obtained in case the dimension of M is larger than or equal to n and M is generic in some neighborhood of each point off its C.R. singularities. The solution is of infinite order and follows from a consideration of the following problem: Given a holomorphic function f and a holomorphic mapping $\Phi$, when does there exist a holomorphic mapping F such that $f = F \circ \Phi$?
Sampling theorems for nonstationary random processes
Alan J.
Lee
225-241
Abstract: Consider a second order stochastic process $\{ X(t),t \in \textbf{R}\} $, and let $H(X)$ be the Hilbert space generated by the random variables of the process. The process is said to be linearly determined by its samples $\{ X(nh),n \in \textbf{Z}\}$ if the random variables $X(nh)$ generate $H(X)$. In this paper we give a sufficient condition for a wide class of nonstationary processes to be determined by their samples, and present sampling theorems for such processes. We also consider similar problems for harmonizable processes indexed by LCA groups having suitable subgroups.
Isomorphic factorisations. I. Complete graphs
Frank
Harary;
Robert W.
Robinson;
Nicholas C.
Wormald
243-260
Abstract: An isomorphic factorisation of the complete graph ${K_p}$ is a partition of the lines of $ {K_p}$ into t isomorphic spanning subgraphs G; we then write $ G\vert{K_p}$, and $G \in {K_p}/t$. If the set of graphs ${K_p}/t$ is not empty, then of course $t\vert p(p - 1)/2$. Our principal purpose is to prove the converse. It was found by Laura Guidotti that the converse does hold whenever $(t,p) = 1$ or $ (t,p - 1) = 1$. We give a new and shorter proof of her result which involves permuting the points and lines of ${K_p}$. The construction developed in our proof happens to give all the graphs in ${K_6}/3$ and ${K_7}/3$. The Divisibility Theorem asserts that there is a factorisation of $ {K_p}$ into t isomorphic parts whenever t divides $p(p - 1)/2$. The proof to be given is based on our proof of Guidotti's Theorem, with embellishments to handle the additional difficulties presented by the cases when t is not relatively prime to p or $p - 1$.
Interpretation of the $p$-adic log gamma function and Euler constants using the Bernoulli measure
Neal
Koblitz
261-269
Abstract: A regularized version of J. Diamond's p-adic log gamma function and his p-adic Euler constants are represented as integrals using B. Mazur's p-adic Bernoulli measure.
The homotopy continuation method: numerically implementable topological procedures
J. C.
Alexander;
James A.
Yorke
271-284
Abstract: The homotopy continuation method involves numerically finding the solution of a problem by starting from the solution of a known problem and continuing the solution as the known problem is homotoped to the given problem. The process is axiomatized and an algebraic topological condition is given that guarantees the method will work. A number of examples are presented that involve fixed points, zeroes of maps, singularities of vector fields, and bifurcation. As an adjunct, proofs using differential rather than algebraic techniques are given for the Borsuk-Ulam Theorem and the Rabinowitz Bifurcation Theorem.
Curves with large tangent space
Joseph
Becker;
Rajendra
Gurjar
285-296
Abstract: Theorem. Let V be a complex analytic variety irreducible at a point $p \in V$. Givén any integer l, there exists an analytic curve $ {C_l}$ on V passing through p and irreducible at p such that the germs of $ {C_l}$ and V at p are isomorphic up to order l.
A probable Hasse principle for pencils of quadrics
William C.
Waterhouse
297-306
Abstract: Let k be a global field, ${\text{char}}(k) \ne 2$. Although pencils of quadrics over k may fail to satisfy a local-to-global equivalence principle, the failures are exceptional in the precise sense of having limiting probability zero. The proof uses the classification of pairs of quadratic forms. It also requires knowing that a square class in a finite extension usually comes from k when it does so locally; the Galois-theoretic criterion for this is determined.
Pullback de Rham cohomology of the free path fibration
Kuo-Tsai
Chen
307-318
Abstract: Let M and N be smooth manifolds and let $\bar B (A)$ be the reduced bar construction on the de Rham complex $ \Lambda (M)$ or a suitable subcomplex A of M. For every smooth map $ f:N \to M \times M$, the tensor product $\Lambda (N) \otimes \bar B(A)$, equipped with a suitable differential, will yield the correct cohomology for the pullback of the free path fibration $P(M) \to M \times M$ via the smooth map F. Moreover, $ \Lambda (N) \otimes \bar B(A)$ can be taken as a de Rham subcomplex of the pullback space.
The product of nonplanar complexes does not imbed in $4$-space
Brian R.
Ummel
319-328
Abstract: We prove that if $ {K_1}$ and ${K_2}$ are nonplanar simplicial complexes, then $ {K_1}\, \times\, {K_2}$ does not imbed in $ {{\textbf{R}}^4}$.
Systems of $n$ partial differential equations in $n$ unknown functions: the conjecture of M. Janet
Joseph
Johnson
329-334
Abstract: It was conjectured by Janet that an analytic solution to a system of n ``independent'' analytic differential equations in n unknown functions if not isolated must depend on at least one unknown function of $m - 1$ variables plus possibly other functions of fewer than m variables. Here m is the dimension of the complex domain on which the equations and the solution are given. An algebraic generalization of the linear form of the conjecture is proven. Also the result is extended to give a nonlinear version.
The ninety-one types of isogonal tilings in the plane
Branko
Grünbaum;
G. C.
Shephard
335-353
Abstract: A tiling of the plane by closed topological disks of isogonal if its symmetries act transitively on the vertices of the tiling. Two isogonal tilings are of the same type provided the symmetries of the tiling relate in the same way every vertex in each to its set of neighbors. Isogonal tilings were considered in 1916 by A. V. Šubnikov and by others since then, without obtaining a complete classification. The isogonal tilings are vaguely dual to the isohedral (tile transitive) tilings, but the duality is not strict. In contrast to the existence of 81 isohedral types of planar tilings we prove the following result: There exist 91 types of isogonal tilings of the plane in which each tile has at least three neighbors.
$R$-separable coordinates for three-dimensional complex Riemannian spaces
C. P.
Boyer;
E. G.
Kalnins;
Willard
Miller
355-376
Abstract: We classify all R-separable coordinate systems for the equations $\Sigma _{i,j = 1}^3\,{g^{ - 1/2}}{\partial _j}({g^{1/2}}{g^{ij}}{\partial _i}\psi ) = 0$ and $\Sigma_{i,j\, = \,1}^3 {{g^{ij}}{\partial _i}W{\partial _j}W\, = \,0}$ with special emphasis on nonorthogonal coordinates, and give a group-theoretic interpretation of the results. We show that for flat space the two equations separate in exactly the same coordinate systems.
Classification of circle actions on $4$-manifolds
Ronald
Fintushel
377-390
Abstract: This article studies locally smooth ${S^1}$-actions on closed oriented 4-manifolds in terms of the orbit space, orbit type data, and the characteristic class of the action which lies in $ {H_1}({M^{\ast}},{S^{\ast}})$ where $ {M^{\ast}}$ is the orbit space and $ {S^{\ast}}$ is the image of a certain collection of singular orbits. It is proved that such actions are determined by their weighted orbit spaces and are in 1-1 correspondence with ``legally-weighted'' 3-manifolds. The information contained in the weighted orbit space is used to give a presentation of the fundamental group of the 4-manifold, and in certain cases the quadratic form is computed.
The Diophantine problem for polynomial rings and fields of rational functions
J.
Denef
391-399
Abstract: We prove that the diophantine problem for a ring of polynomials over an integral domain of characteristic zero or for a field of rational functions over a formally real field is unsolvable.
A theorem of Ahlfors for hyperbolic spaces
Su Shing
Chen
401-406
Abstract: L. Ahlfors has proved that if the Dirichlet fundamental polyhedron of a Kleinian group G in the unit ball ${B^3}$ has finitely many sides, then the normalized Lebesgue measure of $L(G)$ is either zero or one. We generalize this theorem and a theorem of Beardon and Maskit to the n-dimensional case.