Characterizations of the Fischer groups. I, II, III
David
Parrott
303-347
Abstract: B. Fischer, in his work on finite groups which contain a conjugacy class of $ 3$-transpositions, discovered three new sporadic finite simple groups, usually denoted $M(22)$, $M(23)$ and $M(24)'$. In Part I two of these groups, $ M(22)$ and $M(23)$, are characterized by the structure of the centralizer of a central involution. In addition, the simple groups ${U_6}(2)$ (often denoted by $M(21))$ and $ P\Omega (7,3)$, both of which are closely connected with Fischer's groups, are characterized by the same method. The largest of the three Fischer groups $M(24)$ is not simple but contains a simple subgroup $ M(24)'$ of index two. In Part II we give a similar characterization by the centralizer of a central involution of $M(24)$ and also a partial characterization of the simple group $M(24)'$. The purpose of Part III is to complete the characterization of $M(24)'$ by showing that our abstract group $ G$ is isomorphic to $ M(24)'$. We first prove that $G$ contains a subgroup $X \cong M(23)$ and then we construct a graph (on the cosets of $X$) which is shown to be isomorphic to the graph for $ M(24)$.
A unicity theorem for meromorphic mappings between algebraic varieties
S. J.
Drouilhet
349-358
Abstract: Using the techniques of value distribution theory in several complex variables, we obtain a theorem which can be used to determine whether two nondegenerate meromorphic mappings from an affine algebraic variety to a projective algebraic variety of the same or lower dimension are identical. The theorem generalizes a result of $R$. Nevanlinna in one complex variable.
Algebraic invariants of boundary links
Nobuyuki
Sato
359-374
Abstract: In this paper we study the homology of the universal abelian cover of the complement of a boundary link of $n$-spheres in $ {S^{n + 2}}$, as modules over the (free abelian) group of covering transformations. A consequence of our results is a characterization of the polynomial invariants ${p_{i,q}}$ of boundary links for $1 \leqslant q \leqslant [n/2]$. Along the way we address the following algebraic problem: given a homomorphism of commutative rings $ f:R \to S$ and a chain complex ${C_ \ast }$ over $R$, determine when the complex $S{ \otimes _R}{C_ \ast }$ is acyclic. The present work is a step toward the characterization of link modules in general.
Random evolution processes with feedback
Kyle
Siegrist
375-392
Abstract: A general random evolution Markov process is constructed which switches back and forth at random among a given collection of Markov processes ("modes of evolution") defined on a common evolution state space and indexed by an evolution rule space. Feedback is incorporated by allowing the path of the evolution component to influence the changes in evolution rule. The semigroup of the random evolution process is studied and is used to compare the process with the operator random evolutions of Griego and Hersh. Using deterministic modes of evolution, we generalize the Markov processes constructed by Erickson and by Heath. We also study new random evolution processes constructed from Brownian motions and from regular step processes.
Semigroup compactifications of semidirect products
H. D.
Junghenn;
B. T.
Lerner
393-404
Abstract: Let $S$ and $T$ be semigroups, $S\circlebin{\tau} T$ a semidirect product, and $ F$ a ${C^ \ast }$-algebra of bounded, complex-valued functions on $ S\circlebin{\tau} T$. Necessary and sufficient conditions are given for the $ F$-compactification of $ S\circlebin{\tau} T$ to be expressible as a semidirect product of compactifications of $S$ and $T$. This result is used to show that the strongly almost periodic compactification of $S\circlebin{\tau} T$ is a semidirect product and that, in certain general cases, the analogous statement holds for the almost periodic compactification and the left uniformly continuous compactification of $S\circlebin{\tau} T$. Applications are made to wreath products.
A tangential convergence for bounded harmonic functions on a rank one symmetric space
Jacek
Cygan
405-418
Abstract: Let $u$ be a bounded harmonic function on a noncompact rank one symmetric space $M = G/K \approx {N^ - }A,{N^ - }AK$ being a fixed Iwasawa decomposition of $ G$. We prove that if for an ${a_0} \in A$ there exists a limit $u(n{a_0}) \equiv {c_0}$, as $n \in {N^ - }$ goes to infinity, then for any $a \in A$, $u(na) = {c_0}$. For $M = SU(n,1)/S(U(n) \times U(1)) = {B^n}$, the unit ball in $ {{\mathbf{C}}^n}$ with the Bergman metric, this is a result of Hulanicki and Ricci, and in this case it reads (via the Cayley transformation) as a theorem on convergence of a bounded harmonic function to a boundary value at a fixed boundary point, along appropriate, tangent to $\partial {B^n}$, surfaces.
Tensegrity frameworks
B.
Roth;
W.
Whiteley
419-446
Abstract: A tensegrity framework consists of bars which preserve the distance between certain pairs of vertices, cables which provide an upper bound for the distance between some other pairs of vertices and struts which give a lower bound for the distance between still other pairs of vertices. The present paper establishes some basic results concerning the rigidity, flexibility, infinitesimal rigidity and infinitesimal flexibility of tensegrity frameworks. These results are then applied to a number of questions, problems and conjectures regarding tensegrity frameworks in the plane and in space.
R\'ealisation de morphismes donn\'es en cohomologie et suite spectrale d'Eilenberg-Moore
Micheline
Vigué-Poirrier
447-484
Abstract: On construit une suite d'obstructions à la réalisation par une application entre types d'homotopie rationnelle, d'un morphisme donné en cohomologie. On donne, sous des hypothèses de finitude, des conditions simples d'existence de réalisation. On montre aussi que, pour des algèbres différentielles commutatives graduées sur un corps de caractéristique 0, la réalisation d'un morphisme donné en cohomologie dépend, en général, du corps de base. La technique utilisée est la construction du modèle minimal bigradué d'un homomorphisme d'algèbres commutatives graduées, puis du modèle filtré d'une application continue, par déformation des différentielles du modèle bigradué de l'application induite en cohomologie. Cette construction est utilisée pour donner une méthode explicite de calcul de la suite spectrale d'Eilenberg-Moore $({E_i},{d_i})$ d'un carré fibré. On en déduit des critères pour que ${d_i} = 0,i \geqslant 2$.
Unflat connections in $3$-sphere bundles over $S\sp{4}$
Andrzej
Derdziński;
A.
Rigas
485-493
Abstract: The paper concerns connections in $3$-sphere bundles over $4$-manifolds having the property of unflatness, which is a necessary condition in order that a natural construction give a Riemannian metric of positive sectional curvature in the total space. It is shown that, as conjectured by A. Weinstein, the only $3$-sphere bundle over ${S^4}$ with an unflat connection is the Hopf bundle.
The order convergence of martingales indexed by directed sets
Kenneth A.
Astbury
495-510
Abstract: We obtain a condition on the underlying family of $\sigma$-algebras which is properly weaker than the Vitali property but which is also a sufficient condition for the order convergence of martingales of semibounded variation. We also obtain a sufficient condition for the order convergence of martingales of semibounded variation in terms of the finiteness of the extreme order limits of martingales of bounded variation.
Weighted estimates for fractional powers of partial differential operators
Raymond
Johnson
511-525
Abstract: It is shown that fractional powers defined by the wave polynomial $P(\xi ) = \xi _{^1}^2 + \cdots + \xi _p^2 - \xi _{p + 1}^2 - \cdots - \xi _n^2$, defined in terms of Fourier transforms by $\widehat{{T^\lambda }f} = {\left\vert {P(\xi )} \right\vert^\lambda }\hat f$, are in the Bernstein subalgebra of functions with integrable Fourier transforms for $ \lambda > (n - 1)/2$, provided $f \in C_c^m$ with $m$ large enough. The proof uses embedding theorems for Besov spaces and Stein's theorem on interpolation of analytic families of operators.
Recognizing the real line
Yuri
Gurevich;
W. Charles
Holland
527-534
Abstract: A certain elementary statement about the group of automorphisms of the real line $\mathbf{R}$ is sufficient to characterize $\mathbf{R}$ among homogeneous chains. A similar result holds for the chain of rational numbers.
Multiplicatively invariant subspaces of Besov spaces
Per
Nilsson
535-543
Abstract: We study subspaces of Besov spaces $B_p^{s,q}$ which are invariant under pointwise multiplication by characters. The case $s > 0$ is completely described, and for the case $s \leqslant 0$ we extend known results.
Arc-smooth continua
J. B.
Fugate;
G. R.
Gordh;
Lewis
Lum
545-561
Abstract: Continua admitting arc-structures and arc-smooth continua are introduced as higher dimensional analogues of dendroids and smooth dendroids, respectively. These continua include such spaces as: cones over compacta, convex continua in ${l_2}$, strongly convex metric continua, injectively metrizable continua, as well as various topological semigroups, partially ordered spaces, and hyperspaces. The arc-smooth continua are shown to coincide with the freely contractible continua and with the metric $ K$-spaces of Stadtlander. Known characterizations of smoothness in dendroids involving closed partial orders, the set function $ T$, radially convex metrics, continuous selections, and order preserving mappings are extended to the setting of continua with arc-structures. Various consequences of the special contractibility properties of arc-smooth continua are also obtained.
A faithful Hille-Yosida theorem for finite-dimensional evolutions
M. A.
Freedman
563-573
Abstract: As a natural generalization of the classical Hille-Yosida theorem to evolution operators, necessary and sufficient conditions are found for an evolution $U$ acting in ${R^N}$ so that for each $s \geqslant t$, $U(s,t)$ can be uniquely represented as a product integral $ \prod _t^s{[I + V]^{ - 1}}$ for some additive, accretive generator $ V$. Under these conditions, we further show that $ U(\xi ,\zeta )$ is differentiable a.e.
On supercuspidal representations of the metaplectic group
James
Meister
575-598
Abstract: The Weil representations associated to anisotropic quadratic forms in one and three variables are used to study supercuspidal representations of the two-fold metaplectic covering group ${\overline {{\text{GL}}} _2}(k)$, where $ k$ is a local nonarchimedean field of odd residual characteristic. The principal result is the explicit calculation of certain Whittaker functionals for any square-integrable irreducible admissible genuine representation of ${\overline {{\text{GL}}} _2}(k)$. In particular, a recent conjecture of Gelbart and Piatetski-Shapiro is answered by obtaining a bijection between the set of quasicharacters of ${k^ \ast }$ and the set of irreducible admissible genuine distinguished representations of ${\overline {{\text{GL}}} _2}(k)$, i.e. those representations possessing only one Whittaker functional, or, equivalently, those having a unique Whittaker model. The distinguished representations are precisely the representations attached to the Weil representation associated to a one dimensional form. The local piece of the generalized Shimura correspondence between automorphic forms of $ {\overline {{\text{GL}}} _2}({\mathbf{A}})$ and $ {\text{G}}{{\text{L}}_2}({\mathbf{A}})$ is also treated. Based upon a conjecture of the equivalences among the constituents of the Weil representations associated to two nonequivalent ternary forms, evidence for the explicit form of the local piece of this global correspondence, restricted to supercuspidal representations of ${\overline {{\text{GL}}} _2}(k)$, is presented. In this form, the map is shown to be injective and its image is described.
An elementary proof of the local Kronecker-Weber theorem
Michael
Rosen
599-605
Abstract: Let $K$ be a local field. Lubin and Tate have shown how to explicitly construct an abelian extension of $K$ which they prove to be the maximal abelian extension. Their proof of this result uses local class field theory. When $K$ is a $p$-adic field we give an elementary proof which even avoids the use of higher ramification groups. Instead we rely on facts about the principal units in a finite abelian extension of $K$ as a module for the Galois group.
Boolean powers: direct decomposition and isomorphism types
Kenneth
Hickin;
J. M.
Plotkin
607-621
Abstract: We determine properties of Boolean powers of groups and other algebraic structures, and we generalize Jónsson's theorem on Boolean powers of centerless, directly indecomposable groups. We show that every nonabelian, finitely generated group has $ {2^{{\aleph _0}}}$ nonisomorphic countable Boolean, and hence subcartesian, powers. We show that nonabelian groups $G$ such that either (i) $G$ is not the central product of two nonabelian groups or (ii) every pair of nontrivial normal subgroups of $G$ intersect nontrivially yield nonisomorphic Boolean powers with respect to nonisomorphic Boolean algebras.
Uniqueness of invariant means for measure-preserving transformations
Joseph
Rosenblatt
623-636
Abstract: For some compact abelian groups $X$ (e.g. $T^n$, $ n \geqslant 2$, and $ \prod\nolimits_{n = 1}^\infty {{Z_2}}$), the group $G$ of topological automorphisms of $ X$ has the Haar integral as the unique $G$-invariant mean on ${L_\infty }(X,{\lambda _X})$. This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets $\beta$ of ${R^n}$, $ n \geqslant 3$; it is the unique normalized positive finitely-additive measure on $ \beta$ which is invariant under isometries and the transformation of ${R^n}:({x_1}, \ldots ,{x_n}) \mapsto ({x_1} + {x_2},{x_2}, \ldots ,{x_n})$. Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on ${L_\infty }(X,\beta ,p)$, which is invariant by some group of measure-preserving transformations of the probability space $(X,\beta ,p)$, are described.