Binomial enumeration on dissects
Michael
Henle
1-39
Abstract: The Mullin-Rota theory of binomial enumeration is generalized to an abstract context and applied to rook polynomials, order invariants of posets, Tutte invariants of combinatorial geometries, cycle indices and symmetric functions.
Local Jordan algebras
Marvin E.
Camburn
41-50
Abstract: A local Jordan algebra $\mathfrak{J}$ is a unital quadratic Jordan algebra in which $\operatorname{Rad} \mathfrak{J}$ is a maximal ideal, $ \mathfrak{J}/\operatorname{Rad} \mathfrak{J}$ satisfies the DCC, and ${ \cap _k}\operatorname{Rad} {\mathfrak{J}^{(k)}} = 0$ where ${K^{(n + 1)}} = {U_K}(n){K^{(n)}}$. We show that the completion of a local Jordan algebra is also local Jordan, and if $ \mathfrak{J}$ is a complete local Jordan algebra over a field of characteristic not 2, then either (1) $ \mathfrak{J}$ is a complete completely primary Jordan algebra, (2) $\mathfrak{J} \cong {\mathfrak{J}_1} \oplus {\mathfrak{J}_2} \oplus S$ where each ${\mathfrak{J}_i}$ is a completely primary local Jordan algebra, or (3) $\mathfrak{J} \cong \mathfrak{H}({D_n},{J_a})$ where $(D,j)$ is either a not associative alternative algebra with involution or a complete semilocal associative algebra with involution.
Decomposable braids as subgroups of braid groups
H.
Levinson
51-55
Abstract: The group of all decomposable $3$-braids is the commutator subgroup of the group ${I_3}$ of all $3$-braids which leave strand positions invariant. The group of all $2$-decomposable $4$-braids is the commutator subgroup of $ {I_4}$, and the group of all decomposable $4$-braids is explicitly characterized as a subgroup of the second commutator subgroup of ${I_4}$.
Selection theorems and the reduction principle
A.
Maitra;
B. V.
Rao
57-66
Abstract: In this paper we attempt a unification of several selection theorems in the literature. It is proved that the existence of ``nice'' selectors in a certain class of selection problems is essentially equivalent to the fact that certain families of sets satisfy a weak version of Kuratowski's reduction principle. Various special cases are discussed.
Manifolds with no periodic homeomorphisms
Edward M.
Bloomberg
67-78
Abstract: An analysis of the ends of the universal cover of the connected sum of two aspherical manifolds leads to the construction of a class of closed manifolds with no nontrivial periodic homeomorphisms.
The bracket ring of a combinatorial geometry. I
Neil L.
White
79-95
Abstract: The bracket ring is a ring of generalized determinants, called brackets, constructed on an arbitrary combinatorial geometry $G$. The brackets satisfy several familiar properties of determinants, including the syzygies, which are equivalent to Laplace's expansion by minors. We prove that the bracket ring is a universal coordinatization object for $G$ in two senses. First, coordinatizations of $ G$ correspond to homomorphisms of the ring into fields, thus reducing the study of coordinatizations of $G$ to the determination of the prime ideal structure of the bracket ring. Second, $G$ has a coordinatization-like representation over its own bracket ring, which allows an interesting generalization of some familiar results of linear algebra, including Cramer's rule. An alternative form of the syzygies is then derived and applied to the problem of finding a standard form for any element of the bracket ring. Finally, we prove that several important relations between geometries, namely orthogonality, subgeometry, and contraction, are directly reflected in the structure of the bracket ring.
Nil and power-central polynomials in rings
Uri
Leron
97-103
Abstract: A polynomial in noncommuting variables is vanishing, nil or central in a ring, $R$, if its value under every substitution from $ R$ is 0, nilpotent or a central element of $R$, respectively. THEOREM. If $ R$ has no nonvanishing multilinear nil polynomials then neither has the matrix ring ${R_n}$. THEOREM. Let $R$ be a ring satisfying a polynomial identity modulo its nil radical $N$, and let $f$ be a multilinear polynomial. If $ f$ is nil in $ R$ then $f$ is vanishing in $R/N$. Applied to the polynomial $ xy - yx$, this establishes the validity of a conjecture of Herstein's, in the presence of polynomial identity. THEOREM. Let $m$ be a positive integer and let $F$ be a field containing no $m$th roots of unity other than 1. If $ f$ is a multilinear polynomial such that for some $ n > 2{f^m}$ is central in ${F_n}$, then $f$ is central in ${F_n}$. This is related to the (non)existence of noncrossed products among $ {p^2}$-dimensional central division rings.
On symmetrically distributed random measures
Olav
Kallenberg
105-121
Abstract: A random measure $ \xi$ defined on some measurable space $ (S,\mathcal{S})$ is said to be symmetrically distributed with respect to some fixed measure $\omega$ on $S$, if the distribution of $(\xi {A_1}, \cdots ,\xi {A_k})$ for $k \in N$ and disjoint $ {A_1}, \cdots ,{A_k} \in \mathcal{S}$ only depends on $(\omega {A_1}, \cdots ,\omega {A_k})$. The first purpose of the present paper is to extend to such random measures (and then even improve) the results on convergence in distribution and almost surely, previously given for random processes on the line with interchangeable increments, and further to give a new proof of the basic canonical representation. The second purpose is to extend a well-known theorem of Slivnyak by proving that the symmetrically distributed random measures may be characterized by a simple invariance property of the corresponding Palm distributions.
Presentations of $n$-knots
C.
Kearton
123-140
Abstract: The method of critical level embeddings is used to generalize the technique of knot presentations from the classical case to the case of $n$-knots. For $n > 3$, it is shown that an $n$-knot with algebraically simple complement has a correspondingly simple presentation.
Blanchfield duality and simple knots
C.
Kearton
141-160
Abstract: The method of presentation for $n$-knots is used to classify simple $ (2q - 1)$-knots, $ q > 3$, in terms of the Blanchfield duality pairing. As a corollary, we characterize the homology modules and pairings which can arise from classical knots.
$\omega $-cohesive sets
Barbara F.
Ryan
161-171
Abstract: We define and investigate $\omega$-cohesiveness, a strong notion of indecomposability for subsets of the integers and their isols. This notion says, for example, that if $X$ is the isol of an $\omega $-cohesive set then, for any integer $n$ implies that, for some integer $k, \cdot (\begin{array}{*{20}{c}} {X - k} n \end{array} ) \leq Y$ or $Z$. From this it follows that if $f(x) \in {T_1}$, the collection of almost recursive combinatorial polynomials, then the predecessors of $ {f_\Lambda }(X)$ are limited to isols $ {g_\Lambda }(X)$ where $g(X) \in {T_1}$. We show existence of $\omega $-cohesive sets. And we show that the isol of an $\omega$-cohesive set is an $n$-order indecomposable isol as defined by Manaster. This gives an alternate proof to one half of Ellentuck's theorem showing a simple algebraic difference between the isols and cosimple isols. In the last section we study functions of several variables when applied to isols of $\omega$-cohesive sets.
Comparison theorems for bounded solutions of $\triangle u=Pu$
Moses
Glasner
173-179
Abstract: Let $P$ and $Q$ be ${C^1}$ densities on a hyperbolic Riemann surface $ R$. A characterization of isomorphisms between the spaces of bounded solutions of $\Delta u = Pu$ and $\Delta u = Qu$ on $R$ in terms of the Wiener harmonic boundary is given.
Weakly starlike meromorphic univalent functions
Richard J.
Libera;
Albert E.
Livingston
181-191
Abstract: A weakly starlike meromorphic univalent function is one of the form $f(z) = - \rho zg(z){[(z - \rho )(1 - \rho z)]^{ - 1}}$ for $0 < \rho < 1$ and $g(z)$ a meromorphic starlike function. The behavior of coefficients and growth of this class of functions and of a subset are studied.
$H$-closed extensions. II
Jack R.
Porter;
Charles
Votaw
193-209
Abstract: The internal structure and external properties (in terms of other $ H$-closed extensions) of the Fomin extension $\sigma X$ of a Hausdorff space $X$ are investigated. The relationship between $\sigma X$ and the Stone-Čech compactification of the absolute of $X$ is developed and used to prove that a $ \sigma X$-closed subset of $ \sigma X\backslash X$ is compact and to show the existence of a Tychonoff space $Y$ such that $ \sigma X\backslash X$ is homeomorphic to $ \beta Y\backslash Y$. The sequential closure of $X$ in $\sigma X$ is shown to be $X$. It is known that $\sigma X$ is not necessarily projectively larger than any other strict $H$-closed extension of $X$; a necessary and sufficient condition is developed to determine when a $H$-closed extension of $X$ is projectively smaller then $\sigma X$. A theorem by Magill is extended by showing that the sets of $\theta$-isomorphism classes of $H$-closed extensions of locally $ H$-closed spaces $ X$ and $Z$ are lattice isomorphic if and only if $ \sigma X\backslash X$ and $ \sigma Z\backslash Z$ are homeomorphic. Harris has characterized those simple Hausdorff extensions of $X$ which are subextensions of the Katětov extension. Characterizations of Hausdorff (not necessarily simple) extensions of $X$ which are subextensions of $ H$-closed extensions $ \theta$-isomorphic and $ S$-equivalent to the Katětov extension are presented.
Weighted $L\sp{2}$ approximation of entire functions
Devora
Wohlgelernter
211-219
Abstract: Let $S$ be the space of entire functions $ f(z)$ such that $\vert\vert f(z)\vert{\vert^2} = \smallint \smallint \vert f(z){\vert^2}dm(z)$, where $ m$ is a positive measure defined on the Borel sets of the complex plane. Write $ dm(z) = K(z)d{A_z} = K(r,\theta )dAz$. Theorem 1. If $\ln {\inf _\theta }K(r,\theta )$ is asymptotic to $\ln {\sup_\theta} K(r,\theta )$ (together with other mild restrictions) then polynomials are dense in $S$. Theorem 2. Let $K(z) = {e^{ - \phi (z)}}$ where $ \phi (z)$ is a convex function of $z$ such that all exponentials belong to $ S$. Then polynomials are dense in $S$.
Primitive ideals of twisted group algebras
Otha L.
Britton
221-241
Abstract: E. Effros and F. Hahn have conjectured that if $(G,Z)$ is a second countable locally compact transformation group, with $G$ amenable, then every primitive ideal of the associated ${C^\ast }$-algebra arises as the kernel of an irreducible representation induced from a stability subgroup. Results of Effros and Hahn concerning this conjecture are extended to include the twisted group algebra $ {L^1}(G,A;T,\alpha )$, where $A$ is a separable type I ${C^\ast }$-algebra.
Fixed points of pointwise almost periodic homeomorphisms on the two-sphere
W. K.
Mason
243-258
Abstract: A homeomorphism $ f$ of the two-sphere $ {S^2}$ onto itself is defined to be pointwise almost periodic (p.a.p.) if the collection of orbit closures forms a decomposition of $ {S^2}$. It is shown that if $f:{S^2} \to {S^2}$ is p.a.p. and orientation-reversing then the set of fixed points of $ f$ is either empty or a simple closed curve; if $f:{S^2} \to {S^2}$ is p.a.p. orientation-preserving and has a finite number of fixed points, then $ f$ is shown to have exactly two fixed points.
The theory of countable analytical sets
Alexander S.
Kechris
259-297
Abstract: The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd $n$ a largest countable $\Pi _n^1$ set of reals, ${\mathcal{C}_n}$ (this is also true for $ n$ even, replacing $ \Pi _n^1$ by $\Sigma _n^1$ and has been established earlier by Solovay for $n = 2$ and by Moschovakis and the author for all even $n > 2$). The internal structure of the sets ${\mathcal{C}_n}$ is then investigated in detail, the point of departure being the fact that each ${\mathcal{C}_n}$ is a set of $\Delta _n^1$-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, $ \omega$-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc.
One-dimensional polyhedral irregular sets of homomorphisms of $3$-manifolds
L. S.
Husch;
W. H.
Row
299-323
Abstract: Examples are given to show that there exist homeomorphisms of open $ 3$-manifolds whose sets of irregular points are wildly embedded one-dimensional polyhedra. The main result of the paper is that a one-dimensional polyhedral set of irregular points can fail to be locally tame on, at most, a discrete subset of the set of points of order greater than one. Necessary and sufficient conditions are given so that the set of irregular points is locally tame at each point.
Fatou properties of monotone seminorms on Riesz spaces
Theresa K. Y. Chow
Dodds
325-337
Abstract: A monotone seminorm $ \rho$ on a Riesz space $ L$ is called $ \sigma$-Fatou if $ \rho ({u_n}) \uparrow \rho (u)$ holds for every $u \in {L^ + }$ and sequence $\{ {u_n}\}$ in $L$ satisfying $0 \leq {u_n} \uparrow u$. A monotone seminorm $ \rho$ on $L$ is called strong Fatou if $\rho ({u_v}) \uparrow \rho (u)$ holds for every $u \in {L^ + }$ and directed system $\{ {u_v}\}$ in $L$ satisfying $0 \leq {u_v} \uparrow u$. In this paper we determine those Riesz spaces $L$ which have the property that, for any monotone seminorm $\rho$ on $L$, the largest strong Fatou seminorm $ {\rho _m}$ majorized by $ \rho$ is of the form: ${\rho _m}(f) = \inf \{ {\sup _v}\rho ({u_v}):0 \leq {u_v} \uparrow \vert f\vert\}$]> for <![CDATA[$f \in L$. We discuss, in a Riesz space $L$, the condition that a monotone seminorm $\rho$ as well as its Lorentz seminorm $ {\rho _L}$ is $ \sigma$-Fatou in terms of the order and relative uniform topologies on $ L$. A parallel discussion is also given for outer measures on Boolean algebras.
A family of countably compact $P\sb{\ast}$-hypergroups
Charles F.
Dunkl;
Donald E.
Ramirez
339-356
Abstract: An infinite compact group is necessarily uncountable, by the Baire category theorem. A compact ${P_\ast }$-hypergroup, in which the product of two points is a probability measure, is much like a compact group, having an everywhere supported invariant measure, an orthogonal system of characters which span the continuous functions in the uniform topology, and a multiplicative semigroup of positive-definite functions. It is remarkable that a compact ${P_\ast }$-hypergroup can be countably infinite. In this paper a family of such hypergroups, which include the algebra of measures on the $ p$-adic integers which are invariant under the action of the units (for $p = 2,3,5, \cdots )$) is presented. This is an example of the symmetrization technique. It is possible to give a nice characterization of the Fourier algebra in terms of a bounded-variation condition, which shows that the usual Banach algebra questions about the Fourier algebra, such as spectral synthesis, and Helson sets have easily determinable answers. Helson sets are finite, each closed set is a set of synthesis, the maximal ideal space is exactly the underlying hypergroup, and the functions that operate are exactly the Lip 1 functions.
Results on measures of irreducibility and full indecomposability
D. J.
Hartfiel
357-368
Abstract: This paper develops a notion of $k$th measure of irreducibility and $ k$th measure of full indecomposability. The combinatorial properties of these notions, as well as relationships between these notions, are explored. The results are then used in converting results on positive matrices into results on nonnegative matrices.
Converses to the $\Omega $-stability and invariant lamination theorems
Allan
Gottlieb
369-383
Abstract: In 1967 Smale proved that for diffeomorphisms on closed smooth manifolds, Axiom ${\text{A}}$ and no cycles are sufficient conditions for $\Omega$-stability and asserted the analogous theorem for vectorfields. Pugh and Shub have supplied a proof of the latter. Since then a major problem in dynamical systems has been Smale's conjecture that Axiom ${\text{A}}$ (resp.
Flat analytic extensions
Ana M. D.
Viola-Prioli
385-404
Abstract: This paper is concerned, in the first place, with the conditions to be imposed on an ideal $I$ of the power series ring in one indeterminate $ A[[x]]$ ($A$ noetherian) in order that the analytic extension $B = A[[x]]/I$ be a flat $ A$-module. Also the relationship between the projectivity and finiteness of $B$ is found when the content of $I$ (the ideal of $A$ generated by the coefficients of all power series in $I$) equals $A$. A generalization of this result to the power series ring in any finite number of indeterminates is obtained when $A$ is local, noetherian of Krull $\dim \geq 1$, and under certain restrictions on $ I$, for the global case but only for domains. Finally, a contribution to the problem of the finiteness of $I$ when $A[[x]]/I$ is a flat analytic extension is given for $A$ a local ring, not necessarily noetherian.
Abstract: Some theorems in the Benson-Curtis paper [1] were stated subject to possible exceptions in type ${E_7}$, corresponding to the two irreducible characters of the Weyl group of degree 512. An argument due to T. A. Springer($^{1}$) shows that these cases actually are exceptions to the theorems, and also that there are four exceptional cases in type ${E_8}$ (whose possible existence was overlooked in the original version of the paper), corresponding to the characters of the Weyl group of degree 4096.