On Sylow $2$-subgroups with no normal Abelian subgroups of rank $3$, in finite fusion-simple groups
Anne R.
Patterson
1-67
Abstract: Let T be any finite 2-group which has a normal four-group but has no normal Abelian subgroup of rank 3, and assume T is not the dihedral group of order 8. If T is a Sylow 2-subgroup of a finite fusion-simple group G, it follows (Thompson) from Glauberman's ${Z^ \ast }$-theorem that T has exactly one normal four-group, say W. This paper establishes what isomorphism types of T can so occur under the hypothesis that ${{\mathbf{N}}_G}(T) = T{{\mathbf{C}}_G}(T)$ and the three nonidentity elements of W are not all G-conjugate. All T arrived at in this paper are known to so occur. The reason for this hypothesis is that the similar situation for T with a normal four-group and no normal Abelian subgroup of rank 3, where T is a Sylow 2-subgroup of a finite simple group G but without the above hypothesis, had been analyzed earlier by the author (under her maiden name, MacWilliams; Trans. Amer. Math. Soc. 150 (1970), 345-408).
$B$-convexity and reflexivity in Banach spaces
Dean R.
Brown
69-76
Abstract: A proof of James that uniformly nonsquare spaces are reflexive is extended in part to B-convex spaces. A condition sufficient for non-B-convexity and related conditions equivalent to non-B-convexity are given. The following theorem is proved: A Banach space is B-convex if each subspace with basis is B-convex.
$P$-convexity and $B$-convexity in Banach spaces
Dean R.
Brown
77-81
Abstract: Two properties of B-convexity are shown to hold for P-convexity: (1) Under certain conditions, the direct sum of two P-convex spaces is P-convex. (2) A Banach space is P-convex if each subspace having a Schauder decomposition into finite dimensional subspaces is P-convex.
On a compactness property of topological groups
S. P.
Wang
83-88
Abstract: A density theorem of semisimple analytic groups acting on locally compact groups is presented.
Duality theories for metabelian Lie algebras
Michael A.
Gauger
89-102
Abstract: This paper is concerned with duality theories for metabelian (2-step nilpotent) Lie algebras. A duality theory associates to each metabelian Lie algebra N with cod $ {N^2} = g$, another such algebra ${N_D}$ satisfying ${({N_D})_D} \cong N,{N_1} \cong {N_2}$ if and only if $ {({N_1})_D} \cong {({N_2})_D}$, and if $ \dim \,N = g + p$ then $ \dim \,{N_D} = g + (_2^g) - p$. The obvious benefit of such a theory lies in its reduction of the classification problem.
A topology for a lattice-ordered group
R. H.
Redfield
103-125
Abstract: Let G be an arbitrary lattice-ordered group. We define a topology on G, called the $ \mathcal{J}$-topology, which is a group and lattice topology for G and which is preserved by cardinal products. The $\mathcal{J}$-topology is the interval topology on totally ordered groups and is discrete if and only if G is a lexico-sum of lexico-extensions of the integers. We derive necessary and sufficient conditions for the $ \mathcal{J}$-topology to be Hausdorff, and we investigate $\mathcal{J}$-topology convergence.
Structure theory for equational classes generated by quasi-primal algebras
Robert W.
Quackenbush
127-145
Abstract: Quasi-primal algebras (which include finite simple polyadic and cylindric algebras) were introduced by A. S. Pixley. In this paper equational classes generated hy quasi-primal algebras are investigated with respect to the following concepts: the congruence extension property, the amalgamation property and the amalgamation class, weak injectives and weak injective hulls, the standard semigroup of operators. A brief discussion of monadic algebras is included to illustrate the results of the paper.
The theory of $Q$-rings
Eben
Matlis
147-181
Abstract: An integral domain R with quotient field Q is defined to be a Q-ring if $\operatorname{Ext}_R^1(Q,R) \cong Q$. It is shown that R is a Q-ring if and only if there exists an R-module A such that $ {\operatorname{Hom}_R}(A,R) = 0$ and $\operatorname{Ext}_R^1(A,R) \cong Q$. If A is such an R-module and $t(A)$ is its torsion submodule, then it is proved that $A/t(A)$ necessarily has rank one. There are only three kinds of Q-rings, namely, $ {Q_0}{\text{-}},{Q_1}{\text{-}}$, or ${Q_2}$-rings. These are described by the fact that if R is a Q-ring, then $ K = Q/R$ can only have 0, 1, or 2 proper h-divisible submodules. If H is the completion of R in the R-topology, then R is one of the three kinds of Q-rings if and only if $H{ \otimes _R}Q$ is one of the three possible kinds of 2-dimensional commutative Q-algebras. Examples of all three kinds of Q-rings are produced, and the behavior of Q-rings under ring extensions is examined. General conditions are given for a ring not to be a Q-ring. As an application of the theory, necessary and sufficient conditions are found for the integral closure of a non-complete Noetherian domain to be a complete discrete valuation ring.
Weak compactness in the order dual of a vector lattice
Owen
Burkinshaw
183-201
Abstract: A sequence $\{ {x_n}\}$ in a vector lattice E will be called an l'-sequence if there exists an x in E such that $\Sigma _{k = 1}^n\vert{x_k}\vert \leq x$ for all n. Denote the order dual of E by ${E^b}$. For a set $A \subset {E^b}$, let $ {\left\Vert \cdot \right\Vert _{{A^ \circ }}}$ denote the Minkowski functional on E defined by its polar ${A^ \circ }$ in E. A set $A \subset {E^b}$ will be called equi-l'-continuous on E if $\lim {\left\Vert {{x_n}} \right\Vert _{{A^ \circ }}} = 0$ for each l'-sequence $ \{ {x_n}\}$ in E. The main objective of this paper will be to characterize compactness in ${E^b}$ in terms of the order structure on E and ${E^b}$. In particular, the relationship of equi-l'-continuity to compactness is studied. §2 extends to ${E^{\sigma c}}$ the results in Kaplan [8] on vague compactness in ${E^C}$. Then this is used to study vague convergence of sequences in ${E^b}$.
Disjoint meromorphic functions and nonoscillatory differential systems
D.
Aharonov;
M.
Lavie
203-216
Abstract: Conditions satisfied by disjoint meromorphic functions are obtained. These results are applied to nonoscillatory differential systems and disfocal differential equations.
The $p$-adic hull of abelian groups
A.
Mader
217-229
Abstract: In this paper we define ``p-adic hull'' for p-reduced groups K. The p-adic hull ${K^P}$ of K is a module over the ring P of p-adic integers containing K and satisfying certain additional properties. The notion is investigated and then used to prove some known and some new theorems on $\operatorname{Ext}(K,T)$ and $\operatorname{Hom}(K,T)$ for K torsion-free and T a reduced p-group.
Elementary divisor rings and finitely presented modules
Max D.
Larsen;
William J.
Lewis;
Thomas S.
Shores
231-248
Abstract: Throughout, rings are commutative with unit and modules are unital. We prove that R is an elementary divisor ring if and only if every finitely presented module over R is a direct sum of cyclic modules, thus providing a converse to a theorem of Kaplansky and answering a question of Warfield. We show that every Bezout ring with a finite number of minimal prime ideals is Hermite. So, in particular, semilocal Bezout rings are Hermite answering affirmatively a question of Henriksen. We show that every semihereditary Bezout ring is Hermite. Semilocal adequate rings are characterized and a partial converse to a theorem of Henriksen is established.
Smooth partitions of unity on manifolds
John
Lloyd
249-259
Abstract: This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces.
Selfadjoint algebras of unbounded operators. II
Robert T.
Powers
261-293
Abstract: Unbounded selfadjoint representations of $^\ast$-algebras are studied. It is shown that a selfadjoint representation of the enveloping algebra of a Lie algebra can be exponentiated to give a strongly continuous unitary representation of the simply connected Lie group if and only if the representation preserves a certain order structure. This result follows from a generalization of a theorem of Arveson concerning the extensions of completely positive maps of ${C^ \ast }$-algebras. Also with the aid of this generalization of Arveson's theorem it is shown that an operator $ \overline {\pi (A)}$ is affiliated with the commutant $\mathcal{A}$, with $A = {A^ \ast } \in \mathcal{A}$, if and only if $\pi$ preserves a certain order structure associated with A and $ \mathcal{A}$. This result is then applied to obtain a characterization of standard representations of commutative $^\ast $-algebras in terms of an order structure.
Some positive trigonometric sums
Richard
Askey;
John
Steinig
295-307
Abstract: Vietoris found an interesting generalization of the classical inequality $\Sigma _{k = 1}^n(\sin \,k\theta /k) > 0,0 < \theta < \pi$. A simplified proof is given for his inequality and his similar inequality for cosine series. Various new results which follow from these inequalities include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums of ultraspherical polynomials which extend an old inequality of Fejér. Both of Vietoris' inequalities are special cases of a general problem for Jacobi polynomials, and a summary is given of known results on this problem.
Extensions of normal immersions of $S\sp{1}$ into $R\sp{2}$
Morris L.
Marx
309-326
Abstract: Suppose that $f:{S^1} \to {R^2}$ is an immersion, i.e., a $ {C^1}$ map such that $ f'$ is never zero. We call f normal if there are only finitely many self-intersections and these are transverse double points. A normal immersion f can be topologically determined by a finite number of combinatorial invariants. Using these invariants it is possible to give considerable information about extensions of f to $ {D^2}$. In this paper we give a new set of invariants, inspired by the work of S. Blank, to solve several problems concerning the existence of certain kinds of extensions. The problems solved are as follows: (1) When does f have a light open extension $ F:{D^2} \to {R^2}$? (Recall that light means ${F^{ - 1}}(y)$ is totally disconnected for all y and open means F maps open sets of the interior of ${D^2}$ to open sets of ${R^2}$.) Because of the work of Stoïlow, the question is equivalent to the following: when does there exist a homeomorphism $h:{S^1} \to {S^1}$, such that fh has an analytic extension to ${D^2}$? (2) Suppose that $F:{D^2} \to {R^2}$ is light, open, sense preserving, and, at each point of ${S^1}$, F is a local homeomorphism. At each point of the interior of ${D^2}$, F is locally topologically equivalent to the power mapping ${z^m}$ on $ {D^2},m \geq 1$. Points where $m > 1$ are called branch points and $ m - 1$ is the multiplicity of the point. There are only a finite number of branch points. The problem is to discover the minimum number of branch points of any properly interior extension of f. Also we can ask what multiplicities can arise for extensions of a given f. (3) Given a normal f, find the maximum number of properly interior extensions of f that are pairwise inequivalent. Since each immersion of the disk is equivalent to a local homeomorphism, the problem of immersion extensions is a special case of this. It is Blank's solution of the immersion problem that prompted this paper.
Topological dynamics and group theory
Shmuel
Glasner
327-334
Abstract: We prove, using notions and techniques of topological dynamics, that a nonamenable group contains a finitely-generated subgroup of exponential growth. We also show that a group which belongs to a certain class, defined by means of topological dynamical properties, always contains a free subgroup on two generators.
Isolated singularities for solutions of the nonlinear stationary Navier-Stokes equations
Victor L.
Shapiro
335-363
Abstract: The notion for (u, p) to be a distribution solution of the nonlinear stationary Navier-Stokes equations in an open set is defined, and a theorem concerning the removability of isolated singularities for distribution solutions in the punctured open ball $B(0,{r_0}) - \{ 0\}$ is established. This result is then applied to the classical situation to obtain a new theorem for the removability of isolated singularities. In particular, in two dimensions this gives a better than expected result when compared with the theory of removable isolated singularities for harmonic functions.
Egoroff properties and the order topology in Riesz spaces
Theresa K. Y. Chow
Dodds
365-375
Abstract: In this paper we prove that, for a Riesz space L, the order closure of each subset of L coincides with its pseudo order closure if and only if the order closure of each convex subset of L coincides with its pseudo order closure; moreover, each of these statements is equivalent to the strong Egoroff property. For Archimedean Riesz spaces, similar results hold for the relative uniform topology.
One-parameter semigroups holomorphic away from zero
Melinda W.
Certain
377-389
Abstract: Suppose T is a one-parameter semigroup of bounded linear operators on a Banach space, strongly continuous on $[0,\infty )$. It is known that $\lim {\sup _{x \to 0}}\vert T(x) - I\vert < 2$ implies T is holomorphic on $(0,\infty )$. Theorem I is a generalization of this as follows: Suppose $M > 0,0 < r < s$, and $\rho$ is in (1,2). If $\vert{(T(h) - I)^n}\vert \leq M{\rho ^n}$ whenever nh is in $[r,s],n = 1,2, \cdots ,h > 0$, then there exists $b > 0$ such that T is holomorphic on $ [b,\infty )$. Theorem II shows that, in some sense, $b \to 0$ as $r \to 0$. Theorem I is an application of Theorem III: Suppose $ M > 0,0 < r < s,\rho$ is in (1,2), and f is continuous on $[ - 4s,4s]$. If $\vert\sum\nolimits_{q = 0}^n {(\mathop n\limits_q ){{( - 1)}^{n - q}}f(t + qh)\vert \leq M{\rho ^n}}$ whenever nh is in $ [r,s],n = 1,2, \cdots ,h > 0,[t,t + nh] \subset [ - 4s,4s]$, then f has an analytic extension to an ellipse with center zero. Theorem III is a generalization of a theorem of Beurling in which the inequality on the differences is assumed for all nh. An example is given to show the hypothesis of Theorem I does not imply T holomorphic on $ (0,\infty )$.
The existence, characterization and essential uniqueness of solutions of $L\sp{\infty }$ extremal problems
S. D.
Fisher;
J. W.
Jerome
391-404
Abstract: Let $I = (a,b)$ be an interval in R and let ${H^{n,\infty }}$ consist of those real-valued functions f such that $ {f^{(n - 1)}}$ is absolutely continuous on I and ${f^{(n)}} \in {L^\infty }(I)$. Let L be a linear differential operator of order n with leading coefficient $1,a = {x_1} < \cdots < {x_m} = b$ be a partition of I and let the linear functionals $ {L_{ij}}$ on ${H^{n,\infty }}$ be given by $\displaystyle {L_{ij}}f = \sum\limits_{v = 0}^{n - 1} {a_{ij}^{(v)}{f^{(v)}}({x_i}),\quad j = 1, \cdots ,{k_i},i = 1, \cdots ,m,}$ where $1 \leq {k_i} \leq n$ and the ${k_i}$ n-tuples $ (a_{ij}^{(0)}, \cdots ,a_{ij}^{(n - 1)})$ are linearly inde pendent. Let $ {r_{ij}}$ be prescribed real numbers and let $U = \{ f \in {H^{n,\infty }}:{L_{ij}}f = {r_{ij}},j = 1, \cdots ,{k_i},i = 1, \cdots ,m\}$. In this paper we consider the extremal problem $\displaystyle {\left\Vert {Ls} \right\Vert _{{L^\infty }}} = \alpha = \inf \{ {\left\Vert {Lf} \right\Vert _{{L^\infty }}}:f \in U\} .$ ($\ast$) We show that there are, in general, many solutions to $( \ast )$ but that there is, under certain consistency assumptions on L and the ${L_{ij}}$, a fundamental (or core) interval of the form $({x_i},{x_{i + {n_0}}})$ on which all solutions to $ ( \ast )$ agree; $ {n_0}$ is determined by the ${k_i}$ and satisfies $ {n_0} \geq 1$. Further, if s is any solution to $( \ast )$ then on $({x_i},{x_{i + {n_0}}}),\vert Ls\vert = \alpha$ a.e. Further, we show that there is a uniquely determined solution $ {s_ \ast }$ to $ ( \ast )$, found by minimizing ${\left\Vert {Lf} \right\Vert _{{L^\infty }}}$ over all subintervals $({x_j},{x_{j + 1}}),j = 1, \cdots ,m - 1$, with the property that $ \vert L{s_ \ast }\vert$ is constant on each subinterval $({x_j},{x_{j + 1}})$ and $ L{s_ \ast }$ is a step function with at most $n - 1$ discontinuities on $({x_j},{x_{j + 1}})$. When $L = {D^n},{s_ \ast }$ is a piecewise perfect spline. Examples show that the results are essentially best possible.
Jordan algebras and connections on homogeneous spaces
Arthur A.
Sagle
405-427
Abstract: We use the correspondence between G-invariant connections on a reductive homogeneous space $ G/H$ and certain nonassociative algebras to explicitly compute the pseudo-Riemannian connections in terms of a Jordan algebra J of endomorphisms. It is shown that if G and H are semisimple Lie groups, then J is a semisimple Jordan algebra. Also a general method for computing examples of J is given.
Erratum to ``Oscillation, continuation, and uniqueness of solutions of retarded differential equations'' (Trans. Amer. Math. Soc. {\bf 179} (1973), 193--209)
T.
Burton;
R.
Grimmer
429