On $2$-groups with no normal abelian subgroups of rank $3$, and their occurrence as Sylow $2$-subgroups of finite simple groups
Anne R.
MacWilliams
345-408
Abstract: We prove that in a finite $2$-group with no normal Abelian subgroup of rank $ \geqq 3$, every subgroup can be generated by four elements. This result is then used to determine which $2$-groups $T$ with no normal Abelian subgroup of rank $ \geqq 3$ can occur as $ {S_2}$'s of finite simple groups $G$, under certain assumptions on the embedding of $T$ in $G$.
Union extensions of semigroups
L. A. M.
Verbeek
409-423
The Hasse invariant of a vector bundle
Richard R.
Patterson
425-443
Abstract: The object of this work is to define, by analogy with algebra, the Witt group and the graded Brauer group of a topological space $X$. A homomorphism is defined between them analogous to the generalized Hasse invariant. Upon evaluation, the Witt group is seen to be $ \tilde KO(X)$, the graded Brauer group $1 + {H^1}(X;{Z_2}) + {H^2}(X;{Z_2})$ with truncated cup product multiplication, while the homomorphism is given by Stiefel-Whitney classes: $1 + {w_1} + {w_2}$.
Spectral mapping theorems and perturbation theorems for Browder's essential spectrum
Roger D.
Nussbaum
445-455
Abstract: If $T$ is a closed, densely defined linear operator in a Banach space, F. E. Browder has defined the essential spectrum of $T,\operatorname{ess} (T)$ [1]. We derive below spectral mapping theorems and perturbation theorems for Browder's essential spectrum. If $T$ is a bounded linear operator and $f$ is a function analytic on a neighborhood of the spectrum of $T$, we prove that $ f(\operatorname{ess} (T)) = \operatorname{ess} (f(T))$. If $T$ is a closed, densely defined linear operator with nonempty resolvent set and $ f$ is a polynomial, the same theorem holds. For a closed, densely defined linear operator $T$ and a bounded linear operator $B$ which commutes with $T$, we prove that $\operatorname{ess} (T + B) \subseteq \operatorname{ess} (T) + \operatorname{ess} (B) = \{ \mu + v:\mu \in \operatorname{ess} (T),v \in \operatorname{ess} (B)\}$. By making additional assumptions, we obtain an analogous theorem for $B$ unbounded.
Singly generated homogeneous $F$-algebras
Ronn
Carpenter
457-468
Abstract: With each point $ m$ in the spectrum of a singly generated $F$-algebra we associate an algebra ${A_m}$ of germs of functions. It is shown that if ${A_m}$ is isomorphic to the algebra of germs of analytic functions of a single complex variable, then the spectrum of $A$ contains an analytic disc about $m$. The algebra $A$ is called homogeneous if the algebras $ {A_m}$ are all isomorphic. If $A$ is homogeneous and none of the algebras $ {A_m}$ have zero divisors, we show that $A$ is the direct sum of its radical and either an algebra of analytic functions or countably many copies of the complex numbers. If $A$ is a uniform algebra which is homogeneous, then it is shown that $A$ is either the algebra of analytic functions on an open subset of the complex numbers or the algebra of all continuous functions on its spectrum.
An exponential limit formula for nonlinear semigroups
Joel L.
Mermin
469-476
Abstract: In recent papers, many writers have developed the theory of semigroups of operators generated by nonlinear accretive operators. In the present paper, we construct this semigroup by means of an exponential limit formula, and then use this means of obtaining the semigroup to prove an approximation theorem that is a direct generalization of the Kato-Trotter theorem for linear semigroups.
Topology and the duals of certain locally compact groups
I.
Schochetman
477-489
Abstract: We consider some topological questions concerning the dual space of a (separable) extension $G$ of a type I, regularly embedded subgroup $ N$. The dual $ \hat G$ is known to have a fibre-like structure. The fibres are in bijective correspondence with certain subsets of dual spaces of associated stability subgroups. These subsets in turn are in bijective correspondence with certain projective dual spaces. Under varying hypotheses, we give sufficient conditions for these bijections to be homeomorphisms, we determine the support of the induced representation $U^L$ (for $ L \in \hat N$) and we give necessary and sufficient conditions for a union of fibres in $\hat G$ to be closed. In a much more general context we study the Hausdorff and CCR separation properties of the dual of an extension. We then completely describe the dual space topology of the above extension $G$ in an interesting case. The preceding results are then applied to the case where $N$ is abelian and $G/N$ is compact.
A representation of the solutions of the Darboux equation in odd-dimensional spaces
H.
Rhee
491-498
Abstract: It is shown that determining a function from its averages over all spheres passing through the origin leads to an explicit representation of the even solutions of the Darboux equation in the exterior of the characteristic cones in terms of the hyperboloidal means of the boundary data on the cones.
A diophantine problem on groups. I
R. C.
Baker
499-506
Abstract: The following theorem of H. Weyl is generalised to the context of locally compact abelian groups. Theorem. Let ${\lambda _1} < {\lambda _2} < {\lambda _3} \cdots$ be a sequence such that, for some $c > 0,\varepsilon > 0,{\lambda _{n + k}} - {\lambda _n} \geqq c$ whenever $k \geqq n/{(\log n)^{1 + \varepsilon }}(n = 1,2, \ldots )$. Then for almost all real $u$ the sequence ${\lambda _1}u,{\lambda _2}u, \ldots ,{\lambda _n}u\pmod 1$ is uniformly distributed.
$G\sb{2n}$ spaces
Donald O.
Koehler
507-518
Abstract: A complex normed linear space $X$ will be called a ${G_{2n}}$ space if and only if there is a mapping $\left\langle { \cdot , \ldots , \cdot } \right\rangle$ from ${X^{2n}}$ into the complex numbers such that: ${x_k} \to \left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle$ is linear for $k = 1, \ldots ,n;\left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle = {\left\langle {{x_{2n}}, \ldots ,{x_1}} \right\rangle ^ - }$; and ${\left\langle {x, \ldots ,x} \right\rangle ^{1/2n}} = \vert\vert x\vert\vert$. The basic models are the $ {L^{2n}}$ spaces, but one also has that every inner product space is a $ {G_{2n}}$ space for every integer $n$. Hence ${G_{2n}}$ spaces of a given cardinality need not be isometrically isomorphic. It is shown that a complex normed linear space is a ${G_{2n}}$ space if and only if the norm satisfies a generalized parallelogram law. From the proof of this characterization it follows that a linear map $ U$ from $X$ to $X$ is an isometry if and only if $\left\langle {U({x_1}), \ldots ,U({x_{2n}})} \right\rangle = \left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle$ for all ${x_1}, \ldots ,{x_{2n}}$. This then provides a way to construct all of the isometries of a finite dimensional ${G_{2n}}$ space. Of particular interest are the $\operatorname{CBS} {G_{2n}}$ spaces in which $\vert\left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle \vert \leqq \vert\vert{x_1}\vert\vert \cdots \vert\vert{x_{2n}}\vert\vert$. These spaces have many properties similar to inner product spaces. An operator $A$ on a complete $\operatorname{CBS} {G_{2n}}$ space is said to be symmetric if and only if $\left\langle {{x_1}, \ldots ,A({x_i}), \ldots ,{x_{2n}}} \right\rangle = \left\langle {{x_1}, \ldots ,A({x_j}), \ldots ,{x_{2n}}} \right\rangle$ for all $ i$ and $j$. It is easy to show that these operators are scalar and that on ${L^{2n}},n > 1$, they characterize multiplication by a real $ {L^\infty }$ function. The interest in nontrivial symmetric operators is that they exist if and only if the space can be decomposed into the direct sum of nontrivial ${G_{2n}}$ spaces.
On the radius of convexity and boundary distortion of Schlicht functions
David E.
Tepper
519-528
Abstract: Let $w = f(z) = z + \sum\nolimits_{n = 2}^\infty {{a_n}{z^n}}$ be regular and univalent for $ \vert z\vert < 1$ and map $\vert z\vert < 1$ onto a region which is starlike with respect to $w = 0$. If ${r_0}$ denotes the radius of convexity of $w = f(z)$, $d_0 = \min \vert f(z)\vert$ for $\vert z\vert = {r_0}$, and ${d^ \ast } = \inf \vert\beta \vert$ for $f(z) \ne \beta$, then it has been conjectured that $ {d_0}/{d^ \ast } \geqq 2/3$. It is shown here that ${d_0}/{d^ \ast } \geqq 0.343 \ldots$, which improves the old estimate ${d_0}/{d^ \ast } \geqq 0.268 \ldots$. In addition, sharp estimates for ${r_0}$ are given which depend on the value of $ \vert{a_2}\vert$.
Bounded and compact vectorial Hankel operators
Lavon B.
Page
529-539
Abstract: Operators $ H$ satisfying ${S^ \ast }H = HS$ where $S$ is a unilateral shift on Hilbert space are called Hankel operators. For a fixed shift $S$ of arbitrary multiplicity the Banach spaces of bounded Hankel operators and of compact Hankel operators are described, and it is shown that the former is always the second dual of the latter. Representations for bounded and for compact Hankel operators are given in a standard function space model.
Existence and stability of a class of nonlinear Volterra integral equations
Stanley I.
Grossman
541-556
Abstract: In this paper we study the problem of existence and uniqueness to solutions of the nonlinear Volterra integral equation $x = f + {a_1}{g_1}(x) + \cdots + {a_n}{g_n}(x)$, where the ${a_i}$ are continuous linear operators mapping a Fréchet space $ \mathcal{F}$ into itself and the ${g_i}$ are nonlinear operators in that space. Solutions are sought which lie in a Banach subspace of $\mathcal{F}$ having a stronger topology. The equations are studied first when the ${g_i}$ are of the form ${g_i}(x) = x + {h_i}(x)$ where ${h_i}(x)$ is ``small", and then when the $ {g_i}$ are slope restricted. This generalizes certain results in recent papers by Miller, Nohel, Wong, Sandberg, and Beneš.
Automorphism groups on compact Riemann surfaces
W. T.
Kiley
557-563
Abstract: For $g \geqq 2$, let $N(g)$ be the order of the largest automorphism group on a Riemann surface of genus $ g$. In this paper, lower bounds for $N(g)$ for various sequences of $g$'s are obtained. Sequences of appropriate groups are constructed. Each of these groups is then realized as a group of cover transformations of a surface covering the Riemann sphere. The genus of the resulting surface is then found by using the Riemann-Hurwitz formula and the automorphism group of the surface contains the given group. Each lower bound which is found is also shown to be sharp. That is, there are infinitely many $g$'s in the sequence to which the bound applies for which $N(g)$ does not exceed the bound.
Packing and reflexivity in Banach spaces
Clifford A.
Kottman
565-576
Abstract: A measure of the ``massiveness'' of the unit ball of a Banach space is introduced in terms of an efficiency of the tightest packing of balls of equal size in the unit ball. This measure is computed for the ${l_p}$-spaces, and spaces with distinct measures are shown to be not nearly isometric. A new convexity condition, which is compared to $B$-convexity, uniform smoothness, and uniform convexity, is introduced in terms of this measure, and is shown to be a criterion of reflexivity. The property dual to this convexity condition is also exposed and examined.
Finite dimensional inseparable algebras
Shuen
Yuan
577-587
Abstract: We determine the structure of finite dimensional algebras which are differentiably simple with respect to a set of higher derivations.
An alternative proof that Bing's dogbone space is not topologically $E\sp{3}$
E. H.
Anderson
589-609
Harmonic analysis on nilmanifolds
Jonathan
Brezin
611-618
Abstract: We compute, using a device of A. Weil, an explicit decomposition of $ {L^2}$ of a nilmanifold into irreducible translation-invariant subspaces. The results refine previous work of C. C. Moore and L. Green.
Estimates for the number of real-valued continuous functions
W. W.
Comfort;
Anthony W.
Hager
619-631
Abstract: It is a familiar fact that $ \vert C(X)\vert \leqq {2^{\delta X}}$, where $\vert C(X)\vert$ is the cardinal number of the set of real-valued continuous functions on the infinite topological space $X$, and $\delta X$ is the least cardinal of a dense subset of $X$. While for metrizable spaces equality obtains, for some familiar spaces--e.g., the one-point compactification of the discrete space of cardinal $2\aleph 0$--the inequality can be strict, and the problem of more delicate estimates arises. It is hard to conceive of a general upper bound for $ \vert C(X)\vert$ which does not involve a cardinal property of $X$ as an exponent, and therefore we consider exponential combinations of certain natural cardinal numbers associated with $ X$. Among the numbers are $ wX$, the least cardinal of an open basis, and $wcX$, the least $ \mathfrak{m}$ for which each open cover of $X$ has a subfamily with $ \mathfrak{m}$ or fewer elements whose union is dense. We show that $\vert C(X)\vert \leqq {(wX)^{wcX}}$, and that this estimate is best possible among the numbers in question. (In particular, ${(wX)^{wcX}} \leqq {2^{\delta X}}$ always holds.) In fact, it is only with the use of a version of the generalized continuum hypothesis that we succeed in finding an $X$ for which $ \vert C(X)\vert < {(wX)^{wcX}}$.
The strict topology for double centralizer algebras
Donald Curtis
Taylor
633-643
Abstract: Sufficient conditions are given for a double centralizer algebra under the strict topology to be a Mackey space.
Compact riemannian manifolds with essential groups of conformorphisms.
A. J.
Ledger;
Morio
Obata
645-651
Abstract: A solution to the following conjecture: A compact connected riemannian $ n$-manifold $(n > 2)$ with an essential group of conformorphisms is conformorphic to a euclidean $ n$-sphere.