The approximation of linear operators
J. W.
Brace;
P. J.
Richetta
1-21
Abstract: Let $L(E,F)$ be the vector space of all linear maps of $E$ into $F$. Consider a subspace $G$ of $L(E,F)$ such as all continuous maps. In $ G$ distinguish a subspace $ H$ of maps which are to be approximated by members of a smaller subspace $ N$ of $G$. Thus we always have $N \subset H \subset G \subset L(E,F)$. Then the approximation problem which we consider is to find a locally convex linear Hausdorff topology on $ G$ such that $H \subset \bar N,H = \bar N$ or the completion of $N$ is $H$. In the case where $E$ and $F$ are Banach spaces, we have approximation topologies for (i) all linear operators, (ii) all the continuous linear operators, (iii) all weakly compact operators, (iv) all completely continuous operators, (v) all compact operators, and (vi) certain subclasses of the strictly singular operators. Our method is that of considering members of $L(E,F)$ as linear forms on
Entire functions and M\"untz-Sz\'asz type approximation
W. A. J.
Luxemburg;
J.
Korevaar
23-37
Abstract: Let $[a,b]$ be a bounded interval with $a \geqq 0$. Under what conditions on the sequence of exponents $\{ {\lambda _n}\}$ can every function in ${L^p}[a,b]$ or $C[a,b]$ be approximated arbitrarily closely by linear combinations of powers ${x^\lambda }n$? What is the distance between ${x^\lambda }$ and the closed span ${S_c}({x^\lambda }n)$? What is this closed span if not the whole space? Starting with the case of $ {L^2}$, C. H. Müntz and O. Szász considered the first two questions for the interval $[0, 1]$. L. Schwartz, J. A. Clarkson and P. Erdös, and the second author answered the third question for $[0, 1]$ and also considered the interval $ [a,b]$. For the case of $ [0, 1]$, L. Schwartz (and, earlier, in a limited way, T. Carleman) successfully used methods of complex and functional analysis, but until now the case of $[a,b]$ had proved resistant to a direct approach of that kind. In the present paper complex analysis is used to obtain a simple direct treatment for the case of $[a,b]$. The crucial step is the construction of entire functions of exponential type which vanish at prescribed points not too close to the real axis and which, in a sense, are as small on both halves of the real axis as such functions can be. Under suitable conditions on the sequence of complex numbers $\{ {\lambda _n}\}$, the construction leads readily to asymptotic lower bounds for the distances ${d_k} = d\{ {x^{{\lambda _k}}},{S_c}({x^{{\lambda _n}}},n \ne k)\}$. These bounds are used to determine ${S_c}({x^{{\lambda _n}}})$ and to generalize a result for a boundary value problem for the heat equation obtained recently by V. J. Mizel and T. I. Seidman.
A variational method for functions of bounded boundary rotation
H. B.
Coonce
39-51
Abstract: Let $f$ be a function analytic in the unit disc, properly normalized, with bounded boundary rotation. There exists a Stieltjes integral representation for
Incompressible surfaces in knot spaces
Herbert C.
Lyon
53-62
Abstract: The following theorems are proved. Theorem 1. There exist infinitely many distinct, prime, Neuwirth knots, each of which has the property that its complement contains closed, incompressible surfaces of arbitrarily high genus. Theorem 2. There exists a genus one knot which has incompressible spanning surfaces of arbitrarily high genus.
On the complex bordism of Eilenbeg-Mac Lane spaces and connective coverings of ${\rm BU}$
Peter S.
Landweber
63-71
Abstract: Explicit computations show that the universal coefficient spectral sequence from complex bordism to integral homology collapses for the spectra $K(Z)$ and bu, and also for their $ \bmod p$ reductions. Moreover the complex bordism modules of these spectra have infinite projective dimension.
On the solvability of unit groups of group algebras
J. M.
Bateman
73-86
Abstract: Let FG be the group algebra of a finite group $G$ over a field $F$ of characteristic $p \geqq 0$; and let $ U$ be the group of units of FG. We prove that $U$ is solvable if and only if (i) every absolutely irreducible representation of $G$ at characteristic $p$ is of degree one or two and (ii) if any such representation is of degree two, then it is definable in $ F$ and $F = GF(2)$ or $GF(3)$. This result is translated into intrinsic group-theoretic and field-theoretic conditions on $G$ and $F$, respectively. Namely, if ${O_p}(G)$ is the maximum normal $ p$-subgroup of $ G$ and $L = G/{O_p}(G)$, then (i) $L$ is abelian, or (ii) $ F = GF(3)$ and $ L$ is a $2$-group with exactly
On the stability of the cohomology of complex structures
Tapio
Klemola
87-97
Abstract: Let $ \mathcal{V}\mathop \pi \limits_ \to M$ be a differentiable family of compact complex manifolds ${V_t} = {\pi ^{ - 1}}(t)$ on $M = \{ t \in {R^m}\vert\;\vert t\vert < 1\} ,\;\mathcal{B} \to \mathcal{V}$ a differentiable family of holomorphic vector bundles ${B_t} \to {V_t},t \in M$. In this paper we study conditions for the cohomology groups $H_{\bar \partial t}^{r,s}({B_t})$ to be constant in a neighborhood of $0 \in M$.
Deleted products of spaces which are unions of two simplexes
W. T.
Whitley
99-111
Abstract: If $X$ is a space, the deleted product space, ${X^ \ast }$, is $X \times X - D$, where $ D$ is the diagonal. If $ Y$ is a space and $ f$ is a continuous map from $X$ to $Y$, then $X_f^ \ast$ is the inverse image of ${Y^ \ast }$ under the map $f \times f$ taking $ X \times X$ into $Y \times Y$. In this paper, we investigate the following questions: ``What maps $f$ are such that $X_f^ \ast$ is homotopically equivalent to ${X^ \ast }$", and ``What maps $ f$ are such that $ X_f^ \ast$ is homotopically equivalent to $ f{(X)^ \ast }$?'' If $ X$ is the union of two nondisjoint simplexes and $f$ is a simplicial map from $X \times X$ such that $f\vert f(X)$ is one-to-one, we obtain necessary and sufficient conditions for $ X_f^ \ast$ and $f{(X)^ \ast }$ to be homotopically equivalent. If $ X$ is the union of nondisjoint simplexes $A$ and $B$ with $\dim B = 1 + \dim (A \cap B)$, we obtain necessary and sufficient conditions for $ {X^ \ast }$ and $ X_f^ \ast$ to be homotopically equivalent if $f$ is in the class of maps mentioned.
A two-stage Postnikov system where $E\sb{2}\not=E\sb{\infty }$ in the Eilenberg-Moore spectral sequence
Claude
Schochet
113-118
Abstract: Let $\Omega B \to PB \to B$ be the path fibration over the simply-connected space $B$, let $ \Omega B \to E \to X$ be the induced fibration via the map $f:X \to B$, and let $X$ and $B$ be generalized Eilenberg-Mac Lane spaces. G. Hirsch has conjectured that ${H^ \ast }E$ is additively isomorphic to ${\text{Tor}}_{H^ \ast B}({Z_2},{H^ \ast }X)$, where cohomology is with ${Z_2}$ coefficients. Since the Eilenberg-Moore spectral sequence which converges to ${H^ \ast }E$ has ${E_2} = {\text{Tor}_{H^ \ast B}}({Z_2},{H^ \ast }X)$, the conjecture is equivalent to saying $ {E_2} = {E_\infty }$. In the present paper we set $X = K({Z_2} + {Z_2},2),B = K({Z_2},4)$ and ${f^ \ast }i =$ the product of the two fundamental classes, and we prove that ${E_2} \ne {E_3}$, disproving Hirsch's conjecture. The proof involves the use of homology isomorphisms ${C^ \ast }X\mathop \to \limits^g \bar C({H^ \ast }\Omega X)\mathop \to \limits^h {H^ \ast }X$ developed by J. P. May, where $\bar C$ is the reduced cobar construction. The map $ g$ commutes with cup-$ 1$ products. Since the cup-$ 1$ product in $\bar C({H^ \ast }\Omega X)$ is well known, and since differentials in the spectral sequence correspond to certain cup-$1$ products, we may compute ${d_2}$ on specific elements of $ {E_2}$.
A theory of focal points and focal intervals for an elliptic quadratic form on a Hilbert space
John
Gregory
119-128
Abstract: The theory of focal points and conjugate points is an important part of the study of problems in the calculus of variations and control theory. Hestenes has shown that for many problems this theory may be studied by Hilbert space methods. In a previous paper the author has extended the theory of Hestenes to elliptic quadratic forms $J(x;\sigma )$ defined on $\mathfrak{A}(\sigma )$ where $\sigma$ is a member of the metric space $(\Sigma ,\rho )$ and $\mathfrak{A}(\sigma )$ denotes a closed subspace of $\mathfrak{A}$. A fundamental part of this extension is concerned with inequalities dealing with the signature $ s(\sigma )$ and nullity $n(\sigma )$ of $ J(x;\sigma )$ on $\mathfrak{A}(\sigma )$ where $\sigma$ is in a $\rho$ neighborhood of a fixed point ${\sigma _0}$ in $\Sigma$. The purpose of this paper is threefold. The first purpose is to show that the extended theory includes the focal point hypotheses of Hestenes. The second purpose is to obtain a focal point theory much like that of Hestenes. It is interesting to note that our theory is based only on the nonnegative integers $s(\lambda )$ and $ n(\lambda )$. This will facilitate later work on numerical calculations of focal points. Our final purpose is to obtain an abstract focal interval theory in which the usual focal points are (degenerate) focal intervals. While previous authors have considered specific problems, no general results for the focal interval case seem to be contained in the literature. An expression for the number of focal intervals on a subinterval
A class of complete orthogonal sequences of step functions
J. L.
Sox;
W. J.
Harrington
129-135
Abstract: A class of orthogonal sets of step functions is defined and each member is shown to be complete in $ {L_2}(0,1)$. Pointwise convergence theorems are obtained for the Fourier expansions relative to these sets. The classical Haar orthogonal set is shown to be a set of this class and the class itself is seen to be a subclass of the ``generalized Haar systems'' defined recently by Price.
A maximal function characterization of the class $H\sp{p}$
D. L.
Burkholder;
R. F.
Gundy;
M. L.
Silverstein
137-153
Abstract: Let $u$ be harmonic in the upper half-plane and $0 < p < \infty$. Then $u =$ Re$F$ for some analytic function $F$ of the Hardy class ${H^p}$ if and only if the nontangential maximal function of $u$ is in ${L^p}$. A general integral inequality between the nontangential maximal function of $u$ and that of its conjugate function is established.
$p$-solvable linear groups of finite order
David L.
Winter
155-160
Abstract: The purpose of this paper is to prove the following result. Theorem. Let $p$ be an odd prime and let $G$ be a finite $p$-solvable group. Assume that $G$ has a faithful representation of degree $n$ over a field of characteristic zero or over a perfect field of characteristic $ p$. Let $P$ be a Sylow $p$-subgroup of $G$ and let ${O_p}(G)$ be the maximal normal $p$-subgroup of $G$. Then $\vert P:{O_p}(G)\vert \leqq {p^{{\lambda _p}(n)}}$ where \begin{displaymath}\begin{array}{*{20}{c}} {{\lambda _p}(n) = \sum\limits_{i = 0... ...ight]} \quad if\;p\;is\;not\;a\;Fermat\;prime.} \end{array} \end{displaymath}
Locally flat imbeddings of topological manifolds in codimension three.
Glenn P.
Weller
161-178
Abstract: This paper presents an imbedding theorem for one topological manifold $ {M^n}$ in another topological manifold ${Q^q}$, provided that the codimension $(q - n)$ is at least three. The result holds even if the manifolds are of the recently discovered non-piecewise-linear type. Denote the boundaries of $ M$ and $Q$ by $\dot M$ and $\dot Q$ respectively. Suppose that $M$ is $2n - q$ connected and $Q$ is $2n - q + 1$ connected. It is then proved that any map $f:(M,\dot M) \to (Q,\dot Q)$ such that $ f\vert\dot M$ is a locally flat imbedding is homotopic relative to $\dot M$ to a proper locally flat imbedding $g:M \to Q$. It is also shown that if $M$ is closed and $2n - q + 1$ connected and $Q$ is $ 2n - q + 2$ connected, then any two homotopic locally flat imbeddings are locally flatly concordant.
Real vector bundles and spaces with free involutions
Allan L.
Edelson
179-188
Abstract: The functor $ KR(X)$, defined in [4], is a contravariant functor defined in the category of spaces with involutions. It is shown herein that this functor is classified by equivariant maps into the complex Grassmann manifold, which is given the involution induced by complex conjugation. For the case of free involutions it is shown that the classifying maps can be taken to lie outside the fixed point set of the Grassmann manifold. This fixed point set can be identified with the real Grassmann manifold. It is then shown that, for free involutions, $KR(X)$ is an invariant of the homotopy type of the orbit space $X$ modulo its involution. The multiplicative group of real line bundles, real in the sense of [4], is shown to be classified by equivariant maps into a quadric surface $Q$ in complex projective space. $Q$ carries a free involution and this classification is again valid for spaces with free involutions.
A correlation between ${\rm PSU}\sb{4}\,(3)$, the Suzuki group, and the Conway group
J. H.
Lindsey
189-204
Abstract: We shall use a six dimensional projective representation of $PS{U_4}(3)$ of order ${2^7}{3^6}5 \cdot 7$ to construct 12 and $ 24$-dimensional complex projective representations of the Suzuki and Conway groups, respectively, acting on the Leech lattice. The construction makes it easy to show that the Suzuki and Conway simple groups have outer automorphism groups of order two and one, respectively. Also, the simple Suzuki group contains $3 \cdot PS{U_4}(3) \cdot 2,{3^5} \cdot {M_{11}}$, and a group which is probably $ PS{U_5}(2)$, where $A \cdot B$ denotes an extension of the group $A$ by the group $B$.
Coreflective subcategories
Horst
Herrlich;
George E.
Strecker
205-226
Abstract: General morphism factorization criteria are used to investigate categorical reflections and coreflections, and in particular epi-reflections and monocoreflections. It is shown that for most categories with ``reasonable'' smallness and completeness conditions, each coreflection can be ``split'' into the composition of two mono-coreflections and that under these conditions mono-coreflective subcategories can be characterized as those which are closed under the formation of coproducts and extremal quotient objects. The relationship of reflectivity to closure under limits is investigated as well as coreflections in categories which have ``enough'' constant morphisms.
Multicoherence techniques applied to inverse limits
Sam B.
Nadler
227-234
Abstract: Sufficient conditions are given so that the multicoherence degree of continua is not raised when taking inverse limits. These results are then applied to inverse limits of special types of spaces.
Stochastic equations with discontinuous drift
Edward D.
Conway
235-245
Abstract: We study stochastic differential equations, $dx = adt + \sigma d\beta $ where $\beta$ denotes a Brownian motion. By relaxing the definition of solutions we are able to prove existence theorems assuming only that $ a$ is measurable, $ \sigma$ is continuous and that both grow linearly at infinity. Nondegeneracy is not assumed. The relaxed definition of solution is an extension of A. F. Filippov's definition in the deterministic case. When $\sigma$ is constant we prove one-sided uniqueness and approximation theorems under the assumption that $ a$ satisfies a one-sided Lipschitz condition.
A pairing of a class of evolution systems with a class of generators.
J. V.
Herod
247-260
Abstract: Suppose that $ S$ is a Banach space and that $A$ and $M$ are functions such that if $x$ and $y$ are numbers, $x \geqq y$, and $P$ is in $S$ then each of $M(x,y)P$ and $A(y,P)$ is in $S$. This paper studies the relation $\displaystyle M(x,y)P = P + \int_x^y {A(t,M(t,y)P)dt.}$ Classes OM and OA will be described and a correspondence will be established which pairs members of the two classes which are connected as $ M$ and $A$ are by the relation indicated above.
Euclidean $(q+r)$-space modulo an $r$-plane of collapsible $p$-complexes
Leslie C.
Glaser
261-278
Abstract: The following general decomposition result is obtained: Suppose ${K^p}(p \geqq 1)$ is a finite collapsible $ p$-complex topologically embedded as a subset of a separable metric space $ {X^q}$ where, for some $ r \geqq 1,{X^q} \times {E^r}$ is homeomorphic to Euclidean $(q + r)$-space ${E^{q + r}}$. Then the Cartesian product of the quotient space $ {X^q}/{K^p}$ with $ {E^r}$ is topologically ${E^{q + r}}$ provided that $q \geqq 3$ and, for each simplex $ {\Delta ^k} \in {K^p},({X^q} \times {E^r},{\Delta ^k} \times ({[0,1]^{r - 1}} \times 0))$ is homeomorphic, as pairs, to $\displaystyle ({E^{q + r}},{[0,1]^{k + r - 1}} \times (0, \ldots ,0)).$ It is known that this condition is satisfied if $q - p \geqq 2$ and $q + r \geqq 5$. This result implies that if $ {K^k}$ is a finite collapsible $k$-complex topologically embedded as a subset of Euclidean $n$-space ${E^n}$, then the Cartesian product of the quotient space $ {E^n}/{K^k}$ with $ {E^1}$ is topologically ${E^{n + 1}}$ provided either (i) $n \leqq 3$, (ii) $n - k \geqq 2$, or (iii) each simplex of $ {K^k}$ is flat in ${E^{n + 1}}$.
Asymptotic behavior of solutions of hyperbolic inequalities
Amy C.
Murray
279-296
Abstract: This paper discusses the asymptotic behavior of ${C^2}$ solutions $u = u(t,{x_1}, \ldots ,{x_v})$ of the inequality (1) $\vert Lu\vert \leqq {k_1}(t,x)\vert u\vert + {k_2}(t,x)\vert\vert{\nabla _u}\vert\vert$, in domains in $(t,x)$-space which grow unbounded in $x$ as $t \to \infty$. The operator $ L$ is a second order hyperbolic operator with variable coefficients. The main results establish the maximum rate of decay of nonzero solutions of (1). This rate depends on the asymptotic behavior of $ {k_1},{k_2}$, and the time derivatives of the coefficients of $ L$.
Mappings onto the plane
Dix H.
Pettey
297-309
Abstract: In this paper, we show that if $X$ is a connected, locally connected, locally compact topological space and $f$ is a 1-1 mapping of $X$ onto ${E^2}$, then $f$ is a homeomorphism. Using this result, we obtain theorems concerning the compactness of certain mappings onto ${E^2}$.
$G$-structures on spheres
Peter
Leonard
311-327
Abstract: ${G_n}$ denotes one of the classical groups $SO(n),SU(n)$ or $Sp(n)$ and $H$ a closed connected subgroup of ${G_n}$. We ask whether the principal bundle ${G_n} \to {G_{n + 1}} \to {G_{n + 1}}/{G_n}$ admits a reduction of structure group to $H$. If $n$ is even and ${G_n}$ is $SO(n)$ or $SU(n)$ or if $n \not\equiv 11\bmod 12$ and ${G_n}$ is $Sp(n)$, we prove that there are no such reductions unless $ n = 6,{G_6} = SO(6)$ and $ H = SU(3)$ or $ U(3)$. In the remaining cases we consider the problem for $H$ maximal. We divide the maximal subgroups into three main classes: reducible, nonsimple irreducible and simple irreducible. We find a necessary and sufficient condition for reduction to a reducible maximal subgroup and prove that there are no reductions to the nonsimple irreducible maximal subgroups. The remaining case is unanswered.
Integral representations for continuous linear operators in the setting of convex topological vector spaces
J. R.
Edwards;
S. G.
Wayment
329-345
Abstract: Suppose $ X$ and $Y$ are locally convex Hausdorff spaces, $ H$ is arbitrary and $ \Sigma$ is a ring of subsets of $H$. The authors prove the analog of the theorem stated in [Abstract 672-372, Notices Amer. Math. Soc. 17 (1970), 188] in this setting. A theory of extended integration on function spaces with Lebesgue and non-Lebesgue type convex topologies is then developed. As applications, integral representations for continuous transformations into $Y$ for the following function spaces $ F$ (which have domain $ H$ and range $ X$) are obtained: (1) $ H$ and $\Sigma$ are arbitrary, $\tau$ is a convex topology on the simple functions over $\Sigma ,K$ is a set function on $\Sigma$ with values in $L[X,Y]$, and $F$ is the Lebesgue-type space generated by $ K$; (2) $H$ is a normal space and $ F$ is the space of continuous functions each of whose range is totally bounded, with the topology of uniform convergence; (3) $ H$ is a locally compact Hausdorff space, $F$ is the space of continuous functions of compact support with the topology of uniform convergence; (4) $H$ is a locally compact Hausdorff space and $ F$ is the space of continuous functions with the topology of uniform convergence on compact subsets. In the above $X$ and $Y$ may be replaced by topological Hausdorff spaces under certain additional compensating requirements.
The volume of tubes in complex projective space
Robert A.
Wolf
347-371
Abstract: A formula for the volume of a tube about a compact complex submanifold of complex projective space is derived.
The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in $R\sb{3}$
H. S. G.
Swann
373-397
Abstract: It is shown here that a unique solution to the Navier-Stokes equations exists in ${R_3}$ for a small time interval independent of the viscosity and that the solutions for varying viscosities converge uniformly to a function that is a solution to the equations for ideal flow in ${R_3}$. The existence of the solutions is shown by transforming the Navier-Stokes equations to an equivalent system solvable by applying fixed point methods with estimates derived from using semigroup theory.
Meta-analytic functions
M. S.
Krishna Sastry
399-415
On embedding polyhedra and manifolds
Krešo
Horvatić
417-436
Abstract: It is well known that every $n$-polyhedron PL embeds in a Euclidean $ (2n + 1)$-space, and that for PL manifolds the result can be improved upon by one dimension. In the paper are given some sufficient conditions under which the dimension of the ambient space can be decreased. The main theorem asserts that, for there to exist an embedding of the $ n$-polyhedron $ X$ into $2n$-space, it suffices that the integral cohomology group ${H^n}(X - \operatorname{Int} A) = 0$ for some $n$-simplex $A$ of a triangulation of $X$. A number of interesting corollaries follow from this theorem. Along the line of manifolds the known embedding results for PL manifolds are extended over a larger class containing various kinds of generalized manifolds, such as triangulated manifolds, polyhedral homology manifolds, pseudomanifolds and manifolds with singular boundary. Finally, a notion of strong embeddability is introduced which allows us to prove that some class of $n$-manifolds can be embedded into a $ (2n - 1)$-dimensional ambient space.
The embeddability of a semigroup---Conditions common to Mal'cev and Lambek
George C.
Bush
437-448
Abstract: Two systems of conditions--due to Mal'cev and to Lambek--are known to be necessary and sufficient for a semigroup to be embeddable in a group. This paper shows by means of an example that the conditions common to the two systems are not sufficient to guarantee embeddability.
On mean-periodicity. II
Edwin J.
Akutowicz
449-457
Abstract: This paper is devoted to the problem of representing all solutions of certain homogeneous convolution equations through series of exponential polynomials. This representation is sought in the dual space $\mathcal{M}$, the latter consisting of entire functions satisfying growth conditions in horizontal directions. The space $ \mathcal{M}$ is a Fréchet space, which fact permits a simpler and more thorough treatment than that given in the paper [1]. The technique used here is based upon a method developed by L. Ehrenpreis [5] and V. P. Palamodov [3] in the theory of differential equations with constant coefficients. We map the Fourier transform space $\mathcal{F}\mathcal{M}$ into a space of sequences, $\mathcal{M}'$. The crucial point is to identify the quotient space $\mathcal{F}\mathcal{M}/\ker \rho$.
The logarithmic limit-set of an algebraic variety
George M.
Bergman
459-469
Abstract: Let $C$ be the field of complex numbers and $ V$ a subvariety of ${(C - \{ 0\} )^n}$. To study the ``exponential behavior of $V$ at infinity", we define $V_\infty ^{(a)}$ as the set of limitpoints on the unit sphere ${S^{n - 1}}$ of the set of real $ n$-tuples $ ({u_x}\log \vert{x_1}\vert, \ldots ,{u_x}\log \vert{x_n}\vert)$, where $x \in V$ and $ {u_x} = {(1 + \Sigma {(\log \vert{x_i}\vert)^2})^{ - 1/2}}$. More algebraically, in the case of arbitrary base-field $k$ we can look at places ``at infinity'' on $V$ and use the values of the associated valuations on ${X_1}, \ldots ,{X_n}$ to construct an analogous set $ V_\infty ^{(b)}$. Thirdly, simply by studying the terms occurring in elements of the ideal $I$ defining $V$, we define another closely related set, $V_\infty ^{(c)}$. These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of $GL(n,Z)$ on $k[X_1^{ \pm 1}, \ldots ,X_n^{ \pm 1}]$, then studied further. It is shown among other things that $V_\infty ^{(b)} = V_\infty ^{(c)} \supseteq$ (when defined) $V_\infty ^{(a)}$. If a certain natural conjecture is true, then equality holds where we wrote ``$\supseteq$", and the common set ${V_\infty } \subseteq {S^{n - 1}}$ is a finite union of convex spherical polytopes.
Weak topologies on subspaces of $C(S)$
Joel H.
Shapiro
471-479
Abstract: Let $S$ be a locally compact Hausdorff space, $E$ a linear subspace of $C(S)$. It is shown that the unit ball of $ E$ is compact in the strict topology if and only if both of the following two conditions are satisfied: (1) $E$ is the Banach space dual of $M(S)/{E^0}$ in the integration pairing, and (2) the bounded weak star topology on $E$ coincides with the strict topology. This result is applied to several examples, among which are ${l^\infty }$ and the space of bounded analytic functions on a plane region.
On the existence of strongly series summable Markuschevich bases in Banach spaces
William B.
Johnson
481-486
Abstract: The main result is: Let $X$ be a complex separable Banach space. If the identity operator on ${X^ \ast }$ is the limit in the strong operator topology of a uniformly bounded net of linear operators of finite rank, then $X$ admits a strongly series summable Markuschevich basis.
On the existence of trivial intersection subgroups
Mark P.
Hale
487-493
Abstract: Let $G$ be a transitive nonregular permutation group acting on a set $X$, and let $H$ be the subgroup of $G$ fixing some element of $X$. Suppose each nonidentity element of $ H$ fixes exactly $ b$ elements of $ X$. If $b = 1,G$ is a Frobenius group, and it is well known that $H$ has only trivial intersection with its conjugates. If $b > 1$, it is shown that this conclusion still holds, provided $H$ satisfies certain natural conditions. Applications to the study of Hall subgroups and certain simple groups related to Zassenhaus groups are given.
A rank theorem for coherent analytic sheaves
Günther
Trautmann
495-498
Abstract: Let $S$ be an analytic subvariety in $ {C^n}$ and $\mathcal{F}$ a coherent analytic sheaf on $ {C^n}$, such that $\mathcal{F}$ is locally free on ${C^n} - S$ and $ \Gamma (U,\mathcal{F}) = \Gamma (U - S,\mathcal{F})$ for every open set $U \subset {C^n}$. It is shown that $\mathcal{F}$ is locally free everywhere, if codh$ \mathcal{F} \geqq n - 1$ and $\dim S +$ rank$\mathcal{F} \leqq n - 2$.
Addenda to ``A variational problem related to an optimal filter problem with self-correlated noise''
Leonard D.
Berkovitz;
Harry
Pollard
499-504
Abstract: The explicit solution is given of a nonclassical variational problem that is related to an optimal filter problem.
Errata to ``General product measures''
E. O.
Elliott;
A. P.
Morse
505-506
Errata to ``Concerning arcwise connectedness and the existence of simple closed curves in plane continua''
Charles L.
Hagopian
507-509
Errata to ``Endomorphism rings generated by units''
Paul
Hill
511