The homotopy type of the space of diffeomorphisms. I
Dan
Burghelea;
Richard
Lashof
1-36
Abstract: A new proof is given of the unpublished results of Morlet on the relation between the homeomorphism group and the diffeomorphism group of a smooth manifold. In particular, the result $ {\operatorname{Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text{Top}_n}/{O_n})$ is obtained. The main technique is fibrewise smoothing.
The homotopy type of the space of diffeomorphisms. II
Dan
Burghelea;
Richard
Lashof
37-50
Abstract: The result (proved in Part I) that $ {\operatorname{Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text{PL}_n}/{O_n})$ is used to compute some new homotopy of $ {\operatorname{Diff}}({D^n},\partial {D^n})$. The relation between smooth and PL pseudo-isotopy is explored. Known and new results on the homotopy of $ {\text{PL}_n}$ are summarized.
Mayer-Vietoris sequences and Brauer groups of nonnormal domains
L. N.
Childs
51-67
Abstract: Let R be a Noetherian domain with finite integral closure $ \bar R$. We study the map from the Brauer group of $R,B(R)$, to $B(\bar R)$: first, by embedding $B(R)$ into the Čech etale cohomology group ${H^2}(R,U)$ and using a Mayer-Vietoris sequence for Čech cohomology of commutative rings; second, via Milnor's theorem from algebraic K-theory. We apply our results to show, i.e., that if R is a domain with quotient field K a global field, then the map from $B(R)$ to $B(K)$ is 1-1.
Location of the zeros of polynomials with a prescribed norm
Q. I.
Rahman;
G.
Schmeisser
69-78
Abstract: For monic polynomials ${f_n}(z)$ of degree n with prescribed $ {L^p}$ norm $(1 \leq p \leq \infty )$ on the unit circle or supremum norm on the unit interval we determine bounded regions in the complex plane containing at least $k(1 \leq k \leq n)$ zeros. We deduce our results from some new inequalities which are similar to an inequality of Vicente Gonçalves and relate the zeros of a polynomial to its norm.
The inertial aspects of Stein's condition $H-C\sp{\ast} HC\gg O$
Bryan E.
Cain
79-91
Abstract: To each bounded operator C on the complex Hilbert space $\mathcal{H}$ we associate the vector space ${\mathcal{K}_C}$ consisting of those $x \in \mathcal{H}$ for which ${C^n}x \to 0$ as $n \to \infty$. We let $\alpha (C)$ denote the dimension of the closure of $ {\mathcal{K}_C}$ and we set $\beta (C) = \dim (\mathcal{K}_C^ \bot )$. Our main theorem states that if H is Hermitian and if $H - {C^ \ast }HC$ is positive and invertible then $\alpha (C) \leq \pi (H),\beta (C) = \nu (H)$, and $ \beta (C) \geq \delta (H)$ where $(\pi (H),\nu (H),\delta (H))$ is the inertia of H. (That is, $\pi (H) = \dim \;({\text{Range}}\;E[(0,\infty )])$) where E is the spectral measure of H; $ \nu (H) = \pi ( - H)$; and $\delta (H) = \dim ({\operatorname{Ker}}\;H)$.) We also show (l) that in general no stronger conclusion is possible, (2) that, unlike previous inertia theorems, our theorem allows 1 to lie in $\sigma (C)$, the spectrum of C, and (3) that the main inertial results associated with the hypothesis that $ \operatorname{Re} (HA)$ is positive and invertible can be derived from our theorem. Our theorems (1) characterize C in the extreme cases that either $ \pi (H) = 0$ or $\nu (H) = 0$, and (2) prove that $ \alpha (C) = \pi (H),\beta (C) = \nu (H),\delta (H) = 0$ if either $1 \notin \sigma (C)$ or $\beta (C) < \infty $.
Any infinite-dimensional Fr\'echet space homeomorphic with its countable product is topologically a Hilbert space
Wesley E.
Terry
93-104
Abstract: In this paper we will prove that any infinite-dimensional Fréchet space homeomorphic with its own countable product is topologically a Hilbert space. This will be done in two parts. First we will prove the result for infinite-dimensional Banach spaces, and then we will show that the result for Fréchet spaces follows as a corollary.
On the completion of Hausdorff locally solid Riesz spaces
Charalambos D.
Aliprantis
105-125
Abstract: In this paper we consider Hausdorff locally solid Riesz spaces $ (L,\tau )$ and we denote by $(\hat L,\hat \tau )$ the Hausdorff topological completion of $(L,\tau )$. It is proved that $(\hat L,\hat \tau )$ is a Hausdorff locally solid Riesz space containing L as a Riesz subspace. We study the properties of $(L,\tau )$ which are inherited by $(\hat L,\hat \tau )$.
Maximal quotients of semiprime PI-algebras
Louis Halle
Rowen
127-135
Abstract: J. Fisher [3] initiated the study of maximal quotient rings of semiprime PI-rings by noting that the singular ideal of any semiprime Pi-ring R is 0; hence there is a von Neumann regular maximal quotient ring $Q(R)$ of R. In this paper we characterize $Q(R)$ in terms of essential ideals of C = cent R. This permits immediate reduction of many facets of $Q(R)$ to the commutative case, yielding some new results and some rapid proofs of known results. Direct product decompositions of $Q(R)$ are given, and $Q(R)$ turns out to have an involution when R has an involution.
Involutions preserving an ${\rm SU}$ structure
R. J.
Rowlett
137-147
Abstract: Bordism theories $ S{U_ \ast }({Z_2},all)$ for SU-manifolds with involution and $S{U_ \ast }({Z_2},free)$ for SU-manifolds with free involution are defined. The latter is studied by use of the SU-bordism spectral sequence of $B{Z_2}$, and the orders of the spheres ${S^{4n + 3}}$ with antipodal action are determined. It is shown that $ S{U_{2k}}({Z_2},free) \to S{U_{2k}}({Z_2},all)$ is monic, and that an element of $ S{U_{2k}}({Z_2},all)$ bounds as a unitary involution if and only if it is a multiple of the nonzero class $\alpha \in S{U_1}$.
Algebras over absolutely flat commutative rings
Joseph A.
Wehlen
149-160
Abstract: Let A be a finitely generated algebra over an absolutely flat commutative ring. Using sheaf-theoretic techniques, it is shown that the weak Hochschild dimension of A is equal to the supremum of the Hochschild dimension of ${A_x}$ for x in the decomposition space of R. Using this fact, relations are obtained among the weak Hochschild dimension of A and the weak global dimensions of A and ${A^e}$. It is also shown that a central separable algebra is a biregular ring which is finitely generated over its center. A result of S. Eilenberg concerning the separability of A modulo its Jacobson radical is extended. Finally, it is shown that every homomorphic image of an algebra of weak Hochschild dimension 1 is a type of triangular matrix algebra.
Fixed point iterations using infinite matrices
B. E.
Rhoades
161-176
Abstract: Let E be a closed, bounded, convex subset of a Banach space $X,f:E \to E$. Consider the iteration scheme defined by ${\bar x_0} = {x_0} \in E,{\bar x_{n + 1}} = f({x_n}),{x_n} = \Sigma _{k = 0}^n{a_{nk}}{\bar x_k},\;n \geq 1$, where A is a regular weighted mean matrix. For particular spaces X and functions f we show that this iterative scheme converges to a fixed point of f.
Products of initially $m$-compact spaces
R. M.
Stephenson;
J. E.
Vaughan
177-189
Abstract: The main purpose of this paper is to give several theorems and examples which we hope will be of use in the solution of the following problem. For an infinite cardinal number $\mathfrak{m}$, is initial $ \mathfrak{m}$-compactness preserved by products? We also give some results concerning properties of Stone-Čech compactifications of discrete spaces.
New criteria for freeness in abelian groups. II
Paul
Hill
191-201
Abstract: A new criterion is established for an abelian group to be free. The criterion is in terms of an ascending chain of free subgroups and is dependent upon a new class of torsion-free groups. The result leads to the construction, for each positive integer n, of a group ${G_n}$ of cardinality ${\aleph _n}$ that is not free but is ${\aleph _n}$-free. A conjecture in infinitary logic concerning free abelian groups is also verified.
The slimmest geometric lattices
Thomas A.
Dowling;
Richard M.
Wilson
203-215
Abstract: The Whitney numbers of a finite geometric lattice L of rank r are the numbers ${W_k}$ of elements of rank k and the coefficients ${w_k}$ of the characteristic polynomial of L, for $0 \leq k \leq r$. We establish the following lower bounds for the $ {W_k}$ and the absolute values $ w_k^ + = {( - 1)^k}{w_k}$ and describe the lattices for which equality holds (nontrivially) in each case: $\displaystyle {W_k} \geq \left( {\begin{array}{*{20}{c}} r & - & ... ... \left( {\begin{array}{*{20}{c}} r k \end{array} } \right),$ where $n = {W_1}$ is the number of points of L.
On certain convex sets in the space of locally schlicht functions
Y. J.
Kim;
E. P.
Merkes
217-224
Abstract: Let $H = H{(^ \ast },[ + ])$ denote the real linear space of locally schlicht normalized functions in $ \vert z\vert < 1$ as defined by Hornich. Let K and C respectively be the classes of convex functions and the close-to-convex functions. If $M \subset H$ there is a closed nonempty convex set in the $\alpha \beta$-plane such that for $(\alpha ,\beta )$ in this set ${\alpha ^ \ast }f[ + ]{\beta ^ \ast }g \in C$ (in K) whenever f, $g \in M$. This planar convex set is explicitly given when M is the class K, the class C, and for other classes. Some consequences of these results are that K and C are convex sets in H and that the extreme points of C are not in K.
On the zeros of Dirichlet $L$-functions. I
Akio
Fujii
225-235
Abstract: A mean value theorem for $ \arg \;L({\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.... ...e 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}} + it,\chi )$ is established. This yields mean estimates for the number of zeros of $L(s,\chi )$ in small boxes.
Locally $e$-fine measurable spaces
Zdeněk
Frolík
237-247
Abstract: Hyper-Baire sets and hyper-cozero sets in a uniform space are introduced, and it is shown that for metric-fine spaces the property ``every hypercozero set is a cozero set'' is equivalent to several much stronger properties like being locally e-fine (defined in §1), or having locally determined precompact part (introduced in §2). The metric-fine spaces with these additional properties form a coreflective subcategory of uniform spaces; the coreflection is explicitly described. The theory is applied to measurable uniform spaces. It is shown that measurable spaces with the additional properties mentioned above are coreflective and the coreflection is explicitly described. The two coreflections are not metrically determined.
An internal characterization of paracompact $p$-spaces
R. A.
Stoltenberg
249-263
Abstract: The purpose of this paper is to characterize paracompact p-spaces in terms of spaces with refining sequences $ \bmod\;k$. A space X has a refining sequence $\bmod\;k$ if there exists a sequence $\{ {\mathcal{G}_n}\vert n \in N\}$ of open covers for X such that $\cap _{n = 1}^\infty {\text{St}}(C,{\mathcal{G}_n}) = P_C^1$ is compact for each compact subset C of X and ${\text{\{ St}}{(C,{\mathcal{G}_n})^ - }\vert n \in N\}$ is a neighborhood base for $ P_C^1$. If $P_C^1 = C$ for each compact subset C of X then X is metrizable. On the other hand if we restrict the set C to the family of finite subsets of X in the above definition then we have a characterization for strict p-spaces. Moreover, in this case, if $P_C^1 = C$ for all such sets then X is developable. Thus the concept of a refining sequence $ \bmod\;k$ is natural and it is helpful in understanding paracompact p-spaces.
Analytic equivalence among simply connected domains in $C(X)$
Hugh E.
Warren
265-288
Abstract: This work considers analytic equivalence within the analytic function theory for commutative Banach algebras which was introduced by E. R. Lorch. Necessary conditions of a geometric nature are given for simply connected domains in $ C(X)$. These show that there are a great many equivalence classes. In some important cases, as when one domain is the unit ball, the given conditions are also sufficient. The main technique is the association of a simply connected domain in $C(X)$ with a family of Riemann surfaces over the plane.
$\omega $-linear vector fields on manifolds
William
Perrizo
289-312
Abstract: The classical study of a flow near a fixed point is generalized by composing, at each point in the manifold, the flow derivative with a parallel translation back along the flow. Circumstances under which these compositions form a one-parameter group are studied. From the point of view of the linear frame bundle, the condition is that the canonical lift commute with its horizontal part (with respect to some metric connection). The connection form applied to the lift coincides with the infinitesimal generator of the one-parameter group. Analysis of this matrix provides dynamical information about the flow. For example, if such flows are equicontinuous, they have uniformly bounded derivatives and therefore the enveloping semigroup is a Lie transformation group. Subclasses of ergodic, minimal, and weakly mixing flows with integral invariants are determined according to the eigenvalues of the matrices. Such examples as Lie algebra flows, infinitesimal affine transformations, and the geodesic flows on manifolds of constant negative curvature are examined.
Global dimension of tiled orders over a discrete valuation ring
Vasanti A.
Jategaonkar
313-330
Abstract: Let R be a discrete valuation ring with maximal ideal $\mathfrak{m}$ and the quotient field K. Let $\Lambda = ({\mathfrak{m}^{{\lambda _{ij}}}}) \subseteq {M_n}(K)$ be a tiled R-order, where $ {\lambda _{ij}} \in {\mathbf{Z}}$ and $ {\lambda _{ii}} = 0$ for $1 \leq i \leq n$. The following results are proved. Theorem 1. There are, up to conjugation, only finitely many tiled R-orders in $ {M_n}(K)$ of finite global dimension. Theorem 2. Tiled R-orders in ${M_n}(K)$ of finite global dimension satisfy DCC. Theorem 3. Let $\Lambda \subseteq {M_n}(R)$ and let $ \Gamma$ be obtained from $\Lambda$ by replacing the entries above the main diagonal by arbitrary entries from R. If $\Gamma$ is a ring and if gl $\dim \;\Lambda < \infty $, then gl $\dim \;\Gamma < \infty$. Theorem 4. Let $\Lambda$ be a tiled R-order in $ {M_4}(K)$. Then gl $\dim \;\Lambda < \infty$ if and only if $\Lambda$ is conjugate to a triangular tiled R-order of finite global dimension or is conjugate to the tiled R-order $ \Gamma = ({\mathfrak{m}^{{\lambda _{ij}}}}) \subseteq {M_4}(R)$, where $ {\gamma _{ii}} = {\gamma _{1i}} = 0$ for all i, and ${\gamma _{ij}} = 1$ otherwise.
${\rm SU}(n)$ actions on differentiable manifolds with vanishing first and second integral Pontrjagin classes
Edward A.
Grove
331-350
Abstract: In this paper we determine the connected component of the identity of the isotropy subgroups of a given action of ${\text{SU}}(n)$ on a connected manifold whose first and second integral Pontrjagin classes are zero and whose dimension is less than ${n^2} - 8n/3 - 1$.
Groups, semilattices and inverse semigroups. II
D. B.
McAlister
351-370
Abstract: An inverse semigroup is called proper if the equations $ae = e = {e^2}$ together imply ${a^2} = a$. In a previous paper, with the same title, the author proved that every inverse semigroup is an idempotent separating homomorphic image of a proper inverse semigroup. In this paper a structure theorem is given for all proper inverse semigroups in terms of partially ordered sets and groups acting on them by order automorphisms. As a consequence of these two theorems, and Preston's construction for idempotent separating congruences on inverse semigroups, one can give a structure theorem for all inverse semigroups in terms of groups and partially ordered sets.
An induction principle for spectral and rearrangement inequalitities
Kong Ming
Chong
371-383
Abstract: In this paper, expressions of the form $f \prec g$ or $f \prec \prec g$ (where $ \prec$ and $\prec \prec$ denote the Hardy-Littlewood-Pólya spectral order relations) are called spectral inequalities. Here a general induction principle for spectral and rearrangement inequalities involving a pair of n-tuples in ${R^n}$ as well as their decreasing and increasing rearrangements is developed. This induction principle proves that such spectral or rearrangement inequalities hold iff they hold for the case when $n = 2$, and that, under some mild conditions, this discrete result can be generalized to include measurable functions with integrable positive parts. A similar induction principle for spectral and rearrangement inequalities involving more than two measurable functions is also established. With this induction principle, some well-known spectral or rearrangement inequalities are obtained as particular cases and additional new results given.
Axisymmetric harmonic interpolation polynomials in ${\bf R}\sp{N}$
Morris
Marden
385-402
Abstract: Corresponding to a given function $ F(x,\rho )$ which is axisymnetric harmonic in an axisymmetric region $ \Omega \subset {{\text{R}}^3}$ and to a set of $n + 1$ circles ${C_n}$ in an axisymmetric subregion $A \subset \Omega$, an axisymmetric harmonic polynomial $ {\Lambda _n}(x,\rho ;{C_n})$ is found which on the ${C_n}$ interpolates to $ F(x,\rho )$ or to its partial derivatives with respect to x. An axisymmetric subregion $B \subset \Omega$ is found such that ${\Lambda _n}(x,\rho ;{C_n})$ converges uniformly to $F(x,\rho )$ on the closure of B. Also a ${\Lambda _n}(x,\rho ;{x_0},{\rho _0})$ is determined which, together with its first n partial derivatives with respect to x, coincides with $F(x,\rho )$ on a single circle $({x_0},{\rho _0})$ in $\Omega$ and converges uniformly to $F(x,\rho )$ in a closed torus with $({x_0},{\rho _0})$ as central circle.
Density of parts of algebras on the plane
Anthony G.
O’Farrell
403-414
Abstract: We study the Gleason parts of a uniform algebra A on a compact subset of the plane, where it is assumed that for each point $x \in {\text{C}}$ the functions in A which are analytic in a neighborhood of x are uniformly dense in A. We prove that a part neighborhood N of a nonpeak point x for A satisfies a density condition of Wiener type at $x:\Sigma _{n = 1}^{ + \infty }{2^n}C({A_n}(x)\backslash N) < + \infty$, and if A admits a pth order bounded point derivation at x, then N satisfies a stronger density condition: $\Sigma _{n = 1}^{ + \infty }{2^{(p + 1)n}}C({A_n}(x)\backslash N) < + \infty$. Here C is Newtonian capacity and ${A_n}(x)$ is $ \{ z \in {\text{C}}:{2^{ - n - 1}} \leq \vert z - x\vert \leq {2^{ - n}}\}$. These results strengthen and extend Browder's metric density theorem. The relation with potential theory is examined, and analogous results for the algebra ${H^\infty }(U)$ are obtained as corollaries.
Analytic capacity, H\"older conditions and $\tau $-spikes
Anthony G.
O’Farrell
415-424
Abstract: We consider the uniform algebra $R(X)$, for compact $X \subset {\text{C}}$, in relation to the condition ${I_{p + \alpha }} = \Sigma _1^\infty {2^{(p + \alpha + 1)n}}\gamma ({A_n}(x)\backslash X) < + \infty$, where $0 \leq p \in {\mathbf{Z}},0 < \alpha < 1,\gamma$ is analytic capacity, and ${A_n}(x)$ is the annulus $\{ z \in {\text{C}}:{2^{ - n - 1}} < \vert z - x\vert < {2^{ - n}}\}$. We introduce the notion of $ \tau$-spike for $ \tau > 0$, and show that $ {I_{p + \alpha }} = + \infty$ implies x is a $ p + \alpha$-spike. If $\mathop X\limits^ \circ$ satisfies a cone condition at x, and ${I_{p + \alpha }} < + \infty$, we show that the pth derivatives of the functions in $R(X)$ satisfy a uniform Hölder condition at x for nontangential approach. The structure of the set of non-$\tau$-spikes is examined and the results are applied to rational approximation. A geometric question is settled.
Unitary measures on LCA groups
Lawrence
Corwin
425-430
Abstract: A unitary measure on a locally compact Abelian (LCA) group G is a complex measure whose Fourier transform is of absolute value 1 everywhere. The problem of finding all such measures is known to be closely related to that of finding all invertible measures on G. In this paper, we find all unitary measures when G is the circle or a discrete group. If G is a torsion-free discrete group, the characterization generalizes a theorem of Bohr.