Characteristic classes of real manifolds immersed in complex manifolds
Hon Fei
Lai
1-33
Abstract: Let $M$ be a compact, orientable, $ k$-dimensional real differentiaable manifold and $N$ an $n$-dimensional complex manifold, where $k \geq n$. Given an immersion $\iota :M \to N$, a point $x \in M$ is called an RC-singular point of the immersion if the tangent space to $ \iota (M)$ at $ \iota (x)$ contains a complex subspace of dimension $> k - n$. This paper is devoted to the study of the cohomological properties of the set of RC-singular points of an immersion. When $k = 2n - 2$, the following formula is obtained: $\displaystyle \Omega (M) + \sum\limits_{r = 0}^{n - 1} {\tilde \Omega } {(\iota )^{n - r - 1}}{\iota ^ \ast }{c_r}(N) = 2{t^ \ast }DK,$ where $\Omega (M)$ is the Euler class of $M,\widetilde\Omega (\iota )$ is the Euler class of the normal bundle of the immersion, ${c_r}(N)$ are the Chern classes of $ N$, and ${t^ \ast }DK$ is a cohomology class of degree $ 2n - 2$ in $M$ whose value on the fundamental class of $M$ gives the algebraic number of RC-singular points of $\iota$. Various applications are discussed. For $ n \leq k \leq 2n - 2$, it is shown that, as long as dimensions allow, all Pontrjagin classes and the Euler class of $M$ are carried by subsets of the set of RC-singularities of an immersion $\iota :M \to {{\text{C}}^n}$.
Topological properties of paranormal operators on Hilbert space
Glenn R.
Luecke
35-43
Abstract: Let $B(H)$ be the set of all bounded endomorphisms (operators) on the complex Hilbert space $H.T \in B(H)$ is paranormal if $ \vert\vert{(T - zI)^{ - 1}}\vert\vert = 1/d(z,\sigma (T))$ for all $z \notin \sigma (T)$ where $d(z,\sigma (T))$ is the distance from $z$ to $ \sigma (T)$, the spectrum of $T$. If $ \mathcal{P}$ is the set of all paranormal operators on $H$, then $ \mathcal{P}$ contains the normal operators, $ \mathfrak{N}$, and the hyponormal operators; and $ \mathcal{P}$ is contained in $\mathcal{L}$, the set of all $T \in B(H)$ such that the convex hull of $\sigma (T)$ equals the closure of the numerical range of $T$. Thus, $\mathfrak{N} \subseteq \mathcal{P} \subseteq \mathcal{L} \subseteq B(H)$. Give $B(H)$ the norm topology. The main results in this paper are (1) $ \mathfrak{N},\mathcal{P}$, and $\mathcal{L}$ are nowhere dense subsets of $ B(H)$ when $\dim H \geq 2$, (2) $ \mathfrak{N},\mathcal{P}$, and $\mathcal{L}$ are arcwise connected and closed, and (3) $ \mathfrak{N}$ is a nowhere dense subset of $ \mathcal{P}$ when $\dim H = \infty$.
Global stability in $n$-person games
Louis J.
Billera
45-56
Abstract: A class of bargaining sets, including the bargaining set $\mathfrak{M}_1^{(i)}$ and the kernel, is treated with regard to studying the tendency to reach stability from unstable points. A known discrete procedure is extended, and these results are applied to derive global stability properties for the solutions of certain differential equations. These differential equations are given in terms of the demand functions which define the bargaining sets, and the set of critical points is precisely the bargaining set in question.
Finite commutative subdirectly irreducible semigroups
Phillip E.
McNeil
57-67
Abstract: This paper is devoted to completing the solution to the problem of constructing all finite commutative subdirectly irreducible semigroups. Those semigroups of this type which were formerly unknown are realized as certain permutation group extensions of nilpotent semigroups. The results in this paper extend the efforts in this area by G. Thierrin and B. M. Schein.
Flexible algebras of degree two
Joseph H.
Mayne
69-81
Abstract: All known examples of simple flexible power-associative algebras of degree two are either commutative or noncommutative Jordan. In this paper we construct an algebra which is partially stable but not commutative and not a noncommutative Jordan algebra. We then investigate the multiplicative structure of those algebras which are partially stable over an algebraically closed field of characteristic $p \ne 2,3,5$. The results obtained are then used to develop conditions under which such algebras must be commutative.
Extreme invariant means without minimal support
Lonnie
Fairchild
83-93
Abstract: Let $S$ be a left amenable semigroup. We show that if $S$ has a subset satisfying a certain condition, then there is an extreme left invariant mean on $ S$ whose support is not a minimal closed invariant subset of $\beta S$. Then we show that all infinite solvable groups and countably infinite locally finite groups have such subsets.
Lefschetz duality and topological tubular neighbourhoods
F. E. A.
Johnson
95-110
Abstract: We seek an analogue for topological manifolds of closed tubular neighbourhoods (for smooth imbeddings) and closed regular neighbourhoods (for piecewise linear imbeddings). We succeed when the dimension of the ambient manifold is at least six. The proof uses topological handle theory, the results of Siebenmann's thesis, and a strong version of the Lefschetz Duality Theorem which yields a duality formula for Wall's finiteness obstruction.
The action of the automorphism group of $F\sb{2}$ upon the $A\sb{6}$- and ${\rm PSL}(2,\,7)$-defining subgroups of $F\sb{2}$
Daniel
Stork
111-117
Abstract: In this paper is described a graphical technique for determining the action of the automorphism group ${\Phi _2}$, of the free group $ {F_2}$ of rank 2 upon those normal subgroups of ${F_2}$ with quotient groups isomorphic to $ G$, where $G$ is a group represented faithfully as a permutation group. The procedure is applied with $G = {\text{PSL}}(2,7)$ and $ {A_6}$ (the case $G = {A_5}$ having been treated in an earlier paper) with the following results: Theorem 1. ${\Phi _2}$, acts upon the 57 subgroups of $ {F_2}$ with quotient isomorphic to $ {\text{PSL}}(2,7)$ with orbits of lengths 7, 16, 16, and 18. The action of ${\Phi _2}$ is that of ${A_{16}}$ in one orbit of length 16, and of symmetric groups of appropriate degree in the other three orbits. Theorem 2. $ {\Phi _2}$, acts upon the 53 subgroups of ${F_2}$ with quotients isomorphic to ${A_6}$ with orbits of lengths 10, 12, 15, and 16. The action is that of full symmetric groups of appropriate degree in all orbits.
The group of homeomorphisms of a solenoid
James
Keesling
119-131
Abstract: Let $X$ be a topological space. An $ n$-mean on $X$ is a continuous function $\mu :{X^n} \to X$ which is symmetric and idempotent. In the first part of this paper it is shown that if $ X$ is a compact connected abelian topological group, then $X$ admits an $n$-mean if and only if $ {H^1}(X,Z)$ is $ n$-divisible where ${H^m}(X,Z)$ is $m$-dimensional Čech cohomology with integers $ Z$ as coefficient group. This result is used to show that if ${\Sigma _a}$ is a solenoid and $ \operatorname{Aut} ({\Sigma _a})$ is the group of topological group automorphisms of ${\Sigma _a}$, then $\operatorname{Aut} ({\Sigma _a})$ is algebraically ${Z_2} \times G$ where $G$ is $ \{ 0\} ,{Z^n}$, or $\oplus _{i = 1}^\infty Z$. For a given ${\Sigma _a}$, the structure of $ \operatorname{Aut} ({\Sigma _a})$ is determined by the $n$-means which $ {\Sigma _a}$. admits. Topologically, $\operatorname{Aut} ({\Sigma _a})$ is a discrete space which has two points or is countably infinite. The main result of the paper gives the precise topological structure of the group of homeomorphisms $G({\Sigma _a})$ of a solenoid ${\Sigma _a}$ with the compact open topology. In the last section of the paper it is shown that $G({\Sigma _a})$ is homeomorphic to ${\Sigma _a} \times {l_2} \times \operatorname{Aut} ({\Sigma _a})$ where ${l_2}$ is separable infinite-dimensional Hilbert space. The proof of this result uses recent results in infinite-dimensional topology and some techniques using flows developed by the author in a previous paper.
Cohomology of sheaves of holomorphic functions satisfying boundary conditions on product domains
Alexander
Nagel
133-141
Abstract: This paper considers sheaves of germs of holomorphic functions which satisfy certain boundary conditions on product domains in $ {{\mathbf{C}}^n}$. Very general axioms for boundary behavior are given. This includes as special cases ${L^p}$ boundary behavior, $1 \leq p \leq \infty$; continuous boundary behavior; differentiable boundary behavior of order $m,0 \leq m \leq \infty$, with an additional Hölder condition of order $\alpha ,0 \leq \alpha \leq 1$, on the $m$th derivatives. A fine resolution is constructed for those sheaves considered, and the main result of the paper is that all higher cohomology groups for these sheaves are zeŕo.
Zeros of entire functions in several complex variables
Richard A.
Kramer
143-160
Abstract: A geometric condition on the zero set of an entire function $ f$ in ${{\mathbf{C}}^N}(N \geq 1)$ is presented which is both necessary and sufficient for $ f$ to have the same zeros as some polynomial in $ {{\mathbf{C}}^N}$.
Asymptotic behavior of transforms of distributions
E. O.
Milton
161-176
Abstract: In this paper final and initial value type Abelian theorems for Laplace and Fourier transforms of certain types of distributions are obtained. The class of distributions under consideration contains the singular distributions. Thus we generalize the results previously obtained by A. H. Zemanian in two ways: we add Fourier transforms to those considered, and we also deal with a larger class of distributions.
Groups of linear operators defined by group characters
Marvin
Marcus;
James
Holmes
177-194
Abstract: Some of the recent work on invariance questions can be regarded as follows: Characterize those linear operators on $\operatorname{Hom} (V,V)$ which preserve the character of a given representation of the full linear group. In this paper, for certain rational characters, necessary and sufficient conditions are described that ensure that the set of all such operators forms a group $\mathfrak{L}$. The structure of $\mathfrak{L}$ is also determined. The proofs depend on recent results concerning derivations on symmetry classes of tensors.
Maximal ideals in the group algebra of an extension, with applications
Arnold J.
Insel
195-206
Abstract: Let $A$ be a closed central subgroup of $ E$ (all groups are locally compact and second countable). Let $G = E/A$. For each $a\in \hat A$, the dual of $A$, a multiplication is introduced with respect to which the Banach space ${L^1}(G)$ is a Banach algebra, denoted by ${L^1}(G,a(\sigma ))$. A one-to-one correspondence is established between the maximal closed (right, left, $ 2$-sided) ideals of the group algebra ${L^1}(E)$ and the totality of maximal closed (right, left, $2$-sided) ideals of ${L^1}(G,a(\sigma ))$, where $a$ varies over $\hat A$. Applications include a bound for the spectral norm of an element of ${L^1}(E)$ and the representation of a continuous positive definite function on $E$ as an integral (a 'Bochner' theorem).
Varieties of linear topological spaces
J.
Diestel;
Sidney A.
Morris;
Stephen A.
Saxon
207-230
Abstract: This paper initiates the formal study of those classes of locally convex spaces which are closed under the taking of arbitrary subspaces, separated quotients, cartesian products and isomorphic images. Well-known examples include the class of all nuclear spaces and the class of all Schwartz spaces.
Matrix rings over polynomial identity rings
Elizabeth
Berman
231-239
Abstract: We prove that if $ A$ is an algebra over a field with at least $k$ elements, and $A$ satisfies ${x^k} = 0$, then ${A_n}$, the ring of $n$-by-$n$ matrices over $A$, satisfies ${x^q} = 0$, where $ q = k{n^2} + 1$. Theorem 1.3 generalizes this result to rings: If $A$ is a ring satisfying ${x^k} = 0$, then for all $n$, there exists $q$ such that ${A_n}$ satisfies ${x^q} = 0$. Definitions. A checkered permutation of the first $n$ positive integers is a permutation of them sending even integers into even integers. The docile polynomial of degree $n$ is $\displaystyle \prod\limits_{i = 1}^p {D({x_{i1}}, \cdots ,{x_{ik}}){u_i},}$ athewhere the sum is over all checkered permutations $f$ of the first $k$ positive integers. The docile product polynomial of degree $k,p$is $\displaystyle \prod\limits_{i = 1}^p {D({x_{i1}}, \cdots ,{x_{ik}}){u_i},}$ where the $x$'s and $u$'s are noncommuting variables. Theorem 2.1. Any polynomial identity algebra over a field of characteristic 0 satisfies a docile product polynomial identity. Theorem 2.2. If $ A$ is a ring satisfying the docile product polynomial identity of degree $ 2k,p$, and $n$ is a positive integer, and $q = 2{k^2}{n^2} + 1$; then ${A_n}$ satisfies a product of $ p$ standard identities, each of degree $q$.
Group rings, matrix rings, and polynomial identities
Elizabeth
Berman
241-248
Abstract: This paper studies the question, if $R$ is a ring satisfying a polynomial identity, what polynomial identities are satisfied by group rings and matrix rings over $R$? Theorem 2.6. If $R$ is an algebra over a field with at least $ q$ elements, and $ R$ satisfies $ {x^q} = 0$, and $ G$ is a group with an abelian subgroup of index $k$, then the group ring $R(G)$ satisfies ${x^t} = 0$, where $t = q{k^2} + 2$. Theorem 3.2. If $R$ is a ring satisfying a standard identity, and $G$ is a finite group, then $R(G)$ satisfies a standard identity. Theorem 3.4. If $R$ is an algebra over a field, and $R$ satisfies a standard identity, then the $k$-by-$k$ matrix ring ${R_k}$ satisfies a standard identity. Each theorem specifies the degree of the polynomial identity.
Amalgamations of lattice ordered groups
Keith R.
Pierce
249-260
Abstract: The author considers the problem of determining whether certain classes of lattice ordered groups ($l$-groups) have the amalgamation property. It is shown that the classes of abelian totally ordered groups ($o$-groups) and abelian $l$-groups have the property, but that the class of $l$-groups does not. However, under certain cardinality restrictions one can find an $l$-group which is the ``product'' of $ l$-groups with an amalgamated subgroup whenever (a) the $l$-subgroup is an Archimedian $ o$-group, or (b) the $ l$-subgroup is a direct product of Archimedian $o$-groups and the $l$-groups are representable. This yields a new proof that any $l$-group is embeddable in a divisible $ l$-group, and implies that any $l$-group is embeddable in an $l$-group in which any two positive elements are conjugate.
Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains
Melvyn S.
Berger;
Martin
Schechter
261-278
Abstract: The Sobolev-Kondrachov embedding and compactness theorems are extended to cover general unbounded domains, by introducing appropriate weighted ${L_p}$ norms. These results are then applied to the Dirichlet problem for quasi-linear elliptic partial differential equations and isoperimetric variational problems defined on general unbounded domains in ${{\mathbf{R}}^N}$.
Horn classes and reduced direct products
Richard
Mansfield
279-286
Abstract: Boolean-valued model theory is used to give a direct proof that an $E{C_\Delta }$ model class closed under reduced direct products can be characterized by a set of Horn sentences. Previous proofs by Keisler and Galvin used either the G. C H. or involved axiomatic set theory.
The access theorem for subharmonic functions
R.
Hornblower;
E. S.
Thomas
287-297
Abstract: A chain from a point ${z_0}$ of the open unit disk $\Delta$ to the boundary of $ \Delta$ is a set $\Gamma = \cup \{ {\gamma _n}\vert n = 0,1,2, \cdots \}$ where the $ {\gamma _n}$ are compact, connected subsets of $ \Delta ,{z_0}$ is in ${\gamma _0},{\gamma _n}$ meets ${\gamma _{n + 1}}$ and the ${\gamma _n}$ approach the boundary of $\Delta$. The following ``Access Theorem'' is proved: If $u$ is subharmonic in $\Delta ,{z_0}$ is a point of $\Delta$ and $ M < u({z_0})$, then there is a chain from ${z_0}$ to the boundary of $\Delta$ on which $u \geq M$ and on which $u$ tends to a limit. A refinement, in which the chain is a polygonal arc, is established, and an example is constructed to show that the theorem fails if $M = u({z_0})$ even for bounded, continuous subharmonic functions.
Ultrafilters and independent sets
Kenneth
Kunen
299-306
Abstract: Independent families of sets and of functions are used to prove some theorems about ultrafilters. All of our results are well known to be provable from some form of the generalized continuum hypothesis, but had remained open without such an assumption. Independent sets are used to show that the Rudin-Keisler ordering on ultrafilters is nonlinear. Independent functions are used to prove the existence of good ultrafilters.
Group actions on spin manifolds
G.
Chichilnisky
307-315
Abstract: A generalization of the theorem of V. Bargmann concerning unitary and ray representations is obtained and is applied to the general problem of lifting group actions associated to the extension of structure of a bundle. In particular this is applied to the Poincaré group $\mathcal{P}$ of a Lorentz manifold $M$. It is shown that the topological restrictions needed to lift an action in $\mathcal{P}$ are more stringent than for actions in the proper Poincaré group $\mathcal{P}_ \uparrow ^ + $. Similar results hold for the Euclidean group of a Riemannian manifold.
Stability of group representations and Haar spectrum
Robert
Azencott;
William
Parry
317-327
Abstract: If $U$ and $V$ are commuting unitary representations of locally compact abelian groups $S$ and $T$, new representations of $S$ (perturbations of $U$) can be obtained from composition with images of $ U$ in $V$. If most of these representations are equivalent to $U,U$ is said to be $V$ stable. We investigate conditions which, together with stability, ensure that $U$ has (uniform) Haar spectrum. The principal applications are to dynamical systems which possess auxiliary groups with respect to which motion is stable.
Split and minimal abelian extensions of finite groups
Victor E.
Hill
329-337
Abstract: Criteria for an abelian extension of a group to split are given in terms of a Sylow decomposition of the kernel and of normal series for the Sylow subgroups. An extension is minimal if only the entire extension is carried onto the given group by the canonical homomorphism. Various basic results on minimal extensions are given, and the structure question is related to the case of irreducible kernels of prime exponent. It is proved that an irreducible modular representation of ${\text{SL}}(2,p)$ or $ {\text{PSL}}(2,p)$ for $ p$ prime and $ \geq 5$ afford a minimal extension with kernel of exponent $p$ only when the representation has degree 3, i.e., when the kernel has order ${p^3}$.
On Ess\'en's generalization of the Ahlfors-Heins theorem
John L.
Lewis
339-345
Abstract: Recently, Essén has proven a generalization of the Ahlfors-Heins Theorem. In this paper we use Essén's Theorem to obtain a different generalization of the Ahlfors-Heins Theorem.
Extensions of holomorphic maps
Peter
Kiernan
347-355
Abstract: Several generalizations of the big Picard theorem are obtained. We consider holomorphic maps $f$ from $X - A$ into $M \subset Y$. Under various assumptions on $X,A$, and $M$ we show that $f$ can be extended to a holomorphic or meromorphic map of $X$ into $Y$.
Multipliers on modules over the Fourier algebra
Charles F.
Dunkl;
Donald E.
Ramirez
357-364
Abstract: Let $G$ be an infinite compact group and $ \hat G$ its dual. For $1 \leq p < \infty ,{\mathfrak{L}^p}(\hat G)$ is a module over ${\mathfrak{L}^1}(\hat G) \cong A(G)$, the Fourier algebra of $G$. For $ 1 \leq p,q < \infty$, let $ {\mathfrak{M}_{p,q}} = {\operatorname{Hom} _{A(G)}}({\mathfrak{L}^p}(\hat G),{\mathfrak{L}^q}(\hat G))$. If $G$ is abelian, then ${\mathfrak{M}_{p,p}}$ is the space of ${L^P}(\hat G)$-multipliers. For $1 \leq p < 2$ and $p'$ the conjugate index of $p$, $\displaystyle A(G) \cong {\mathfrak{M}_{1,1}} \subset {\mathfrak{M}_{p,p}} = {\mathfrak{M}_{p',p'}} \subsetneqq {\mathfrak{M}_{2,2}} \cong {L^\infty }(G).$ Further, the space $ {\mathfrak{M}_{p,p}}$ is the dual of a space called $\mathcal{A}_p$, a subspace of ${\mathcal{C}_0}(\hat G)$. Using a method of J. F. Price we observe that $\displaystyle \cup \{ {\mathfrak{M}_{q,q}}:1 \leq q < p\} \subsetneqq {\mathfrak{M}_{p,p}} \subsetneqq \cap \{ {\mathfrak{M}_{q,q}}:p < q < 2\}$ (where $1 < p < 2$). Finally, $ {\mathfrak{M}_{q,p}} = \{ 0\}$ for $ 1 \leq p < q < \infty$.
On Lie's theorem in operator algebras
F.-H.
Vasilescu
365-372
Abstract: This work contains some algebraic results concerning infinite dimensional Lie algebras, as well as further statements within a topological background. Natural generalizations of the notion of radical, solvable and semisimple Lie algebra are introduced. The last part deals with variants of a Lie's theorem in operator algebras.
Integral representation theorems in topological vector spaces
Alan H.
Shuchat
373-397
Abstract: We present a theory of measure and integration in topological vector spaces and generalize the Fichtenholz-Kantorovich-Hildebrandt and Riesz representation theorems to this setting, using strong integrals. As an application, we find the containing Banach space of the space of continuous $p$-normed space-valued functions. It is known that Bochner integration in $p$-normed spaces, using Lebesgue measure, is not well behaved and several authors have developed integration theories for restricted classes of functions. We find conditions under which scalar measures do give well-behaved vector integrals and give a method for constructing examples.
Complete multipliers
J. S.
Byrnes
399-403
Abstract: We investigate whether the completeness of a complete orthonormal sequence for $ {L^2}( - \pi ,\pi )$ is preserved if the sequence is perturbed by multiplying a portion of it by a fixed function. For the particular sequence $\{ {(2\pi )^{ - 1/2}}{e^{inx}}\}$ we show that given any $\psi \in {L^\infty }( - \pi ,\pi )$, except $ \psi = 0$ a.e., there is a nontrivial portion of $\{ {(2\pi )^{ - 1/2}}{e^{inx}}\}$ which will maintain completeness under this perturbation.
The nonstandard theory of topological vector spaces
C. Ward
Henson;
L. C.
Moore
405-435
Abstract: In this paper the nonstandard theory of topological vector spaces is developed, with three main objectives: (1) creation of the basic nonstandard concepts and tools; (2) use of these tools to give nonstandard treatments of some major standard theorems; (3) construction of the nonstandard hull of an arbitrary topological vector space, and the beginning of the study of the class of spaces which results.
On the nonstandard representation of measures
C. Ward
Henson
437-446
Abstract: In this paper it is shown that every finitely additive probability measure $ \mu$ on $S$ which assigns 0 to finite sets can be given a nonstandard representation using the counting measure for some $^ \ast$-finite subset $F$ of $^ \ast S$. Moreover, if $\mu$ is countably additive, then $ F$ can be chosen so that $\displaystyle \int {fd\mu } = {\text{st( }}\frac{1}{{\vert\vert F\vert\vert}}\sum _{p \in F} ^\ast f(p))$ for every $\mu$-integrable function $f$. An application is given of such representations. Also, a simple nonstandard method for constructing invariant measures is presented.
Hyponormal operators having real parts with simple spectra
C. R.
Putnam
447-464
Abstract: Let ${T^ \ast }T - T{T^ \ast } = D \geq 0$ and suppose that the real part of $T$ has a simple spectrum. Then $D$ is of trace class and $\pi$ trace$(D)$ is a lower bound for the measure of the spectrum of $T$. This latter set is specified in terms of the real and imaginary parts of $T$. In addition, the spectra are determined of self-adjoint singular integral operators on ${L^2}(E)$ of the form $ A(x)f(x) + \Sigma {b_j}(x)H[f{\bar b_j}](x)$, where $E \ne ( - \infty ,\infty ),A(x)$ is real and bounded, $ \Sigma \vert{b_j}(x){\vert^2}$ is positive and bounded, and $H$ denotes the Hilbert transform.
Piecewise monotone polynomial approximation
D. J.
Newman;
Eli
Passow;
Louis
Raymon
465-472
Abstract: Given a real function $f$ satisfying a Lipschitz condition of order 1 on $ [a,b]$, there exists a sequence of approximating polynomials $\{ {P_n}\}$ such that the sequence ${E_n} = \vert\vert{P_n} - f\vert\vert$ (sup norm) has order of magnitude $1/n$ (D. Jackson). We investigate the possibility of selecting polynomials ${P_n}$ having the same local monotonicity as $ f$ without affecting the order of magnitude of the error. In particular, we establish that if $f$ has a finite number of maxima and minima on $ [a,b]$ and $S$ is a closed subset of $ [a,b]$ not containing any of the extreme points of $f$, then there is a sequence of polynomials $ {P_n}$ such that $ {E_n}$ has order of magnitude $1/n$ and such that for $n$ sufficiently large ${P_n}$ and $f$ have the same monotonicity at each point of $S$. The methods are classical.
Homological dimensions of stable homotopy modules and their geometric characterizations
T. Y.
Lin
473-490
Abstract: Projective dimensions of modules over the stable homotopy ring are shown to be either 0, 1 or $\infty$; weak dimensions are shown to be 0 or $ \infty$. Also geometric charactetizations are obtained for projective dimensions 0, 1 and weak dimension 0. The geometric characterizations are interesting; for projective modules they are about the cohomology of geometric realization; while for flat modules they are about homology. This shows that the algebraic duality between ``projective'' and ``flat'' is strongly connected with the topological duality between ``cohomology'' and ``homology". Finally, all the homological numerical invariants of the stable homotopy ring--the so-called finitistic dimensions--are completely computed except the one on injective dimension.
Vector valued absolutely continuous functions on idempotent semigroups
Richard A.
Alò;
André
de Korvin;
Richard J.
Easton
491-500
Abstract: In this paper the concept of vector valued, absolutely continuous functions on an idempotent semigroup is studied. For $ F$ a function of bounded variation on the semigroup $S$ of semicharacters with values of $F$ in the Banach space $X$, let $A = {\text{AC}}(S,X,F)$ be all those functions of bounded variation which are absolutely continuous with respect to $F$. A representation theorem is obtained for linear transformations from the space $A$ to a Banach space which are continuous in the BV-norm. A characterization is also obtained fot the collection of functions of $A$ which are Lipschitz with respect to $ F$. With regards to the new integral being utilized it is shown that all absolutely continuous functions are integrable.
Zero points of Killing vector fields, geodesic orbits, curvature, and cut locus
Walter C.
Lynge
501-506
Abstract: Let $(M,g)$ be a compact, connected, Riemannian manifold. Let $X$ be a Killing vector field on $M$. $f = g(X,X)$ is called the length function of $X$. Let $D$ denote the minimum of the distances from points to their cut loci on $M$. We derive an inequality involving $ f$ which enables us to prove facts relating $D$, the zero ponts of $X$, orbits of $X$ which are closed geodesics, and, applying theorems of Klingenberg, the curvature of $M$. Then we use these results together with a further analysis of $f$ to describe the nature of a Killing vector field in a neighborhood of an isolated zero point.
Erratum to: ``Automorphisms of group extensions''
Charles
Wells
507
Erratum to: ``Oscillation of an operator''
Robert
Whitley
507
Erratum to ``The $(\phi^{2n})_2$ field Hamiltonian for complex coupling constant
Lon
Rosen;
Barry
Simon
508