Polars and their applications in directed interpolation groups
A. M. W.
Glass
1-25
Abstract: In the study of l-groups, as in many other branches of mathematics, use is made of the concept of ``orthogonal elements". The purpose of this paper is to show that this concept can be extended to directed, interpolation groups and that most of the theorems in l-groups concerning polars hold in the more general setting of directed, interpolation groups. As consequences, generalisations of Holland's and Lorenzen's theorems are obtained and a result on o-simple abelian, directed, interpolation groups.
Skew products of dynamical systems
Eijun
Kin
27-43
Abstract: In 1950-1951, H. Anzai introduced a method of skew products of dynamical systems in connection with isomorphism problems in ergodic theory. There is a problem to give a necessary and sufficient condition under which an ergodic skew product dynamical system has pure point spectrum. For the special case, translations on the torus, he gave a partial answer for this question. However, this problem has been open in the general case. In the present paper, we generalize the notion of skew products proposed by Anzai and give a complete answer for this problem.
Automorphism groups of bounded domains in Banach spaces
Stephen J.
Greenfield;
Nolan R.
Wallach
45-57
Abstract: We prove a weak Schwarz lemma in Banach space and use it to show that in Hilbert space a Siegel domain of type II is not necessarily biholomorphic to a bounded domain. We use a strong Schwarz lemma of L. Harris to find the full group of automorphisms of the infinite dimensional versions of the Cartan domains of type I. We then show that all domains of type I are holomorphically inequivalent, and are different from k-fold products of unit balls $(k \geqq 2)$. Other generalizations and comments are given.
A class of regular functions containing spirallike and close-to-convex functions
M. R.
Ziegler
59-70
Abstract: A class of functions $\mathcal{A}$ is defined which contains the spirallike and close-to-convex functions. By decomposing $ \mathcal{A}$ into subclasses in a natural way, some basic properties of $\mathcal{A}$ and these subclasses are determined, including solutions to extremal problems; distortion theorems; coefficient inequalities; and the radii of convexity and close-to-convexity.
Complex structures on real product bundles with applications to differential geometry
Richard S.
Millman
71-99
Abstract: The purpose of this paper is to classify holomorphic principal fibre bundles which admit a smooth section (i.e. are real product bundles). This is accomplished if the structure group is solvable of type (E). In the general case, a sufficient condition is obtained for a real product bundle to be equivalent to the complex product bundle. A necessary and sufficient condition for the existence of a holomorphic connection on a real product bundle is also obtained. Using this criterion in the case where the structure group is abelian, a generalization of a theorem due to Atiyah (in the case the structure group is ${C^ \ast }$) is obtained.
On inverse scattering for the Klein-Gordon equation
Tomas P.
Schonbek
101-123
Abstract: A scattering operator $S = S(V)$ is set up for the Klein-Gordon equation $ \square u = {m^2}u(m > 0)$ perturbed by a linear potential $V = V(x)$ to $\square u = {m^2}u + Vu$. It is found that for each $R > 0$ there exists a constant $c(R)$ (of order $ {R^{2 - n}}$ as $R \to + \infty$, n = space dimension) such that if the ${L_1}$ and the ${L_q}$ norm of V and $V'$ are bounded by $S(V') \ne S(V)$ or $V' = V$. Here $q > n/2$, and $c(R)$ may also depend on q.
A Sturmian theorem for first order partial differential equations
Pui Kei
Wong
125-131
Abstract: A pair of first order partial differential equations is considered. The system is transformed into a single nonlinear scalar equation of the Riccati type from which some Wirtinger type integral inequalities for functions of several variables are derived. A comparison theorem for two such pairs of first order equations is then proved using the Wirtinger inequalities.
Current valued measures and Ge\"ocze area
Ronald
Gariepy
133-146
Abstract: If f is a continuous mapping of finite Geöcze area from a polyhedral region $X \subset {R^k}$ into ${R^n},2 \leqq k \leqq n$, then, under suitable hypotheses, one can associate with f, by means of the Cesari-Weierstrass integral, a current valued measure T over the middle space of f. In particular, if either $k = 2$ or the $k + 1$-dimensional Hausdorff measure of $f(X)$ is zero, then T is essentially the same as a current valued measure defined by H. Federer and hence serves to describe the tangential properties of f and the multiplicities with which f assumes its values. Further, the total variation of T is equal to the Geöcze area of f.
Extending congruences on semigroups
A. R.
Stralka
147-161
Abstract: The two main results are: (1) Let S be a semigroup which satisfies the relation $abcd = acbd$, let A be a subsemigroup of Reg S which is a band of groups and let $[\varphi ]$ be a congruence on A. Then $[\varphi ]$ can be extended to a congruence on S. (2) Let S be a compact topological semigroup which satisfies the relation $abcd = acbd$, let A be a closed subsemigroup of Reg S and let $[\varphi ]$ be a closed congruence on A such that $\dim \,\varphi (A)\vert\mathcal{H} = 0$. Then $[\varphi ]$ can be extended to a closed congruence on S.
Tangential limits of functions orthogonal to invariant subspaces
David
Protas
163-172
Abstract: For any inner function $\varphi$, let $ {M^ \bot }$ be the orthogonal complement of $ \varphi {H^2}$, in $ {H^2}$, where $ {H^2}$ is the usual Hardy space. The relationship between the tangential convergence of all functions in $ {M^ \bot }$ and the finiteness of certain sums and integrals involving $ \varphi$ is studied. In particular, it is shown that the tangential convergence of all functions in $ {M^ \bot }$ is a stronger condition than the tangential convergence of $ \varphi$, itself.
On the rank of a space
Christopher
Allday
173-185
Abstract: The rank of a space is defined as the dimension of the highest dimensional torus which can act almost-freely on the space. (By an almost-free action is meant one for which all the isotropy subgroups are finite.) This definition is shown to extend the classical definition of the rank of a Lie group. A conjecture giving an upper bound for the rank of a space in terms of its rational homotopy is investigated.
$G\sb{0}$ of a graded ring
Leslie G.
Roberts
187-195
Abstract: We consider the Grothendieck group ${G_0}$ of various graded rings, including ${G_0}(A_n^r)$ where A is a commutative noetherian ring, and $A_n^r$ is the A-subalgebra of the polynomial ring $A[{X_0}, \ldots ,{X_n}]$ generated by monomials of degree r. If A is regular, then ${G_0}(A_n^r)$ has a ring structure. The ideal class groups of these rings are also considered.
Cobordism Massey products
J. C.
Alexander
197-214
Abstract: The structure of Massey products is introduced into the bordism ring ${\Omega ^S}$ of manifolds with structure S and machinery is developed to investigate it. The product is changed to one in homotopy via the Pontrjagin-Thom map and methods for computation via the Adams spectral sequence are developed. To illustrate the methods, some products in ${\Omega ^{SU}}$ and $ {\Omega ^{Sp}}$ are computed.
A general class of factors of $E^4$
Leonard R.
Rubin
215-224
Abstract: In this paper we prove that any upper semicontinuous decomposition of $E^n$ which is generated by a trivial defining sequence of cubes with handles determines a factor of $ E^{n + 1}$. An important corollary to this result is that every 0-dimensional point-like decomposition of $E^3$ determines a factor of $E^4$. In our approach we have simplified the construction of the sequence of shrinking homeomorphisms by eliminating the necessity of shrinking sets piecewise in a collection of n-cells, the technique employed by R. H. Bing in the original result of this type.
A continuity theorem for Fuchsian groups
C. K.
Wong
225-239
Abstract: On a given Riemann surface, fix a discrete (finite or infinite) sequence of points $\{ {P_k}\} ,k = 1,2,3, \ldots ,$ and associate to each ${P_k}$ an ``integer'' ${\nu _k}$ (which may be $1,2,3, \ldots ,{\text{or}}\;\infty )$. This sequence of points and ``integers'' is called a ``signature'' on the Riemann surface. With only a few exceptions, a Riemann surface with signature can always be represented by a Fuchsian group. We investigate here the dependence of the group on the number ${\nu _k}$. More precisely, keeping the points ${P_k}$ fixed, we vary the numbers ${\nu _k}$ in such a way that the signature tends to a limit signature. We shall prove that the corresponding representing Fuchsian group converges to the Fuchsian group which corresponds to the limit signature.
Analytic sets as branched coverings
John
Stutz
241-259
Abstract: In this paper we study the relation between the tangent structure of an analytic set V at a point p and the local representation of V as a branched covering. A prototype for our type of result is the fact that one obtains a covering of minimal degree by projecting transverse to the Zariski tangent cone $ {C_3}(V,p)$. We show, for instance, that one obtains the smallest possible branch locus for a branched covering if one projects transverse to the cone $ {C_4}(V,p)$. This and similar results show that points where the various tangent cones ${C_i}(V,p),i = 4,5,6$, have minimal dimension give rise to the simplest branched coverings. This observation leads to the idea of ``Puiseux series normalization", generalizing the situation in one dimension. These Puiseux series allow us to strengthen some results of Hironaka and Whitney on the local structure of certain types of singularities.
On replacing proper Dehn maps with proper embeddings
C. D.
Feustel
261-267
Abstract: In this paper we develop algebraic and geometric conditions which imply that a given proper Dehn map can be replaced by an embedding. The embedding, whose existence is implied by our theorem, retains most of the algebraic and geometric properties required in the original proper Dehn map.
Zeros of partial sums and remainders of power series
J. D.
Buckholtz;
J. K.
Shaw
269-284
Abstract: For a power series $ f(z) = \Sigma _{k = 0}^\infty {a_k}{z^k}$ let ${s_n}(f)$ denote the maximum modulus of the zeros of the nth partial sum of f and let $ {r_n}(f)$ denote the smallest modulus of a zero of the nth normalized remainder $\Sigma _{k = n}^\infty {a_k}{z^{k - n}}$. The present paper investigates the relationships between the growth of the analytic function f and the behavior of the sequences $ \{ {s_n}(f)\}$ and $\{ {r_n}(f)\}$. The principal growth measure used is that of R-type: if $R = \{ {R_n}\}$ is a nondecreasing sequence of positive numbers such that $\lim ({R_{n + 1}}/{R_n}) = 1$, then the R-type of f is $ {\tau _R}(f) = \lim \sup \vert{a_n}{R_1}{R_2} \cdots {R_n}{\vert^{1/n}}$. We prove that there is a constant P such that $\displaystyle {\tau _R}(f)\lim \inf ({s_n}(f)/{R_n}) \leqq P\quad {\text{and}}\quad {\tau _R}(f)\lim \sup ({r_n}(f)/{R_n}) \geqq (1/P)$ for functions f of positive finite R-type. The constant P cannot be replaced by a smaller number in either inequality; P is called the power series constant.
The gliding humps technique for $FK$-spaces
G.
Bennett
285-292
Abstract: The gliging humps technique has been used by various authors to establish the existence of bounded divergent sequences in certain summability domains. The purpose of this paper is to extend these results and to obtain analogous ones for sequence spaces other than c and m. This serves to unify and improve many known results and to obtain several new ones--applications include extensions to theorems of Dawson, Lorentz-Zeller, Snyder-Wilansky and Yurimyae. Improving another result of Wilansky allows us to consider countable collections of sequence spaces--applications including the proof of a conjecture of Hill and Sledd and extensions to theorems of Berg and Brudno. A related result of Petersen is also considered and a simple proof using the Baire category theorem is given.
A class of complete orthogonal sequences of broken line functions
J. L.
Sox
293-296
Abstract: A class of orthonormal sets of continuous broken line functions is defined. Each member is shown to be complete in ${L_2}(0,1)$ and pointwise convergence theorems are obtained for the Fourier expansions relative to these sets.
A new characterization of the $F$ set of a rational function
Marilyn K.
Oba;
Tom S.
Pitcher
297-308
Abstract: In the early part of this century G. Julia and P. Fatou extensively studied the iteration of functions on the complex plane. More recently Hans Brolin reopened the investigation. In this paper, we are interested in the F set which is the set of points at which the family of iterates of a given rational function R is not normal and in a measure which is in some sense naturally imposed on the F set by the iterates of R. We construct a sequence of probability measures via the inverse functions of the iterates of R and almost any starting point. The measure of primary interest is the weak limit of such sequences. These weak limits are supported by F and have certain invariance properties. We establish that this weak limit measure is unique and is ergodic with respect to the transformation R on the F set for a large class of rational functions. In the course of the proof of uniqueness we develop expressions for the logarithmic potential function and for the energy integral of F. We also establish inequalities for the capacity of the F set which become equalities for the polynomial case.
A unified approach to uniform real approximation by polynomials with linear restrictions
Bruce L.
Chalmers
309-316
Abstract: Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, $ \varepsilon$-interpolator approximation, and polynomial approximation with interpolation fit into our scheme.
Products of complexes and Fr\'echet spaces which are manifolds
James E.
West
317-337
Abstract: It is shown that if a locally finite-dimensional simplicial complex is given the ``barycentric'' metric, then its product with any Fréchet space X of suitably high weight is a manifold modelled on X, provided that X is homeomorphic to its countably infinite Cartesian power. It is then shown that if X is Banach, all paracompact X-manifolds may be represented (topologically) by such products.
Hereditary properties and maximality conditions with respect to essential extensions of lattice group orders
Jorge
Martinez
339-350
Abstract: An l-group will be denoted by the pair (G, P), where G is the group and P is the positive cone. The cone Q is an essential extension of P if every convex l-subgroup of (G, Q) is a convex l-subgroup of (G, P). Q is very essential over P if it is essential over P and for each $0 \ne x \in G$ and each Q-value D of x, there is a unique P-value C of x containing D. We seek conditions which are preserved by essential extensions; normal valuedness and the existence of a finite basis are so preserved. We then investigate l-groups which have the property that their positive cone has no proper very essential extensions. Q is a c-essential extension of P if Q is essential over P and every closed convex l-subgroup of (G, Q) is closed in (G, P). We show that a wreath product of totally ordered groups has no proper very c-essential extensions. We derive a sufficient condition for the nonexistence of such extensions in case the l-group has property (F): no positive element exceeds an infinite set of pairwise disjoint elements.
Segment-preserving maps of partial orders
Geert
Prins
351-360
Abstract: A bijective map from a partial order P to a partial order Q is defined to be segment-preserving if the image of every segment in P is a segment in Q. It is proved that a partial order P with 0-element admits nontrivial segment-preserving maps if and only if P is decomposable in a certain sense. By introducing the concept of ``strong'' segment-preserving maps further insight into the relations between segment-preserving maps and decompositions of partial orders is obtained.
Regular functions $f(z)$ for which $z f'(z)$ is $\alpha$-spiral
Richard J.
Libera;
Michael R.
Ziegler
361-370
Abstract: A function $ f(z) = z + \Sigma _{n = 2}^\infty {a_n}{z^n}$ regular in the open unit disk $\Delta = \{ z:\vert z\vert < 1\}$ is a (univalent) $\alpha$-spiral function for real $\alpha ,\vert\alpha \vert < \pi /2$, if $f(z) \in {\mathcal{F}_\alpha }$. A fundamental result of this paper shows that the transformation $\displaystyle {f_ \ast }(z) = \frac{{azf((z + a)/(1 + \bar az))}}{{f(a)(z + a){{(1 + \bar az)}^{{e^{ - 2i\alpha }}}}}}$ defines a function in $ {\mathcal{F}_\alpha }$ whenever $f(z)$ is in $ {\mathcal{F}_\alpha }$ and a is in $\Delta$. If $g(z)$ is regular in $ \Delta ,g(0) = 0$ and $ g'(0) = 1$, then $ g(z)$ is in ${\mathcal{G}_\alpha }$ if and only if $ zg'(z)$ is in ${\mathcal{F}_\alpha }$. The main result of the paper is the derivation of the sharp radius of close-to-convexity for each class ${\mathcal{G}_\alpha }$; it is given as the solution of an equation in r which is dependent only on $ \alpha$. (Approximate solutions of this equation were made by computer and these suggest that the radius of close-to-convexity of the class $\mathcal{G} = { \cup _\alpha }{\mathcal{G}_\alpha }$ is approximately $ .99097^{+}$.) Additional results are also obtained such as the radius of convexity of $ {\mathcal{G}_\alpha }$, a range of $\alpha$ for which $g(z)$ in $ {\mathcal{G}_\alpha }$ is always univalent is given, etc. These conclusions all depend heavily on the transformation cited above and its analogue for ${\mathcal{G}_\alpha }$.
Two methods of integrating Monge-Amp\`ere's equations. II
Michihiko
Matsuda
371-386
Abstract: Generalizing the notion of an integrable system given in the previous note [2], we shall define an integrable system of higher order, and obtain the following results: 1. A linear hyperbolic equation is solved by integrable systems of order n if and only if its $(n + 1)$th Laplace invariant $ {H_n}$ vanishes. 2. An equation of Laplace type is solved by integrable systems of the second order if and only if the transformed equation by the associated Imschenetsky transformation is solved by integrable systems of the first order.
Operator and dual operator bases in linear topological spaces
William B.
Johnson
387-400
Abstract: A net $\{ {S_d}:d \in D\}$ of continuous linear projections of finite range on a Hausdorff linear topological space V is said to be a Schauder operator basis--S.O.B. --(resp. Schauder dual operator basis--S.D.O.B.) provided it is pointwise bounded and converges pointwise to the identity operator on V, and ${S_e}{S_d} = {S_d}$ (resp. ${S_d}{S_e} = {S_d}$) whenever $e \geqq d$. S.O.B.'s and S.D.O.B.'s are natural generalizations of finite dimensional Schauder bases of subspaces. In fact, a sequence of operators is both a S.O.B. and S.D.O.B. iff it is the sequence of partial sum operators associated with a finite dimensional Schauder basis of subspaces. We show that many duality-theory results concerning Schauder bases can be extended to S.O.B.'s or S.D.O.B.'s. In particular, a space with a S.D.O.B. is semi-reflexive if and only if the S.D.O.B. is shrinking and boundedly complete. Several results on S.O.B.'s and S.D.O.B.'s were previously unknown even in the case of Schauder bases. For example, Corollary IV.2 implies that the strong dual of an evaluable space which admits a shrinking Schauder basis is a complete barrelled space.
Groups with finite dimensional irreducible representations
Calvin C.
Moore
401-410
Abstract: It will be shown that a locally compact group has a finite bound for the dimensions of its irreducible unitary representations if and only if it has a closed abelian subgroup of finite index. It will further be shown that a locally compact group has all of its irreducible representations of finite dimension if and only if it is a projective limit of Lie groups with the same property, and finally that a Lie group has this property if and only if it has a closed subgroup H of finite index such that H modulo its center is compact.
Diffusion semigroups on abstract Wiener space
M. Ann
Piech
411-430
Abstract: The existence of a semigroup of solution operators associated with a second order infinite dimensional parabolic equation of the form $\partial u/\partial t = {L_x}u$ was previously established by the author. The present paper investigates the relationship between ${L_x}$ and the infinitesimal generator $\mathcal{U}$ of the semigroup. In particular, it is shown that $ \mathcal{U}$ is the closure of ${L_x}$ in a natural sense. This in turn implies certain uniqueness results for both the semigroup and for solutions of the parabolic equation.
Uniformly bounded representations for the Lorentz groups
Edward N.
Wilson
431-438
Abstract: A family of uniformly bounded class 1 representations of the Lorentz groups is constructed. This family of representations includes, but is larger than, a similar family of representations constructed by Lipsman. The construction technique relies on a multiplicative analysis of various operators under a Mellin transform.
Categorical $W\sp{\ast} $-tensor product
John
Dauns
439-456
Abstract: If A and B are von Neumann algebras and $A\bar \otimes B$ denotes their categorical ${C^ \ast }$-tensor product with the universal property, then the von Neumann tensor product $ A\nabla B$ of A and B is defined as $\displaystyle A\nabla B = {(A\bar \otimes B)^{ \ast \ast }}/J,$ where $J \subset {(A\bar \otimes B)^{\ast \ast}}$ is an appropriate ideal. It has the universal property.
Simple modules and centralizers
John
Dauns
457-477
Abstract: A class of modules generalizing the simple ones is constructed. Submodule structure and centralizers of quotient modules are completely determined. The above class of modules is used to study the primitive ideal structure of the tensor products of algebras.
Slicing theorems for $n$-spheres in Euclidean $(n+1)$-space
Robert J.
Daverman
479-489
Abstract: This paper describes conditions on the intersection of an n-sphere $\Sigma$ in Euclidean $(n + 1)$-space $ {E^{n + 1}}$ with the horizontal hyperplanes of $ {E^{n + 1}}$ sufficient to determine that the sphere be nicely embedded. The results generally are pointed towards showing that the complement of $\Sigma$ is 1-ULC (uniformly locally 1-connected) rather than towards establishing the stronger property that $\Sigma$ is locally flat. For instance, the main theorem indicates that ${E^{n + 1}} - \Sigma$ is 1-ULC provided each non-degenerate intersection of $\Sigma$ and a horizontal hyperplane be an $ (n - 1)$-sphere bicollared both in that hyperplane and in $\Sigma$ itself $(n \ne 4)$.
An extension of the theorem of Hartogs
L. R.
Hunt
491-495
Abstract: Hartogs proved that every function which is holomorphic on the boundary of the unit ball in $ {C^n},n > 1$, can be extended to a function holomorphic on the ball itself. It is conjectured that a real k-dimensional $ {\mathcal{C}^\infty }$ compact submanifold of $ {C^n},k > n$, is extendible over a manifold of real dimension $(k + 1)$. This is known for hypersurfaces (i.e., $k = 2n - 1$) and submanifolds of real codimension 2. It is the purpose of this paper to prove this conjecture and to show that we actually get C-R extendibility.