On the homotopy index for infinite-dimensional semiflows
Krzysztof P.
Rybakowski
351-382
Abstract: In this paper we consider semiflows whose solution operator is eventually a conditional $\alpha$-contraction. Such semiflows include solutions of retarded and neutral functional differential equations, of parabolic and certain other classes of partial differential equations. We prove existence of (nonsmooth) isolating blocks and index pairs for such semiflows, via the construction of special Lyapunov functionals. We show that index pairs enjoy all the properties needed to define the notion of a homotopy index, thus generalizing earlier results of Conley [2]. Finally, using a result of Mañé [9], we prove that, under additional smoothness assumptions on the semiflow, the homotopy index is essentially a finite-dimensional concept. This gives a formal justification of the applicability of Ważewski's Principle to infinite-dimensional problems. Several examples illustrate the theory.
Finiteness theorems for approximate fibrations
D. S.
Coram;
P. F.
Duvall
383-394
Abstract: This paper concerns conditions on the point inverses of a mapping between manifolds which insure that it is an approximate fibration almost everywhere. The primary condition is $ {\pi _i}$-movability, which says roughly that nearby point inverses include isomorphically on the $i$th shape group into a mutual neighborhood. Suppose $f:{M^m} \to {N^n}$ is a $U{V^1}$ mapping which is $ {\pi _i}$-movable for $i \leqslant k - 1$, and $n \geqslant k + 1$. An earlier paper proved that $ f$ is an approximate fibration when $ m \leqslant 2k - 1$. If instead $m = 2k$, this paper proves that there is a locally finite set $ S \subset N$ such that $f\vert{f^{ - 1}}(N - S)$ is an approximate fibration. Also if $ m = 2k + 1$ and all of the point inverses are FANR's with the same shape, then there is a locally finite set $ E \subset N$ such that $f\vert{f^{ - 1}}(N - E)$ is an approximate fibration.
On an extension of localization theorem and generalized Conner conjecture
Satya
Deo;
Tej Bahadur
Singh;
Ram Anugrah
Shukla
395-402
Abstract: Let $G$ be a compact Lie group. Then Borel-Segal-Quillen-Hsiang localization theorems are known for any $G$-space $X$ where $X$ is any compact Hausdorff space or a paracompact Hausdorff space of finite cohomology dimension. The Conner conjecture proved by Oliver and its various generalizations by Skjelbred are also known for only these two classes of spaces. In this paper we extend all of these results for the equivariant category of all finitistic $G$-spaces. For the case when $G = {Z_p}$ or $G = T$ (torus) some of these results were already proved by Bredon.
The automorphism group of a composition of quadratic forms
C.
Riehm
403-414
Abstract: Let $U \times X \to X$ be a (bilinear) composition $(u,\,x) \mapsto ux$ of two quadratic spaces $U$ and $X$ over a field $F$ of characteristic $\ne 2$ and assume there is a vector in $ U$ which induces the identity map on $X$ via this composition. Define $G$ to be the subgroup of $O(U) \times O(X)$ consisting of those pairs $(\phi ,\,\psi )$ satisfying $\phi (u)\psi (x) = \psi (ux)$ identically and define ${G_X}$ to be the projection of $G$ on $O(X)$. The group $G$ is investigated and in particular it is shown that its connected component, as an algebraic group, is isogenous to a product of two or three classical groups and so is reductive. Necessary and sufficient conditions are given for ${G_X}$ to be transitive on the unit sphere of $ X$ when $U$ and $X$ are Euclidean spaces.
Weakly Ramsey $P$ points
Ned I.
Rosen
415-427
Abstract: If the continuum hypothesis (CH) holds, then for any $n$ Ramsey $P$ point $D$ and any $k \geqslant 1$ there exist many $n + k$ Ramsey $P$ points which are immediate Rudin-Keisler successors of $D$. There exist (CH) many 5 Ramsey $P$ points whose constellations are not linearly ordered.
Structure of complex linear differential equations
Margalit
Ronen
429-444
Abstract: In this paper homogeneous linear differential equations in the complex domain are considered. Relations between (a) properties of the zeros of solutions, (b) factorization of the equation into linear factors, and (c) nonvanishing of corresponding Wronskians are proved.
On the fullness of surjective maps of an interval
Harold
Proppe;
Abraham
Boyarsky
445-452
Abstract: Let $I = [0,\,1]$, $ \mathcal{B}$ = Lebesgue measurable subsets of $[0,\,1]$, and let $\lambda$ denote the Lebesgue measure on $(I,\,\mathcal{B})$. Let $\tau :I \to I$ be measurable and surjective. We say $\tau$ is full, if for all $A \in \mathcal{B}$, $ \lambda (A) > 0$, $ \tau (A),\,{\tau ^2}(A), \ldots$, measurable, the condition (1) $\displaystyle \mathop {\lim }\limits_{n \to \infty } \lambda ({\tau ^n}(A)) = 1$ holds. We say $\tau$ is interval full if (1) holds for any interval $ A \subset I$. In this note, we give an example of $ \tau :I \to I$ which is continuous and interval full, but not full. We also show that for a class of transformations $ \tau$ satisfying Renyi's condition, interval fullness implies fullness. Finally, we show that fullness is not preserved under limits on the surjections.
Probabilistic and deterministic averaging
N. H.
Bingham;
Charles M.
Goldie
453-480
Abstract: Let $\{ {S_n}\}$ be a random walk whose step distribution has positive mean $\mu$ and an absolutely continuous component. For any bounded measurable function $f$, a Marcinkiewicz-Zygmund strong law in an $r$-quick version (a 'Lai strong law') is proved for $ f({S_n})$, assuming existence of a suitable higher moment of the step distribution. This is extended to show ${n^{ - \alpha }}\{ \sum\nolimits_1^n {f({S_k})} - \int_0^n {f(\mu t)dt\} \to 0}$ ($r$-quickly). These results remain true when the step distribution is lattice, provided $ f$ is constant between lattice points. Certain intermediate results on renewal theory, mixing, local limit theory, ladder height, and a strong law of Lai for mixing random variables are of independent interest.
Focal sets and real hypersurfaces in complex projective space
Thomas E.
Cecil;
Patrick J.
Ryan
481-499
Abstract: Let $M$ be a real submanifold of $ C{P^m}$, and let $ J$ denote the complex structure. We begin by finding a formula for the location of the focal points of $M$ in terms of its second fundamental form. This takes a particularly tractable form when $M$ is a complex submanifold or a real hypersurface on which $J\xi$ is a principal vector for each unit normal $ \xi$ to $M$. The rank of the focal map onto a sheet of the focal set of $M$ is also computed in terms of the second fundamental form. In the case of a real hypersurface on which $ J\xi$ is principal with corresponding principal curvature $\mu$, if the map onto a sheet of the focal set corresponding to $\mu$ has constant rank, then that sheet is a complex submanifold over which $M$ is a tube of constant radius (Theorem 1). The other sheets of the focal set of such a hypersurface are given a real manifold structure in Theorem 2. These results are then employed as major tools in obtaining two classifications of real hypersurfaces in $ C{P^m}$. First, there are no totally umbilic real hypersurfaces in $ C{P^m}$, but we show: Theorem 3. Let $M$ be a connected real hypersurface in $ C{P^m}$, $m \geqslant 3$, with at most two distinct principal curvatures at each point. Then $M$ is an open subset of a geodesic hypersphere. Secondly, we show that there are no Einstein real hypersurfaces in $C{P^m}$ and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the Ricci tensor in Theorem 4.
Ramsey numbers for the pair sparse graph-path or cycle
S. A.
Burr;
P.
Erdős;
R. J.
Faudree;
C. C.
Rousseau;
R. H.
Schelp
501-512
Abstract: Let $G$ be a connected graph on $ n$ vertices with no more than $ n(1 + \varepsilon )$ edges, and ${P_k}$ or ${C_k}$ a path or cycle with $k$ vertices. In this paper we will show that if $n$ is sufficiently large and $\varepsilon$ is sufficiently small then for $ k$ odd $\displaystyle r(G,\,{C_k}) = 2n - 1.$ Also, for $k \geqslant 2$, $\displaystyle r(G,\,{P_k}) = \max \{ n + [k/2] - 1,\,n + k - 2 - \alpha \prime - \delta \} ,$ where $\alpha \prime$ is the independence number of an appropriate subgraph of $G$ and $\delta$ is 0 or $1$ depending upon $n$, $k$ and $\alpha \prime $.
Picard's theorem
Douglas
Bridges;
Allan
Calder;
William
Julian;
Ray
Mines;
Fred
Richman
513-520
Abstract: This paper deals with the numerical content of Picard's Theorem. Two classically equivalent versions of this theorem are proved which are distinct from a computational point of view. The proofs are elementary, and constructive in the sense of Bishop. A Brouwerian counterexample is given to the original version of the theorem.
A Loeb-measure approach to theorems by Prohorov, Sazonov and Gross
Tom L.
Lindstrøm
521-534
Abstract: We use the Loeb-measure of nonstandard analysis to prove three classical results on limit measures: Let ${\{ {\mu _i}\} _{i \in I}}$ be a projective system of Radon measures, we use the Loeb-measure $L({\tilde \mu _E})$ for an infinite $E \in {}^{\ast}I$ and a standard part map to construct a Radon limit measure on the projective limit (Prohorov's Theorem). Using the Loeb-measures on hyperfinite dimensional linear spaces, we characterize the Fourier-transforms of measures on Hilbert spaces (Sazonov's Theorem), and extend cylindrical measures on Hilbert spaces to $\sigma$-additive measures on Banach spaces (Gross' Theorem).
Geometry and the Pettis integral
Robert F.
Geitz
535-548
Abstract: Convex sets involving the range of a vector-valued function are constructed. These constructions provide a complete characterization of the bounded Pettis integrable functions.
Continuous measures and lacunarity on hypergroups
Richard C.
Vrem
549-556
Abstract: The relationship between measures on a compact hypergroup $K$ whose Fourier-Stieltjes transforms vanish at infinity and the space ${M_c}(K)$ of continuous measures is studied. Examples are provided of measures $\mu$ with $\hat \mu$ vanishing at infinity and $\mu \in {M_c}(K)$. Sufficient conditions are given for $ \hat \mu \in {c_0}(\hat K)$ to imply $ \mu \in {M_c}(K)$. An investigation of Helson sets on compact abelian hypergroups is initiated and the study of Sidon sets on compact abelian hypergroups is continued. A class of compact abelian hypergroups is shown to have no infinite Helson sets and no infinite Sidon sets. This result generalizes results of D. L. Ragozin and D. Rider on central Sidon sets for compact connected Lie groups.
Weak $P$-points in compact CCC $F$-spaces
Alan
Dow
557-565
Abstract: Using a technique due to van Mill we show that each compact ccc $ F$-space of weight greater than ${2^\omega }$ contains a weak $ P$-point, i.e. a point $ x \in X$ such that $x \notin \overline F$ for each countable $ F \subset X - \{ x\}$. We show that, assuming $BF(c)$, each nowhere separable compact $ F$-space has a weak $ P$-point. We show the existence of points which are not limit points of any countable nowhere dense set in compact $F$-spaces of weight ${\aleph _1}$. We also discuss remote points and points not the limit point of any countable discrete set.
An alternating sum formula for multiplicities in $L\sp{2}(\Gamma \backslash G)$
Roberto J.
Miatello
567-574
Abstract: We prove an alternating sum formula for multiplicities in ${L^2}(\Gamma \backslash G)$, where $ G$ is a semisimple Lie group of split rank one with finite center and $ \Gamma$ is a discrete cocompact torsion free subgroup.
Collections of subsets with the Sperner property
Jerrold R.
Griggs
575-591
Abstract: Let $X = \{ 1, \ldots ,n\}$ and $Y = \{ 1, \ldots ,k\}$, $k \leqslant n$. Let $C(n,\,k)$ be the subsets of $X$ which intersect $Y$, ordered by inclusion. Lih showed that $ C(n\,,k)$ has the Sperner property. Here it is shown that $C(n,\,k)$ has several stronger properties. A nested chain decomposition is constructed for $C(n,\,k)$ by bracketing. $C(n,\,k)$ is shown to have the LYM property. A more general class of collections of subsets is studied: Let $X$ be partitioned into parts ${X_1}, \ldots ,{X_m}$, let ${I_1}, \ldots ,{I_m}$ be subsets of $\{ 0,\,1, \ldots ,\,n\} $, and let $P = \{ Z \subset X\vert\vert Z \cap {X_i}\vert\, \in {I_i},\,1 \leqslant i \leqslant m\}$. Sufficient conditions on the ${I_i}$ are given for $P$ to be LYM, or at least Sperner, and examples are provided in which $P$ is not Sperner. Other results related to Sperner's theorem, the Kruskal-Katona theorem, and the LYM inequality are presented.
A generalization of Torres' second relation
Lorenzo
Traldi
593-610
Abstract: Let $L = {K_1} \cup \cdots \cup {K_\mu }$ be a tame link in ${S^3}$ of $ \mu \geqslant 2$ components, and let ${L_\mu }$ be its sublink ${L_\mu } = L - {K_\mu }$. Let $H$ and ${H_\mu }$ be the abelianizations of ${\pi _1}({S^3} - L)$ and ${\pi _1}({S^3} - {L_\mu })$, respectively, and let $ {t_1}, \ldots ,{t_\mu }$ (resp., $ {t_1}, \ldots ,{t_{\mu - 1}}$) be the usual generators of $H$ (resp., ${H_\mu }$). If $\phi :{\mathbf{Z}}H \to {\mathbf{Z}}{H_\mu }$ is the (unique) ring homomorphism with $\phi ({t_i}) = {t_i}$ for $1 \leqslant i < \mu $, and $\phi ({t_\mu }) = 1$, then Torres' second relation is equivalent to the statement that $\phi {E_1}(L) = (({\prod _{i < \mu }}t_i^{{l_i}}) - 1) \cdot {E_1}({L_\mu })$, where for $1 \leqslant i < \mu $, ${l_i}$ is the linking number ${l_i} = l({K_i},\,{K_\mu })$. We prove that if $I{H_\mu }$ is the augmentation ideal of $ {\mathbf{Z}}{H_\mu }$, then for any $ k \geqslant 2$, $\displaystyle {E_{k - 1}}({L_\mu }) + \left( {\left( {\prod\limits_{i < \mu } {... ...q \phi {E_k}(L) \subseteq {E_{k - 1}}({L_\mu }) + I{H_\mu }\cdot{E_k}({L_\mu })$ and examples are given to indicate that either of these inclusions may be an equality. This theorem is used to generalize certain known properties of ${E_1}$ to the higher ideals.
Products of two Borel measures
Roy A.
Johnson
611-625
Abstract: Let $\mu$ and $\nu$ be finite Borel measures on Hausdorff spaces $ X$ and $Y$, respectively, and suppose product measures $ \mu \times {}_1\nu$ and $\mu \times {}_2\nu$ are defined on the Borel sets of $X \times Y$ by integrating vertical and horizontal cross-section measure, respectively. Sufficient conditions are given so that $\mu \times {}_1\nu = \mu \times {}_2\nu$ and so that the usual product measure $\mu \times \nu$ can be extended to a Borel measure on $X \times Y$ by means of completion. Examples are given to illustrate these ideas.
Classes of Baire functions
Gregory V.
Cox;
Paul D.
Humke
627-635
Abstract: Let $\mathcal{A}$ and $ \mathcal{P}$ denote the sets of approximately continuous and almost everywhere continuous functions, and ${B_1}(F)$ denote Baire's first class generated by $F$. The classes ${B_1}(\mathcal{A})$, ${B_1}(\mathcal{P})$, ${B_1}(\mathcal{A} \cap \mathcal{P})$, and Grande's class $ \mathcal{A}{\mathcal{P}_1}$ are investigated in some detail. Although Grande's question of whether ${B_1}(\mathcal{A} \cap \mathcal{P}) = {B_1}(\mathcal{A}) \cap {B_1}(\mathcal{A}) \cap \mathcal{A}{\mathcal{P}_1}$ is not settled, we do show, among other results, that $ \mathcal{A}{\mathcal{P}_1} \subset {B_1}(\mathcal{P})$.
Nonexponential leaves at finite level
John
Cantwell;
Lawrence
Conlon
637-661
Abstract: Previous examples of leaves with nonexponential and nonpolynomial growth (due to G. Hector) have occurred at infinite level. Here the same growth types are produced at finite level in open, saturated sets of leaves without holonomy. Such sets consist of leaves with only one or two locally dense ends, and it is shown that the exotic growth types only occur in the case of one locally dense end. Finally, ${C^1}$-foliations are produced with open, saturated sets as above in which the leaves have strictly fractional growth.
The semicellularity theorem
Gene G.
Garza
663-676
Abstract: In this paper are proved several theorems concerning semicellularity of subsets of $2$-spheres in ${E^3}$. In particular, it is shown that a cellular arc or disk on a $2$-sphere which has no nonpiercing points is semicellular in both complementary domains of the $ 2$-sphere. The proof is entirely geometrical and involves the idea of piercing points.
A class of $L\sp{1}$-convergence
R.
Bojanić;
Č. V.
Stanojević
677-683
Abstract: It is proved that if the Fourier coefficients $ \{ {a_n}\}$ of $f \in {L^1}(0,\,\pi )$ satisfy $({\ast}){n^{ - 1}}\sum\nolimits_{k = n}^{2n} {{k^p}\vert\Delta {a_n}\vert p = o(1)}$, for some $1 < p \leqslant 2$, then $\vert\vert{s_n} - f\vert\vert = o(1)$, if and only if $ {a_n}\lg n = o(1)$. For cosine trigonometric series with coefficients of bounded variation and satisfying $({\ast})$ it is proved that a necessary and sufficient condition for the series to be a Fourier series is $ \{ {a_n}\} \in \mathcal{C}$, where $ \mathcal{C}$ is the Garrett-Stanojević [4] class.