Oscillation and asymptotic behavior of systems of ordinary linear differential equations
Carl H.
Rasmussen
1-47
Abstract: Conditions are established for oscillatory and asymptotic behavior for first-order matrix systems of ordinary differential equations, including Hamiltonian systems in the selfadjoint case. Asymptotic results of Hille, Shreve, and Hartman are generalized. Disconjugacy criteria of Ahlbrandt, Tomastik, and Reid are extended.
Global analysis on PL-manifolds
Nicolae
Teleman
49-88
Abstract: The paper deals mainly with combinatorial structures; in some cases we need refinements of combinatorial structures. Riemannian metrics are defined on any combinatorial manifold M. The existence of distance functions and of Riemannian metrics with ``constant volume density'' implies smoothing. A geometric realization of $ {\text{PL}}\left( m \right){\text{/O}}\left( m \right)$ is given in terms of Riemannian metrics. A graded differential complex $ {\Omega ^ {\ast} }( M )$ is constructed: it appears as a subcomplex of Sullivan's complex of piecewise differentiable forms. In the complex $ {\Omega ^{\ast}}( M )$ the operators $d$, $\ast$, $\delta$, $\Delta$ are defined. A Rellich chain of Sobolev spaces is presented. We obtain a Hodge-type decomposition theorem, and the Hodge homomorphism is defined and studied. We study also the combinatorial analogue of the signature operator.
${\bf Z}\sb{(2)}$-knot cobordism in codimension two, and involutions on homotopy spheres
Chao Chu
Liang
89-97
Abstract: Let ${Z_{(2)}}$ denote the ring of rational 2-adic integers. In this paper, we consider the group $ {\Psi _k}$ of $ {Z_{(2)}}$-cobordism classes of ${Z_{(2)}} - \operatorname{knot} (\Sigma ^{k + 2}, \,{K^k})$, where $\Sigma$ is a 1-connected ${Z_{(2)}}$-sphere ${Z_{(2)}}$-cobordant to $ {S^{k + 2}}$, and K is a 1-connected ${Z_{(2)}}$-sphere embedded in $\Sigma$ with trivial normal bundle. For $n \geqslant 3$, we will prove that ${\Psi _{2n}} = 0$ and $ {\Psi _{2n - 1}} = {C_\varepsilon }({Z_{(2)}})$, $\varepsilon = {( - 1)^n}$. Also, we will show that the group $\Theta _{4m - 1}^{4m + 1}$ of L-equivalence classes of differentiable involutions on $ (4m + 1)$-homotopy spheres with codimension two fixed point sets defined by Bredon contains infinitely many copies of Z.
Continuous functions on countable compact ordered sets as sums of their increments
Gadi
Moran
99-112
Abstract: Every continuous function from a countable compact linearly ordered set A into a Banach space V (vanishing at the least element of A ) admits a representation as a sum of a series of its increments (in the topology of uniform convergence). This series converges to no other sum under rearrangements of its terms. A uniqueness result to the problem of representation of a regulated real function on the unit interval as a sum of a continuous and a steplike function is derived.
On the Littlewood-Paley theory for mixed norm spaces
John A.
Gosselin
113-124
Abstract: An inequality of Littlewood-Paley type is proved for the mixed norm spaces $ {L_P}({l_r})$, $1 < p$, $r < \infty$, on the interval $[0,1]$. This result makes use of recent work by C. Fefferman and A. Cordoba on the boundedness of singular integrals on these spaces. As an application of this inequality, boundedness of the lacunary maximal partial sum operator for Walsh-Fourier series on ${L_p}({l_r})$ is established. This result can be viewed as an extension of a similar result for the Hardy-Littlewood maximal function due to C. Fefferman and E. M. Stein.
Generic differentiability of Lipschitzian functions
G.
Lebourg
125-144
Abstract: It is shown that, in separable topological vector spaces which are Baire spaces, the usual properties that have been introduced to study the local ``first order'' behaviour of real-valued functions which satisfy a Lipschitz type condition are ``generically'' equivalent and thus lead to a unique class of ``generically smooth'' functions. These functions are characterized in terms of tangent cones and directional derivatives and their ``generic'' differentiability properties are studied. The results extend some of the well-known differentiability properties of continuous convex functions.
Subnormal operators quasisimilar to an isometry
William W.
Hastings
145-161
Abstract: Let $V = {V_0} \oplus {V_1}$ be an isometry, where $ {V_0}$ is unitary and $ {V_1}$ is a unilateral shift of finite multiplicity n. Let $S = {S_0} \oplus {S_1}$ be a subnormal operator where ${S_0} \oplus {S_1}$ is the normal decomposition of S into a normal operator $ {S_0}$ and a completely nonnormal operator ${S_1}$. It is shown that S is quasisimilar to V if and only if ${S_0}$ is unitarily equivalent to ${V_0}$ and ${S_1}$ is quasisimilar to ${V_1}$. To prove this, a standard representation is developed for n-cyclic subnormal operators. Using this representation, the class of subnormal operators which are quasisimilar to ${V_1}$ is completely characterized.
Formal and convergent power series solutions of singular partial differential equations
Stanley
Kaplan
163-183
Abstract: A class of singular first-order partial differential equations is described for which an analogue of a theorem of M. Artin on the solutions of analytic equations holds: given any formal power series solution and any nonnegative integer v, a convergent power series solution may be found which agrees with the given formal solution up to all terms of order $\leqslant v$.
On the embedding problem for $1$-convex spaces
Vo Van
Tan
185-197
Abstract: In this paper we provide a necessary and sufficient condition for 1-convex spaces (i.e., strongly pseudoconvex spaces) which can be realized as closed analytic subvarieties in some ${C^N} \times {P_M}$. A construction of some normal 3-dimensional 1-convex space which cannot be embedded in any $ {C^N} \times {P_M}$ is given. Furthermore, we construct explicitly a non-kählerian 3-dimensional 1-convex manifold which answers a question posed by Grauert.
On natural radii of $p$-adic convergence
B.
Dwork;
P.
Robba
199-213
Abstract: We study the radius of p-adic convergence of power series which represent algebraic functions. We apply the p-adic theory of ordinary linear differential equations to show that the radius of convergence is the natural one, provided the degree of the function is less than p. The study of similar questions for solutions of linear differential equations is indicated.
The theorems of Beth and Craig in abstract model theory. I. The abstract setting
J. A.
Makowsky;
S.
Shelah
215-239
Abstract: In the context of abstract model theory various definability properties, their interrelations and their relation to compactness are investigated.
Global ideal theory of meromorphic function fields
Norman L.
Alling
241-266
Abstract: It is shown that the ideal theories of the fields of all meromorphic functions on any two noncompact Riemann surfaces are isomorphic. Further, various new representation and factorization theorems are proved.
On $2$-blocks with semidihedral defect groups
Karin
Erdmann
267-287
Abstract: Let G be one of the following groups: ${L_3}(q)$ and $GL(2,\,q)$ with $q\, \equiv \,3\,(\bmod \,4)$, ${U_3}(q)$ and $GU(2,\,q)$ with $q\, \equiv \,1\,(\bmod \,4)$. This paper is concerned with 2-blocks B of G having semidihedral defect groups. In particular, vertices, sources and Green correspondents of the simple modules in B are determined and used to obtain the submodule structure of the indecomposable projective modules.
Reparametrization of $n$-flows of zero entropy
J.
Feldman;
D.
Nadler
289-304
Abstract: Let $\phi$, $\psi$ be two ergodic n-parameter flows which preserve finite probability measures on their spaces X, Y. Let T be a nullset-preserving map: $X\, \to \,Y$ sending each $\phi$-orbit homeomorphically to a $ \phi$-orbit. Then $ \phi$, $\psi$ are called homeomorphically orbit-equivalent. For $n\, = \,1$, there has been developed a theory of such equivalence: ``Loosely Bernoulli'' theory. A completely parallel theory exists for higher dimensions, except that it is necessary to impose a certain natural ``growth'' restriction on T, a restriction which is vacuous in the case $n\, = \,1$. In this paper we carry out this program, but only for the case of zero entropy.
An equivariant Wall obstruction theory
Jenny A.
Baglivo
305-324
Abstract: Let G be a finite group. For a certain class of CW-complexes with a G-action which are equivariantly dominated by a finite complex we define algebraic invariants to decide when the space is equivariantly homotopy or homology equivalent to a finite complex.
Hypoconvexity and essentially $n$-normal operators
Norberto
Salinas
325-351
Abstract: In this paper a classifying structure for the class of essentially n-normal operators on a separable Hilbert space is introduced, and various invariance properties of this classifying structure are studied. The notion of a hypoconvex subset of the algebra ${\mathcal{M}_n}$ of all complex $n\, \times \,n$ matrices is defined, and it is shown that the set of all equivalence classes of essentially n-normal operators (under a natural equivalence relation), whose reducing essential $n\, \times \,n$ matricial spectrum is a given hypoconvex set, forms an abelian group. It is also shown that this correspondence between hypoconvex subsets of $ {\mathcal{M}_n}$ and abelian groups is a homotopy invariant, covariant functor. This result is then used to prove that Toeplitz operators (on strongly pseudoconvex domains) which have homotopic continuous matricial symbols, are unitarily equivalent up to compact perturbation.
Some Ramsey-type numbers and the independence ratio
William
Staton
353-370
Abstract: If each of k, m, and n is a positive integer, there is a smallest positive integer $r\, = \,{r_k}\,(m,\,n)$ with the property that each graph G with at least r vertices, and with maximum degree not exceeding k, has either a complete subgraph with m vertices, or an independent subgraph with n vertices. In this paper we determine ${r_3}(3,\,n)\, = \,r(n)$, for all n. As a corollary we obtain the largest possible lower bound for the independence ratio of graphs with maximum degree three containing no triangles.
Geometric properties of a class of support points of univalent functions
Johnny E.
Brown
371-382
Abstract: Let S denote the set of functions $f(z)$ analytic and univalent in $\vert z\vert\, < \,1$, normalized by $f(0)\, = \,0$ and $ \operatorname{Re} \,L(g)$, $g \in S$. The support points corresponding to the point-evaluation functionals are determined explicitly and are shown to also be extreme points of S. New geometric properties of their omitte $\operatorname {arcs}\,\Gamma$ are found. In particular, it is shown that for each such support point $\Gamma$ lies entirely in a certain half-strip, $ \Gamma$ has monotonic argument, and the angle between radius and tangent vectors increases from zero at infinity to a finite maximum value at the tip of the $\operatorname{arc}\,\Gamma $. Numerical calculations appear to indicate that the known bound $\pi /4$ for the angle between radius and tangent vectors is actually best possible.
Approximation theory in the space of sections of a vector bundle
David
Handel
383-394
Abstract: Let $p:\,E\, \to \,B$ be a real m-plane bundle and S an n-dimensional subspace of the space of sections $ \Gamma (E)$ of E. S is said to be k-regular if whenever ${x_1},\, \ldots ,\,{x_k}$ are distinct points of B and ${\upsilon _i}\, \in \,{p^{ - 1}}({x_i})$, $ 1\, \leqslant \,i\, \leqslant \,k$, there exists a $\sigma \, \in \,S$ such that $\sigma ({x_i})\, = \,{\upsilon _i}$ for $ 1\, \leqslant \,i\, \leqslant k$. It is proved that if E has a Riemannian metric and B is compact Hausdorff with at least $ k\, + \,1$ points, then S is k-regular if and only if for each $ \varphi \, \in \,\Gamma (E)$, the set of best approximations to $ \varphi$ by elements of S has dimension at most n - km. This extends a classical theorem of Haar, Kolmogorov, and Rubinstein (the case of the product line bundle). Complex and quaternionic analogues of the above are obtained simultaneously. Existence and nonexistence of k-regular subspaces of a given dimension are obtained in special cases via cohomological methods involving configuration spaces. For example, if E is the product real $ (2m\, - \,1)$-plane bundle over a 2-dimensional disk, then $\Gamma (E)$ contains a k-regular subspace of dimension $2km\, - \,1$, but not one of dimension $2km\, - \,1\,\alpha (k)$, where $\alpha (k)$ denotes the number of ones in the dyadic expansion of k.
The dynamics of Morse-Smale diffeomorphisms on the torus
Steve
Batterson
395-403
Abstract: For orientation preserving diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior.
Holomorphic sectional curvatures of bounded homogeneous domains and related questions
J. E.
D’Atri
405-413
Abstract: This paper considers a class of homogeneous Kähler metrics which include the Bergman metrics on homogeneous bounded domains. We obtain various necessary conditions for (a) nonpositive holomorphic sectional curvature, (b) nonpositive sectional curvature, and (c) covariant constant curvature (symmetric metric). In particular, we give examples showing that there exist homogeneous bounded domains which in the Bergman metric have some positive holomorphic sectional curvature.