Nonsmooth analysis: differential calculus of nondifferentiable mappings
A. D.
Ioffe
1-56
Abstract: A new approach to local analysis of nonsmooth mappings from one Banach space into another is suggested. The approach is essentially based on the use of set-valued mappings of a special kind, called fans, for local approximation. Convex sets of linear operators provide an example of fans. Generally, fans can be considered a natural set-valued extension of linear operators. The first part of the paper presents a study of fans; the second is devoted to calculus and includes extensions of the main theorems of classical calculus.
On two-dimensional normal singularities of type ${}\sb\ast A\sb{n}$, ${}\sb\ast D\sb{n}$ and ${}\sb\ast E\sb{n}$
Shigeki
Ohyanagi
57-69
Abstract: Let $G$ be the weighted dual graph associated with a contractible curve $A = \cup {A_i}$. There are many combinations of the weights $ {A_i} \cdot {A_i}$ which make the graph contractible. If $G$ is a graph which is the weighted dual graph for a rational singularity with any combination of the weights, then $G$ is either $ {}_ \ast {A_n}$, ${}_ \ast {D_n}$ or $ _\ast{E_n}$.
Standard $3$-components of type ${\rm Sp}(6,\,2)$
Larry
Finkelstein;
Daniel
Frohardt
71-92
Abstract: It is shown that if $ G$ is a finite simple group with a standard $3$-component of type ${\text{Sp}}(6,2)$ and $G$ satisfies certain $2$-local and $3$-local conditions then either $G$ is isomorphic to ${\text{Sp}}(8,2)$ or $G$ is isomorphic to ${F_4}(2)$.
Some properties of measure and category
Arnold W.
Miller
93-114
Abstract: Several elementary cardinal properties of measure and category on the real line are studied. For example, one property is that every set of real numbers of cardinality less than the continuum has measure zero. All of the properties are true if the continuum hypothesis is assumed. Several of the properties are shown to be connected with the properties of the set of functions from integers to integers partially ordered by eventual dominance. Several, but not all, combinations of these properties are shown to be consistent with the usual axioms of set theory. The main technique used is iterated forcing.
Neighborhood fixed pendant vertices
S. E.
Anacker;
G. N.
Robertson
115-128
Abstract: If $x$ is pendant in $G$, then ${x^ \ast }$ denotes the unique vertex of $ G$ adjacent to $ x$. Such an $x$ is said to be neighborhood-fixed whenever ${x^ \ast }$ is fixed by $A(G - x)$. It is shown that if $ G$ is not a tree and has a pendant vertex, but no *-fixed pendant vertex, then there is a subgraph ${G^\char93 }$ of $G$ such that for some $y \in V({G^\char93 })$, $O(A{({G^\char93 })_y}) \geqslant t!$ where $ t$ is the maximum number of edges in a tree rooted in ${G^\char93 }$.
Fredholm and invertible $n$-tuples of operators. The deformation problem
Raul E.
Curto
129-159
Abstract: Using J. L. Taylor's definition of joint spectrum, we study Fredholm and invertible $n$-tuples of operators on a Hilbert space. We give the foundations for a "several variables" theory, including a natural generalization of Atkinson's theorem and an index which well behaves. We obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length. We then consider the deformation problem and solve it for the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate.
Time-ordered operators. I. Foundations for an alternative view of reality
Tepper L.
Gill
161-181
Abstract: The purpose of this paper is to present the proper framework for the mathematical foundations of time-ordered operators. We introduce a new mathematical process which we call the chronological process. This process generalizes the notion of a limit and allows us to recapture many properties lost in time-ordering. We then construct time-ordered integrals and evolution operators. We show that under reasonable assumptions, the time-ordered sum of two generators of contraction semigroups is a generator. This result resolves a question that has been debated in physics for forty years.
Lifting cycles to deformations of two-dimensional pseudoconvex manifolds
Henry B.
Laufer
183-202
Abstract: Let $M$ be a strictly pseudoconvex manifold with exceptional set $A$. Let $D \geqslant 0$ be a cycle on $ A$. Let $\omega :\mathfrak{M} \to Q$ be a deformation of $ M$. Kodaira's theory for deforming submanifolds of $ \mathfrak{M}$ is extended to the subspace $D$. Let $ \mathfrak{J}$ be the sheaf of germs of infinitesimal deformations of $ D$. Suppose that $ {H^1}(D,\mathfrak{J}) = 0$. If $\omega$ is the versal deformation, then $ D$ lifts to above a submanifold of $Q$. This lifting is a complete deformation of $ D$ with a smooth generic fiber. If all of the fibers of $\mathfrak{M}$ are isomorphic, then $ \omega$ is the trivial deformation. If $M$ has no exceptional curves of the first kind, then there exists $\omega$ such that only any given irreducible component of $A$ disappears as part of the exceptional set.
Nonseparable approximate equivalence
Donald W.
Hadwin
203-231
Abstract: This paper extends Voiculescu's theorem on approximate equivalence to the case of nonseparable representations of nonseparable ${C^ \ast }$-algebras. The main result states that two representations $ f$ and $g$ are approximately equivalent if and only if ${\text{rank}}f(x) = {\text{rank}}g(x)$ for every $x$. For representations of separable ${C^ \ast }$-algebras a multiplicity theory is developed that characterizes approximate equivalence. Thus for a separable $ {C^ \ast }$-algebra, the space of representations modulo approximate equivalence can be identified with a class of cardinal-valued functions on the primitive ideal space of the algebra. Nonseparable extensions of Voiculescu's reflexivity theorem for subalgebras of the Calkin algebra are also obtained.
Absolute continuity and uniqueness of measures on metric spaces
Pertti
Mattila
233-242
Abstract: We show that if two Borel regular measures on a separable metric space are in a suitable sense homogeneous, then they are mutually absolutely continuous. We use such absolute continuity theorems together with some density theorems to prove the uniqueness of measures more general than Haar measures.
Hulls of deformations in ${\bf C}\sp{n}$
H.
Alexander
243-257
Abstract: A problem of ${\text{E}}$. Bishop on the polynomially convex hulls of deformations of the torus is considered. Let the torus ${T^2}$ be the distinguished boundary of the unit polydisc in $ {{\mathbf{C}}^2}$. If $t \mapsto T_t^2$ is a smooth deformation of $ {T^2}$ in ${{\mathbf{C}}^2}$ and ${g_0}$ is an analytic disc in ${{\mathbf{C}}^2}$ with boundary in ${T^2}$, a smooth family of analytic discs $t \mapsto {g_t}$, is constructed with the property that the boundary of ${g_t}$ lies in $T_t^2$. This construction has implications for the polynomially convex hulls of the tori $ T_t^2$. An analogous problem for a $2$-sphere in $ {{\mathbf{C}}^2}$ is also considered.
Algebraic fiber bundles
Steven E.
Landsburg
259-273
Abstract: When $X$ is a finite simplicial complex and $ G$ is any of a certain class of groups, a classification of $ G$-principal bundles over $ X$ in terms of projective modules over a ring $R(G,X)$ is given. This generalizes Swan's classification of vector bundles and uses the results of Mulvey. Often, $R$ can be taken to be noetherian; in this case $ {\text{Spec}}(R)$ is usually reducible with "cohomologically trivial" irreducible components. Information is derived concerning the nature of projective modules over such rings, and some results are obtained indicating how such information reflects information about $ X$.
On the convergence of closed-valued measurable multifunctions
Gabriella
Salinetti;
Roger J.-B.
Wets
275-289
Abstract: In this paper we study the convergence almost everywhere and in measure of sequences of closed-valued multifunctions. We first give a number of criteria for the convergence of sequences of closed subsets. These results are used to obtain various characterizations for the convergence of measurable multifunctions. In particular we are interested in the convergence properties of (measurable) selections.
$C\sp{\ast} $-extreme points
Alan
Hopenwasser;
Robert L.
Moore;
V. I.
Paulsen
291-307
Abstract: Let $\mathcal{A}$ be a $ {C^ \ast }$-algebra and let $\mathcal{S}$ be a subset of $\mathcal{A}$. $ \mathcal{S}$ is ${C^ \ast }$-convex if whenever ${T_1},{T_2}, \ldots ,{T_n}$ are in $\mathcal{S}$ and ${A_1}, \ldots ,{A_n}$ are in $\mathcal{A}$ with $\sum\nolimits_{i = 1}^n {A_i^ \ast {A_i} = I}$, then $\sum\nolimits_{i = 1}^n {A_i^ \ast {T_i}{A_i}}$ is in $\mathcal{S}$. An element $T$ in $ \mathcal{S}$ is called ${C^ \ast }$-extreme in $ \mathcal{S}$ if whenever $T = \sum\nolimits_{i = 1}^n {A_i^ \ast {T_i}{A_i}}$ with ${T_i}$ and ${A_i}$ as above and with ${A_i}$ invertible, then ${T_i}$ is unitarily equivalent to $T$ for each $i$. We investigate the linear extreme points and ${C^ \ast }$-extreme points for three sets: first, the unit ball of operators in Hilbert space; next, the set of $2 \times 2$ matrices with numerical radius bounded by $1$; and last, the unit interval of positive operators on Hilbert space. In particular we find that for the second set, the linear and ${C^ \ast }$-extreme points are different.
Power series methods of summability: positivity and gap perfectness
A.
Jakimovski;
W.
Meyer-König;
K.
Zeller
309-317
Abstract: A class of power series methods of summability is defined. By means of a positivity argument (Bohman-Korovkin) it is shown that each method of the class is gap perfect. This fact facilitates the proof of Tauberian gap theorems. Several examples are given.
Projective geometry on partially ordered sets
Ulrich
Faigle;
Christian
Herrmann
319-332
Abstract: A set of axioms is presented for a projective geometry as an incidence structure on partially ordered sets of "points" and "lines". The axioms reduce to the axioms of classical projective geometry in the case where the points and lines are unordered. It is shown that the lattice of linear subsets of a projective geometry is modular and that every modular lattice of finite length is isomorphic to the lattice of linear subsets of some finite-dimensional projective geometry. Primary geometries are introduced as the incidence-geometric counterpart of primary lattices. Thus the theory of finite-dimensional projective geometries includes the theory of finite-dimensional projective Hjelmslev-spaces of level $n$ as a special case. Finally, projective geometries are characterized by incidence properties of points and hyperplanes.
Erratum to: ``Permutation-partition pairs: a combinatorial generalization of graph embeddings''
Saul
Stahl
333